From 6f65afe0d8e0b7662f74d9a445d6edd82fd473ec Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 20 Jul 2015 17:58:34 -0700 Subject: [PATCH 01/57] ENH: Add constraint for multivariate stationary --- statsmodels/tsa/statespace/tools.py | 259 ++++++++++++++++++++++++++++ 1 file changed, 259 insertions(+) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 54b8011e266..27f40a21f06 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -287,6 +287,265 @@ def unconstrain_stationary_univariate(constrained): return x +def _constrain_sv_less_than_one(unconstrained, order=None, k_endog=None): + """ + Transform arbitrary matrices to matrices with singular values less than + one. + + Corresponds to Lemma 2.2 in Ansley and Kohn (1986). See + `constrain_stationary_multivariate` for more details. + """ + from scipy import linalg + + constrained = [] # P_s, s = 1, ..., p + if order is None: + order = len(unconstrained) + if k_endog is None: + k_endog = unconstrained[0].shape[0] + + eye = np.eye(k_endog) + for i in range(order): + A = unconstrained[i] + B, lower = linalg.cho_factor(eye + np.dot(A, A.T), lower=True) + constrained.append(linalg.solve_triangular(B, A, lower=lower)) + return constrained + + +def _constrain_stationary_multivariate(sv_constrained, variance, order=None, + k_endog=None): + """ + Transform matrices with singular values less than one to matrices + corresponding to a stationary (or invertible) process. + + Corresponds to Lemma 2.1 in Ansley and Kohn (1986). See + `constrain_stationary_multivariate` for more details. + """ + from scipy import linalg + + if order is None: + order = len(sv_constrained) + if k_endog is None: + k_endog = sv_constrained[0].shape[0] + + forward_variances = [variance] # \Sigma_s + backward_variances = [variance] # \Sigma_s^*, s = 0, ..., p + variances = [variance] # \Gamma_s + # \phi_{s,k}, s = 1, ..., p + # k = 1, ..., s+1 + constrained = [] + # \phi_{s,k}^* + backwards = [] + + variance_factor = linalg.cholesky(variance, lower=True) + + forward_factors = [variance_factor] + backward_factors = [variance_factor] + + # We fill in the entries as follows: + # [1,1] + # [2,2], [2,1] + # [3,3], [3,1], [3,2] + # ... + # [p,p], [p,1], ..., [p,p-1] + # the last row, correctly ordered, is then used as the coefficients + for s in range(order): # s = 0, ..., p-1 + prev_constrained = constrained + prev_backwards = backwards + constrained = [] + backwards = [] + + # Create the "last" (k = s+1) matrix + # Note: this is for k = s+1. However, below we then have to fill + # in for k = 1, ..., s in order. + # P L^{-1} = x + # x L = P + # L' x' = P' + constrained.append(linalg.solve_triangular( + backward_factors[s], sv_constrained[s].T, lower=True, trans='T')) + constrained[0] = np.dot(forward_factors[s], constrained[0].T) + + # P' L^{-1} = x + # x L = P' + # L' x' = P + backwards.append(linalg.solve_triangular( + forward_factors[s], sv_constrained[s], lower=True, trans='T')) + backwards[0] = np.dot(backward_factors[s], backwards[0].T) + + # Update the variance + # Note: if s >= 1, this will be further updated in the for loop + # below + # Also, this calculation will be re-used in the forward variance + tmp = np.dot(constrained[0], backward_variances[s]) + variances.append(tmp.copy()) + + # Create the remaining k = 1, ..., s matrices, + # only has an effect if s >= 1 + for k in range(s): + constrained.insert(k, prev_constrained[k] - np.dot( + constrained[k], prev_backwards[s-(k+1)])) + + backwards.insert(k, prev_backwards[k] - np.dot( + backwards[k], prev_constrained[s-(k+1)])) + + variances[s+1] += np.dot(prev_constrained[s-(k+1)], + variances[k+1]) + + # Create forward and backwards variances + forward_variances.append( + forward_variances[s] - np.dot(tmp, constrained[s].T) + ) + backward_variances.append( + backward_variances[s] - + np.dot( + np.dot(backwards[s], forward_variances[s]), + backwards[s].T + ) + ) + + # Cholesky factors + forward_factors.append( + linalg.cholesky(forward_variances[s+1], lower=True) + ) + backward_factors.append( + linalg.cholesky(backward_variances[s+1], lower=True) + ) + + return constrained, forward_variances[-1] + + +def constrain_stationary_multivariate(unconstrained, variance, + transform_variance=False): + """ + Transform unconstrained parameters used by the optimizer to constrained + parameters used in likelihood evaluation for a vector autoregression. + + Parameters + ---------- + unconstrained : iterable + Arbitrary matrices to be transformed to stationary coefficient matrices + of the VAR. + variance : array, 2-dim + Variance matrix corresponding to the error term. This is used as + input in the algorithm even if is not transformed by it (when + `transform_variance` is False. The error term variance is required + input when transformation is used either to force an autoregressive + component to be stationary or to force a moving average component to + be invertible. + transform_variance : boolean, optional + Whether or not to transform the error variance term. This option is + not typically used, and the default is False. + + Returns + ------- + constrained : tuple + A tuple of coefficient matrices which lead to a stationary VAR. + + Notes + ----- + In the notation of [1]_, the arguments `(variance, unconstrained)` are + written as :math:`(\Sigma, A_1, \dots, A_p)`, where :math:`p` is the order + of the vector autoregression, and is here determined by the length of + the `unconstrained` argument. + + There are two steps in the constraining algorithm. + + First, :math:`(A_1, \dots, A_p)` are transformed into + :math:`(P_1, \dots, P_p)` via Lemma 2.2 of [1]_. + + Second, :math:`(\Sigma, P_1, \dots, P_p)` are transformed into + :math:`(\Sigma, \phi_1, \dots, \phi_p)` via Lemmas 2.1 and 2.3 of [1]_. + + If `transform_variance=True`, then only Lemma 2.1 is applied in the second + step. + + References + ---------- + Ansley, Craig F., and Robert Kohn. 1986. + "A Note on Reparameterizing a Vector Autoregressive Moving Average Model to + Enforce Stationarity." + Journal of Statistical Computation and Simulation 24 (2): 99-106. + + """ + from scipy import linalg + + order = len(unconstrained) + k_endog = unconstrained[0].shape[0] + + # Step 1: convert from arbitrary matrices to those with singular values + # less than one. + sv_constrained = _constrain_sv_less_than_one(unconstrained, order, k_endog) + + # Step 2: convert matrices from our "partial autocorrelation matrix" space + # (matrices with singular values less than one) to the space of stationary + # coefficient matrices + if transform_variance: + input_variance = variance + else: + # Need to make the input variance large enough that the recursions + # don't lead to zero-matrices due to roundoff error, which would case + # exceptions from the Cholesky decompositions. + # Note that this will still not always ensure positive definiteness, + # and for k_endog, order large enough an exception may still be raised + input_variance = np.eye(k_endog) * (order + k_endog)**10 + constrained, transformed_variance = ( + _constrain_stationary_multivariate(sv_constrained, input_variance, + order, k_endog) + ) + + # Step 3: If we do not want to use the transformed variance, we need to + # adjust the constrained matrices, as presented in Lemma 2.3, see Notes + if not transform_variance: + # Here, we need to construct T such that: + # variance = T * initial_variance * T' + # To do that, consider the Cholesky of variance (L) and + # input_variance (M) to get: + # L L' = T M M' T' = (TM) (TM)' + # => L = T M + # => L M^{-1} = T + variance_factor = np.linalg.cholesky(variance) + input_variance_factor = np.linalg.cholesky(input_variance) + transform = np.dot(variance_factor, + np.linalg.inv(input_variance_factor)) + inv_transform = np.linalg.inv(transform) + + for i in range(order): + constrained[i] = ( + np.dot(np.dot(transform, constrained[i]), inv_transform) + ) + + return tuple(constrained), variance + + +def unconstrain_stationary_multivariate(constrained): + """ + Transform constrained parameters used in likelihood evaluation + to unconstrained parameters used by the optimizer + + Parameters + ---------- + constrained : array + Constrained parameters of, e.g., an autoregressive or moving average + component, to be transformed to arbitrary parameters used by the + optimizer. + + Returns + ------- + unconstrained : array + Unconstrained parameters used by the optimizer, to be transformed to + stationary coefficients of, e.g., an autoregressive or moving average + component. + + References + ---------- + Ansley, Craig F., and Robert Kohn. 1986. + "A Note on Reparameterizing a Vector Autoregressive Moving Average Model to + Enforce Stationarity." + Journal of Statistical Computation and Simulation 24 (2): 99-106. + + """ + pass + + def validate_matrix_shape(name, shape, nrows, ncols, nobs): """ Validate the shape of a possibly time-varying matrix, or raise an exception From fd0d3df72ad756dd191f8f01f42c4d91545e6867 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 20 Jul 2015 20:36:35 -0700 Subject: [PATCH 02/57] ENH: Multivariate is_invertible, companion_matrix --- statsmodels/tsa/statespace/tools.py | 79 ++++++++++++++++++++++------- 1 file changed, 61 insertions(+), 18 deletions(-) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 27f40a21f06..675c680b0e7 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -42,11 +42,16 @@ def companion_matrix(polynomial): Parameters ---------- - polynomial : array_like, optional. + polynomial : array_like or list If an iterable, interpreted as the coefficients of the polynomial from which to form the companion matrix. Polynomial coefficients are in - order of increasing degree. If an integer, the size of the companion - matrix (the polynomial coefficients are then set to zeros). + order of increasing degree, and may be either scalars (as in an AR(p) + model) or coefficient matrices (as in a VAR(p) model). If an integer, + it is interpereted as the size of a companion matrix of a scalar + polynomial, where the polynomial coefficients are initialized to zeros. + If a matrix polynomial is passed, :math:`C_0` may be set to the scalar + value 1 to indicate an identity matrix (doing so will improve the speed + of the companion matrix creation). Returns ------- @@ -69,24 +74,56 @@ def companion_matrix(polynomial): where some or all of the :math:`\phi_i` may be non-zero (if `polynomial` is None, then all are equal to zero). - If the coefficients provided are :math:`(c_0, c_1, \dots, c_{n})`, + If the coefficients provided are scalars :math:`(c_0, c_1, \dots, c_{n})`, then the companion matrix is an :math:`n \times n` matrix formed with the elements in the first column defined as :math:`\phi_i = -\frac{c_i}{c_0}, i \in 1, \dots, n`. + + If the coefficients provided are matrices :math:`(C_0, C_1, \dots, C_{n})`, + each of shape :math:`(m, m)`, then the companion matrix is an + :math:`nm \times nm` matrix formed with the elements in the first column + defined as :math:`\phi_i = -C_0^{-1} C_i', i \in 1, \dots, n`. """ + identity_matrix = False if isinstance(polynomial, int): n = polynomial polynomial = None else: n = len(polynomial) - 1 - polynomial = np.asanyarray(polynomial) - matrix = np.zeros((n, n)) - idx = np.diag_indices(n - 1) - idx = (idx[0], idx[1] + 1) + if n < 1: + raise ValueError("Companion matrix polynomials must include at" + " least two terms.") + + if isinstance(polynomial, list) or isinstance(polynomial, tuple): + m = len(polynomial[1]) + + # Check if we just have a scalar polynomial + if m == 1: + polynomial = np.asanyarray(polynomial) + # Check if 1 was passed as the first argument (indicating an + # identity matrix) + elif polynomial[0] == 1: + polynomial[0] = np.eye(m) + identity_matrix = True + else: + m = 1 + polynomial = np.asanyarray(polynomial) + + matrix = np.zeros((n * m, n * m)) + idx = np.diag_indices((n - 1) * m) + idx = (idx[0], idx[1] + m) matrix[idx] = 1 if polynomial is not None and n > 0: - matrix[:, 0] = -polynomial[1:] / polynomial[0] + if m == 1: + matrix[:, 0] = -polynomial[1:] / polynomial[0] + elif identity_matrix: + for i in range(n): + matrix[i * m:(i + 1) * m, :m] = polynomial[i+1].T + else: + inv = np.linalg.inv(polynomial[0]) + for i in range(n): + matrix[i * m:(i + 1) * m, :m] = np.dot(inv, polynomial[i+1]).T return matrix @@ -154,18 +191,24 @@ def is_invertible(polynomial, threshold=1.): Parameters ---------- - polynomial : array_like + polynomial : array_like or tuple, list Coefficients of a polynomial, in order of increasing degree. For example, `polynomial=[1, -0.5]` corresponds to the polynomial - :math:`1 - 0.5x` which has root :math:`2`. + :math:`1 - 0.5x` which has root :math:`2`. If it is a matrix + polynomial (in which case the coefficients are coefficient matrices), + a tuple or list of matrices should be passed. threshold : number Allowed threshold for `is_invertible` to return True. Default is 1. Notes ----- - If the coefficients provided are :math:`(c_0, c_1, \dots, c_n)`, then - the corresponding polynomial is :math:`c_0 + c_1 L + \dots + c_n L^n`. + If the coefficients provided are scalars :math:`(c_0, c_1, \dots, c_n)`, + then the corresponding polynomial is :math:`c_0 + c_1 L + \dots + c_n L^n`. + + + If the coefficients provided are matrices :math:`(C_0, C_1, \dots, C_n)`, + then the corresponding polynomial is :math:`C_0 + C_1 L + \dots + C_n L^n`. There are three equivalent methods of determining if the polynomial represented by the coefficients is invertible: @@ -188,7 +231,7 @@ def is_invertible(polynomial, threshold=1.): .. math:: C(L) & = c_0 + c_1 L + \dots + c_n L^n \\ - & = constant (L - \zeta_1 L) (L - \zeta_2) \dots (L - \zeta_3) + & = constant (L - \zeta_1) (L - \zeta_2) \dots (L - \zeta_3) The condition is now :math:`|\zeta_i| > 1`, where :math:`\zeta_i` is a root of the polynomial with reversed coefficients and @@ -437,8 +480,8 @@ def constrain_stationary_multivariate(unconstrained, variance, Returns ------- - constrained : tuple - A tuple of coefficient matrices which lead to a stationary VAR. + constrained : list + A list of coefficient matrices which lead to a stationary VAR. Notes ----- @@ -513,7 +556,7 @@ def constrain_stationary_multivariate(unconstrained, variance, np.dot(np.dot(transform, constrained[i]), inv_transform) ) - return tuple(constrained), variance + return constrained, variance def unconstrain_stationary_multivariate(constrained): @@ -543,7 +586,7 @@ def unconstrain_stationary_multivariate(constrained): Journal of Statistical Computation and Simulation 24 (2): 99-106. """ - pass + raise NotImplementedError def validate_matrix_shape(name, shape, nrows, ncols, nobs): From 155588d9c467c32669943f83dffed8dbbae46bb8 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Tue, 21 Jul 2015 10:34:19 -0700 Subject: [PATCH 03/57] DOC: Add note on constraint function performance. --- statsmodels/tsa/statespace/tools.py | 3 +++ 1 file changed, 3 insertions(+) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 675c680b0e7..7fee5ee2349 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -501,6 +501,9 @@ def constrain_stationary_multivariate(unconstrained, variance, If `transform_variance=True`, then only Lemma 2.1 is applied in the second step. + While this function can be used even in the univariate case, it is much + slower, so in that case `constrain_stationary_univariate` is preferred. + References ---------- Ansley, Craig F., and Robert Kohn. 1986. From 61608b8f90eeca5d0407f12b965781662c79a86e Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Tue, 21 Jul 2015 10:58:12 -0700 Subject: [PATCH 04/57] FIX: Bugs in companion_matrix for scalar case. --- statsmodels/tsa/statespace/tools.py | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 7fee5ee2349..ea5e8c5cb4d 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -87,6 +87,7 @@ def companion_matrix(polynomial): identity_matrix = False if isinstance(polynomial, int): n = polynomial + m = 1 polynomial = None else: n = len(polynomial) - 1 @@ -96,7 +97,13 @@ def companion_matrix(polynomial): " least two terms.") if isinstance(polynomial, list) or isinstance(polynomial, tuple): - m = len(polynomial[1]) + try: + # Note: can't use polynomial[0] because of the special behavior + # associated with matrix polynomials and the constant 1, see + # below. + m = len(polynomial[1]) + except TypeError: + m = 1 # Check if we just have a scalar polynomial if m == 1: From e74465ce06a47d0a2a14415c30db41a33252d52a Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Tue, 21 Jul 2015 14:26:44 -0700 Subject: [PATCH 05/57] ENH: Add unconstrain for SV < 1 matrices. --- statsmodels/tsa/statespace/tools.py | 27 +++++++++++++++++++++++++++ 1 file changed, 27 insertions(+) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index ea5e8c5cb4d..5d7b5377196 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -569,6 +569,33 @@ def constrain_stationary_multivariate(unconstrained, variance, return constrained, variance +def _unconstrain_sv_less_than_one(constrained, order=None, k_endog=None): + """ + Transform matrices with singular values less than one to arbitrary + matrices. + + Corresponds to the inverse of Lemma 2.2 in Ansley and Kohn (1986). See + `unconstrain_stationary_multivariate` for more details. + """ + from scipy import linalg + + unconstrained = [] # A_s, s = 1, ..., p + if order is None: + order = len(constrained) + if k_endog is None: + k_endog = constrained[0].shape[0] + + eye = np.eye(k_endog) + for i in range(order): + P = constrained[i] + # B^{-1} B^{-1}' = I - P P' + B_inv, lower = linalg.cho_factor(eye - np.dot(P, P.T), lower=True) + # A = BP + # B^{-1} A = P + unconstrained.append(linalg.solve_triangular(B_inv, P, lower=lower)) + return unconstrained + + def unconstrain_stationary_multivariate(constrained): """ Transform constrained parameters used in likelihood evaluation From 2b422876d5e2080cd34d5008ac7c59983774e30e Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 22 Jul 2015 15:06:48 -0700 Subject: [PATCH 06/57] ENH: Add unconstrain for multivariate stationary --- statsmodels/tsa/statespace/tools.py | 291 ++++++++++++++++++++++------ 1 file changed, 228 insertions(+), 63 deletions(-) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 5d7b5377196..528da7a2da9 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -361,8 +361,10 @@ def _constrain_sv_less_than_one(unconstrained, order=None, k_endog=None): return constrained -def _constrain_stationary_multivariate(sv_constrained, variance, order=None, - k_endog=None): +def _compute_coefficients_from_multivariate_pacf(partial_autocorrelations, + error_variance, + transform_variance=False, + order=None, k_endog=None): """ Transform matrices with singular values less than one to matrices corresponding to a stationary (or invertible) process. @@ -373,23 +375,35 @@ def _constrain_stationary_multivariate(sv_constrained, variance, order=None, from scipy import linalg if order is None: - order = len(sv_constrained) + order = len(partial_autocorrelations) if k_endog is None: - k_endog = sv_constrained[0].shape[0] + k_endog = partial_autocorrelations[0].shape[0] - forward_variances = [variance] # \Sigma_s - backward_variances = [variance] # \Sigma_s^*, s = 0, ..., p - variances = [variance] # \Gamma_s + # If we want to keep the provided variance but with the constrained + # coefficient matrices, we need to make a copy here, and then after the + # main loop we will transform the coefficients to match the passed variance + if not transform_variance: + initial_variance = error_variance + # Need to make the input variance large enough that the recursions + # don't lead to zero-matrices due to roundoff error, which would case + # exceptions from the Cholesky decompositions. + # Note that this will still not always ensure positive definiteness, + # and for k_endog, order large enough an exception may still be raised + error_variance = np.eye(k_endog) * (order + k_endog)**10 + + forward_variances = [error_variance] # \Sigma_s + backward_variances = [error_variance] # \Sigma_s^*, s = 0, ..., p + autocovariances = [error_variance] # \Gamma_s # \phi_{s,k}, s = 1, ..., p # k = 1, ..., s+1 - constrained = [] + forwards = [] # \phi_{s,k}^* backwards = [] - variance_factor = linalg.cholesky(variance, lower=True) + error_variance_factor = linalg.cholesky(error_variance, lower=True) - forward_factors = [variance_factor] - backward_factors = [variance_factor] + forward_factors = [error_variance_factor] + backward_factors = [error_variance_factor] # We fill in the entries as follows: # [1,1] @@ -399,9 +413,9 @@ def _constrain_stationary_multivariate(sv_constrained, variance, order=None, # [p,p], [p,1], ..., [p,p-1] # the last row, correctly ordered, is then used as the coefficients for s in range(order): # s = 0, ..., p-1 - prev_constrained = constrained + prev_forwards = forwards prev_backwards = backwards - constrained = [] + forwards = [] backwards = [] # Create the "last" (k = s+1) matrix @@ -410,39 +424,43 @@ def _constrain_stationary_multivariate(sv_constrained, variance, order=None, # P L^{-1} = x # x L = P # L' x' = P' - constrained.append(linalg.solve_triangular( - backward_factors[s], sv_constrained[s].T, lower=True, trans='T')) - constrained[0] = np.dot(forward_factors[s], constrained[0].T) + forwards.append( + linalg.solve_triangular( + backward_factors[s], partial_autocorrelations[s].T, + lower=True, trans='T')) + forwards[0] = np.dot(forward_factors[s], forwards[0].T) # P' L^{-1} = x # x L = P' # L' x' = P - backwards.append(linalg.solve_triangular( - forward_factors[s], sv_constrained[s], lower=True, trans='T')) + backwards.append( + linalg.solve_triangular( + forward_factors[s], partial_autocorrelations[s], + lower=True, trans='T')) backwards[0] = np.dot(backward_factors[s], backwards[0].T) # Update the variance # Note: if s >= 1, this will be further updated in the for loop # below # Also, this calculation will be re-used in the forward variance - tmp = np.dot(constrained[0], backward_variances[s]) - variances.append(tmp.copy()) + tmp = np.dot(forwards[0], backward_variances[s]) + autocovariances.append(tmp.copy()) # Create the remaining k = 1, ..., s matrices, # only has an effect if s >= 1 for k in range(s): - constrained.insert(k, prev_constrained[k] - np.dot( - constrained[k], prev_backwards[s-(k+1)])) + forwards.insert(k, prev_forwards[k] - np.dot( + forwards[k], prev_backwards[s-(k+1)])) backwards.insert(k, prev_backwards[k] - np.dot( - backwards[k], prev_constrained[s-(k+1)])) + backwards[k], prev_forwards[s-(k+1)])) - variances[s+1] += np.dot(prev_constrained[s-(k+1)], - variances[k+1]) + autocovariances[s+1] += np.dot(prev_forwards[s-(k+1)], + autocovariances[k+1]) # Create forward and backwards variances forward_variances.append( - forward_variances[s] - np.dot(tmp, constrained[s].T) + forward_variances[s] - np.dot(tmp, forwards[s].T) ) backward_variances.append( backward_variances[s] - @@ -460,7 +478,29 @@ def _constrain_stationary_multivariate(sv_constrained, variance, order=None, linalg.cholesky(backward_variances[s+1], lower=True) ) - return constrained, forward_variances[-1] + # If we do not want to use the transformed variance, we need to + # adjust the constrained matrices, as presented in Lemma 2.3, see above + variance = forward_variances[-1] + if not transform_variance: + # Here, we need to construct T such that: + # variance = T * initial_variance * T' + # To do that, consider the Cholesky of variance (L) and + # input_variance (M) to get: + # L L' = T M M' T' = (TM) (TM)' + # => L = T M + # => L M^{-1} = T + initial_variance_factor = np.linalg.cholesky(initial_variance) + transformed_variance_factor = np.linalg.cholesky(variance) + transform = np.dot(initial_variance_factor, + np.linalg.inv(transformed_variance_factor)) + inv_transform = np.linalg.inv(transform) + + for i in range(order): + forwards[i] = ( + np.dot(np.dot(transform, forwards[i]), inv_transform) + ) + + return forwards, variance def constrain_stationary_multivariate(unconstrained, variance, @@ -531,40 +571,8 @@ def constrain_stationary_multivariate(unconstrained, variance, # Step 2: convert matrices from our "partial autocorrelation matrix" space # (matrices with singular values less than one) to the space of stationary # coefficient matrices - if transform_variance: - input_variance = variance - else: - # Need to make the input variance large enough that the recursions - # don't lead to zero-matrices due to roundoff error, which would case - # exceptions from the Cholesky decompositions. - # Note that this will still not always ensure positive definiteness, - # and for k_endog, order large enough an exception may still be raised - input_variance = np.eye(k_endog) * (order + k_endog)**10 - constrained, transformed_variance = ( - _constrain_stationary_multivariate(sv_constrained, input_variance, - order, k_endog) - ) - - # Step 3: If we do not want to use the transformed variance, we need to - # adjust the constrained matrices, as presented in Lemma 2.3, see Notes - if not transform_variance: - # Here, we need to construct T such that: - # variance = T * initial_variance * T' - # To do that, consider the Cholesky of variance (L) and - # input_variance (M) to get: - # L L' = T M M' T' = (TM) (TM)' - # => L = T M - # => L M^{-1} = T - variance_factor = np.linalg.cholesky(variance) - input_variance_factor = np.linalg.cholesky(input_variance) - transform = np.dot(variance_factor, - np.linalg.inv(input_variance_factor)) - inv_transform = np.linalg.inv(transform) - - for i in range(order): - constrained[i] = ( - np.dot(np.dot(transform, constrained[i]), inv_transform) - ) + constrained, variance = _compute_coefficients_from_multivariate_pacf( + sv_constrained, variance, transform_variance, order, k_endog) return constrained, variance @@ -596,7 +604,148 @@ def _unconstrain_sv_less_than_one(constrained, order=None, k_endog=None): return unconstrained -def unconstrain_stationary_multivariate(constrained): +def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, + order=None, k_endog=None): + """ + Transform matrices corresponding to a stationary (or invertible) process + to matrices with singular values less than one. + + Note that this computes multivariate partial autocorrelations. + + Corresponds to the inverse of Lemma 2.1 in Ansley and Kohn (1986). See + `unconstrain_stationary_multivariate` for more details. + """ + from scipy import linalg + + if order is None: + order = len(constrained) + if k_endog is None: + k_endog = constrained[0].shape[0] + + # Start with VAR(p): w_{t+1} = phi_1 w_t + ... + phi_p w_{t-p+1} + u_{t+1} + # Then stack the VAR(p) into a VAR(1) in companion matrix form: + # z_{t+1} = F z_t + v_t + companion = companion_matrix( + [1] + [-constrained[i] for i in range(order)] + ).T + + # Compute the error variance matrix for the stacked form: E v_t v_t' + selected_variance = np.zeros(companion.shape) + selected_variance[:k_endog, :k_endog] = error_variance + + # Compute the unconditional variance of z_t: E z_t z_t' + stacked_cov = linalg.solve_discrete_lyapunov(companion, selected_variance) + + # The first (block) row of the variance of z_t gives the first p-1 + # autocovariances of w_t: \Gamma_i = E w_t w_t+i with \Gamma_0 = Var(w_t) + # Note: these are okay, checked against ArmaProcess + autocovariances = [ + stacked_cov[:k_endog, i*k_endog:(i+1)*k_endog] + for i in range(order) + ] + + # Now apply the Ansley and Kohn (1986) algorithm, except that instead of + # calculating phi_{s+1, s+1} = L_s P_{s+1} {L_s^*}^{-1} (which requires + # the partial autocorrelation P_{s+1} which is what we're trying to + # calculate here), we calculate it as in Ansley and Newbold (1979), using + # the autocovariances \Gamma_s and the forwards and backwards residual + # variances \Sigma_s, \Sigma_s^*: + # phi_{s+1, s+1} = [ \Gamma_{s+1}' - \phi_{s,1} \Gamma_s' - ... - + # \phi_{s,s} \Gamma_1' ] {\Sigma_s^*}^{-1} + + # Forward and backward variances + forward_variances = [] # \Sigma_s + backward_variances = [] # \Sigma_s^*, s = 0, ..., p + # \phi_{s,k}, s = 1, ..., p + # k = 1, ..., s+1 + forwards = [] + # \phi_{s,k}^* + backwards = [] + + forward_factors = [] # L_s + backward_factors = [] # L_s^*, s = 0, ..., p + + # Ultimately we want to construct the partial autocorrelation matrices + # Note that this is "1-indexed" in the sense that it stores P_1, ... P_p + # rather than starting with P_0. + partial_autocorrelations = [] + + # We fill in the entries of phi_{s,k} as follows: + # [1,1] + # [2,2], [2,1] + # [3,3], [3,1], [3,2] + # ... + # [p,p], [p,1], ..., [p,p-1] + # the last row, correctly ordered, should be the same as the coefficient + # matrices provided in the argument `constrained` + for s in range(order): # s = 0, ..., p-1 + prev_forwards = list(forwards) + prev_backwards = list(backwards) + forwards = [] + backwards = [] + + # Create forward and backwards variances Sigma_s, Sigma*_s + forward_variance = autocovariances[0].copy() + backward_variance = autocovariances[0].copy() + + for k in range(s): + forward_variance -= np.dot(prev_forwards[k], autocovariances[k+1]) + backward_variance -= np.dot(prev_backwards[k], autocovariances[k+1].T) + + forward_variances.append(forward_variance) + backward_variances.append(backward_variance) + + # Cholesky factors + forward_factors.append( + linalg.cholesky(forward_variances[s], lower=True) + ) + backward_factors.append( + linalg.cholesky(backward_variances[s], lower=True) + ) + + if s == order-1: + forwards = constrained + else: + # Create the intermediate sum term + # G := \Gamma_{s+1}' - \phi_{s,1} \Gamma_s' - .. - \phi_{s,s} \Gamma_1' + tmp_sum = autocovariances[s+1].T.copy() + for k in range(s): + tmp_sum -= np.dot(prev_forwards[k], autocovariances[s-k].T) + + # Create the "last" (k = s+1) matrix + # Note: this is for k = s+1. However, below we then have to fill + # in for k = 1, ..., s in order. + # phi = G Sigma*^{-1} + # phi Sigma* = G + # Sigma*' phi' = G' + forwards.append(linalg.solve_triangular( + backward_variances[s], tmp_sum.T, lower=True, trans='T').T) + + # phi = G' Sigma^{-1} + # phi Sigma = G' + # Sigma' phi' = G + backwards.append(linalg.solve_triangular( + forward_variances[s], tmp_sum, lower=True, trans='T').T) + + # Create the remaining k = 1, ..., s matrices, + # only has an effect if s >= 1 + for k in range(s): + forwards.insert(k, prev_forwards[k] - np.dot( + forwards[-1], prev_backwards[s-(k+1)])) + backwards.insert(k, prev_backwards[k] - np.dot( + backwards[-1], prev_forwards[s-(k+1)])) + + # Partial autocorrelation matrix: P_{s+1} + # P = L^{-1} phi L* + # L P = (phi L*) + partial_autocorrelations.append(linalg.solve_triangular( + forward_factors[s], np.dot(forwards[s], backward_factors[s]), lower=True)) + + return partial_autocorrelations + + +def unconstrain_stationary_multivariate(constrained, error_variance, + transform_variance=False): """ Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer @@ -623,7 +772,23 @@ def unconstrain_stationary_multivariate(constrained): Journal of Statistical Computation and Simulation 24 (2): 99-106. """ - raise NotImplementedError + + from scipy import linalg + + order = len(constrained) + k_endog = constrained[0].shape[0] + + # Step 1: convert matrices from the space of stationary + # coefficient matrices to our "partial autocorrelation matrix" space + # (matrices with singular values less than one) + partial_autocorrelations = _compute_multivariate_pacf_from_coefficients( + constrained, error_variance, order, k_endog) + + # Step 2: convert from arbitrary matrices to those with singular values + # less than one. + unconstrained = _unconstrain_sv_less_than_one(partial_autocorrelations, order, k_endog) + + return unconstrained, error_variance def validate_matrix_shape(name, shape, nrows, ncols, nobs): From b39d025c0da9f07e02907142b34f2fcbd3adb520 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Tue, 28 Jul 2015 16:27:38 -0700 Subject: [PATCH 07/57] BUG: Fix multivariate constrain and unconstrain. --- statsmodels/tsa/statespace/tools.py | 189 ++++++++++++++++++++-------- 1 file changed, 139 insertions(+), 50 deletions(-) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 528da7a2da9..c3043b824c3 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -59,8 +59,12 @@ def companion_matrix(polynomial): Notes ----- + Given coefficients of a lag polynomial of the form: - Returns a matrix of the form + .. math:: + c(L) = c_0 + c_1 L + \dots + c_p L^p + + returns a matrix of the form .. math:: \begin{bmatrix} @@ -74,15 +78,33 @@ def companion_matrix(polynomial): where some or all of the :math:`\phi_i` may be non-zero (if `polynomial` is None, then all are equal to zero). - If the coefficients provided are scalars :math:`(c_0, c_1, \dots, c_{n})`, + If the coefficients provided are scalars :math:`(c_0, c_1, \dots, c_p)`, then the companion matrix is an :math:`n \times n` matrix formed with the elements in the first column defined as - :math:`\phi_i = -\frac{c_i}{c_0}, i \in 1, \dots, n`. + :math:`\phi_i = -\frac{c_i}{c_0}, i \in 1, \dots, p`. - If the coefficients provided are matrices :math:`(C_0, C_1, \dots, C_{n})`, + If the coefficients provided are matrices :math:`(C_0, C_1, \dots, C_p)`, each of shape :math:`(m, m)`, then the companion matrix is an :math:`nm \times nm` matrix formed with the elements in the first column - defined as :math:`\phi_i = -C_0^{-1} C_i', i \in 1, \dots, n`. + defined as :math:`\phi_i = -C_0^{-1} C_i', i \in 1, \dots, p`. + + It is important to understand the expected signs of the coefficients. A + typical AR(p) model is written as: + + .. math:: + y_t = a_1 y_{t-1} + \dots + a_p y_{t-p} + \varepsilon_t + + This can be rewritten as: + + .. math:: + (1 - a_1 L - \dots - a_p L^p )y_t = \varepsilon_t \\ + (1 + c_1 L + \dots + c_p L^p )y_t = \varepsilon_t \\ + c(L) y_t = \varepsilon_t + + The coefficients from this form are defined to be :math:`c_i = - a_i`, and + it is the :math:`c_i` coefficients that this function expects to be + provided. + """ identity_matrix = False if isinstance(polynomial, int): @@ -126,11 +148,11 @@ def companion_matrix(polynomial): matrix[:, 0] = -polynomial[1:] / polynomial[0] elif identity_matrix: for i in range(n): - matrix[i * m:(i + 1) * m, :m] = polynomial[i+1].T + matrix[i * m:(i + 1) * m, :m] = -polynomial[i+1].T else: inv = np.linalg.inv(polynomial[0]) for i in range(n): - matrix[i * m:(i + 1) * m, :m] = np.dot(inv, polynomial[i+1]).T + matrix[i * m:(i + 1) * m, :m] = -np.dot(inv, polynomial[i+1]).T return matrix @@ -421,9 +443,9 @@ def _compute_coefficients_from_multivariate_pacf(partial_autocorrelations, # Create the "last" (k = s+1) matrix # Note: this is for k = s+1. However, below we then have to fill # in for k = 1, ..., s in order. - # P L^{-1} = x - # x L = P - # L' x' = P' + # P L*^{-1} = x + # x L* = P + # L*' x' = P' forwards.append( linalg.solve_triangular( backward_factors[s], partial_autocorrelations[s].T, @@ -444,19 +466,19 @@ def _compute_coefficients_from_multivariate_pacf(partial_autocorrelations, # below # Also, this calculation will be re-used in the forward variance tmp = np.dot(forwards[0], backward_variances[s]) - autocovariances.append(tmp.copy()) + autocovariances.append(tmp.copy().T) # Create the remaining k = 1, ..., s matrices, # only has an effect if s >= 1 for k in range(s): forwards.insert(k, prev_forwards[k] - np.dot( - forwards[k], prev_backwards[s-(k+1)])) + forwards[-1], prev_backwards[s-(k+1)])) backwards.insert(k, prev_backwards[k] - np.dot( - backwards[k], prev_forwards[s-(k+1)])) + backwards[-1], prev_forwards[s-(k+1)])) - autocovariances[s+1] += np.dot(prev_forwards[s-(k+1)], - autocovariances[k+1]) + autocovariances[s+1] += np.dot(autocovariances[k+1], + prev_forwards[s-(k+1)].T) # Create forward and backwards variances forward_variances.append( @@ -604,29 +626,32 @@ def _unconstrain_sv_less_than_one(constrained, order=None, k_endog=None): return unconstrained -def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, - order=None, k_endog=None): +def _compute_multivariate_acovf_from_coefficients(coefficients, error_variance, + order=None, k_endog=None, + maxlag=None): """ - Transform matrices corresponding to a stationary (or invertible) process - to matrices with singular values less than one. + Notes + ----- - Note that this computes multivariate partial autocorrelations. + Coefficients are assumed to be provided from the VAR model: - Corresponds to the inverse of Lemma 2.1 in Ansley and Kohn (1986). See - `unconstrain_stationary_multivariate` for more details. + .. math:: + y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + \varepsilon_t """ from scipy import linalg if order is None: - order = len(constrained) + order = len(coefficients) if k_endog is None: - k_endog = constrained[0].shape[0] + k_endog = coefficients[0].shape[0] + if maxlag is None: + maxlag = order-1 # Start with VAR(p): w_{t+1} = phi_1 w_t + ... + phi_p w_{t-p+1} + u_{t+1} # Then stack the VAR(p) into a VAR(1) in companion matrix form: # z_{t+1} = F z_t + v_t companion = companion_matrix( - [1] + [-constrained[i] for i in range(order)] + [1] + [-coefficients[i] for i in range(order)] ).T # Compute the error variance matrix for the stacked form: E v_t v_t' @@ -641,9 +666,51 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, # Note: these are okay, checked against ArmaProcess autocovariances = [ stacked_cov[:k_endog, i*k_endog:(i+1)*k_endog] - for i in range(order) + for i in range(min(order, maxlag+1)) ] + for i in range(maxlag - (order-1)): + stacked_cov = np.dot(companion, stacked_cov) + autocovariances += [ + stacked_cov[:k_endog, -k_endog:] + ] + + return autocovariances + + +def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, + order=None, k_endog=None): + """ + Transform matrices corresponding to a stationary (or invertible) process + to matrices with singular values less than one. + + Note that this computes multivariate partial autocorrelations. + + Corresponds to the inverse of Lemma 2.1 in Ansley and Kohn (1986). See + `unconstrain_stationary_multivariate` for more details. + + Notes + ----- + + Coefficients are assumed to be provided from the VAR model: + + .. math:: + y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + \varepsilon_t + """ + from scipy import linalg + + if order is None: + order = len(constrained) + if k_endog is None: + k_endog = constrained[0].shape[0] + + # Get autocovariances for the process; these are defined to be + # E z_t z_{t-j}' + # However, we want E z_t z_{t+j}' = (E z_t z_{t-j}')' + _acovf = _compute_multivariate_acovf_from_coefficients + autocovariances = [autocovariance.T for autocovariance in + _acovf(constrained, error_variance, order, k_endog, maxlag=order)] + # Now apply the Ansley and Kohn (1986) algorithm, except that instead of # calculating phi_{s+1, s+1} = L_s P_{s+1} {L_s^*}^{-1} (which requires # the partial autocorrelation P_{s+1} which is what we're trying to @@ -686,11 +753,13 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, # Create forward and backwards variances Sigma_s, Sigma*_s forward_variance = autocovariances[0].copy() - backward_variance = autocovariances[0].copy() + backward_variance = autocovariances[0].T.copy() for k in range(s): - forward_variance -= np.dot(prev_forwards[k], autocovariances[k+1]) - backward_variance -= np.dot(prev_backwards[k], autocovariances[k+1].T) + forward_variance -= np.dot(prev_forwards[k], + autocovariances[k+1]) + backward_variance -= np.dot(prev_backwards[k], + autocovariances[k+1].T) forward_variances.append(forward_variance) backward_variances.append(backward_variance) @@ -703,29 +772,48 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, linalg.cholesky(backward_variances[s], lower=True) ) - if s == order-1: + if False and s == order-1: forwards = constrained else: # Create the intermediate sum term - # G := \Gamma_{s+1}' - \phi_{s,1} \Gamma_s' - .. - \phi_{s,s} \Gamma_1' - tmp_sum = autocovariances[s+1].T.copy() - for k in range(s): - tmp_sum -= np.dot(prev_forwards[k], autocovariances[s-k].T) - - # Create the "last" (k = s+1) matrix - # Note: this is for k = s+1. However, below we then have to fill - # in for k = 1, ..., s in order. - # phi = G Sigma*^{-1} - # phi Sigma* = G - # Sigma*' phi' = G' - forwards.append(linalg.solve_triangular( - backward_variances[s], tmp_sum.T, lower=True, trans='T').T) - - # phi = G' Sigma^{-1} - # phi Sigma = G' - # Sigma' phi' = G - backwards.append(linalg.solve_triangular( - forward_variances[s], tmp_sum, lower=True, trans='T').T) + if s == 0: + # phi_11 = \Gamma_1' \Gamma_0^{-1} + # phi_11 \Gamma_0 = \Gamma_1' + # \Gamma_0 phi_11' = \Gamma_1 + forwards.append(linalg.cho_solve( + (forward_factors[0], True), autocovariances[1]).T) + # backwards.append(forwards[-1]) + # phi_11_star = \Gamma_1 \Gamma_0^{-1} + # phi_11_star \Gamma_0 = \Gamma_1 + # \Gamma_0 phi_11_star' = \Gamma_1' + backwards.append(linalg.cho_solve( + (backward_factors[0], True), autocovariances[1].T).T) + else: + # G := \Gamma_{s+1}' - + # \phi_{s,1} \Gamma_s' - .. - \phi_{s,s} \Gamma_1' + tmp_sum = autocovariances[s+1].T.copy() + + for k in range(s): + tmp_sum -= np.dot(prev_forwards[k], autocovariances[s-k].T) + + # Create the "last" (k = s+1) matrix + # Note: this is for k = s+1. However, below we then have to + # fill in for k = 1, ..., s in order. + # phi = G Sigma*^{-1} + # phi Sigma* = G + # Sigma*' phi' = G' + # Sigma* phi' = G' + # (because Sigma* is symmetric) + forwards.append(linalg.cho_solve( + (backward_factors[s], True), tmp_sum.T).T) + + # phi = G' Sigma^{-1} + # phi Sigma = G' + # Sigma' phi' = G + # Sigma phi' = G + # (because Sigma is symmetric) + backwards.append(linalg.cho_solve( + (forward_factors[s], True), tmp_sum).T) # Create the remaining k = 1, ..., s matrices, # only has an effect if s >= 1 @@ -739,7 +827,8 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, # P = L^{-1} phi L* # L P = (phi L*) partial_autocorrelations.append(linalg.solve_triangular( - forward_factors[s], np.dot(forwards[s], backward_factors[s]), lower=True)) + forward_factors[s], np.dot(forwards[s], backward_factors[s]), + lower=True)) return partial_autocorrelations From 2a57d087ec9e06133f6282ca766851adc3c6af32 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Tue, 28 Jul 2015 16:28:32 -0700 Subject: [PATCH 08/57] TST: Initial tests for multivariate un/constraint --- .../tsa/statespace/tests/test_tools.py | 80 ++++++++++++++++++- 1 file changed, 77 insertions(+), 3 deletions(-) diff --git a/statsmodels/tsa/statespace/tests/test_tools.py b/statsmodels/tsa/statespace/tests/test_tools.py index afbd55d8650..a26f833eccb 100644 --- a/statsmodels/tsa/statespace/tests/test_tools.py +++ b/statsmodels/tsa/statespace/tests/test_tools.py @@ -12,15 +12,24 @@ from statsmodels.tsa.statespace import tools # from .results import results_sarimax from numpy.testing import ( - assert_equal, assert_array_equal, assert_almost_equal, assert_raises + assert_allclose, assert_equal, assert_array_equal, assert_almost_equal, + assert_raises ) class TestCompanionMatrix(object): cases = [ (2, np.array([[0,1],[0,0]])), - ([1,-1,-2], np.array([[1,1],[2,0]])), - ([1,-1,-2,-3], np.array([[1,1,0],[2,0,1],[3,0,0]])) + ([1,-1,-2], np.array([[1,1], + [2,0]])), + ([1,-1,-2,-3], np.array([[1,1,0], + [2,0,1], + [3,0,0]])), + ([1,-np.array([[1,2],[3,4]]),-np.array([[5,6],[7,8]])], + np.array([[1,2,5,6], + [3,4,7,8], + [1,0,0,0], + [0,1,0,0]]).T) ] def test_cases(self): @@ -147,3 +156,68 @@ def test_invalid_cases(self): assert_raises( ValueError, tools.validate_vector_shape, *args ) + +def test_multivariate_acovf(): + _acovf = tools._compute_multivariate_acovf_from_coefficients + + # Test for a VAR(1) process. From Lutkepohl (2007), pages 27-28. + # See (2.1.14) for Phi_1, (2.1.33) for Sigma_u, and (2.1.34) for Gamma_0 + Sigma_u = np.array([[2.25, 0, 0], + [0, 1.0, 0.5], + [0, 0.5, 0.74]]) + Phi_1 = np.array([[0.5, 0, 0], + [0.1, 0.1, 0.3], + [0, 0.2, 0.3]]) + Gamma_0 = np.array([[3.0, 0.161, 0.019], + [0.161, 1.172, 0.674], + [0.019, 0.674, 0.954]]) + # assert_allclose(_acovf([Phi_1], Sigma_u)[0], Gamma_0, atol=1e-3) + + # Test for a VAR(2) process. From Lutkepohl (2007), pages 28-29 + # See (2.1.40) for Phi_1, Phi_2, (2.1.14) for Sigma_u, and (2.1.42) for + # Gamma_0, Gamma_1 + Sigma_u = np.diag([0.09, 0.04]) + Phi_1 = np.array([[0.5, 0.1], + [0.4, 0.5]]) + Phi_2 = np.array([[0, 0], + [0.25, 0]]) + Gamma_0 = np.array([[0.131, 0.066], + [0.066, 0.181]]) + Gamma_1 = np.array([[0.072, 0.051], + [0.104, 0.143]]) + Gamma_2 = np.array([[0.046, 0.040], + [0.113, 0.108]]) + Gamma_3 = np.array([[0.035, 0.031], + [0.093, 0.083]]) + + assert_allclose( + _acovf([Phi_1, Phi_2], Sigma_u, maxlag=0), + [Gamma_0], atol=1e-3) + + assert_allclose( + _acovf([Phi_1, Phi_2], Sigma_u, maxlag=1), + [Gamma_0, Gamma_1], atol=1e-3) + + assert_allclose( + _acovf([Phi_1, Phi_2], Sigma_u), + [Gamma_0, Gamma_1], atol=1e-3) + + assert_allclose( + _acovf([Phi_1, Phi_2], Sigma_u, maxlag=2), + [Gamma_0, Gamma_1, Gamma_2], atol=1e-3) + + assert_allclose( + _acovf([Phi_1, Phi_2], Sigma_u, maxlag=3), + [Gamma_0, Gamma_1, Gamma_2, Gamma_3], atol=1e-3) + +class TestConstrainStationaryMultivariate(object): + + cases = [ + ([np.array([[2.]])], [np.array([[2./((1+2.**2)**0.5)]])]) + ] + + def test_cases(self): + for unconstrained, constrained in self.cases: + result = tools.constrain_stationary_multivariate( + unconstrained, np.eye(len(unconstrained[0]))) + assert_allclose(result[0], constrained) From 9c310e230f62bd73aa14833143265387460f00b6 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Tue, 28 Jul 2015 16:28:46 -0700 Subject: [PATCH 09/57] ENH: Add VARMAX model. --- statsmodels/tsa/statespace/varmax.py | 529 +++++++++++++++++++++++++++ 1 file changed, 529 insertions(+) create mode 100644 statsmodels/tsa/statespace/varmax.py diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py new file mode 100644 index 00000000000..6fcac5c17df --- /dev/null +++ b/statsmodels/tsa/statespace/varmax.py @@ -0,0 +1,529 @@ +""" +Vector Autoregressive Moving Average with eXogenous regressors model + +Author: Chad Fulton +License: Simplified-BSD +""" +from __future__ import division, absolute_import, print_function + +from warnings import warn +from statsmodels.compat.collections import OrderedDict + +import numpy as np +from .kalman_filter import ( + KalmanFilter, FilterResults, INVERT_UNIVARIATE, SOLVE_LU +) +from .mlemodel import MLEModel, MLEResults, MLEResultsWrapper +from .tools import ( + companion_matrix, diff, is_invertible, + constrain_stationary_multivariate, unconstrain_stationary_multivariate +) +from statsmodels.tools.tools import Bunch +from statsmodels.tools.data import _is_using_pandas +from statsmodels.tsa.tsatools import lagmat +from statsmodels.tsa.vector_ar import var_model + + +class VARMAX(MLEModel): + r""" + Vector Autoregressive Moving Average with eXogenous regressors model + + Parameters + ---------- + endog : array_like + The observed time-series process :math:`y`, , shaped nobs x k_endog. + exog : array_like, optional + Array of exogenous regressors, shaped nobs x k. + order : iterable + The (p,q) order of the model for the number of AR and MA parameters to + use. + trend : {'nc', 'c'}, optional + Parameter controlling the deterministic trend polynomial. + Can be specified as a string where 'c' indicates a constant intercept + and 'nc' indicates no intercept term. + error_cov_type : {'diagonal', 'unstructured'}, optional + The structure of the covariance matrix of the error term, where + "unstructured" puts no restrictions on the matrix and "diagonal" + requires it to be a diagonal matrix (uncorrelated errors). Default is + "unstructured". + measurement_error : boolean, optional + Whether or not to assume the endogenous observations `endog` were + measured with error. Default is False. + enforce_stationarity : boolean, optional + Whether or not to transform the AR parameters to enforce stationarity + in the autoregressive component of the model. Default is True. + enforce_invertibility : boolean, optional + Whether or not to transform the MA parameters to enforce invertibility + in the moving average component of the model. Default is True. + **kwargs + Keyword arguments may be used to provide default values for state space + matrices or for Kalman filtering options. See `Representation`, and + `KalmanFilter` for more details. + """ + + def __init__(self, endog, exog=None, order=(1,0), trend='c', + error_cov_type='unstructured', measurement_error=False, + enforce_stationarity=True, enforce_invertibility=True, + **kwargs): + + # Model parameters + self.error_cov_type = error_cov_type + self.measurement_error = measurement_error + self.enforce_stationarity = enforce_stationarity + self.enforce_invertibility = enforce_invertibility + + # Save the given orders + self.order = order + self.trend = trend + + # Model orders + self.k_ar = int(order[0]) + self.k_ma = int(order[1]) + self.k_trend = int(self.trend == 'c') + + # Check for valid model + if trend not in ['c','nc']: + raise ValueError('Invalid trend specification.') + if error_cov_type not in ['diagonal', 'unstructured']: + raise ValueError('Invalid error covariance matrix type' + ' specification.') + if self.k_ar == 0 and self.k_ma == 0: + raise ValueError('Invalid VARMAX(p,q) specification; at least one' + ' p,q must be greater than zero.') + + # Warn for VARMA model + if self.k_ar > 0 and self.k_ma > 0: + warn('Estimation of VARMA(p,q) models is not generically robust,' + ' due especially to identification issues.') + + # Exogenous data + self.k_exog = 0 + if exog is not None: + exog_is_using_pandas = _is_using_pandas(exog, None) + if not exog_is_using_pandas: + exog = np.asarray(exog) + + # Make sure we have 2-dimensional array + if exog.ndim == 1: + if not exog_is_using_pandas: + exog = exog[:, None] + else: + exog = pd.DataFrame(exog) + + self.k_exog = exog.shape[1] + + # Note: at some point in the future might add state regression, as in + # SARIMAX. + self.mle_regression = self.k_exog > 0 + + # We need to have an array or pandas at this point + if not _is_using_pandas(endog, None): + endog = np.asanyarray(endog) + + # Model order + # Used internally in various places + self._k_order = self.k_ar + self.k_ma + + # Number of states + k_endog = endog.shape[1] + k_posdef = k_endog + k_states = k_endog * self._k_order + + # By default, initialize as stationary + kwargs.setdefault('initialization', 'stationary') + + # By default, use LU decomposition + kwargs.setdefault('inversion_method', INVERT_UNIVARIATE | SOLVE_LU) + + # Initialize the state space model + super(VARMAX, self).__init__( + endog, exog=exog, k_states=k_states, k_posdef=k_posdef, **kwargs + ) + + # Initialize the parameters + self.parameters = OrderedDict() + self.parameters['trend'] = self.k_endog * self.k_trend + self.parameters['ar'] = self.k_endog**2 * self.k_ar + self.parameters['ma'] = self.k_endog**2 * self.k_ma + self.parameters['regression'] = self.k_endog * self.k_exog + if self.error_cov_type == 'diagonal': + self.parameters['state_cov'] = self.k_endog + # These parameters fill in a lower-triangular matrix which is then + # dotted with itself to get a positive definite matrix. + elif self.error_cov_type == 'unstructured': + self.parameters['state_cov'] = ( + int(self.k_endog * (self.k_endog + 1) / 2) + ) + self.parameters['obs_cov'] = self.k_endog * self.measurement_error + self.k_params = sum(self.parameters.values()) + + # Initialize known elements of the state space matrices + + # If we have exog effects, then the state intercept needs to be + # time-varying + if self.k_exog > 0: + self.ssm['state_intercept'] = np.zeros((self.k_states, self.nobs)) + + # The design matrix is just an identity for the first k_endog states + idx = np.diag_indices(self.k_endog) + self.ssm[('design',) + idx] = 1 + + # The transition matrix is described in four blocks, where the upper + # left block is in companion form with the autoregressive coefficient + # matrices (so it is shaped k_endog * k_ar x k_endog * k_ar) ... + idx = np.diag_indices((self.k_ar - 1) * self.k_endog) + idx = idx[0] + self.k_endog, idx[1] + self.ssm[('transition',) + idx] = 1 + # ... and the lower right block is in companion form with zeros as the + # coefficient matrices (it is shaped k_endog * k_ma x k_endog * k_ma). + idx = np.diag_indices((self.k_ma - 1) * self.k_endog) + idx = idx[0] + (self.k_ar + 1) * self.k_endog, idx[1] + self.ssm[('transition',) + idx] = 1 + + # The selection matrix is described in two blocks, where the upper + # block selects the all k_posdef errors in the first k_endog rows + # (the upper block is shaped k_endog * k_ar x k) and the lower block + # also selects all k_posdef errors in the first k_endog rows (the lower + # block is shaped k_endog * k_ma x k). + idx = np.diag_indices(self.k_endog) + if self.k_ar > 0: + self.ssm[('selection',) + idx] = 1 + idx = idx[0] + self.k_ar * self.k_endog, idx[1] + if self.k_ma > 0: + self.ssm[('selection',) + idx] = 1 + + # Cache some indices + if self.trend == 'c' and self.k_exog == 0: + self._idx_state_intercept = np.s_['state_intercept', :k_endog] + elif self.k_exog > 0: + self._idx_state_intercept = np.s_['state_intercept', :k_endog, :] + self._idx_transition = np.s_['transition', :k_endog, :] + if self.error_cov_type == 'diagonal': + self._idx_state_cov = ( + ('state_cov',) + np.diag_indices(self.k_endog)) + elif self.error_cov_type == 'unstructured': + self._idx_lower_state_cov = np.tril_indices(self.k_endog) + if self.measurement_error: + self._idx_obs_cov = ('obs_cov',) + np.diag_indices(self.k_endog) + + # Cache some slices + def _slice(key, offset): + length = self.parameters[key] + param_slice = np.s_[offset:offset + length] + offset += length + return param_slice, offset + + offset = 0 + self._params_trend, offset = _slice('trend', offset) + self._params_ar, offset = _slice('ar', offset) + self._params_ma, offset = _slice('ma', offset) + self._params_regression, offset = _slice('regression', offset) + self._params_state_cov, offset = _slice('state_cov', offset) + self._params_obs_cov, offset = _slice('obs_cov', offset) + + @property + def start_params(self): + params = np.zeros(self.k_params, dtype=np.float64) + + # A. Run a multivariate regression to get beta estimates + endog = self.endog.copy() + exog = self.exog.copy() if self.k_exog > 0 else None + + # Although the Kalman filter can deal with missing values in endog, + # conditional sum of squares cannot + if np.any(np.isnan(endog)): + endog = endog[~np.isnan(endog)] + if exog is not None: + exog = exog[~np.isnan(endog)] + + # Regression effects via OLS + exog_params = np.zeros(0) + if self.k_exog > 0: + exog_params = np.linalg.pinv(exog).dot(endog) + endog -= np.dot(exog, exog_params) + + # B. Run a VAR model on endog to get trend, AR parameters + ar_params = [] + if self.k_ar > 0 or self.trend == 'c': + mod_ar = var_model.VAR(endog) + res_ar = mod_ar.fit(maxlags=self.k_ar, ic=None, trend=self.trend) + ar_params = np.array(res_ar.params.T) + if self.trend == 'c': + trend_params = ar_params[:,0] + ar_params = ar_params[:,1:].ravel() + else: + ar_params = ar_params.ravel() + endog = res_ar.resid + + # C. Run a VAR model on the residuals to get MA parameters + ma_params = [] + if self.k_ma > 0: + mod_ma = var_model.VAR(endog) + res_ma = mod_ma.fit(maxlags=self.k_ma, ic=None, trend='nc') + ma_params = np.array(res_ma.params.T).ravel() + + # 1. Intercept terms + if self.trend == 'c': + params[self._params_trend] = trend_params + + # 2. AR terms + params[self._params_ar] = ar_params + + # 3. MA terms + params[self._params_ma] = ma_params + + # 4. Regression terms + if self.mle_regression: + params[self._params_regression] = exog_params.ravel() + + # 5. State covariance terms + if self.error_cov_type == 'diagonal': + params[self._params_state_cov] = res_ar.sigma_u.diagonal() + elif self.error_cov_type == 'unstructured': + cov_factor = np.linalg.cholesky(res_ar.sigma_u) + params[self._params_state_cov] = ( + cov_factor[self._idx_lower_state_cov].ravel()) + + # 5. Measurement error variance terms + if self.measurement_error: + params[self._params_obs_cov] = res_ma.sigma_u.diagonal() + + return params + + @property + def param_names(self): + param_names = [] + + # 1. Intercept terms + if self.trend == 'c': + param_names += ['const.y%d' % (i+1) for i in range(self.k_endog)] + + # 2. AR terms + param_names += [ + 'L%dy%d.y%d' % (i+1, k+1, j+1) + for i in range(self.k_ar) + for j in range(self.k_endog) + for k in range(self.k_endog) + ] + + # 3. MA terms + param_names += [ + 'L%de%d.y%d' % (i+1, k+1, j+1) + for i in range(self.k_ma) + for j in range(self.k_endog) + for k in range(self.k_endog) + ] + + # 4. Regression terms + # TODO replace "beta" with the name of the exog, when possible + param_names += [ + 'beta%d.y%d' % (j+1, i+1) + for i in range(self.k_endog) + for j in range(self.k_exog) + ] + + # 5. State covariance terms + if self.error_cov_type == 'diagonal': + param_names += ['sigma2.y%d' % i for i in range(self.k_endog)] + elif self.error_cov_type == 'unstructured': + param_names += [ + 'sqrt.cov.y%d.y%d' % (j+1, i+1) + for i in range(self.k_endog) + for j in range(i+1) + ] + + # 5. Measurement error variance terms + if self.measurement_error: + param_names += [ + 'measurement_variance.y%d' % (i+1) + for i in range(self.k_endog) + ] + + return param_names + + def transform_params(self, unconstrained): + """ + Transform unconstrained parameters used by the optimizer to constrained + parameters used in likelihood evaluation + + Parameters + ---------- + unconstrained : array_like + Array of unconstrained parameters used by the optimizer, to be + transformed. + + Returns + ------- + constrained : array_like + Array of constrained parameters which may be used in likelihood + evalation. + + Notes + ----- + Constrains the factor transition to be stationary and variances to be + positive. + """ + unconstrained = np.array(unconstrained, ndmin=1) + constrained = np.zeros(unconstrained.shape, dtype=unconstrained.dtype) + + # 1. Intercept terms: nothing to do + constrained[self._params_trend] = unconstrained[self._params_trend] + + # 2. AR terms: optionally force to be stationary + if self.k_ar > 0 and self.enforce_stationarity: + # Create the state covariance matrix + if self.error_cov_type == 'diagonal': + state_cov = np.diag(unconstrained[self._params_state_cov]**2) + elif self.error_cov_type == 'unstructured': + state_cov_lower = np.zeros(self.ssm['state_cov'].shape, dtype=unconstrained.dtype) + state_cov_lower[self._idx_lower_state_cov] = unconstrained[self._params_state_cov] + state_cov = np.dot(state_cov_lower, state_cov_lower.T) + + # Transform the parameters + coefficients = unconstrained[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) + unconstrained_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ar)] + coefficient_matrices, variance = ( + constrain_stationary_multivariate(unconstrained_matrices, state_cov)) + constrained[self._params_ar] = np.concatenate(coefficient_matrices, axis=1).ravel() + else: + constrained[self._params_ar] = unconstrained[self._params_ar] + + # 3. MA terms: optionally force to be invertible + if self.k_ma > 0 and self.enforce_invertibility: + # Transform the parameters, using an identity variance matrix + coefficients = unconstrained[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) + unconstrained_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ma)] + # TODO cache this np.eye call + coefficient_matrices, variance = ( + constrain_stationary_multivariate(unconstrained_matrices, np.eye(self.k_endog, dtype=constrained.dtype))) + constrained[self._params_ma] = np.concatenate(coefficient_matrices, axis=1).ravel() + else: + constrained[self._params_ma] = unconstrained[self._params_ma] + + # 4. Regression terms: nothing to do + constrained[self._params_regression] = ( + unconstrained[self._params_regression]) + + # 5. State covariance terms + # If we have variances, force them to be positive + if self.error_cov_type == 'diagonal': + constrained[self._params_state_cov] = ( + unconstrained[self._params_state_cov]**2) + # Otherwise, nothing needs to be done + elif self.error_cov_type == 'unstructured': + constrained[self._params_state_cov] = ( + unconstrained[self._params_state_cov]) + + # 5. Measurement error variance terms + if self.measurement_error: + # Force these to be positive + constrained[self._params_obs_cov] = ( + constrained[self._params_obs_cov]**2) + + return constrained + + def untransform_params(self, constrained): + """ + Transform constrained parameters used in likelihood evaluation + to unconstrained parameters used by the optimizer. + + Parameters + ---------- + constrained : array_like + Array of constrained parameters used in likelihood evalution, to be + transformed. + + Returns + ------- + unconstrained : array_like + Array of unconstrained parameters used by the optimizer. + """ + constrained = np.array(constrained, ndmin=1) + unconstrained = np.zeros(constrained.shape, dtype=constrained.dtype) + + # 1. Intercept terms: nothing to do + unconstrained[self._params_trend] = constrained[self._params_trend] + + # 2. AR terms: optionally were forced to be stationary + if self.k_ar > 0 and self.enforce_stationarity: + # Create the state covariance matrix + if self.error_cov_type == 'diagonal': + state_cov = np.diag(constrained[self._params_state_cov]**2) + elif self.error_cov_type == 'unstructured': + state_cov_lower = np.zeros(self.ssm['state_cov'].shape, dtype=constrained.dtype) + state_cov_lower[self._idx_lower_state_cov] = constrained[self._params_state_cov] + state_cov = np.dot(state_cov_lower, state_cov_lower.T) + + # Transform the parameters + coefficients = constrained[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) + coefficient_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ar)] + unconstrained_matrices, variance = ( + unconstrain_stationary_multivariate(coefficient_matrices, state_cov)) + unconstrained[self._params_ar] = np.concatenate(unconstrained_matrices, axis=1).ravel() + else: + unconstrained[self._params_ar] = constrained[self._params_ar] + + # 3. MA terms: optionally were forced to be invertible + if self.k_ma > 0 and self.enforce_invertibility: + # Transform the parameters, using an identity variance matrix + coefficients = constrained[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) + coefficient_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ma)] + unconstrained_matrices, variance = ( + unconstrain_stationary_multivariate(coefficient_matrices, np.eye(self.k_endog, dtype=constrained.dtype))) + unconstrained[self._params_ma] = np.concatenate(unconstrained_matrices, axis=1).ravel() + else: + unconstrained[self._params_ma] = constrained[self._params_ma] + + # 4. Regression terms: nothing to do + unconstrained[self._params_regression] = ( + constrained[self._params_regression]) + + # 5. State covariance terms + # If we have variances, then these were forced to be positive + if self.error_cov_type == 'diagonal': + unconstrained[self._params_state_cov] = ( + constrained[self._params_state_cov]**0.5) + # Otherwise, nothing needs to be done + elif self.error_cov_type == 'unstructured': + unconstrained[self._params_state_cov] = ( + constrained[self._params_state_cov]) + + # 5. Measurement error variance terms + if self.measurement_error: + # These were forced to be positive + unconstrained[self._params_obs_cov] = ( + constrained[self._params_obs_cov]**0.5) + + return unconstrained + + def update(self, params, *args, **kwargs): + params = super(VARMAX, self).update(params, *args, **kwargs) + + # 1. State intercept + if self.mle_regression: + exog_params = params[self._params_regression].reshape(self.k_exog, self.k_endog) + intercept = np.dot(self.exog, exog_params) + if self.trend == 'c': + intercept += params[self._params_trend] + self.ssm[self._idx_state_intercept] = intercept.T + elif self.trend == 'c': + self.ssm[self._idx_state_intercept] = params[self._params_trend] + + # 2. Transition + ar = params[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) + ma = params[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) + self.ssm[self._idx_transition] = np.c_[ar, ma] + + # 3. State covariance + if self.error_cov_type == 'diagonal': + self.ssm[self._idx_state_cov] = ( + params[self._params_state_cov] + ) + elif self.error_cov_type == 'unstructured': + state_cov_lower = np.zeros(self.ssm['state_cov'].shape, dtype=params.dtype) + state_cov_lower[self._idx_lower_state_cov] = params[self._params_state_cov] + self.ssm['state_cov'] = np.dot(state_cov_lower, state_cov_lower.T) + + # 4. Observation covariance + if self.measurement_error: + self.ssm[self._idx_obs_cov] = params[self._params_obs_cov] From 97f0f6e55e11df7516baa22ad01e982a0e65de31 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Tue, 28 Jul 2015 16:49:19 -0700 Subject: [PATCH 10/57] ENH: Add Dynamic factor model. --- statsmodels/tsa/statespace/dynamic_factor.py | 333 +++++++++++++++++++ 1 file changed, 333 insertions(+) create mode 100644 statsmodels/tsa/statespace/dynamic_factor.py diff --git a/statsmodels/tsa/statespace/dynamic_factor.py b/statsmodels/tsa/statespace/dynamic_factor.py new file mode 100644 index 00000000000..de884d8805d --- /dev/null +++ b/statsmodels/tsa/statespace/dynamic_factor.py @@ -0,0 +1,333 @@ +""" +Dynamic factor model + +Author: Chad Fulton +License: Simplified-BSD +""" +from __future__ import division, absolute_import, print_function + +from warnings import warn +from statsmodels.compat.collections import OrderedDict + +import numpy as np +import pandas as pd +from .kalman_filter import KalmanFilter, FilterResults +from .mlemodel import MLEModel, MLEResults, MLEResultsWrapper +from .tools import ( + companion_matrix, diff, is_invertible, + constrain_stationary_multivariate, unconstrain_stationary_multivariate +) +from scipy.linalg import solve_discrete_lyapunov +from statsmodels.tools.pca import PCA +from statsmodels.tsa.arima_model import ARMA +from statsmodels.tsa.vector_ar.var_model import VAR +from statsmodels.tools.tools import Bunch +from statsmodels.tools.data import _is_using_pandas +from statsmodels.tsa.tsatools import lagmat +from statsmodels.tools.decorators import cache_readonly +import statsmodels.base.wrapper as wrap + + +class StaticFactors(MLEModel): + r""" + Static form of the dynamic factor model + + Parameters + ---------- + endog : array_like + The observed time-series process :math:`y` + k_factors : int + The number of unobserved factors. + factor_order : int + The order of the vector autoregression followed by the factors. + enforce_stationarity : boolean, optional + Whether or not to transform the AR parameters to enforce stationarity + in the autoregressive component of the model. Default is True. + **kwargs + Keyword arguments may be used to provide default values for state space + matrices or for Kalman filtering options. See `Representation`, and + `KalmanFilter` for more details. + + Attributes + ---------- + + Notes + ----- + The static form of the dynamic factor model is specified: + + .. math:: + + X_t & = \Lambda F_t + \varepsilon_t \\ + F_t & = \Psi(L) F_{t-1} + \eta_t + + where there are :math:`N` observed series and :math:`q` unobserved factors + so that :math:`X_t, \varepsilon_t` are :math:`(N x 1)` and + :math:`F_t, \eta_t` are :math:`(q x 1)`. + + We assume that :math:`\varepsilon_{it} \sim N(0, \sigma_{\varepsilon_i}^2)` + and :math:`\eta_t \sim N(0, I)` + + References + ---------- + .. [1] Durbin, James, and Siem Jan Koopman. 2012. + Time Series Analysis by State Space Methods: Second Edition. + Oxford University Press. + .. [2] Lutkepohl, Helmut. 2007. + New Introduction to Multiple Time Series Analysis. + Berlin: Springer. + + """ + + def __init__(self, endog, k_factors, factor_order, + enforce_stationarity=True, **kwargs): + + # Model parameters + self.k_factors = k_factors + self.enforce_stationarity = enforce_stationarity + + # Save given orders + self.factor_order = factor_order + + # Calculate the number of states + k_states = self.factor_order * self.k_factors + k_posdef = self.k_factors + + # By default, initialize as stationary + kwargs.setdefault('initialization', 'stationary') + + # Initialize the state space model + super(StaticFactors, self).__init__( + endog, k_states=k_states, k_posdef=k_posdef, **kwargs + ) + + # Initialize the parameters + self.parameters = OrderedDict() + self.parameters['factor_loadings'] = self.k_endog * self.k_factors + self.parameters['idiosyncratic'] = self.k_endog + self.parameters['transition'] = self.k_factors**2 * self.factor_order + self.k_params = sum(self.parameters.values()) + + # Setup fixed components of state space matrices + self.ssm['selection', :k_posdef, :k_posdef] = np.eye(k_posdef) + self.ssm['transition', k_factors:, :k_factors * (factor_order - 1)] = ( + np.eye(k_factors * (factor_order - 1))) + self.ssm['state_cov'] = np.eye(self.k_factors) + + # Setup indices of state space matrices + idx = np.diag_indices(self.k_endog) + self._idx_obs_cov = ('obs_cov', idx[0], idx[1]) + self._idx_transition = np.s_['transition', :self.k_factors, :] + + # Cache some slices + def _slice(key, offset): + length = self.parameters[key] + param_slice = np.s_[offset:offset + length] + offset += length + return param_slice, offset + + offset = 0 + self._params_loadings, offset = _slice('factor_loadings', offset) + self._params_idiosyncratic, offset = _slice('idiosyncratic', offset) + self._params_transition, offset = _slice('transition', offset) + + @property + def start_params(self): + params = np.zeros(self.k_params, dtype=np.float64) + + # Use principal components as starting values + res_pca = PCA(self.endog, ncomp=self.k_factors) + + # 1. Factor loadings (estimated via PCA) + params[self._params_loadings] = res_pca.loadings.ravel() + + # 2. Idiosyncratic variances + resid = self.endog - np.dot(res_pca.factors, res_pca.loadings.T) + params[self._params_idiosyncratic] = resid.var(axis=0) + + if self.k_factors > 1: + # 3a. VAR transition (OLS on factors estimated via PCA) + mod_var = VAR(res_pca.factors) + res_var = mod_var.fit(maxlags=self.factor_order, ic=None, + trend='nc') + params[self._params_transition] = res_var.params.T.ravel() + else: + # 3b. AR transition + Y = res_pca.factors[self.factor_order:] + X = lagmat(res_pca.factors, self.factor_order, trim='both') + params_ar = np.linalg.pinv(X).dot(Y) + resid = Y - np.dot(X, params_ar) + params[self._params_transition] = params_ar[:,0] + + return params + + @property + def param_names(self): + param_names = [] + + # 1. Factor loadings (estimated via PCA) + param_names += [ + 'loading%d.y%d' % (j+1, i+1) + for i in range(self.k_endog) + for j in range(self.k_factors) + ] + + # 2. Idiosyncratic variances + param_names += ['sigma2.y%d' % (i+1) for i in range(self.k_endog)] + + # 3. VAR transition (OLS on factors estimated via PCA) + param_names += [ + 'L%df%d.f%d' % (i+1, k+1, j+1) + for i in range(self.factor_order) + for j in range(self.k_factors) + for k in range(self.k_factors) + ] + + return param_names + + def transform_params(self, unconstrained): + """ + Transform unconstrained parameters used by the optimizer to constrained + parameters used in likelihood evaluation + + Parameters + ---------- + unconstrained : array_like + Array of unconstrained parameters used by the optimizer, to be + transformed. + + Returns + ------- + constrained : array_like + Array of constrained parameters which may be used in likelihood + evalation. + + Notes + ----- + Constrains the factor transition to be stationary and variances to be + positive. + """ + unconstrained = np.array(unconstrained, ndmin=1) + constrained = np.zeros(unconstrained.shape, dtype=unconstrained.dtype) + + # The factor loadings do not need to be adjusted + constrained[self._params_loadings] = ( + unconstrained[self._params_loadings]) + + # The observation variances must be positive + constrained[self._params_idiosyncratic] = ( + unconstrained[self._params_idiosyncratic]**2) + + # VAR transition: optionally force to be stationary + if self.enforce_stationarity: + # Transform the parameters + coefficients = unconstrained[self._params_transition].reshape(self.k_factors, self.k_factors * self.factor_order) + unconstrained_matrices = [coefficients[:,i*self.k_factors:(i+1)*self.k_factors] for i in range(self.factor_order)] + coefficient_matrices, variance = ( + constrain_stationary_multivariate(unconstrained_matrices, self.ssm['state_cov'])) + constrained[self._params_transition] = np.concatenate(coefficient_matrices, axis=1).ravel() + else: + constrained[self._params_transition] = unconstrained[self._params_transition] + + return constrained + + def untransform_params(self, constrained): + """ + Transform constrained parameters used in likelihood evaluation + to unconstrained parameters used by the optimizer. + + Parameters + ---------- + constrained : array_like + Array of constrained parameters used in likelihood evalution, to be + transformed. + + Returns + ------- + unconstrained : array_like + Array of unconstrained parameters used by the optimizer. + """ + constrained = np.array(constrained, ndmin=1) + unconstrained = np.zeros(constrained.shape, dtype=constrained.dtype) + + # The factor loadings do not need to be adjusted + unconstrained[self._params_loadings] = ( + constrained[self._params_loadings]) + + # The observation variances must have been positive + unconstrained[self._params_idiosyncratic] = ( + constrained[self._params_idiosyncratic]**0.5) + + # VAR transition: optionally were forced to be stationary + if self.enforce_stationarity: + coefficients = constrained[self._params_transition].reshape(self.k_factors, self.k_factors * self.factor_order) + coefficient_matrices = [coefficients[:,i*self.k_factors:(i+1)*self.k_factors] for i in range(self.factor_order)] + unconstrained_matrices, variance = ( + unconstrain_stationary_multivariate(coefficient_matrices, self.ssm['state_cov'])) + unconstrained[self._params_transition] = np.concatenate(unconstrained_matrices, axis=1).ravel() + else: + unconstrained[self._params_transition] = constrained[self._params_transition] + + return unconstrained + + def update(self, params, transformed=True): + """ + Update the parameters of the model + + Updates the representation matrices to fill in the new parameter + values. + + Parameters + ---------- + params : array_like + Array of new parameters. + transformed : boolean, optional + Whether or not `params` is already transformed. If set to False, + `transform_params` is called. Default is True.. + + Returns + ------- + params : array_like + Array of parameters. + + Notes + ----- + Let `n = k_endog`, `m = k_factors`, and `p = factor_order`. Then the + `params` vector has length + :math:`[n \times m] + [n] + [m^2 \times p] + [m \times (m + 1) / 2]`. + It is expanded in the following way: + + - The first :math:`n \times m` parameters fill out the factor loading + matrix, starting from the [0,0] entry and then proceeding along rows. + These parameters are not modified in `transform_params`. + - The next :math:`n` parameters provide variances for the idiosyncratic + errors in the observation equation. They fill in the diagonal of the + observation covariance matrix, and are constrained to be positive by + `transofrm_params`. + - The next :math:`m^2 \times p` parameters are used to create the `p` + coefficient matrices for the vector autoregression describing the + factor transition. They are transformed in `transform_params` to + enforce stationarity of the VAR(p). They are placed so as to make + the transition matrix a companion matrix for the VAR. In particular, + we assume that the first :math:`m^2` parameters fill the first + coefficient matrix (starting at [0,0] and filling along rows), the + second :math:`m^2` parameters fill the second matrix, etc. + - The last :math:`m \times (m + 1) / 2` parameters are used to fill in + a lower-triangular matrix, which is multipled with its transpose to + create a positive definite variance / covariance matrix for the + factor's VAR. The are not transformed in `transform_params` because + the matrix multiplication procedure ensures the variance terms are + positive and the matrix is positive definite. + + """ + params = super(StaticFactors, self).update(params, transformed) + + # Update the design / factor loading matrix + self.ssm['design', :, :self.k_factors] = ( + params[self._params_loadings].reshape(self.k_endog, self.k_factors) + ) + + # Update the observation covariance + self.ssm[self._idx_obs_cov] = params[self._params_idiosyncratic] + + # Update the transition matrix + self.ssm[self._idx_transition] = params[self._params_transition].reshape(self.k_factors, self.k_factors * self.factor_order) From 71964a0013391186e731b91567104fb2e7197b9d Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 3 Aug 2015 19:17:46 -0700 Subject: [PATCH 11/57] TST: VARMAX tests for VAR(1) model. --- .../tests/results/results_varmax.py | 97 ++++++++++++++ .../tests/results/results_varmax_stata.csv | 93 +++++++++++++ .../tests/results/test_varmax_stata.do | 48 +++++++ .../tsa/statespace/tests/test_varmax.py | 122 ++++++++++++++++++ 4 files changed, 360 insertions(+) create mode 100644 statsmodels/tsa/statespace/tests/results/results_varmax.py create mode 100644 statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv create mode 100644 statsmodels/tsa/statespace/tests/results/test_varmax_stata.do create mode 100644 statsmodels/tsa/statespace/tests/test_varmax.py diff --git a/statsmodels/tsa/statespace/tests/results/results_varmax.py b/statsmodels/tsa/statespace/tests/results/results_varmax.py new file mode 100644 index 00000000000..9483df56924 --- /dev/null +++ b/statsmodels/tsa/statespace/tests/results/results_varmax.py @@ -0,0 +1,97 @@ +""" +Results for VARMAX tests + +Results from Stata using script `test_varmax_stata.do`. +See also Stata time series documentation, in particular `dfactor`. + +Data from: + +http://www.jmulti.de/download/datasets/e1.dat + +Author: Chad Fulton +License: Simplified-BSD +""" + +# See http://www.jmulti.de/download/datasets/e1.dat +# 1960:Q1 - 1982Q4 +lutkepohl_data = [ + [180, 451, 415], [179, 465, 421], [185, 485, 434], [192, 493, 448], + [211, 509, 459], [202, 520, 458], [207, 521, 479], [214, 540, 487], + [231, 548, 497], [229, 558, 510], [234, 574, 516], [237, 583, 525], + [206, 591, 529], [250, 599, 538], [259, 610, 546], [263, 627, 555], + [264, 642, 574], [280, 653, 574], [282, 660, 586], [292, 694, 602], + [286, 709, 617], [302, 734, 639], [304, 751, 653], [307, 763, 668], + [317, 766, 679], [314, 779, 686], [306, 808, 697], [304, 785, 688], + [292, 794, 704], [275, 799, 699], [273, 799, 709], [301, 812, 715], + [280, 837, 724], [289, 853, 746], [303, 876, 758], [322, 897, 779], + [315, 922, 798], [339, 949, 816], [364, 979, 837], [371, 988, 858], + [375, 1025, 881], [432, 1063, 905], [453, 1104, 934], [460, 1131, 968], + [475, 1137, 983], [496, 1178, 1013], [494, 1211, 1034], [498, 1256, 1064], + [526, 1290, 1101], [519, 1314, 1102], [516, 1346, 1145], [531, 1385, 1173], + [573, 1416, 1216], [551, 1436, 1229], [538, 1462, 1242], [532, 1493, 1267], + [558, 1516, 1295], [524, 1557, 1317], [525, 1613, 1355], [519, 1642, 1371], + [526, 1690, 1402], [510, 1759, 1452], [519, 1756, 1485], [538, 1780, 1516], + [549, 1807, 1549], [570, 1831, 1567], [559, 1873, 1588], [584, 1897, 1631], + [611, 1910, 1650], [597, 1943, 1685], [603, 1976, 1722], [619, 2018, 1752], + [635, 2040, 1774], [658, 2070, 1807], [675, 2121, 1831], [700, 2132, 1842], + [692, 2199, 1890], [759, 2253, 1958], [782, 2276, 1948], [816, 2318, 1994], + [844, 2369, 2061], [830, 2423, 2056], [853, 2457, 2102], [852, 2470, 2121], + [833, 2521, 2145], [860, 2545, 2164], [870, 2580, 2206], [830, 2620, 2225], + [801, 2639, 2235], [824, 2618, 2237], [831, 2628, 2250], [830, 2651, 2271], +] + +lutkepohl_ar1 = { + 'params': [ + -0.25034303, 0.28759168, 0.81626475, # Phi, row 1 + 0.023383, 0.19048278, 0.66502259, # Phi, row 2 + -0.01492992, 0.53796097, 0.28114733, # Phi, row 3 + 0.04464236, # Covariance, lower triangle + 0.00136552, 0.01354125, + 0.0029089 , 0.00834324, 0.00915471], + 'var_oim': [ + .01319669, .19413864, .2386643, + .0012437, .01829378, .02234399, + .00107749, .01584584, .01938099, + 1.061e-07, + 4.981e-09, 4.549e-09, + 9.211e-10, 5.825e-10, 7.016e-10], + 'loglike': 587.8481018831948, + 'aic': -1145.696, + 'bic': -1110.934, +} + +lutkepohl_ar1_diag = { + 'params': [ + -0.24817904, 0.29283012, 0.80794938, # Phi, row 1 + 0.02282985, 0.19672157, 0.66329776, # Phi, row 2 + -0.01522531, 0.53500874, 0.28859213, # Phi, row 3 + 0.00199106, 0.00018529, 0.00016179 # Variances, diagonal + ], + 'var_oim': [ + .01314245, .1902972, .23400828, + .00124336, .01840132, .02229946, + .00107322, .01558391, .01909303, + 1.057e-07, 9.233e-10, 7.011e-10], + 'loglike': 562.8168476509002, + 'aic': -1101.634, + 'bic': -1073.824 +} + +lutkepohl_ar1_obs_intercept = { + 'params': [ + -.24762, .25961003, .75992623, # Phi, row 1 + .03186854, -.07271862, .23697765, # Phi, row 2 + -.0053055, .2362571, -.19438311, # Phi, row 3 + .00199116, .00013515, .00009937 # Variances, diagonal + ], + 'obs_intercept': [.01799302, .02065458, .01987525], # Intercepts + 'var_oim': [ + .01317874, .2311403, .33481866, + .00090084, .0157839, .0229119, + .00065737, .01149729, .01661236, + # .00001802, 1.818e-06, 1.086e-06, # Intercept parameters + 1.057e-07, 4.869e-10, 2.630e-10], + 'loglike': 593.5252693885262, + 'aic': -1101.634, + 'bic': -1073.824 +} diff --git a/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv b/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv new file mode 100644 index 00000000000..cc7cd7eac90 --- /dev/null +++ b/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv @@ -0,0 +1,93 @@ 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equation(dln_inc) +predict predict_3, dynamic(tq(1979q1)) equation(dln_consump) + +// These are predict in-sample for first 3-observations (1960q2-1960q4), then +// dynamic predict for the rest of in-sample observations (1961q1-1978q4), then +// forecast for the remaining periods 1979q1 - 1982q4 +predict dyn_predict_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_3, dynamic(tq(1961q1)) equation(dln_consump) + +// VAR(1), diagonal covariance +dfactor (dln_inv dln_inc dln_consump = , ar(1) arstructure(general) noconstant covstructure(diagonal)) if qtr<=tq(1978q4) +estat ic + +// predict + forecast, see above +predict predict_diag1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_diag2, dynamic(tq(1979q1)) equation(dln_inc) +predict predict_diag3, dynamic(tq(1979q1)) equation(dln_consump) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_diag1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_diag2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_diag3, dynamic(tq(1961q1)) equation(dln_consump) + +// VAR(1), diagonal covariance + observation intercept +dfactor (dln_inv dln_inc dln_consump = , ar(1) arstructure(general) covstructure(diagonal)) if qtr<=tq(1978q4) +estat ic + +// predict + forecast, see above +predict predict_int1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_int2, dynamic(tq(1979q1)) equation(dln_inc) +predict predict_int3, dynamic(tq(1979q1)) equation(dln_consump) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_int1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_int2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_int3, dynamic(tq(1961q1)) equation(dln_consump) + +outsheet pred* dyn* using results_varmax_stata.csv, comma replace diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py new file mode 100644 index 00000000000..b500ef20482 --- /dev/null +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -0,0 +1,122 @@ +""" +Tests for VARMAX models + +Author: Chad Fulton +License: Simplified-BSD +""" +from __future__ import division, absolute_import, print_function + +import numpy as np +import pandas as pd +import os + +import warnings +from statsmodels.datasets import webuse +from statsmodels.tsa.statespace import varmax +from .results import results_varmax +from numpy.testing import assert_equal, assert_almost_equal, assert_raises, assert_allclose + +current_path = os.path.dirname(os.path.abspath(__file__)) + +output_path = 'results' + os.sep + 'results_varmax_stata.csv' +output_results = pd.read_csv(current_path + os.sep + output_path) + + + +class CheckVAR(object): + """ + Test Vector Autoregression against Stata's `dfactor` code (Stata's + `var` function uses OLS and not state space / MLE, so we can't get + equivalent log-likelihoods) + """ + + def __init__(self, true, order, trend, error_cov_type, cov_type='oim', + **kwargs): + self.true = true + # 1960:Q1 - 1982:Q4 + dta = pd.DataFrame( + results_varmax.lutkepohl_data, columns=['inv', 'inc', 'consump'], + index=pd.date_range('1960-01-01', '1982-10-01', freq='QS')) + + dta['dln_inv'] = np.log(dta['inv']).diff() + dta['dln_inc'] = np.log(dta['inc']).diff() + dta['dln_consump'] = np.log(dta['consump']).diff() + + endog = dta.ix['1960-04-01':'1978-10-01', + ['dln_inv', 'dln_inc', 'dln_consump']] + + self.model = varmax.VARMAX(endog, order=order, trend=trend, + error_cov_type=error_cov_type, **kwargs) + + self.results = self.model.filter(true['params'], cov_type=cov_type) + + def test_mle(self): + results = self.model.fit(method='powell', disp=-1) + results = self.model.fit(results.params, disp=False) + assert_allclose(results.llf, self.results.llf, rtol=1e-4) + + def test_loglike(self): + assert_allclose(self.results.llf, self.true['loglike'], rtol=1e-6) + + def test_bse_oim(self): + assert_allclose(self.results.bse**2, self.true['var_oim'], atol=1e-2) + + def test_aic(self): + # We only get 3 digits from Stata + assert_allclose(self.results.aic, self.true['aic'], atol=3) + + def test_bic(self): + # We only get 3 digits from Stata + assert_allclose(self.results.bic, self.true['bic'], atol=3) + + def test_predict(self): + # Tests predict + forecast + assert_allclose( + self.results.predict(end='1982-10-01'), + self.true['predict'].T, + atol=1e-6) + + def test_dynamic_predict(self): + # Tests predict + dynamic predict + forecast + assert_allclose( + self.results.predict(end='1982-10-01', dynamic='1961-01-01'), + self.true['dynamic_predict'].T, + atol=1e-6) + + +class TestVAR(CheckVAR): + def __init__(self): + true = results_varmax.lutkepohl_ar1.copy() + true['predict'] = output_results.ix[1:, ['predict_1', 'predict_2', 'predict_3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_1', 'dyn_predict_2', 'dyn_predict_3']] + super(TestVAR, self).__init__( + true, order=(1,0), trend='nc', + error_cov_type="unstructured") + +class TestVAR_diagonal(CheckVAR): + def __init__(self): + true = results_varmax.lutkepohl_ar1_diag.copy() + true['predict'] = output_results.ix[1:, ['predict_diag1', 'predict_diag2', 'predict_diag3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_diag1', 'dyn_predict_diag2', 'dyn_predict_diag3']] + super(TestVAR_diagonal, self).__init__( + true, order=(1,0), trend='nc', + error_cov_type="diagonal") + +class TestVAR_obs_intercept(CheckVAR): + def __init__(self): + true = results_varmax.lutkepohl_ar1_obs_intercept.copy() + true['predict'] = output_results.ix[1:, ['predict_int1', 'predict_int2', 'predict_int3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_int1', 'dyn_predict_int2', 'dyn_predict_int3']] + super(TestVAR_obs_intercept, self).__init__( + true, order=(1,0), trend='nc', + error_cov_type="diagonal", obs_intercept=true['obs_intercept']) + + def test_aic(self): + # Since the obs_intercept is added in in an ad-hoc way here, the number + # of parameters, and hence the aic and bic, will be off + pass + + def test_bic(self): + # Since the obs_intercept is added in in an ad-hoc way here, the number + # of parameters, and hence the aic and bic, will be off + pass \ No newline at end of file From e666bcd5c97532e806e65a810869daa02410bcca Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 5 Aug 2015 00:53:28 -0700 Subject: [PATCH 12/57] ENH: Add triangular solver ?trtrs. --- statsmodels/src/blas_lapack.pxd | 68 +++++++++++++++++++++++++++++++++ 1 file changed, 68 insertions(+) diff --git a/statsmodels/src/blas_lapack.pxd b/statsmodels/src/blas_lapack.pxd index cf6076edf37..03c1001bd2e 100644 --- a/statsmodels/src/blas_lapack.pxd +++ b/statsmodels/src/blas_lapack.pxd @@ -593,6 +593,23 @@ ctypedef int spotrs_t( int *info # 0 if success, otherwise an error code (integer) ) +ctypedef int strtrs_t( + # STRTRS solves a triangular system of the form + # A * X = B, A**T * X = B, or A**H * X = B, + # where A is a triangular matrix of order N, and B is an N-by-NRHS + # matrix. A check is made to verify that A is nonsingular. + char *uplo, # 'U': A is upper triangular + char *trans, # N: A * X = B; T: A**T * X = B; C: A**H * X = B + char *diag, # {'U','N'}: unit triangular or not + int *n, # The order of the matrix A. n >= 0. + int *nrhs, # The number of right hand sides + np.float32_t *a, # Matrix A: nxn + int *lda, # The size of the first dimension of A (in memory) + np.float32_t *b, # Matrix B: nxnrhs + int *ldb, # The size of the first dimension of B (in memory) + int *info # 0 if success, otherwise an error code (integer) +) + ctypedef int dgetrf_t( # DGETRF - compute an LU factorization of a general M-by-N # matrix A using partial pivoting with row interchanges @@ -666,6 +683,23 @@ ctypedef int dpotrs_t( int *info # 0 if success, otherwise an error code (integer) ) +ctypedef int dtrtrs_t( + # DTRTRS solves a triangular system of the form + # A * X = B, A**T * X = B, or A**H * X = B, + # where A is a triangular matrix of order N, and B is an N-by-NRHS + # matrix. A check is made to verify that A is nonsingular. + char *uplo, # 'U': A is upper triangular + char *trans, # N: A * X = B; T: A**T * X = B; C: A**H * X = B + char *diag, # {'U','N'}: unit triangular or not + int *n, # The order of the matrix A. n >= 0. + int *nrhs, # The number of right hand sides + np.float64_t *a, # Matrix A: nxn + int *lda, # The size of the first dimension of A (in memory) + np.float64_t *b, # Matrix B: nxnrhs + int *ldb, # The size of the first dimension of B (in memory) + int *info # 0 if success, otherwise an error code (integer) +) + ctypedef int cgetrf_t( # CGETRF - compute an LU factorization of a general M-by-N # matrix A using partial pivoting with row interchanges @@ -739,6 +773,23 @@ ctypedef int cpotrs_t( int *info # 0 if success, otherwise an error code (integer) ) +ctypedef int ctrtrs_t( + # CTRTRS solves a triangular system of the form + # A * X = B, A**T * X = B, or A**H * X = B, + # where A is a triangular matrix of order N, and B is an N-by-NRHS + # matrix. A check is made to verify that A is nonsingular. + char *uplo, # 'U': A is upper triangular + char *trans, # N: A * X = B; T: A**T * X = B; C: A**H * X = B + char *diag, # {'U','N'}: unit triangular or not + int *n, # The order of the matrix A. n >= 0. + int *nrhs, # The number of right hand sides + np.complex64_t *a, # Matrix A: nxn + int *lda, # The size of the first dimension of A (in memory) + np.complex64_t *b, # Matrix B: nxnrhs + int *ldb, # The size of the first dimension of B (in memory) + int *info # 0 if success, otherwise an error code (integer) +) + ctypedef int zgetrf_t( # ZGETRF - compute an LU factorization of a general M-by-N # matrix A using partial pivoting with row interchanges @@ -810,4 +861,21 @@ ctypedef int zpotrs_t( np.complex128_t *b, # Matrix B: nxnrhs int *ldb, # The size of the first dimension of B (in memory) int *info # 0 if success, otherwise an error code (integer) +) + +ctypedef int ztrtrs_t( + # ZTRTRS solves a triangular system of the form + # A * X = B, A**T * X = B, or A**H * X = B, + # where A is a triangular matrix of order N, and B is an N-by-NRHS + # matrix. A check is made to verify that A is nonsingular. + char *uplo, # 'U': A is upper triangular + char *trans, # N: A * X = B; T: A**T * X = B; C: A**H * X = B + char *diag, # {'U','N'}: unit triangular or not + int *n, # The order of the matrix A. n >= 0. + int *nrhs, # The number of right hand sides + np.complex128_t *a, # Matrix A: nxn + int *lda, # The size of the first dimension of A (in memory) + np.complex128_t *b, # Matrix B: nxnrhs + int *ldb, # The size of the first dimension of B (in memory) + int *info # 0 if success, otherwise an error code (integer) ) \ No newline at end of file From 1e19f27b3522d0a72b15d711e73141885adaf89a Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 5 Aug 2015 00:54:45 -0700 Subject: [PATCH 13/57] ENH: Add Cython constrain_stationary_multivariate. --- statsmodels/tsa/statespace/_statespace.pyx.in | 355 +++++++++++++++++- statsmodels/tsa/statespace/tools.py | 259 ++++++++++++- statsmodels/tsa/statespace/varmax.py | 42 ++- 3 files changed, 633 insertions(+), 23 deletions(-) diff --git a/statsmodels/tsa/statespace/_statespace.pyx.in b/statsmodels/tsa/statespace/_statespace.pyx.in index f341c81d464..b858f70598e 100644 --- a/statsmodels/tsa/statespace/_statespace.pyx.in +++ b/statsmodels/tsa/statespace/_statespace.pyx.in @@ -17,8 +17,8 @@ TYPES = { "z": ("np.complex128_t", "complex", "np.NPY_COMPLEX128"), } -BLAS_FUNCTIONS = ["gemm", "gemv", "copy", "axpy", "scal"] -LAPACK_FUNCTIONS = ["getrf", "getri", "getrs", "potrf", "potri", "potrs"] +BLAS_FUNCTIONS = ["gemm", "gemv", "copy", "axpy", "scal", "trmm", "symm"] +LAPACK_FUNCTIONS = ["getrf", "getri", "getrs", "potrf", "potri", "potrs", "trtrs"] }} @@ -1755,4 +1755,355 @@ cdef class {{prefix}}KalmanFilter(object): {{prefix}}copy(&self.k_states, &self.predicted_state[0, 2], &inc, &self.predicted_state[0, 1], &inc) {{prefix}}copy(&self.k_states2, &self.predicted_state_cov[0, 0, 2], &inc, &self.predicted_state_cov[0, 0, 1], &inc) +cpdef _{{prefix}}compute_coefficients_from_multivariate_pacf({{cython_type}} [::1,:] partial_autocorrelations, + {{cython_type}} [::1,:] error_variance, + int order, int k_endog, + int transform_variance=False): + """ + Notes + ----- + + This uses the ?trmm BLAS functions which are not available in + Scipy v0.11.0 + """ + # Constants + cdef: + int inc = 1 + {{cython_type}} alpha = 1.0 + {{cython_type}} beta = 0.0 + {{cython_type}} gamma = -1.0 + int k_endog2 = k_endog**2 + int k_endog_order = k_endog * order + int k_endog_order1 = k_endog * (order+1) + int info, s, k + # Local variables + cdef: + np.npy_intp dim2[2] + {{cython_type}} [::1, :] initial_variance + {{cython_type}} [::1, :] forward_variance + {{cython_type}} [::1, :] backward_variance + {{cython_type}} [::1, :] autocovariances + {{cython_type}} [::1, :] forwards1 + {{cython_type}} [::1, :] forwards2 + {{cython_type}} [::1, :] backwards1 + {{cython_type}} [::1, :] backwards2 + {{cython_type}} [::1, :] forward_factors + {{cython_type}} [::1, :] backward_factors + {{cython_type}} [::1, :] tmp + {{cython_type}} [::1, :] tmp2 + # Pointers + cdef: + {{cython_type}} * forwards + {{cython_type}} * prev_forwards + {{cython_type}} * backwards + {{cython_type}} * prev_backwards + + # dim2[0] = self.k_endog; dim2[1] = storage; + # self.forecast = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + + # If we want to keep the provided variance but with the constrained + # coefficient matrices, we need to make a copy here, and then after the + # main loop we will transform the coefficients to match the passed variance + if not transform_variance: + initial_variance = np.asfortranarray(error_variance.copy()) + # Need to make the input variance large enough that the recursions + # don't lead to zero-matrices due to roundoff error, which would case + # exceptions from the Cholesky decompositions. + # Note that this will still not always ensure positive definiteness, + # and for k_endog, order large enough an exception may still be raised + error_variance = np.asfortranarray(np.eye(k_endog) * (order + k_endog)**10) + + # Initialize matrices + dim2[0] = k_endog; dim2[1] = k_endog; + forward_variance = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + backward_variance = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + forward_factors = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + backward_factors = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + tmp = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + tmp2 = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + + dim2[0] = k_endog; dim2[1] = k_endog_order; + # \phi_{s,k}, s = 1, ..., p + # k = 1, ..., s+1 + forwards1 = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + forwards2 = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + # \phi_{s,k}^* + backwards1 = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + backwards2 = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + + dim2[0] = k_endog; dim2[1] = k_endog_order1; + autocovariances = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + + {{prefix}}copy(&k_endog2, &error_variance[0,0], &inc, &forward_variance[0,0], &inc) # \Sigma_s + {{prefix}}copy(&k_endog2, &error_variance[0,0], &inc, &backward_variance[0,0], &inc) # \Sigma_s^*, s = 0, ..., p + {{prefix}}copy(&k_endog2, &error_variance[0,0], &inc, &autocovariances[0,0], &inc) # \Gamma_s + + # error_variance_factor = linalg.cholesky(error_variance, lower=True) + {{prefix}}copy(&k_endog2, &error_variance[0,0], &inc, &forward_factors[0,0], &inc) + {{prefix}}potrf("L", &k_endog, &forward_factors[0,0], &k_endog, &info) + {{prefix}}copy(&k_endog2, &forward_factors[0,0], &inc, &backward_factors[0,0], &inc) + + # We fill in the entries as follows: + # [1,1] + # [2,2], [2,1] + # [3,3], [3,1], [3,2] + # ... + # [p,p], [p,1], ..., [p,p-1] + # the last row, correctly ordered, is then used as the coefficients + for s in range(order): # s = 0, ..., p-1 + if s % 2 == 0: + forwards = &forwards1[0, 0] + prev_forwards = &forwards2[0, 0] + backwards = &backwards1[0, 0] + prev_backwards = &backwards2[0, 0] + else: + forwards = &forwards2[0, 0] + prev_forwards = &forwards1[0, 0] + backwards = &backwards2[0, 0] + prev_backwards = &backwards1[0, 0] + + # Create the "last" (k = s+1) matrix + # Note: this is for k = s+1. However, below we then have to fill + # in for k = 1, ..., s in order. + # P L*^{-1} = x + # x L* = P + # L*' x' = P' + # forwards[:, s*k_endog:(s+1)*k_endog] = np.dot( + # forward_factors, + # linalg.solve_triangular( + # backward_factors, partial_autocorrelations[:, s*k_endog:(s+1)*k_endog].T, + # lower=True, trans='T').T + # ) + for k in range(k_endog): + {{prefix}}copy(&k_endog, &partial_autocorrelations[k,s*k_endog], &k_endog, &tmp[0, k], &inc) + {{prefix}}trtrs("L", "T", "N", &k_endog, &k_endog, &backward_factors[0,0], &k_endog, + &tmp[0, 0], &k_endog, &info) + # {{prefix}}gemm("N", "T", &k_endog, &k_endog, &k_endog, + # &alpha, &forward_factors[0,0], &k_endog, + # &tmp[0, 0], &k_endog, + # &beta, &forwards[s*k_endog2], &k_endog) + {{prefix}}trmm("R", "L", "T", "N", &k_endog, &k_endog, + &alpha, &forward_factors[0,0], &k_endog, + &tmp[0, 0], &k_endog) + for k in range(k_endog): + {{prefix}}copy(&k_endog, &tmp[k,0], &k_endog, &forwards[s*k_endog2 + k*k_endog], &inc) + + # P' L^{-1} = x + # x L = P' + # L' x' = P + # backwards[:, s*k_endog:(s+1)*k_endog] = np.dot( + # backward_factors, + # linalg.solve_triangular( + # forward_factors, partial_autocorrelations[:, s*k_endog:(s+1)*k_endog], + # lower=True, trans='T').T + # ) + {{prefix}}copy(&k_endog2, &partial_autocorrelations[0, s*k_endog], &inc, &tmp[0, 0], &inc) + {{prefix}}trtrs("L", "T", "N", &k_endog, &k_endog, &forward_factors[0, 0], &k_endog, + &tmp[0, 0], &k_endog, &info) + # {{prefix}}gemm("N", "T", &k_endog, &k_endog, &k_endog, + # &alpha, &backward_factors[0, 0], &k_endog, + # &tmp[0, 0], &k_endog, + # &beta, &backwards[s * k_endog2], &k_endog) + {{prefix}}trmm("R", "L", "T", "N", &k_endog, &k_endog, + &alpha, &backward_factors[0,0], &k_endog, + &tmp[0, 0], &k_endog) + for k in range(k_endog): + {{prefix}}copy(&k_endog, &tmp[k,0], &k_endog, &backwards[s*k_endog2 + k*k_endog], &inc) + + # Update the variance + # Note: if s >= 1, this will be further updated in the for loop + # below + # Also, this calculation will be re-used in the forward variance + # tmp = np.dot(forwards[:, s*k_endog:(s+1)*k_endog], backward_variance) + # tmpT = np.dot(backward_variance.T, forwards[:, s*k_endog:(s+1)*k_endog].T) + {{prefix}}gemm("T", "T", &k_endog, &k_endog, &k_endog, + &alpha, &backward_variance[0, 0], &k_endog, + &forwards[s * k_endog2], &k_endog, + &beta, &tmp[0, 0], &k_endog) + # autocovariances[:, (s+1)*k_endog:(s+2)*k_endog] = tmp.copy().T + {{prefix}}copy(&k_endog2, &tmp[0, 0], &inc, &autocovariances[0, (s+1)*k_endog], &inc) + + # Create the remaining k = 1, ..., s matrices, + # only has an effect if s >= 1 + for k in range(s): + # forwards[:, k*k_endog:(k+1)*k_endog] = ( + # prev_forwards[:, k*k_endog:(k+1)*k_endog] - + # np.dot( + # forwards[:, s*k_endog:(s+1)*k_endog], + # prev_backwards[:, (s-k-1)*k_endog:(s-k)*k_endog] + # ) + # ) + {{prefix}}copy(&k_endog2, &prev_forwards[k * k_endog2], &inc, &forwards[k * k_endog2], &inc) + {{prefix}}gemm("N", "N", &k_endog, &k_endog, &k_endog, + &gamma, &forwards[s * k_endog2], &k_endog, + &prev_backwards[(s - k - 1) * k_endog2], &k_endog, + &alpha, &forwards[k * k_endog2], &k_endog) + + # backwards[:, k*k_endog:(k+1)*k_endog] = ( + # prev_backwards[:, k*k_endog:(k+1)*k_endog] - + # np.dot( + # backwards[:, s*k_endog:(s+1)*k_endog], + # prev_forwards[:, (s-k-1)*k_endog:(s-k)*k_endog] + # ) + # ) + {{prefix}}copy(&k_endog2, &prev_backwards[k * k_endog2], &inc, &backwards[k * k_endog2], &inc) + {{prefix}}gemm("N", "N", &k_endog, &k_endog, &k_endog, + &gamma, &backwards[s * k_endog2], &k_endog, + &prev_forwards[(s - k - 1) * k_endog2], &k_endog, + &alpha, &backwards[k * k_endog2], &k_endog) + + # autocovariances[:, (s+1)*k_endog:(s+2)*k_endog] += np.dot( + # autocovariances[:, (k+1)*k_endog:(k+2)*k_endog], + # prev_forwards[:, (s-k-1)*k_endog:(s-k)*k_endog].T + # ) + {{prefix}}gemm("N", "T", &k_endog, &k_endog, &k_endog, + &alpha, &autocovariances[0, (k+1)*k_endog], &k_endog, + &prev_forwards[(s - k - 1) * k_endog2], &k_endog, + &alpha, &autocovariances[0, (s+1)*k_endog], &k_endog) + + # Create forward and backwards variances + # backward_variance = ( + # backward_variance - + # np.dot( + # np.dot(backwards[:, s*k_endog:(s+1)*k_endog], forward_variance), + # backwards[:, s*k_endog:(s+1)*k_endog].T + # ) + # ) + {{prefix}}gemm("N", "N", &k_endog, &k_endog, &k_endog, + &alpha, &backwards[s * k_endog2], &k_endog, + &forward_variance[0, 0], &k_endog, + &beta, &tmp2[0, 0], &k_endog) + {{prefix}}gemm("N", "T", &k_endog, &k_endog, &k_endog, + &gamma, &tmp2[0, 0], &k_endog, + &backwards[s * k_endog2], &k_endog, + &alpha, &backward_variance[0, 0], &k_endog) + # forward_variance = ( + # forward_variance - + # np.dot(tmp, forwards[:, s*k_endog:(s+1)*k_endog].T) + # ) + # forward_variance = ( + # forward_variance - + # np.dot(tmpT.T, forwards[:, s*k_endog:(s+1)*k_endog].T) + # ) + {{prefix}}gemm("T", "T", &k_endog, &k_endog, &k_endog, + &gamma, &tmp[0, 0], &k_endog, + &forwards[s * k_endog2], &k_endog, + &alpha, &forward_variance[0, 0], &k_endog) + + # Cholesky factors + # forward_factors = linalg.cholesky(forward_variance, lower=True) + # backward_factors = linalg.cholesky(backward_variance, lower=True) + {{prefix}}copy(&k_endog2, &forward_variance[0,0], &inc, &forward_factors[0,0], &inc) + {{prefix}}potrf("L", &k_endog, &forward_factors[0,0], &k_endog, &info) + {{prefix}}copy(&k_endog2, &backward_variance[0,0], &inc, &backward_factors[0,0], &inc) + {{prefix}}potrf("L", &k_endog, &backward_factors[0,0], &k_endog, &info) + + + # If we do not want to use the transformed variance, we need to + # adjust the constrained matrices, as presented in Lemma 2.3, see above + if not transform_variance: + if order % 2 == 0: + forwards = &forwards2[0,0] + else: + forwards = &forwards1[0,0] + + # Here, we need to construct T such that: + # variance = T * initial_variance * T' + # To do that, consider the Cholesky of variance (L) and + # input_variance (M) to get: + # L L' = T M M' T' = (TM) (TM)' + # => L = T M + # => L M^{-1} = T + # initial_variance_factor = np.linalg.cholesky(initial_variance) + # L' + {{prefix}}potrf("U", &k_endog, &initial_variance[0,0], &k_endog, &info) + # transformed_variance_factor = np.linalg.cholesky(variance) + # M' + {{prefix}}copy(&k_endog2, &forward_variance[0,0], &inc, &tmp[0,0], &inc) + {{prefix}}potrf("U", &k_endog, &tmp[0,0], &k_endog, &info) + # {{prefix}}potri("L", &k_endog, &tmp[0,0], &k_endog, &info) + + # Note that T is lower triangular + # L M^{-1} = T + # M' T' = L' + # transform = np.dot(initial_variance_factor, + # np.linalg.inv(transformed_variance_factor)) + {{prefix}}trtrs("U", "N", "N", &k_endog, &k_endog, &tmp[0,0], &k_endog, + &initial_variance[0, 0], &k_endog, &info) + # Now: + # initial_variance = T' + + for s in range(order): + # forwards[:, s*k_endog:(s+1)*k_endog] = ( + # np.dot( + # np.dot(transform, forwards[:, s*k_endog:(s+1)*k_endog]), + # inv_transform + # ) + # ) + # TF T^{-1} = x + # TF = x T + # (TF)' = T' x' + + # Get TF + {{prefix}}copy(&k_endog2, &forwards[s * k_endog2], &inc, &tmp2[0,0], &inc) + {{prefix}}trmm("L", "U", "T", "N", &k_endog, &k_endog, + &alpha, &initial_variance[0, 0], &k_endog, + &tmp2[0, 0], &k_endog) + for k in range(k_endog): + {{prefix}}copy(&k_endog, &tmp2[k,0], &k_endog, &tmp[0, k], &inc) + # Get x' + {{prefix}}trtrs("U", "N", "N", &k_endog, &k_endog, &initial_variance[0,0], &k_endog, + &tmp[0, 0], &k_endog, &info) + # Get x + for k in range(k_endog): + {{prefix}}copy(&k_endog, &tmp[k,0], &k_endog, &forwards[s * k_endog2 + k*k_endog], &inc) + + + if order % 2 == 0: + return forwards2, forward_variance + else: + return forwards1, forward_variance + +cpdef _{{prefix}}constrain_sv_less_than_one({{cython_type}} [::1,:] unconstrained, int order, int k_endog): + """ + Transform arbitrary matrices to matrices with singular values less than + one. + + Corresponds to Lemma 2.2 in Ansley and Kohn (1986). See + `constrain_stationary_multivariate` for more details. + """ + # Constants + cdef: + int inc = 1 + {{cython_type}} alpha = 1.0 + int k_endog2 = k_endog**2 + int info, i + # Local variables + cdef: + np.npy_intp dim2[2] + {{cython_type}} [::1, :] constrained + {{cython_type}} [::1, :] tmp + {{cython_type}} [::1, :] eye + + dim2[0] = k_endog; dim2[1] = k_endog * order; + constrained = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + dim2[0] = k_endog; dim2[1] = k_endog; + tmp = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + eye = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) + + eye = np.asfortranarray(np.eye(k_endog)) + for i in range(order): + {{prefix}}copy(&k_endog2, &eye[0, 0], &inc, &tmp[0, 0], &inc) + {{prefix}}gemm("N", "T", &k_endog, &k_endog, &k_endog, + &alpha, &unconstrained[0, i*k_endog], &k_endog, + &unconstrained[0, i*k_endog], &k_endog, + &alpha, &tmp[0, 0], &k_endog) + {{prefix}}potrf("L", &k_endog, &tmp[0, 0], &k_endog, &info) + + {{prefix}}copy(&k_endog2, &unconstrained[0, i*k_endog], &inc, &constrained[0, i*k_endog], &inc) + # constrained.append(linalg.solve_triangular(B, A, lower=lower)) + {{prefix}}trtrs("L", "N", "N", &k_endog, &k_endog, &tmp[0, 0], &k_endog, + &constrained[0, i*k_endog], &k_endog, &info) + return constrained + {{endfor}} \ No newline at end of file diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index c3043b824c3..3bd1d866417 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -10,9 +10,11 @@ from statsmodels.tools.data import _is_using_pandas from . import _statespace +old_scipy_compat = False try: from scipy.linalg.blas import find_best_blas_type except ImportError: # pragma: no cover + old_scipy_compat = True # Shim for SciPy 0.11, derived from tag=0.11 scipy.linalg.blas _type_conv = {'f': 's', 'd': 'd', 'F': 'c', 'D': 'z', 'G': 'z'} @@ -34,6 +36,18 @@ def find_best_blas_type(arrays): 's': _statespace.sKalmanFilter, 'd': _statespace.dKalmanFilter, 'c': _statespace.cKalmanFilter, 'z': _statespace.zKalmanFilter } +prefix_pacf_map = { + 's': _statespace._scompute_coefficients_from_multivariate_pacf, + 'd': _statespace._dcompute_coefficients_from_multivariate_pacf, + 'c': _statespace._ccompute_coefficients_from_multivariate_pacf, + 'z': _statespace._zcompute_coefficients_from_multivariate_pacf +} +prefix_sv_map = { + 's': _statespace._sconstrain_sv_less_than_one, + 'd': _statespace._dconstrain_sv_less_than_one, + 'c': _statespace._cconstrain_sv_less_than_one, + 'z': _statespace._zconstrain_sv_less_than_one +} def companion_matrix(polynomial): @@ -359,7 +373,8 @@ def unconstrain_stationary_univariate(constrained): return x -def _constrain_sv_less_than_one(unconstrained, order=None, k_endog=None): +def _constrain_sv_less_than_one_python(unconstrained, order=None, + k_endog=None): """ Transform arbitrary matrices to matrices with singular values less than one. @@ -383,10 +398,161 @@ def _constrain_sv_less_than_one(unconstrained, order=None, k_endog=None): return constrained -def _compute_coefficients_from_multivariate_pacf(partial_autocorrelations, - error_variance, - transform_variance=False, - order=None, k_endog=None): +def _compute_coefficients_from_multivariate_pacf_python( + partial_autocorrelations, error_variance, order, k_endog, + transform_variance=False): + """ + Transform matrices with singular values less than one to matrices + corresponding to a stationary (or invertible) process. + + Corresponds to Lemma 2.1 in Ansley and Kohn (1986). See + `constrain_stationary_multivariate` for more details. + """ + from scipy import linalg + + partial_autocorrelations = np.concatenate(partial_autocorrelations, axis=1) + + # If we want to keep the provided variance but with the constrained + # coefficient matrices, we need to make a copy here, and then after the + # main loop we will transform the coefficients to match the passed variance + if not transform_variance: + initial_variance = error_variance + # Need to make the input variance large enough that the recursions + # don't lead to zero-matrices due to roundoff error, which would case + # exceptions from the Cholesky decompositions. + # Note that this will still not always ensure positive definiteness, + # and for k_endog, order large enough an exception may still be raised + error_variance = np.eye(k_endog) * (order + k_endog)**10 + + forward_variance = error_variance # \Sigma_s + backward_variance = error_variance # \Sigma_s^*, s = 0, ..., p + autocovariances = np.zeros((k_endog, k_endog * (order+1))) # \Gamma_s + autocovariances[:, :k_endog] = error_variance + # \phi_{s,k}, s = 1, ..., p + # k = 1, ..., s+1 + forwards = np.zeros((k_endog, k_endog * order)) + prev_forwards = np.zeros((k_endog, k_endog * order)) + # \phi_{s,k}^* + backwards = np.zeros((k_endog, k_endog * order)) + prev_backwards = np.zeros((k_endog, k_endog * order)) + + error_variance_factor = linalg.cholesky(error_variance, lower=True) + + forward_factors = error_variance_factor + backward_factors = error_variance_factor + + tmp = np.zeros((k_endog, k_endog)) + + # We fill in the entries as follows: + # [1,1] + # [2,2], [2,1] + # [3,3], [3,1], [3,2] + # ... + # [p,p], [p,1], ..., [p,p-1] + # the last row, correctly ordered, is then used as the coefficients + for s in range(order): # s = 0, ..., p-1 + prev_forwards = forwards.copy() + prev_backwards = backwards.copy() + + # Create the "last" (k = s+1) matrix + # Note: this is for k = s+1. However, below we then have to fill + # in for k = 1, ..., s in order. + # P L*^{-1} = x + # x L* = P + # L*' x' = P' + forwards[:, s*k_endog:(s+1)*k_endog] = np.dot( + forward_factors, + linalg.solve_triangular( + backward_factors, partial_autocorrelations[:, s*k_endog:(s+1)*k_endog].T, + lower=True, trans='T').T + ) + + # P' L^{-1} = x + # x L = P' + # L' x' = P + backwards[:, s*k_endog:(s+1)*k_endog] = np.dot( + backward_factors, + linalg.solve_triangular( + forward_factors, partial_autocorrelations[:, s*k_endog:(s+1)*k_endog], + lower=True, trans='T').T + ) + + # Update the variance + # Note: if s >= 1, this will be further updated in the for loop + # below + # Also, this calculation will be re-used in the forward variance + tmp = np.dot(forwards[:, s*k_endog:(s+1)*k_endog], backward_variance) + autocovariances[:, (s+1)*k_endog:(s+2)*k_endog] = tmp.copy().T + + # Create the remaining k = 1, ..., s matrices, + # only has an effect if s >= 1 + for k in range(s): + forwards[:, k*k_endog:(k+1)*k_endog] = ( + prev_forwards[:, k*k_endog:(k+1)*k_endog] - + np.dot( + forwards[:, s*k_endog:(s+1)*k_endog], + prev_backwards[:, (s-k-1)*k_endog:(s-k)*k_endog] + ) + ) + + backwards[:, k*k_endog:(k+1)*k_endog] = ( + prev_backwards[:, k*k_endog:(k+1)*k_endog] - + np.dot( + backwards[:, s*k_endog:(s+1)*k_endog], + prev_forwards[:, (s-k-1)*k_endog:(s-k)*k_endog] + ) + ) + + autocovariances[:, (s+1)*k_endog:(s+2)*k_endog] += np.dot( + autocovariances[:, (k+1)*k_endog:(k+2)*k_endog], + prev_forwards[:, (s-k-1)*k_endog:(s-k)*k_endog].T + ) + + # Create forward and backwards variances + backward_variance = ( + backward_variance - + np.dot( + np.dot(backwards[:, s*k_endog:(s+1)*k_endog], forward_variance), + backwards[:, s*k_endog:(s+1)*k_endog].T + ) + ) + forward_variance = ( + forward_variance - + np.dot(tmp, forwards[:, s*k_endog:(s+1)*k_endog].T) + ) + + # Cholesky factors + forward_factors = linalg.cholesky(forward_variance, lower=True) + backward_factors = linalg.cholesky(backward_variance, lower=True) + + # If we do not want to use the transformed variance, we need to + # adjust the constrained matrices, as presented in Lemma 2.3, see above + variance = forward_variance + if not transform_variance: + # Here, we need to construct T such that: + # variance = T * initial_variance * T' + # To do that, consider the Cholesky of variance (L) and + # input_variance (M) to get: + # L L' = T M M' T' = (TM) (TM)' + # => L = T M + # => L M^{-1} = T + initial_variance_factor = np.linalg.cholesky(initial_variance) + transformed_variance_factor = np.linalg.cholesky(variance) + transform = np.dot(initial_variance_factor, + np.linalg.inv(transformed_variance_factor)) + inv_transform = np.linalg.inv(transform) + + for s in range(order): + forwards[:, s*k_endog:(s+1)*k_endog] = ( + np.dot(np.dot(transform, forwards[:, s*k_endog:(s+1)*k_endog]), inv_transform) + ) + + return forwards, variance + + +def _compute_coefficients_from_multivariate_pacf_python2( + partial_autocorrelations, error_variance, order=None, k_endog=None, + transform_variance=False): """ Transform matrices with singular values less than one to matrices corresponding to a stationary (or invertible) process. @@ -525,8 +691,59 @@ def _compute_coefficients_from_multivariate_pacf(partial_autocorrelations, return forwards, variance -def constrain_stationary_multivariate(unconstrained, variance, - transform_variance=False): +if not old_scipy_compat: + + def constrain_stationary_multivariate(unconstrained, variance, + transform_variance=False, + prefix=None): + + use_list = type(unconstrained) == list + if use_list: + unconstrained = np.concatenate(unconstrained, axis=1) + + unconstrained = np.asfortranarray(unconstrained) + variance = np.asfortranarray(variance) + + k_endog, order = unconstrained.shape + order //= k_endog + + if prefix is None: + prefix, dtype, _ = find_best_blas_type( + [unconstrained, variance]) + + # Step 1: convert from arbitrary matrices to those with singular values + # less than one. + # sv_constrained = _constrain_sv_less_than_one(unconstrained, order, + # k_endog, prefix) + sv_constrained = prefix_sv_map[prefix](unconstrained, order, k_endog) + + # Step 2: convert matrices from our "partial autocorrelation matrix" space + # (matrices with singular values less than one) to the space of stationary + # coefficient matrices + constrained, variance = prefix_pacf_map[prefix]( + sv_constrained, variance, order, k_endog, transform_variance) + + constrained = np.array(constrained) + variance = np.array(variance) + + if use_list: + constrained = [ + constrained[:k_endog, i*k_endog:(i+1)*k_endog] + for i in range(order) + ] + + return constrained, variance + +else: + _compute_coefficients_from_multivariate_pacf = ( + _compute_coefficients_from_multivariate_pacf_python) + _constrain_sv_less_than_one = _constrain_sv_less_than_one_python + constrain_stationary_multivariate = constrain_stationary_multivariate_python + + +def constrain_stationary_multivariate_python(unconstrained, variance, + transform_variance=False, + prefix=None): """ Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation for a vector autoregression. @@ -583,6 +800,16 @@ def constrain_stationary_multivariate(unconstrained, variance, """ from scipy import linalg + use_list = type(unconstrained) == list + if not use_list: + k_endog, order = unconstrained.shape + order //= k_endog + + unconstrained = [ + unconstrained[:k_endog, i*k_endog:(i+1)*k_endog] + for i in range(order) + ] + order = len(unconstrained) k_endog = unconstrained[0].shape[0] @@ -594,7 +821,10 @@ def constrain_stationary_multivariate(unconstrained, variance, # (matrices with singular values less than one) to the space of stationary # coefficient matrices constrained, variance = _compute_coefficients_from_multivariate_pacf( - sv_constrained, variance, transform_variance, order, k_endog) + sv_constrained, variance, order, k_endog, transform_variance) + + if not use_list: + constrained = np.concatenate(constrained, axis=1) return constrained, variance @@ -864,6 +1094,16 @@ def unconstrain_stationary_multivariate(constrained, error_variance, from scipy import linalg + use_list = type(constrained) == list + if not use_list: + k_endog, order = constrained.shape + order //= k_endog + + constrained = [ + constrained[:k_endog, i*k_endog:(i+1)*k_endog] + for i in range(order) + ] + order = len(constrained) k_endog = constrained[0].shape[0] @@ -877,6 +1117,9 @@ def unconstrain_stationary_multivariate(constrained, error_variance, # less than one. unconstrained = _unconstrain_sv_less_than_one(partial_autocorrelations, order, k_endog) + if not use_list: + unconstrained = np.concatenate(unconstrained, axis=1) + return unconstrained, error_variance diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index 6fcac5c17df..18ca48eec7e 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -380,23 +380,31 @@ def transform_params(self, unconstrained): state_cov = np.dot(state_cov_lower, state_cov_lower.T) # Transform the parameters + # coefficients = unconstrained[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) + # unconstrained_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ar)] + # coefficient_matrices, variance = ( + # constrain_stationary_multivariate(unconstrained_matrices, state_cov)) + # constrained[self._params_ar] = np.concatenate(coefficient_matrices, axis=1).ravel() coefficients = unconstrained[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) - unconstrained_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ar)] coefficient_matrices, variance = ( - constrain_stationary_multivariate(unconstrained_matrices, state_cov)) - constrained[self._params_ar] = np.concatenate(coefficient_matrices, axis=1).ravel() + constrain_stationary_multivariate(coefficients, state_cov)) + constrained[self._params_ar] = coefficient_matrices.ravel() else: constrained[self._params_ar] = unconstrained[self._params_ar] # 3. MA terms: optionally force to be invertible if self.k_ma > 0 and self.enforce_invertibility: # Transform the parameters, using an identity variance matrix + # coefficients = unconstrained[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) + # unconstrained_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ma)] + # # TODO cache this np.eye call + # coefficient_matrices, variance = ( + # constrain_stationary_multivariate(unconstrained_matrices, np.eye(self.k_endog, dtype=constrained.dtype))) + # constrained[self._params_ma] = np.concatenate(coefficient_matrices, axis=1).ravel() coefficients = unconstrained[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) - unconstrained_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ma)] - # TODO cache this np.eye call coefficient_matrices, variance = ( - constrain_stationary_multivariate(unconstrained_matrices, np.eye(self.k_endog, dtype=constrained.dtype))) - constrained[self._params_ma] = np.concatenate(coefficient_matrices, axis=1).ravel() + constrain_stationary_multivariate(coefficients, np.eye(self.k_endog, dtype=constrained.dtype))) + constrained[self._params_ma] = coefficient_matrices.ravel() else: constrained[self._params_ma] = unconstrained[self._params_ma] @@ -455,22 +463,30 @@ def untransform_params(self, constrained): state_cov = np.dot(state_cov_lower, state_cov_lower.T) # Transform the parameters + # coefficients = constrained[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) + # coefficient_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ar)] + # unconstrained_matrices, variance = ( + # unconstrain_stationary_multivariate(coefficient_matrices, state_cov)) + # unconstrained[self._params_ar] = np.concatenate(unconstrained_matrices, axis=1).ravel() coefficients = constrained[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) - coefficient_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ar)] unconstrained_matrices, variance = ( - unconstrain_stationary_multivariate(coefficient_matrices, state_cov)) - unconstrained[self._params_ar] = np.concatenate(unconstrained_matrices, axis=1).ravel() + unconstrain_stationary_multivariate(coefficients, state_cov)) + unconstrained[self._params_ar] = unconstrained_matrices.ravel() else: unconstrained[self._params_ar] = constrained[self._params_ar] # 3. MA terms: optionally were forced to be invertible if self.k_ma > 0 and self.enforce_invertibility: # Transform the parameters, using an identity variance matrix + # coefficients = constrained[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) + # coefficient_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ma)] + # unconstrained_matrices, variance = ( + # unconstrain_stationary_multivariate(coefficient_matrices, np.eye(self.k_endog, dtype=constrained.dtype))) + # unconstrained[self._params_ma] = np.concatenate(unconstrained_matrices, axis=1).ravel() coefficients = constrained[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) - coefficient_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ma)] unconstrained_matrices, variance = ( - unconstrain_stationary_multivariate(coefficient_matrices, np.eye(self.k_endog, dtype=constrained.dtype))) - unconstrained[self._params_ma] = np.concatenate(unconstrained_matrices, axis=1).ravel() + unconstrain_stationary_multivariate(coefficients, np.eye(self.k_endog, dtype=constrained.dtype))) + unconstrained[self._params_ma] = unconstrained_matrices.ravel() else: unconstrained[self._params_ma] = constrained[self._params_ma] From ac578bb2f2146e3ac08b0a4e54326432a2411670 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 5 Aug 2015 10:38:14 -0700 Subject: [PATCH 14/57] TST: Add VAR(2) test. --- .../tests/results/results_varmax.py | 24 +++ .../tests/results/results_varmax_stata.csv | 186 +++++++++--------- .../tests/results/test_varmax_stata.do | 15 ++ .../tsa/statespace/tests/test_varmax.py | 29 ++- 4 files changed, 158 insertions(+), 96 deletions(-) diff --git a/statsmodels/tsa/statespace/tests/results/results_varmax.py b/statsmodels/tsa/statespace/tests/results/results_varmax.py index 9483df56924..42ac9fb9295 100644 --- a/statsmodels/tsa/statespace/tests/results/results_varmax.py +++ b/statsmodels/tsa/statespace/tests/results/results_varmax.py @@ -95,3 +95,27 @@ 'aic': -1101.634, 'bic': -1073.824 } + +lutkepohl_ar2 = { + 'params': [ + -.25244981, .62528114, # Phi_1, row 1 + -.13011679, .58173748, # Phi_1, row 2 + .05369178, .35716349, # Phi_2, row 1 + .03861472, .43812606, # Phi_2, row 2 + # .00197786, # Covariance, lower triangle + # .00008091, .00018269 + 0.04447314, # Covariance cholesky, lower triangle + 0.0018193, 0.01339329 + ], + 'var_oim': [ + .01315844, .11805816, # Phi_1, row 1 + .01321036, .11300702, # Phi_1, row 2 + .00122666, .01064478, # Phi_2, row 1 + .0012571, .0106738, # Phi_2, row 2 + 1.048e-07, # Covariance, lower triangle + 4.994e-09, 8.940e-10 + ], + 'loglike': 343.3149718445623, + 'aic': -664.6299, + 'bic': -639.1376 +} \ No newline at end of file diff --git a/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv b/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv index cc7cd7eac90..df045b2e476 100644 --- a/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv +++ b/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv @@ -1,93 +1,93 @@ -predict_1,predict_2,predict_3,predict_diag1,predict_diag2,predict_diag3,predict_int1,predict_int2,predict_int3,dyn_predict_1,dyn_predict_2,dyn_predict_3,dyn_predict_diag1,dyn_predict_diag2,dyn_predict_diag3,dyn_predict_int1,dyn_predict_int2,dyn_predict_int3 -,,,,,,,,,,,,,,,,, -0,0,0,0,0,0,.017993,.0206546,.0198753,0,0,0,0,0,0,.017993,.0206546,.0198753 -.0219036,.015239,.0205644,.0219323,.015408,.0205827,.0222069,.0178744,.023416,.0219036,.015239,.0205644,.0219323,.015408,.0205827,.0222069,.0178744,.023416 -.0286807,.0290166,.0307121,.0287198,.0292086,.0308044,.0278614,.0220684,.0228171,.0286807,.0290166,.0307121,.0287198,.0292086,.0308044,.0278614,.0220684,.0228171 -.0213229,.0250984,.0171728,.021225,.0251252,.0173499,.0211601,.0243908,.0164511,.0213229,.0250984,.0171728,.021225,.0251252,.0173499,.0211601,.0243908,.0164511 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a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do index ba5235b1bd1..6e79a45a6bb 100644 --- a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do +++ b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do @@ -45,4 +45,19 @@ predict dyn_predict_int1, dynamic(tq(1961q1)) equation(dln_inv) predict dyn_predict_int2, dynamic(tq(1961q1)) equation(dln_inc) predict dyn_predict_int3, dynamic(tq(1961q1)) equation(dln_consump) + +// VAR(2) +dfactor (dln_inv dln_inc = , ar(1/2) arstructure(general) noconstant covstructure(unstructured)) if qtr<=tq(1978q4) +estat ic + +// predict + forecast, see above +predict predict_var2_1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_var2_2, dynamic(tq(1979q1)) equation(dln_inc) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_var2_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_var2_2, dynamic(tq(1961q1)) equation(dln_inc) + + outsheet pred* dyn* using results_varmax_stata.csv, comma replace + diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index b500ef20482..1a7c043beb4 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -31,6 +31,7 @@ class CheckVAR(object): """ def __init__(self, true, order, trend, error_cov_type, cov_type='oim', + included_vars=['dln_inv', 'dln_inc', 'dln_consump'], **kwargs): self.true = true # 1960:Q1 - 1982:Q4 @@ -42,8 +43,7 @@ def __init__(self, true, order, trend, error_cov_type, cov_type='oim', dta['dln_inc'] = np.log(dta['inc']).diff() dta['dln_consump'] = np.log(dta['consump']).diff() - endog = dta.ix['1960-04-01':'1978-10-01', - ['dln_inv', 'dln_inc', 'dln_consump']] + endog = dta.ix['1960-04-01':'1978-10-01', included_vars] self.model = varmax.VARMAX(endog, order=order, trend=trend, error_cov_type=error_cov_type, **kwargs) @@ -93,6 +93,7 @@ def __init__(self): true, order=(1,0), trend='nc', error_cov_type="unstructured") + class TestVAR_diagonal(CheckVAR): def __init__(self): true = results_varmax.lutkepohl_ar1_diag.copy() @@ -102,6 +103,7 @@ def __init__(self): true, order=(1,0), trend='nc', error_cov_type="diagonal") + class TestVAR_obs_intercept(CheckVAR): def __init__(self): true = results_varmax.lutkepohl_ar1_obs_intercept.copy() @@ -119,4 +121,25 @@ def test_aic(self): def test_bic(self): # Since the obs_intercept is added in in an ad-hoc way here, the number # of parameters, and hence the aic and bic, will be off - pass \ No newline at end of file + pass + +class TestVAR2(CheckVAR): + def __init__(self): + true = results_varmax.lutkepohl_ar2.copy() + true['predict'] = output_results.ix[1:, ['predict_var2_1', 'predict_var2_2']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_var2_1', 'dyn_predict_var2_2']] + super(TestVAR2, self).__init__( + true, order=(2,0), trend='nc', error_cov_type='unstructured', + included_vars=['dln_inv', 'dln_inc']) + + def test_bse_oim(self): + # Exclude the covariance cholesky terms + assert_allclose( + self.results.bse[:-3]**2, self.true['var_oim'][:-3], atol=1e-2) + + def test_mle(self): + # Found maximum needs lower tolerance here than other tests + results = self.model.fit(method='powell', disp=-1) + results = self.model.fit(results.params, disp=False) + assert_allclose(results.llf, self.results.llf, rtol=1e-3) + From 3f8071ddef7612a4e2112a004c434fe39f4c676b Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 5 Aug 2015 11:27:57 -0700 Subject: [PATCH 15/57] BUG: Missing dtype in np.eye --- statsmodels/tsa/statespace/_statespace.pyx.in | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/statsmodels/tsa/statespace/_statespace.pyx.in b/statsmodels/tsa/statespace/_statespace.pyx.in index b858f70598e..d14e9eb203a 100644 --- a/statsmodels/tsa/statespace/_statespace.pyx.in +++ b/statsmodels/tsa/statespace/_statespace.pyx.in @@ -1811,7 +1811,7 @@ cpdef _{{prefix}}compute_coefficients_from_multivariate_pacf({{cython_type}} [:: # exceptions from the Cholesky decompositions. # Note that this will still not always ensure positive definiteness, # and for k_endog, order large enough an exception may still be raised - error_variance = np.asfortranarray(np.eye(k_endog) * (order + k_endog)**10) + error_variance = np.asfortranarray(np.eye(k_endog, dtype={{dtype}}) * (order + k_endog)**10) # Initialize matrices dim2[0] = k_endog; dim2[1] = k_endog; @@ -2091,7 +2091,7 @@ cpdef _{{prefix}}constrain_sv_less_than_one({{cython_type}} [::1,:] unconstraine tmp = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) eye = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) - eye = np.asfortranarray(np.eye(k_endog)) + eye = np.asfortranarray(np.eye(k_endog, dtype={{dtype}})) for i in range(order): {{prefix}}copy(&k_endog2, &eye[0, 0], &inc, &tmp[0, 0], &inc) {{prefix}}gemm("N", "T", &k_endog, &k_endog, &k_endog, From 9ec88fcfa42c6c54c37e42501087efe712818947 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 5 Aug 2015 11:38:07 -0700 Subject: [PATCH 16/57] TST: Improve MLE tests. --- .../tsa/statespace/tests/test_varmax.py | 19 +++++++++---------- 1 file changed, 9 insertions(+), 10 deletions(-) diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index 1a7c043beb4..9985619095e 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -51,9 +51,15 @@ def __init__(self, true, order, trend, error_cov_type, cov_type='oim', self.results = self.model.filter(true['params'], cov_type=cov_type) def test_mle(self): - results = self.model.fit(method='powell', disp=-1) - results = self.model.fit(results.params, disp=False) - assert_allclose(results.llf, self.results.llf, rtol=1e-4) + # Fit with all transformations + # results = self.model.fit(method='powell', disp=-1) + results = self.model.fit(maxiter=100, disp=False) + # Fit now without transformations + self.model.enforce_stationarity = False + self.model.enforce_invertibility = False + results = self.model.fit(results.params, method='nm', maxiter=1000, + disp=False) + assert_allclose(results.llf, self.results.llf) def test_loglike(self): assert_allclose(self.results.llf, self.true['loglike'], rtol=1e-6) @@ -136,10 +142,3 @@ def test_bse_oim(self): # Exclude the covariance cholesky terms assert_allclose( self.results.bse[:-3]**2, self.true['var_oim'][:-3], atol=1e-2) - - def test_mle(self): - # Found maximum needs lower tolerance here than other tests - results = self.model.fit(method='powell', disp=-1) - results = self.model.fit(results.params, disp=False) - assert_allclose(results.llf, self.results.llf, rtol=1e-3) - From 6c098ea7fc164f4e849e0b42fef827c0f95dfe5c Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 5 Aug 2015 21:31:22 -0700 Subject: [PATCH 17/57] TST: Add dynamic factor test. --- .../tests/results/results_dynamic_factor.py | 29 ++++++ .../results/results_dynamic_factor_stata.csv | 93 +++++++++++++++++++ .../results/test_dynamic_factor_stata.do | 18 ++++ .../statespace/tests/test_dynamic_factor.py | 92 ++++++++++++++++++ 4 files changed, 232 insertions(+) create mode 100644 statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py create mode 100644 statsmodels/tsa/statespace/tests/results/results_dynamic_factor_stata.csv create mode 100644 statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do create mode 100644 statsmodels/tsa/statespace/tests/test_dynamic_factor.py diff --git a/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py new file mode 100644 index 00000000000..72ff37f9913 --- /dev/null +++ b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py @@ -0,0 +1,29 @@ +""" +Results for VARMAX tests + +Results from Stata using script `test_varmax_stata.do`. +See also Stata time series documentation, in particular `dfactor`. + +Data from: + +http://www.jmulti.de/download/datasets/e1.dat + +Author: Chad Fulton +License: Simplified-BSD +""" + +lutkepohl_dfm = { + 'params': [ + .0063728, .00660177, .00636009, # Factor loadings + .00203899, .00009016, .00005348, # Idiosyncratic variances + .33101874, .63927819, # Factor transitions + ], + 'bse_oim': [ + .002006, .0012514, .0012128, # Factor loadings + .0003359, .0000184, .0000141, # Idiosyncratic variances + .1196637, .1218577, # Factor transitions + ], + 'loglike': 594.0902026190786, + 'aic': -1172.18, + 'bic': -1153.641, +} diff --git a/statsmodels/tsa/statespace/tests/results/results_dynamic_factor_stata.csv b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor_stata.csv new file mode 100644 index 00000000000..a5df40f10f0 --- /dev/null +++ b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor_stata.csv 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+.0092597,.0095923,.0092412,.0056568,.0058601,.0056456 +.009101,.009428,.0090829,.0055551,.0057547,.005544 +.0089321,.009253,.0089143,.0054551,.0056511,.0054442 diff --git a/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do b/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do new file mode 100644 index 00000000000..2fadb939fa3 --- /dev/null +++ b/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do @@ -0,0 +1,18 @@ +webuse lutkepohl2, clear +tsset + +// Dynamic Factors: Static Factor model +dfactor (dln_inv dln_inc dln_consump = , noconstant ) (f = , ar(1/2)) if qtr<=tq(1978q4) + +// predict + forecast, see above +predict predict_dfm_1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_dfm_2, dynamic(tq(1979q1)) equation(dln_inc) +predict predict_dfm_3, dynamic(tq(1979q1)) equation(dln_consump) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_dfm_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_dfm_2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_dfm_3, dynamic(tq(1961q1)) equation(dln_consump) + +outsheet pred* dyn* using results_dynamic_factor_stata.csv, comma replace + diff --git a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py new file mode 100644 index 00000000000..dcbc3b3fcaa --- /dev/null +++ b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py @@ -0,0 +1,92 @@ +""" +Tests for VARMAX models + +Author: Chad Fulton +License: Simplified-BSD +""" +from __future__ import division, absolute_import, print_function + +import numpy as np +import pandas as pd +import os + +import warnings +from statsmodels.datasets import webuse +from statsmodels.tsa.statespace import dynamic_factor +from .results import results_varmax, results_dynamic_factor +from numpy.testing import assert_equal, assert_almost_equal, assert_raises, assert_allclose +from nose.exc import SkipTest + +current_path = os.path.dirname(os.path.abspath(__file__)) + +output_path = 'results' + os.sep + 'results_dynamic_factor_stata.csv' +output_results = pd.read_csv(current_path + os.sep + output_path) + + +class CheckStaticFactor(object): + def __init__(self, true, k_factors, factor_order, cov_type='oim', + included_vars=['dln_inv', 'dln_inc', 'dln_consump'], + **kwargs): + self.true = true + # 1960:Q1 - 1982:Q4 + dta = pd.DataFrame( + results_varmax.lutkepohl_data, columns=['inv', 'inc', 'consump'], + index=pd.date_range('1960-01-01', '1982-10-01', freq='QS')) + + dta['dln_inv'] = np.log(dta['inv']).diff() + dta['dln_inc'] = np.log(dta['inc']).diff() + dta['dln_consump'] = np.log(dta['consump']).diff() + + endog = dta.ix['1960-04-01':'1978-10-01', included_vars] + + self.model = dynamic_factor.StaticFactors(endog, k_factors=k_factors, + factor_order=factor_order, + **kwargs) + + self.results = self.model.filter(true['params'], cov_type=cov_type) + + def test_mle(self): + # Fit with all transformations + # results = self.model.fit(method='powell', disp=-1) + results = self.model.fit(maxiter=100, disp=False) + # Fit now without transformations + self.model.enforce_stationarity = False + results = self.model.fit(results.params, method='nm', maxiter=1000, + disp=False) + assert_allclose(results.llf, self.results.llf, rtol=1e-5) + + def test_loglike(self): + assert_allclose(self.results.llf, self.true['loglike'], rtol=1e-6) + + def test_bse_oim(self): + raise SkipTest('Known failure: standard errors do not match.') + # assert_allclose(self.results.bse, self.true['bse_oim'], atol=1e-2) + + def test_aic(self): + # We only get 3 digits from Stata + assert_allclose(self.results.aic, self.true['aic'], atol=3) + + def test_bic(self): + # We only get 3 digits from Stata + assert_allclose(self.results.bic, self.true['bic'], atol=3) + + def test_predict(self): + # Tests predict + forecast + assert_allclose( + self.results.predict(end='1982-10-01'), + self.true['predict'].T, + atol=1e-6) + + def test_dynamic_predict(self): + # Tests predict + dynamic predict + forecast + assert_allclose( + self.results.predict(end='1982-10-01', dynamic='1961-01-01'), + self.true['dynamic_predict'].T, + atol=1e-6) + +class TestStaticFactor(CheckStaticFactor): + def __init__(self): + true = results_dynamic_factor.lutkepohl_dfm.copy() + true['predict'] = output_results.ix[1:, ['predict_dfm_1', 'predict_dfm_2', 'predict_dfm_3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_dfm_1', 'dyn_predict_dfm_2', 'dyn_predict_dfm_3']] + super(TestStaticFactor, self).__init__(true, k_factors=1, factor_order=2) From 352f5c5414b4199ff3f36e73ed2338d1aaf7138b Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 5 Aug 2015 21:53:00 -0700 Subject: [PATCH 18/57] TST: Reduce MLE rtol from 1e-7 to 1e-5 --- statsmodels/tsa/statespace/tests/test_varmax.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index 9985619095e..426406ed7f4 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -22,7 +22,6 @@ output_results = pd.read_csv(current_path + os.sep + output_path) - class CheckVAR(object): """ Test Vector Autoregression against Stata's `dfactor` code (Stata's @@ -59,7 +58,7 @@ def test_mle(self): self.model.enforce_invertibility = False results = self.model.fit(results.params, method='nm', maxiter=1000, disp=False) - assert_allclose(results.llf, self.results.llf) + assert_allclose(results.llf, self.results.llf, rtol=1e-5) def test_loglike(self): assert_allclose(self.results.llf, self.true['loglike'], rtol=1e-6) @@ -142,3 +141,4 @@ def test_bse_oim(self): # Exclude the covariance cholesky terms assert_allclose( self.results.bse[:-3]**2, self.true['var_oim'][:-3], atol=1e-2) + From 41aadee8a7cb6b6afcebeab2885260841e859296 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 5 Aug 2015 22:29:58 -0700 Subject: [PATCH 19/57] REF: Improved ?trmm feature detection. --- statsmodels/tsa/statespace/_statespace.pyx.in | 4 +- .../tsa/statespace/tests/test_tools.py | 4 +- statsmodels/tsa/statespace/tools.py | 175 ++++++++++-------- 3 files changed, 100 insertions(+), 83 deletions(-) diff --git a/statsmodels/tsa/statespace/_statespace.pyx.in b/statsmodels/tsa/statespace/_statespace.pyx.in index d14e9eb203a..8547d7cde44 100644 --- a/statsmodels/tsa/statespace/_statespace.pyx.in +++ b/statsmodels/tsa/statespace/_statespace.pyx.in @@ -17,7 +17,7 @@ TYPES = { "z": ("np.complex128_t", "complex", "np.NPY_COMPLEX128"), } -BLAS_FUNCTIONS = ["gemm", "gemv", "copy", "axpy", "scal", "trmm", "symm"] +BLAS_FUNCTIONS = ["gemm", "gemv", "copy", "axpy", "scal"] LAPACK_FUNCTIONS = ["getrf", "getri", "getrs", "potrf", "potri", "potrs", "trtrs"] }} @@ -1797,6 +1797,8 @@ cpdef _{{prefix}}compute_coefficients_from_multivariate_pacf({{cython_type}} [:: {{cython_type}} * prev_forwards {{cython_type}} * backwards {{cython_type}} * prev_backwards + # ?trmm + cdef {{prefix}}trmm_t *{{prefix}}trmm = <{{prefix}}trmm_t*>Capsule_AsVoidPtr(blas.{{prefix}}trmm._cpointer) # dim2[0] = self.k_endog; dim2[1] = storage; # self.forecast = np.PyArray_ZEROS(2, dim2, {{typenum}}, FORTRAN) diff --git a/statsmodels/tsa/statespace/tests/test_tools.py b/statsmodels/tsa/statespace/tests/test_tools.py index a26f833eccb..b409d928d68 100644 --- a/statsmodels/tsa/statespace/tests/test_tools.py +++ b/statsmodels/tsa/statespace/tests/test_tools.py @@ -213,11 +213,11 @@ def test_multivariate_acovf(): class TestConstrainStationaryMultivariate(object): cases = [ - ([np.array([[2.]])], [np.array([[2./((1+2.**2)**0.5)]])]) + (np.array([[2.]]), np.array([[2./((1+2.**2)**0.5)]])) ] def test_cases(self): for unconstrained, constrained in self.cases: result = tools.constrain_stationary_multivariate( - unconstrained, np.eye(len(unconstrained[0]))) + unconstrained, np.eye(unconstrained.shape[0])) assert_allclose(result[0], constrained) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 3bd1d866417..2efa11e4cf8 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -10,11 +10,11 @@ from statsmodels.tools.data import _is_using_pandas from . import _statespace -old_scipy_compat = False +has_find_best_blas_type = True try: from scipy.linalg.blas import find_best_blas_type except ImportError: # pragma: no cover - old_scipy_compat = True + has_find_best_blas_type = False # Shim for SciPy 0.11, derived from tag=0.11 scipy.linalg.blas _type_conv = {'f': 's', 'd': 'd', 'F': 'c', 'D': 'z', 'G': 'z'} @@ -24,6 +24,12 @@ def find_best_blas_type(arrays): prefix = _type_conv.get(dtype.char, 'd') return prefix, dtype, None +has_trmm = True +try: + from scipy.linalg.blas import dtrmm +except ImportError: + has_trmm = False + prefix_dtype_map = { 's': np.float32, 'd': np.float64, 'c': np.complex64, 'z': np.complex128 @@ -36,18 +42,19 @@ def find_best_blas_type(arrays): 's': _statespace.sKalmanFilter, 'd': _statespace.dKalmanFilter, 'c': _statespace.cKalmanFilter, 'z': _statespace.zKalmanFilter } -prefix_pacf_map = { - 's': _statespace._scompute_coefficients_from_multivariate_pacf, - 'd': _statespace._dcompute_coefficients_from_multivariate_pacf, - 'c': _statespace._ccompute_coefficients_from_multivariate_pacf, - 'z': _statespace._zcompute_coefficients_from_multivariate_pacf -} -prefix_sv_map = { - 's': _statespace._sconstrain_sv_less_than_one, - 'd': _statespace._dconstrain_sv_less_than_one, - 'c': _statespace._cconstrain_sv_less_than_one, - 'z': _statespace._zconstrain_sv_less_than_one -} +if has_trmm: + prefix_pacf_map = { + 's': _statespace._scompute_coefficients_from_multivariate_pacf, + 'd': _statespace._dcompute_coefficients_from_multivariate_pacf, + 'c': _statespace._ccompute_coefficients_from_multivariate_pacf, + 'z': _statespace._zcompute_coefficients_from_multivariate_pacf + } + prefix_sv_map = { + 's': _statespace._sconstrain_sv_less_than_one, + 'd': _statespace._dconstrain_sv_less_than_one, + 'c': _statespace._cconstrain_sv_less_than_one, + 'z': _statespace._zconstrain_sv_less_than_one + } def companion_matrix(polynomial): @@ -399,8 +406,8 @@ def _constrain_sv_less_than_one_python(unconstrained, order=None, def _compute_coefficients_from_multivariate_pacf_python( - partial_autocorrelations, error_variance, order, k_endog, - transform_variance=False): + partial_autocorrelations, error_variance, order, k_endog, + transform_variance=False): """ Transform matrices with singular values less than one to matrices corresponding to a stationary (or invertible) process. @@ -463,7 +470,8 @@ def _compute_coefficients_from_multivariate_pacf_python( forwards[:, s*k_endog:(s+1)*k_endog] = np.dot( forward_factors, linalg.solve_triangular( - backward_factors, partial_autocorrelations[:, s*k_endog:(s+1)*k_endog].T, + backward_factors, + partial_autocorrelations[:, s*k_endog:(s+1)*k_endog].T, lower=True, trans='T').T ) @@ -473,7 +481,8 @@ def _compute_coefficients_from_multivariate_pacf_python( backwards[:, s*k_endog:(s+1)*k_endog] = np.dot( backward_factors, linalg.solve_triangular( - forward_factors, partial_autocorrelations[:, s*k_endog:(s+1)*k_endog], + forward_factors, + partial_autocorrelations[:, s*k_endog:(s+1)*k_endog], lower=True, trans='T').T ) @@ -512,7 +521,8 @@ def _compute_coefficients_from_multivariate_pacf_python( backward_variance = ( backward_variance - np.dot( - np.dot(backwards[:, s*k_endog:(s+1)*k_endog], forward_variance), + np.dot( + backwards[:, s*k_endog:(s+1)*k_endog], forward_variance), backwards[:, s*k_endog:(s+1)*k_endog].T ) ) @@ -523,7 +533,7 @@ def _compute_coefficients_from_multivariate_pacf_python( # Cholesky factors forward_factors = linalg.cholesky(forward_variance, lower=True) - backward_factors = linalg.cholesky(backward_variance, lower=True) + backward_factors = linalg.cholesky(backward_variance, lower=True) # If we do not want to use the transformed variance, we need to # adjust the constrained matrices, as presented in Lemma 2.3, see above @@ -544,15 +554,16 @@ def _compute_coefficients_from_multivariate_pacf_python( for s in range(order): forwards[:, s*k_endog:(s+1)*k_endog] = ( - np.dot(np.dot(transform, forwards[:, s*k_endog:(s+1)*k_endog]), inv_transform) - ) + np.dot( + np.dot(transform, forwards[:, s*k_endog:(s+1)*k_endog]), + inv_transform)) return forwards, variance def _compute_coefficients_from_multivariate_pacf_python2( - partial_autocorrelations, error_variance, order=None, k_endog=None, - transform_variance=False): + partial_autocorrelations, error_variance, order=None, k_endog=None, + transform_variance=False): """ Transform matrices with singular values less than one to matrices corresponding to a stationary (or invertible) process. @@ -691,56 +702,6 @@ def _compute_coefficients_from_multivariate_pacf_python2( return forwards, variance -if not old_scipy_compat: - - def constrain_stationary_multivariate(unconstrained, variance, - transform_variance=False, - prefix=None): - - use_list = type(unconstrained) == list - if use_list: - unconstrained = np.concatenate(unconstrained, axis=1) - - unconstrained = np.asfortranarray(unconstrained) - variance = np.asfortranarray(variance) - - k_endog, order = unconstrained.shape - order //= k_endog - - if prefix is None: - prefix, dtype, _ = find_best_blas_type( - [unconstrained, variance]) - - # Step 1: convert from arbitrary matrices to those with singular values - # less than one. - # sv_constrained = _constrain_sv_less_than_one(unconstrained, order, - # k_endog, prefix) - sv_constrained = prefix_sv_map[prefix](unconstrained, order, k_endog) - - # Step 2: convert matrices from our "partial autocorrelation matrix" space - # (matrices with singular values less than one) to the space of stationary - # coefficient matrices - constrained, variance = prefix_pacf_map[prefix]( - sv_constrained, variance, order, k_endog, transform_variance) - - constrained = np.array(constrained) - variance = np.array(variance) - - if use_list: - constrained = [ - constrained[:k_endog, i*k_endog:(i+1)*k_endog] - for i in range(order) - ] - - return constrained, variance - -else: - _compute_coefficients_from_multivariate_pacf = ( - _compute_coefficients_from_multivariate_pacf_python) - _constrain_sv_less_than_one = _constrain_sv_less_than_one_python - constrain_stationary_multivariate = constrain_stationary_multivariate_python - - def constrain_stationary_multivariate_python(unconstrained, variance, transform_variance=False, prefix=None): @@ -824,7 +785,8 @@ def constrain_stationary_multivariate_python(unconstrained, variance, sv_constrained, variance, order, k_endog, transform_variance) if not use_list: - constrained = np.concatenate(constrained, axis=1) + constrained = np.concatenate(constrained, axis=1).reshape( + k_endog, k_endog * order) return constrained, variance @@ -909,7 +871,7 @@ def _compute_multivariate_acovf_from_coefficients(coefficients, error_variance, def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, - order=None, k_endog=None): + order=None, k_endog=None): """ Transform matrices corresponding to a stationary (or invertible) process to matrices with singular values less than one. @@ -938,7 +900,8 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, # E z_t z_{t-j}' # However, we want E z_t z_{t+j}' = (E z_t z_{t-j}')' _acovf = _compute_multivariate_acovf_from_coefficients - autocovariances = [autocovariance.T for autocovariance in + autocovariances = [ + autocovariance.T for autocovariance in _acovf(constrained, error_variance, order, k_endog, maxlag=order)] # Now apply the Ansley and Kohn (1986) algorithm, except that instead of @@ -1002,7 +965,7 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, linalg.cholesky(backward_variances[s], lower=True) ) - if False and s == order-1: + if s == order-1: forwards = constrained else: # Create the intermediate sum term @@ -1063,6 +1026,57 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, return partial_autocorrelations +if has_trmm: + + def constrain_stationary_multivariate(unconstrained, variance, + transform_variance=False, + prefix=None): + + use_list = type(unconstrained) == list + if use_list: + unconstrained = np.concatenate(unconstrained, axis=1) + + unconstrained = np.asfortranarray(unconstrained) + variance = np.asfortranarray(variance) + + k_endog, order = unconstrained.shape + order //= k_endog + + if prefix is None: + prefix, dtype, _ = find_best_blas_type( + [unconstrained, variance]) + + # Step 1: convert from arbitrary matrices to those with singular values + # less than one. + # sv_constrained = _constrain_sv_less_than_one(unconstrained, order, + # k_endog, prefix) + sv_constrained = prefix_sv_map[prefix](unconstrained, order, k_endog) + + # Step 2: convert matrices from our "partial autocorrelation matrix" + # space (matrices with singular values less than one) to the space of + # stationary coefficient matrices + constrained, variance = prefix_pacf_map[prefix]( + sv_constrained, variance, order, k_endog, transform_variance) + + constrained = np.array(constrained) + variance = np.array(variance) + + if use_list: + constrained = [ + constrained[:k_endog, i*k_endog:(i+1)*k_endog] + for i in range(order) + ] + + return constrained, variance + +else: + _compute_coefficients_from_multivariate_pacf = ( + _compute_coefficients_from_multivariate_pacf_python) + _constrain_sv_less_than_one = _constrain_sv_less_than_one_python + constrain_stationary_multivariate = ( + constrain_stationary_multivariate_python) + + def unconstrain_stationary_multivariate(constrained, error_variance, transform_variance=False): """ @@ -1091,7 +1105,7 @@ def unconstrain_stationary_multivariate(constrained, error_variance, Journal of Statistical Computation and Simulation 24 (2): 99-106. """ - + from scipy import linalg use_list = type(constrained) == list @@ -1115,7 +1129,8 @@ def unconstrain_stationary_multivariate(constrained, error_variance, # Step 2: convert from arbitrary matrices to those with singular values # less than one. - unconstrained = _unconstrain_sv_less_than_one(partial_autocorrelations, order, k_endog) + unconstrained = _unconstrain_sv_less_than_one( + partial_autocorrelations, order, k_endog) if not use_list: unconstrained = np.concatenate(unconstrained, axis=1) From 3224645bbf650543dc7abcec638897dc43145a65 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Fri, 7 Aug 2015 09:56:56 -0700 Subject: [PATCH 20/57] TST: More robust MLE test. --- statsmodels/tsa/statespace/tests/test_dynamic_factor.py | 4 ---- 1 file changed, 4 deletions(-) diff --git a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py index dcbc3b3fcaa..eb2df36f0ae 100644 --- a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py @@ -46,11 +46,7 @@ def __init__(self, true, k_factors, factor_order, cov_type='oim', self.results = self.model.filter(true['params'], cov_type=cov_type) def test_mle(self): - # Fit with all transformations - # results = self.model.fit(method='powell', disp=-1) results = self.model.fit(maxiter=100, disp=False) - # Fit now without transformations - self.model.enforce_stationarity = False results = self.model.fit(results.params, method='nm', maxiter=1000, disp=False) assert_allclose(results.llf, self.results.llf, rtol=1e-5) From adf57656e1805d2fa403c9b021d8f350d193a271 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Fri, 7 Aug 2015 15:38:17 -0700 Subject: [PATCH 21/57] ENH: Improved tools structure, tests, consistency --- statsmodels/tsa/statespace/_statespace.pyx.in | 4 +- .../tsa/statespace/tests/test_tools.py | 139 +++- statsmodels/tsa/statespace/tools.py | 764 +++++++++++------- 3 files changed, 591 insertions(+), 316 deletions(-) diff --git a/statsmodels/tsa/statespace/_statespace.pyx.in b/statsmodels/tsa/statespace/_statespace.pyx.in index 8547d7cde44..b7c47d140e2 100644 --- a/statsmodels/tsa/statespace/_statespace.pyx.in +++ b/statsmodels/tsa/statespace/_statespace.pyx.in @@ -1757,8 +1757,8 @@ cdef class {{prefix}}KalmanFilter(object): cpdef _{{prefix}}compute_coefficients_from_multivariate_pacf({{cython_type}} [::1,:] partial_autocorrelations, {{cython_type}} [::1,:] error_variance, - int order, int k_endog, - int transform_variance=False): + int transform_variance, + int order, int k_endog): """ Notes ----- diff --git a/statsmodels/tsa/statespace/tests/test_tools.py b/statsmodels/tsa/statespace/tests/test_tools.py index b409d928d68..93bcbff6a62 100644 --- a/statsmodels/tsa/statespace/tests/test_tools.py +++ b/statsmodels/tsa/statespace/tests/test_tools.py @@ -10,6 +10,7 @@ import pandas as pd from statsmodels.tsa.statespace import tools +from statsmodels.tsa.api import acovf # from .results import results_sarimax from numpy.testing import ( assert_allclose, assert_equal, assert_array_equal, assert_almost_equal, @@ -105,6 +106,36 @@ def test_cases(self): result = tools.constrain_stationary_univariate(unconstrained) assert_equal(result, constrained) +class TestUnconstrainStationaryUnivariate(object): + + cases = [ + (np.array([-2./((1+2.**2)**0.5)]), np.array([2.])) + ] + + def test_cases(self): + for constrained, unconstrained in self.cases: + result = tools.unconstrain_stationary_univariate(constrained) + assert_allclose(result, unconstrained) + +class TestStationaryUnivariate(object): + # Test that the constraint and unconstraint functions are inverses + + constrained_cases = [ + np.array([0]), np.array([0.1]), np.array([-0.5]), np.array([0.999])] + unconstrained_cases = [ + np.array([10.]), np.array([-40.42]), np.array([0.123])] + + def test_cases(self): + for constrained in self.constrained_cases: + unconstrained = tools.unconstrain_stationary_univariate(constrained) + reconstrained = tools.constrain_stationary_univariate(unconstrained) + assert_allclose(reconstrained, constrained) + + for unconstrained in self.unconstrained_cases: + constrained = tools.constrain_stationary_univariate(unconstrained) + reunconstrained = tools.unconstrain_stationary_univariate(constrained) + assert_allclose(reunconstrained, unconstrained) + class TestValidateMatrixShape(object): # name, shape, nrows, ncols, nobs valid = [ @@ -171,7 +202,7 @@ def test_multivariate_acovf(): Gamma_0 = np.array([[3.0, 0.161, 0.019], [0.161, 1.172, 0.674], [0.019, 0.674, 0.954]]) - # assert_allclose(_acovf([Phi_1], Sigma_u)[0], Gamma_0, atol=1e-3) + assert_allclose(_acovf([Phi_1], Sigma_u)[0], Gamma_0, atol=1e-3) # Test for a VAR(2) process. From Lutkepohl (2007), pages 28-29 # See (2.1.40) for Phi_1, Phi_2, (2.1.14) for Sigma_u, and (2.1.42) for @@ -210,14 +241,114 @@ def test_multivariate_acovf(): _acovf([Phi_1, Phi_2], Sigma_u, maxlag=3), [Gamma_0, Gamma_1, Gamma_2, Gamma_3], atol=1e-3) + # Test sample acovf in the univariate case against sm.tsa.acovf + x = np.arange(20)*1.0 + assert_allclose( + np.squeeze(tools._compute_multivariate_sample_acovf(x, maxlag=4)), + acovf(x)[:5]) + + +def test_multivariate_pacf(): + # Test sample acovf in the univariate case against sm.tsa.acovf + np.random.seed(1234) + x = np.arange(10000) + y = np.random.normal(size=10000) + # Note: could make this test more precise with higher nobs, but no need to + assert_allclose( + tools._compute_multivariate_sample_pacf(np.c_[x, y], maxlag=1)[0], + np.diag([1, 0]), atol=1e-2) + class TestConstrainStationaryMultivariate(object): cases = [ - (np.array([[2.]]), np.array([[2./((1+2.**2)**0.5)]])) + # This is the same test as the univariate case above, except notice + # the sign difference; this is an array input / output + (np.array([[2.]]), np.eye(1), np.array([[2./((1+2.**2)**0.5)]])), + # Same as above, but now a list input / output + ([np.array([[2.]])], np.eye(1), [np.array([[2./((1+2.**2)**0.5)]])]) + ] + + eigval_cases = [ + [np.array([[0]])], + [np.array([[100]]), np.array([[50]])], + [np.array([[30, 1], [-23, 15]]), np.array([[10, .3], [.5, -30]])], ] def test_cases(self): - for unconstrained, constrained in self.cases: + # Test against known results + for unconstrained, error_variance, constrained in self.cases: result = tools.constrain_stationary_multivariate( - unconstrained, np.eye(unconstrained.shape[0])) + unconstrained, error_variance) assert_allclose(result[0], constrained) + + # Test that the constrained results correspond to companion matrices + # with eigenvalues less than 1 in modulus + for unconstrained in self.eigval_cases: + if type(unconstrained) == list: + cov = np.eye(unconstrained[0].shape[0]) + else: + cov = np.eye(unconstrained.shape[0]) + constrained, _ = tools.constrain_stationary_multivariate(unconstrained, cov) + companion = tools.companion_matrix( + [1] + [-constrained[i] for i in range(len(constrained))] + ).T + assert_equal(np.max(np.abs(np.linalg.eigvals(companion))) < 1, True) + + + +class TestUnconstrainStationaryMultivariate(object): + + cases = [ + # This is the same test as the univariate case above, except notice + # the sign difference; this is an array input / output + (np.array([[2./((1+2.**2)**0.5)]]), np.eye(1), np.array([[2.]])), + # Same as above, but now a list input / output + ([np.array([[2./((1+2.**2)**0.5)]])], np.eye(1), [np.array([[2.]])]) + ] + + def test_cases(self): + for constrained, error_variance, unconstrained in self.cases: + result = tools.unconstrain_stationary_multivariate( + constrained, error_variance) + assert_allclose(result[0], unconstrained) + +class TestStationaryMultivariate(object): + # Test that the constraint and unconstraint functions are inverses + + constrained_cases = [ + np.array([[0]]), np.array([[0.1]]), np.array([[-0.5]]), np.array([[0.999]]), + [np.array([[0]])], + np.array([[0.8, -0.2]]), + [np.array([[0.8]]), np.array([[-0.2]])], + [np.array([[0.3, 0.01], [-0.23, 0.15]]), np.array([[0.1, 0.03], [0.05, -0.3]])], + np.array([[0.3, 0.01, 0.1, 0.03], [-0.23, 0.15, 0.05, -0.3]]) + ] + unconstrained_cases = [ + np.array([[0]]), np.array([[-40.42]]), np.array([[0.123]]), + [np.array([[0]])], + np.array([[100, 50]]), + [np.array([[100]]), np.array([[50]])], + [np.array([[30, 1], [-23, 15]]), np.array([[10, .3], [.5, -30]])], + np.array([[30, 1, 10, .3], [-23, 15, .5, -30]]) + ] + + def test_cases(self): + for constrained in self.constrained_cases: + if type(constrained) == list: + cov = np.eye(constrained[0].shape[0]) + else: + cov = np.eye(constrained.shape[0]) + unconstrained, _ = tools.unconstrain_stationary_multivariate(constrained, cov) + reconstrained, _ = tools.constrain_stationary_multivariate(unconstrained, cov) + assert_allclose(reconstrained, constrained) + + for unconstrained in self.unconstrained_cases: + if type(unconstrained) == list: + cov = np.eye(unconstrained[0].shape[0]) + else: + cov = np.eye(unconstrained.shape[0]) + constrained, _ = tools.constrain_stationary_multivariate(unconstrained, cov) + reunconstrained, _ = tools.unconstrain_stationary_multivariate(constrained, cov) + # Note: low tolerance comes from last example in unconstrained_cases, + # but is not a real problem + assert_allclose(reunconstrained, unconstrained, atol=1e-4) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 2efa11e4cf8..72dc11757ed 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -325,11 +325,10 @@ def constrain_stationary_univariate(unconstrained): References ---------- - - Monahan, John F. 1984. - "A Note on Enforcing Stationarity in - Autoregressive-moving Average Models." - Biometrika 71 (2) (August 1): 403-404. + .. [1] Monahan, John F. 1984. + "A Note on Enforcing Stationarity in + Autoregressive-moving Average Models." + Biometrika 71 (2) (August 1): 403-404. """ n = unconstrained.shape[0] @@ -363,11 +362,10 @@ def unconstrain_stationary_univariate(constrained): References ---------- - - Monahan, John F. 1984. - "A Note on Enforcing Stationarity in - Autoregressive-moving Average Models." - Biometrika 71 (2) (August 1): 403-404. + .. [1] Monahan, John F. 1984. + "A Note on Enforcing Stationarity in + Autoregressive-moving Average Models." + Biometrika 71 (2) (August 1): 403-404. """ n = constrained.shape[0] y = np.zeros((n, n), dtype=constrained.dtype) @@ -386,9 +384,33 @@ def _constrain_sv_less_than_one_python(unconstrained, order=None, Transform arbitrary matrices to matrices with singular values less than one. + Parameters + ---------- + unconstrained : list + Arbitrary matrices. Should be a list of length `order`, where each + element is an array sized `k_endog` x `k_endog`. + order : integer, optional + The order of the autoregression. + k_endog : integer, optional + The dimension of the data vector. + + Returns + ------- + constrained : list + Partial autocorrelation matrices. Should be a list of length + `order`, where each element is an array sized `k_endog` x `k_endog`. + + Notes + ----- Corresponds to Lemma 2.2 in Ansley and Kohn (1986). See `constrain_stationary_multivariate` for more details. + + There is a Cython implementation of this function that can be much faster, + but which requires SciPy 0.14.0 or greater. See + `constrain_stationary_multivariate` for details. + """ + from scipy import linalg constrained = [] # P_s, s = 1, ..., p @@ -406,170 +428,46 @@ def _constrain_sv_less_than_one_python(unconstrained, order=None, def _compute_coefficients_from_multivariate_pacf_python( - partial_autocorrelations, error_variance, order, k_endog, - transform_variance=False): + partial_autocorrelations, error_variance, transform_variance=False, + order=None, k_endog=None): """ Transform matrices with singular values less than one to matrices corresponding to a stationary (or invertible) process. - Corresponds to Lemma 2.1 in Ansley and Kohn (1986). See - `constrain_stationary_multivariate` for more details. - """ - from scipy import linalg - - partial_autocorrelations = np.concatenate(partial_autocorrelations, axis=1) - - # If we want to keep the provided variance but with the constrained - # coefficient matrices, we need to make a copy here, and then after the - # main loop we will transform the coefficients to match the passed variance - if not transform_variance: - initial_variance = error_variance - # Need to make the input variance large enough that the recursions - # don't lead to zero-matrices due to roundoff error, which would case - # exceptions from the Cholesky decompositions. - # Note that this will still not always ensure positive definiteness, - # and for k_endog, order large enough an exception may still be raised - error_variance = np.eye(k_endog) * (order + k_endog)**10 - - forward_variance = error_variance # \Sigma_s - backward_variance = error_variance # \Sigma_s^*, s = 0, ..., p - autocovariances = np.zeros((k_endog, k_endog * (order+1))) # \Gamma_s - autocovariances[:, :k_endog] = error_variance - # \phi_{s,k}, s = 1, ..., p - # k = 1, ..., s+1 - forwards = np.zeros((k_endog, k_endog * order)) - prev_forwards = np.zeros((k_endog, k_endog * order)) - # \phi_{s,k}^* - backwards = np.zeros((k_endog, k_endog * order)) - prev_backwards = np.zeros((k_endog, k_endog * order)) - - error_variance_factor = linalg.cholesky(error_variance, lower=True) - - forward_factors = error_variance_factor - backward_factors = error_variance_factor - - tmp = np.zeros((k_endog, k_endog)) - - # We fill in the entries as follows: - # [1,1] - # [2,2], [2,1] - # [3,3], [3,1], [3,2] - # ... - # [p,p], [p,1], ..., [p,p-1] - # the last row, correctly ordered, is then used as the coefficients - for s in range(order): # s = 0, ..., p-1 - prev_forwards = forwards.copy() - prev_backwards = backwards.copy() - - # Create the "last" (k = s+1) matrix - # Note: this is for k = s+1. However, below we then have to fill - # in for k = 1, ..., s in order. - # P L*^{-1} = x - # x L* = P - # L*' x' = P' - forwards[:, s*k_endog:(s+1)*k_endog] = np.dot( - forward_factors, - linalg.solve_triangular( - backward_factors, - partial_autocorrelations[:, s*k_endog:(s+1)*k_endog].T, - lower=True, trans='T').T - ) - - # P' L^{-1} = x - # x L = P' - # L' x' = P - backwards[:, s*k_endog:(s+1)*k_endog] = np.dot( - backward_factors, - linalg.solve_triangular( - forward_factors, - partial_autocorrelations[:, s*k_endog:(s+1)*k_endog], - lower=True, trans='T').T - ) - - # Update the variance - # Note: if s >= 1, this will be further updated in the for loop - # below - # Also, this calculation will be re-used in the forward variance - tmp = np.dot(forwards[:, s*k_endog:(s+1)*k_endog], backward_variance) - autocovariances[:, (s+1)*k_endog:(s+2)*k_endog] = tmp.copy().T - - # Create the remaining k = 1, ..., s matrices, - # only has an effect if s >= 1 - for k in range(s): - forwards[:, k*k_endog:(k+1)*k_endog] = ( - prev_forwards[:, k*k_endog:(k+1)*k_endog] - - np.dot( - forwards[:, s*k_endog:(s+1)*k_endog], - prev_backwards[:, (s-k-1)*k_endog:(s-k)*k_endog] - ) - ) - - backwards[:, k*k_endog:(k+1)*k_endog] = ( - prev_backwards[:, k*k_endog:(k+1)*k_endog] - - np.dot( - backwards[:, s*k_endog:(s+1)*k_endog], - prev_forwards[:, (s-k-1)*k_endog:(s-k)*k_endog] - ) - ) - - autocovariances[:, (s+1)*k_endog:(s+2)*k_endog] += np.dot( - autocovariances[:, (k+1)*k_endog:(k+2)*k_endog], - prev_forwards[:, (s-k-1)*k_endog:(s-k)*k_endog].T - ) - - # Create forward and backwards variances - backward_variance = ( - backward_variance - - np.dot( - np.dot( - backwards[:, s*k_endog:(s+1)*k_endog], forward_variance), - backwards[:, s*k_endog:(s+1)*k_endog].T - ) - ) - forward_variance = ( - forward_variance - - np.dot(tmp, forwards[:, s*k_endog:(s+1)*k_endog].T) - ) - - # Cholesky factors - forward_factors = linalg.cholesky(forward_variance, lower=True) - backward_factors = linalg.cholesky(backward_variance, lower=True) - - # If we do not want to use the transformed variance, we need to - # adjust the constrained matrices, as presented in Lemma 2.3, see above - variance = forward_variance - if not transform_variance: - # Here, we need to construct T such that: - # variance = T * initial_variance * T' - # To do that, consider the Cholesky of variance (L) and - # input_variance (M) to get: - # L L' = T M M' T' = (TM) (TM)' - # => L = T M - # => L M^{-1} = T - initial_variance_factor = np.linalg.cholesky(initial_variance) - transformed_variance_factor = np.linalg.cholesky(variance) - transform = np.dot(initial_variance_factor, - np.linalg.inv(transformed_variance_factor)) - inv_transform = np.linalg.inv(transform) - - for s in range(order): - forwards[:, s*k_endog:(s+1)*k_endog] = ( - np.dot( - np.dot(transform, forwards[:, s*k_endog:(s+1)*k_endog]), - inv_transform)) - - return forwards, variance - + Parameters + ---------- + partial_autocorrelations : list + Partial autocorrelation matrices. Should be a list of length `order`, + where each element is an array sized `k_endog` x `k_endog`. + error_variance : array + The variance / covariance matrix of the error term. Should be sized + `k_endog` x `k_endog`. This is used as input in the algorithm even if + is not transformed by it (when `transform_variance` is False). The + error term variance is required input when transformation is used + either to force an autoregressive component to be stationary or to + force a moving average component to be invertible. + transform_variance : boolean, optional + Whether or not to transform the error variance term. This option is + not typically used, and the default is False. + order : integer, optional + The order of the autoregression. + k_endog : integer, optional + The dimension of the data vector. -def _compute_coefficients_from_multivariate_pacf_python2( - partial_autocorrelations, error_variance, order=None, k_endog=None, - transform_variance=False): - """ - Transform matrices with singular values less than one to matrices - corresponding to a stationary (or invertible) process. + Returns + ------- + coefficient_matrices : list + Transformed coefficient matrices leading to a stationary VAR + representation. + Notes + ----- Corresponds to Lemma 2.1 in Ansley and Kohn (1986). See `constrain_stationary_multivariate` for more details. + + There is a Cython implementation of this function that can be much faster, + but which requires SciPy 0.14.0 or greater. See + `constrain_stationary_multivariate` for details. """ from scipy import linalg @@ -702,7 +600,7 @@ def _compute_coefficients_from_multivariate_pacf_python2( return forwards, variance -def constrain_stationary_multivariate_python(unconstrained, variance, +def constrain_stationary_multivariate_python(unconstrained, error_variance, transform_variance=False, prefix=None): """ @@ -711,24 +609,33 @@ def constrain_stationary_multivariate_python(unconstrained, variance, Parameters ---------- - unconstrained : iterable + unconstrained : array or list Arbitrary matrices to be transformed to stationary coefficient matrices - of the VAR. - variance : array, 2-dim - Variance matrix corresponding to the error term. This is used as - input in the algorithm even if is not transformed by it (when - `transform_variance` is False. The error term variance is required - input when transformation is used either to force an autoregressive - component to be stationary or to force a moving average component to - be invertible. + of the VAR. If a list, should be a list of length `order`, where each + element is an array sized `k_endog` x `k_endog`. If an array, should be + the matrices horizontally concatenated and sized + `k_endog` x `k_endog * order`. + error_variance : array + The variance / covariance matrix of the error term. Should be sized + `k_endog` x `k_endog`. This is used as input in the algorithm even if + is not transformed by it (when `transform_variance` is False). The + error term variance is required input when transformation is used + either to force an autoregressive component to be stationary or to + force a moving average component to be invertible. transform_variance : boolean, optional Whether or not to transform the error variance term. This option is not typically used, and the default is False. + prefix : {'s','d','c','z'}, optional + The appropriate BLAS prefix to use for the passed datatypes. Only + use if absolutely sure that the prefix is correct or an error will + result. Returns ------- - constrained : list - A list of coefficient matrices which lead to a stationary VAR. + constrained : array or list + Transformed coefficient matrices leading to a stationary VAR + representation. Will match the type of the passed `unconstrained` + variable (so if a list was passed, a list will be returned). Notes ----- @@ -753,13 +660,16 @@ def constrain_stationary_multivariate_python(unconstrained, variance, References ---------- - Ansley, Craig F., and Robert Kohn. 1986. - "A Note on Reparameterizing a Vector Autoregressive Moving Average Model to - Enforce Stationarity." - Journal of Statistical Computation and Simulation 24 (2): 99-106. + .. [1] Ansley, Craig F., and Robert Kohn. 1986. + "A Note on Reparameterizing a Vector Autoregressive Moving Average Model + to Enforce Stationarity." + Journal of Statistical Computation and Simulation 24 (2): 99-106. + .. [2] Ansley, Craig F, and Paul Newbold. 1979. + "Multivariate Partial Autocorrelations." + In Proceedings of the Business and Economic Statistics Section, 349-53. + American Statistical Association """ - from scipy import linalg use_list = type(unconstrained) == list if not use_list: @@ -776,19 +686,78 @@ def constrain_stationary_multivariate_python(unconstrained, variance, # Step 1: convert from arbitrary matrices to those with singular values # less than one. - sv_constrained = _constrain_sv_less_than_one(unconstrained, order, k_endog) + sv_constrained = _constrain_sv_less_than_one_python( + unconstrained, order, k_endog) # Step 2: convert matrices from our "partial autocorrelation matrix" space # (matrices with singular values less than one) to the space of stationary # coefficient matrices - constrained, variance = _compute_coefficients_from_multivariate_pacf( - sv_constrained, variance, order, k_endog, transform_variance) + constrained, var = _compute_coefficients_from_multivariate_pacf_python( + sv_constrained, error_variance, transform_variance, order, k_endog) if not use_list: constrained = np.concatenate(constrained, axis=1).reshape( k_endog, k_endog * order) - return constrained, variance + return constrained, var + + +# Conditionally use the Cython versions of the multivariate constraint if +# possible (i.e. if Scipy >= 0.14.0 is available.) +if False and has_trmm: + + def constrain_stationary_multivariate(unconstrained, variance, + transform_variance=False, + prefix=None): + + use_list = type(unconstrained) == list + if use_list: + unconstrained = np.concatenate(unconstrained, axis=1) + + k_endog, order = unconstrained.shape + order //= k_endog + + if order < 1: + raise ValueError('Must have order at least 1') + if k_endog < 1: + raise ValueError('Must have at least 1 endogenous variable') + + if prefix is None: + prefix, dtype, _ = find_best_blas_type( + [unconstrained, variance]) + dtype = prefix_dtype_map[prefix] + + unconstrained = np.asfortranarray(unconstrained, dtype=dtype) + variance = np.asfortranarray(variance, dtype=dtype) + + # Step 1: convert from arbitrary matrices to those with singular values + # less than one. + # sv_constrained = _constrain_sv_less_than_one(unconstrained, order, + # k_endog, prefix) + sv_constrained = prefix_sv_map[prefix](unconstrained, order, k_endog) + + # Step 2: convert matrices from our "partial autocorrelation matrix" + # space (matrices with singular values less than one) to the space of + # stationary coefficient matrices + constrained, variance = prefix_pacf_map[prefix]( + sv_constrained, variance, transform_variance, order, k_endog) + + constrained = np.array(constrained) + variance = np.array(variance) + + if use_list: + constrained = [ + constrained[:k_endog, i*k_endog:(i+1)*k_endog] + for i in range(order) + ] + + return constrained, variance + constrain_stationary_multivariate.__doc__ = ( + constrain_stationary_multivariate_python.__doc__) + +else: + constrain_stationary_multivariate = ( + constrain_stationary_multivariate_python) def _unconstrain_sv_less_than_one(constrained, order=None, k_endog=None): @@ -796,8 +765,27 @@ def _unconstrain_sv_less_than_one(constrained, order=None, k_endog=None): Transform matrices with singular values less than one to arbitrary matrices. + Parameters + ---------- + constrained : list + The partial autocorrelation matrices. Should be a list of length + `order`, where each element is an array sized `k_endog` x `k_endog`. + order : integer, optional + The order of the autoregression. + k_endog : integer, optional + The dimension of the data vector. + + Returns + ------- + unconstrained : list + Unconstrained matrices. A list of length `order`, where each element is + an array sized `k_endog` x `k_endog`. + + Notes + ----- Corresponds to the inverse of Lemma 2.2 in Ansley and Kohn (1986). See `unconstrain_stationary_multivariate` for more details. + """ from scipy import linalg @@ -818,24 +806,130 @@ def _unconstrain_sv_less_than_one(constrained, order=None, k_endog=None): return unconstrained -def _compute_multivariate_acovf_from_coefficients(coefficients, error_variance, - order=None, k_endog=None, - maxlag=None): +def _compute_multivariate_sample_acovf(endog, maxlag): + """ + Computer multivariate sample autocovariances + + Parameters + ---------- + endog : array_like + Sample data on which to compute sample autocovariances. Shaped + `nobs` x `k_endog`. + + Returns + ------- + sample_autocovariances : list + A list of the first `maxlag` sample autocovariance matrices. Each + matrix is shaped `k_endog` x `k_endog`. + + Notes + ----- + This function computes the forward sample autocovariances: + + .. math:: + + \hat \Gamma(s) = \frac{1}{n} \sum_{t=1}^{n-s} + (Z_t - \bar Z) (Z_{t+s} - \bar Z)' + + See page 353 of Wei (1990). This function is primarily implemented for + checking the partial autocorrelation functions below, and so is quite slow. + + References + ---------- + .. [1] Wei, William. 1990. + Time Series Analysis : Univariate and Multivariate Methods. + Boston: Pearson. """ + # Get the (demeaned) data as an array + endog = np.array(endog) + if endog.ndim == 1: + endog = endog[:, np.newaxis] + endog -= np.mean(endog, axis=0) + + # Dimensions + nobs, k_endog = endog.shape + + sample_autocovariances = [] + for s in range(maxlag + 1): + sample_autocovariances.append(np.zeros((k_endog, k_endog))) + for t in range(nobs - s): + sample_autocovariances[s] += np.outer(endog[t], endog[t+s]) + sample_autocovariances[s] /= nobs + + return sample_autocovariances + + +def _compute_multivariate_acovf_from_coefficients( + coefficients, error_variance, maxlag=None, + forward_autocovariances=False): + """ + Compute multivariate autocovariances from vector autoregression coefficient + matrices + + Parameters + ---------- + coefficients : array or list + The coefficients matrices. If a list, should be a list of length + `order`, where each element is an array sized `k_endog` x `k_endog`. If + an array, should be the coefficient matrices horizontally concatenated + and sized `k_endog` x `k_endog * order`. + error_variance : array + The variance / covariance matrix of the error term. Should be sized + `k_endog` x `k_endog`. + maxlag : integer, optional + The maximum autocovariance to compute. Default is `order`-1. Can be + zero, in which case it returns the variance. + forward_autocovariances : boolean, optional + Whether or not to compute forward autocovariances + :math:`E(y_t y_{t+j}')`. Default is False, so that backward + autocovariances :math:`E(y_t y_{t-j}')` are returned. + + Returns + ------- + autocovariances : list + A list of the first `maxlag` autocovariance matrices. Each matrix is + shaped `k_endog` x `k_endog`. + Notes ----- + Computes + + ..math:: + + \Gamma(j) = E(y_t y_{t-j}') + + for j = 1, ..., `maxlag`, unless `forward_autocovariances` is specified, + in which case it computes: + + ..math:: + + E(y_t y_{t+j}') = \Gamma(j)' Coefficients are assumed to be provided from the VAR model: .. math:: y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + \varepsilon_t + + Autocovariances are calculated by solving the associated discrete Lyapunov + equation of the state space representation of the VAR process. + """ from scipy import linalg - if order is None: + # Convert coefficients to a list of matrices, for use in + # `companion_matrix`; get dimensions + if type(coefficients) == list: order = len(coefficients) - if k_endog is None: k_endog = coefficients[0].shape[0] + else: + k_endog, order = coefficients.shape + order //= k_endog + + coefficients = [ + coefficients[:k_endog, i*k_endog:(i+1)*k_endog] + for i in range(order) + ] + if maxlag is None: maxlag = order-1 @@ -867,15 +961,58 @@ def _compute_multivariate_acovf_from_coefficients(coefficients, error_variance, stacked_cov[:k_endog, -k_endog:] ] + if forward_autocovariances: + for i in range(len(autocovariances)): + autocovariances[i] = autocovariances[i].T + return autocovariances +def _compute_multivariate_sample_pacf(endog, maxlag): + """ + Computer multivariate sample partial autocorrelations + + Parameters + ---------- + endog : array_like + Sample data on which to compute sample autocovariances. Shaped + `nobs` x `k_endog`. + maxlag : integer + Maximum lag for which to calculate sample partial autocorrelations. + + Returns + ------- + sample_pacf : list + A list of the first `maxlag` sample partial autocorrelation matrices. + Each matrix is shaped `k_endog` x `k_endog`. -def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, - order=None, k_endog=None): """ - Transform matrices corresponding to a stationary (or invertible) process - to matrices with singular values less than one. + sample_autocovariances = _compute_multivariate_sample_acovf(endog, maxlag) + + return _compute_multivariate_pacf_from_autocovariances( + sample_autocovariances) + +def _compute_multivariate_pacf_from_autocovariances(autocovariances, + order=None, k_endog=None): + """ + Compute multivariate partial autocorrelations from autocovariances. + + Parameters + ---------- + autocovariances : list + Autocorrelations matrices. Should be a list of length `order` + 1, + where each element is an array sized `k_endog` x `k_endog`. + order : integer, optional + The order of the autoregression. + k_endog : integer, optional + The dimension of the data vector. + + Returns + ------- + pacf : list + List of first `order` multivariate partial autocorrelations. + Notes + ----- Note that this computes multivariate partial autocorrelations. Corresponds to the inverse of Lemma 2.1 in Ansley and Kohn (1986). See @@ -883,26 +1020,16 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, Notes ----- + Computes sample partial autocorrelations if sample autocovariances are + given. - Coefficients are assumed to be provided from the VAR model: - - .. math:: - y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + \varepsilon_t """ from scipy import linalg if order is None: - order = len(constrained) + order = len(autocovariances)-1 if k_endog is None: - k_endog = constrained[0].shape[0] - - # Get autocovariances for the process; these are defined to be - # E z_t z_{t-j}' - # However, we want E z_t z_{t+j}' = (E z_t z_{t-j}')' - _acovf = _compute_multivariate_acovf_from_coefficients - autocovariances = [ - autocovariance.T for autocovariance in - _acovf(constrained, error_variance, order, k_endog, maxlag=order)] + k_endog = autocovariances[0].shape[0] # Now apply the Ansley and Kohn (1986) algorithm, except that instead of # calculating phi_{s+1, s+1} = L_s P_{s+1} {L_s^*}^{-1} (which requires @@ -965,56 +1092,53 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, linalg.cholesky(backward_variances[s], lower=True) ) - if s == order-1: - forwards = constrained + # Create the intermediate sum term + if s == 0: + # phi_11 = \Gamma_1' \Gamma_0^{-1} + # phi_11 \Gamma_0 = \Gamma_1' + # \Gamma_0 phi_11' = \Gamma_1 + forwards.append(linalg.cho_solve( + (forward_factors[0], True), autocovariances[1]).T) + # backwards.append(forwards[-1]) + # phi_11_star = \Gamma_1 \Gamma_0^{-1} + # phi_11_star \Gamma_0 = \Gamma_1 + # \Gamma_0 phi_11_star' = \Gamma_1' + backwards.append(linalg.cho_solve( + (backward_factors[0], True), autocovariances[1].T).T) else: - # Create the intermediate sum term - if s == 0: - # phi_11 = \Gamma_1' \Gamma_0^{-1} - # phi_11 \Gamma_0 = \Gamma_1' - # \Gamma_0 phi_11' = \Gamma_1 - forwards.append(linalg.cho_solve( - (forward_factors[0], True), autocovariances[1]).T) - # backwards.append(forwards[-1]) - # phi_11_star = \Gamma_1 \Gamma_0^{-1} - # phi_11_star \Gamma_0 = \Gamma_1 - # \Gamma_0 phi_11_star' = \Gamma_1' - backwards.append(linalg.cho_solve( - (backward_factors[0], True), autocovariances[1].T).T) - else: - # G := \Gamma_{s+1}' - - # \phi_{s,1} \Gamma_s' - .. - \phi_{s,s} \Gamma_1' - tmp_sum = autocovariances[s+1].T.copy() - - for k in range(s): - tmp_sum -= np.dot(prev_forwards[k], autocovariances[s-k].T) - - # Create the "last" (k = s+1) matrix - # Note: this is for k = s+1. However, below we then have to - # fill in for k = 1, ..., s in order. - # phi = G Sigma*^{-1} - # phi Sigma* = G - # Sigma*' phi' = G' - # Sigma* phi' = G' - # (because Sigma* is symmetric) - forwards.append(linalg.cho_solve( - (backward_factors[s], True), tmp_sum.T).T) - - # phi = G' Sigma^{-1} - # phi Sigma = G' - # Sigma' phi' = G - # Sigma phi' = G - # (because Sigma is symmetric) - backwards.append(linalg.cho_solve( - (forward_factors[s], True), tmp_sum).T) - - # Create the remaining k = 1, ..., s matrices, - # only has an effect if s >= 1 + # G := \Gamma_{s+1}' - + # \phi_{s,1} \Gamma_s' - .. - \phi_{s,s} \Gamma_1' + tmp_sum = autocovariances[s+1].T.copy() + for k in range(s): - forwards.insert(k, prev_forwards[k] - np.dot( - forwards[-1], prev_backwards[s-(k+1)])) - backwards.insert(k, prev_backwards[k] - np.dot( - backwards[-1], prev_forwards[s-(k+1)])) + tmp_sum -= np.dot(prev_forwards[k], autocovariances[s-k].T) + + # Create the "last" (k = s+1) matrix + # Note: this is for k = s+1. However, below we then have to + # fill in for k = 1, ..., s in order. + # phi = G Sigma*^{-1} + # phi Sigma* = G + # Sigma*' phi' = G' + # Sigma* phi' = G' + # (because Sigma* is symmetric) + forwards.append(linalg.cho_solve( + (backward_factors[s], True), tmp_sum.T).T) + + # phi = G' Sigma^{-1} + # phi Sigma = G' + # Sigma' phi' = G + # Sigma phi' = G + # (because Sigma is symmetric) + backwards.append(linalg.cho_solve( + (forward_factors[s], True), tmp_sum).T) + + # Create the remaining k = 1, ..., s matrices, + # only has an effect if s >= 1 + for k in range(s): + forwards.insert(k, prev_forwards[k] - np.dot( + forwards[-1], prev_backwards[s-(k+1)])) + backwards.insert(k, prev_backwards[k] - np.dot( + backwards[-1], prev_forwards[s-(k+1)])) # Partial autocorrelation matrix: P_{s+1} # P = L^{-1} phi L* @@ -1025,84 +1149,104 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, return partial_autocorrelations +def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, + order=None, k_endog=None): + """ + Transform matrices corresponding to a stationary (or invertible) process + to matrices with singular values less than one. -if has_trmm: - - def constrain_stationary_multivariate(unconstrained, variance, - transform_variance=False, - prefix=None): - - use_list = type(unconstrained) == list - if use_list: - unconstrained = np.concatenate(unconstrained, axis=1) + Parameters + ---------- + constrained : array or list + The coefficients matrices. If a list, should be a list of length + `order`, where each element is an array sized `k_endog` x `k_endog`. If + an array, should be the coefficient matrices horizontally concatenated + and sized `k_endog` x `k_endog * order`. + error_variance : array + The variance / covariance matrix of the error term. Should be sized + `k_endog` x `k_endog`. + order : integer, optional + The order of the autoregression. + k_endog : integer, optional + The dimension of the data vector. - unconstrained = np.asfortranarray(unconstrained) - variance = np.asfortranarray(variance) + Returns + ------- + pacf : list + List of first `order` multivariate partial autocorrelations. - k_endog, order = unconstrained.shape - order //= k_endog + Notes + ----- + Note that this computes multivariate partial autocorrelations. - if prefix is None: - prefix, dtype, _ = find_best_blas_type( - [unconstrained, variance]) + Corresponds to the inverse of Lemma 2.1 in Ansley and Kohn (1986). See + `unconstrain_stationary_multivariate` for more details. - # Step 1: convert from arbitrary matrices to those with singular values - # less than one. - # sv_constrained = _constrain_sv_less_than_one(unconstrained, order, - # k_endog, prefix) - sv_constrained = prefix_sv_map[prefix](unconstrained, order, k_endog) + Notes + ----- - # Step 2: convert matrices from our "partial autocorrelation matrix" - # space (matrices with singular values less than one) to the space of - # stationary coefficient matrices - constrained, variance = prefix_pacf_map[prefix]( - sv_constrained, variance, order, k_endog, transform_variance) + Coefficients are assumed to be provided from the VAR model: - constrained = np.array(constrained) - variance = np.array(variance) + .. math:: + y_t = A_1 y_{t-1} + \dots + A_p y_{t-p} + \varepsilon_t + """ - if use_list: - constrained = [ - constrained[:k_endog, i*k_endog:(i+1)*k_endog] - for i in range(order) - ] + if type(constrained) == list: + order = len(constrained) + k_endog = constrained[0].shape[0] + else: + k_endog, order = constrained.shape + order //= k_endog + + # Get autocovariances for the process; these are defined to be + # E z_t z_{t-j}' + # However, we want E z_t z_{t+j}' = (E z_t z_{t-j}')' + _acovf = _compute_multivariate_acovf_from_coefficients - return constrained, variance + autocovariances = [ + autocovariance.T for autocovariance in + _acovf(constrained, error_variance, maxlag=order)] -else: - _compute_coefficients_from_multivariate_pacf = ( - _compute_coefficients_from_multivariate_pacf_python) - _constrain_sv_less_than_one = _constrain_sv_less_than_one_python - constrain_stationary_multivariate = ( - constrain_stationary_multivariate_python) + return _compute_multivariate_pacf_from_autocovariances(autocovariances) -def unconstrain_stationary_multivariate(constrained, error_variance, - transform_variance=False): +def unconstrain_stationary_multivariate(constrained, error_variance): """ Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer Parameters ---------- - constrained : array + constrained : array or list Constrained parameters of, e.g., an autoregressive or moving average component, to be transformed to arbitrary parameters used by the - optimizer. + optimizer. If a list, should be a list of length `order`, where each + element is an array sized `k_endog` x `k_endog`. If an array, should be + the coefficient matrices horizontally concatenated and sized + `k_endog` x `k_endog * order`. + error_variance : array + The variance / covariance matrix of the error term. Should be sized + `k_endog` x `k_endog`. This is used as input in the algorithm even if + is not transformed by it (when `transform_variance` is False). Returns ------- unconstrained : array Unconstrained parameters used by the optimizer, to be transformed to stationary coefficients of, e.g., an autoregressive or moving average - component. + component. Will match the type of the passed `constrained` + variable (so if a list was passed, a list will be returned). + + Notes + ----- + Uses the list representation internally, even if an array is passed. References ---------- - Ansley, Craig F., and Robert Kohn. 1986. - "A Note on Reparameterizing a Vector Autoregressive Moving Average Model to - Enforce Stationarity." - Journal of Statistical Computation and Simulation 24 (2): 99-106. + .. [1] Ansley, Craig F., and Robert Kohn. 1986. + "A Note on Reparameterizing a Vector Autoregressive Moving Average Model + to Enforce Stationarity." + Journal of Statistical Computation and Simulation 24 (2): 99-106. """ @@ -1117,9 +1261,9 @@ def unconstrain_stationary_multivariate(constrained, error_variance, constrained[:k_endog, i*k_endog:(i+1)*k_endog] for i in range(order) ] - - order = len(constrained) - k_endog = constrained[0].shape[0] + else: + order = len(constrained) + k_endog = constrained[0].shape[0] # Step 1: convert matrices from the space of stationary # coefficient matrices to our "partial autocorrelation matrix" space From 181ce01b5f3e974d36948a4a71d151494250098c Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Fri, 7 Aug 2015 15:44:29 -0700 Subject: [PATCH 22/57] CLEAN: pep8, whitespace --- statsmodels/tsa/statespace/dynamic_factor.py | 37 +++++---- .../tests/results/test_varmax_stata.do | 1 - statsmodels/tsa/statespace/tools.py | 9 ++- statsmodels/tsa/statespace/varmax.py | 78 +++++++++---------- 4 files changed, 65 insertions(+), 60 deletions(-) diff --git a/statsmodels/tsa/statespace/dynamic_factor.py b/statsmodels/tsa/statespace/dynamic_factor.py index de884d8805d..0110633ca8e 100644 --- a/statsmodels/tsa/statespace/dynamic_factor.py +++ b/statsmodels/tsa/statespace/dynamic_factor.py @@ -66,7 +66,7 @@ class StaticFactors(MLEModel): We assume that :math:`\varepsilon_{it} \sim N(0, \sigma_{\varepsilon_i}^2)` and :math:`\eta_t \sim N(0, I)` - + References ---------- .. [1] Durbin, James, and Siem Jan Koopman. 2012. @@ -124,7 +124,7 @@ def _slice(key, offset): param_slice = np.s_[offset:offset + length] offset += length return param_slice, offset - + offset = 0 self._params_loadings, offset = _slice('factor_loadings', offset) self._params_idiosyncratic, offset = _slice('idiosyncratic', offset) @@ -156,7 +156,7 @@ def start_params(self): X = lagmat(res_pca.factors, self.factor_order, trim='both') params_ar = np.linalg.pinv(X).dot(Y) resid = Y - np.dot(X, params_ar) - params[self._params_transition] = params_ar[:,0] + params[self._params_transition] = params_ar[:, 0] return params @@ -220,13 +220,17 @@ def transform_params(self, unconstrained): # VAR transition: optionally force to be stationary if self.enforce_stationarity: # Transform the parameters - coefficients = unconstrained[self._params_transition].reshape(self.k_factors, self.k_factors * self.factor_order) - unconstrained_matrices = [coefficients[:,i*self.k_factors:(i+1)*self.k_factors] for i in range(self.factor_order)] + unconstrained_matrices = ( + unconstrained[self._params_transition].reshape( + self.k_factors, self.k_factors * self.factor_order)) coefficient_matrices, variance = ( - constrain_stationary_multivariate(unconstrained_matrices, self.ssm['state_cov'])) - constrained[self._params_transition] = np.concatenate(coefficient_matrices, axis=1).ravel() + constrain_stationary_multivariate(unconstrained_matrices, + self.ssm['state_cov'])) + constrained[self._params_transition] = ( + coefficient_matrices.ravel()) else: - constrained[self._params_transition] = unconstrained[self._params_transition] + constrained[self._params_transition] = ( + unconstrained[self._params_transition]) return constrained @@ -259,13 +263,16 @@ def untransform_params(self, constrained): # VAR transition: optionally were forced to be stationary if self.enforce_stationarity: - coefficients = constrained[self._params_transition].reshape(self.k_factors, self.k_factors * self.factor_order) - coefficient_matrices = [coefficients[:,i*self.k_factors:(i+1)*self.k_factors] for i in range(self.factor_order)] + coefficients = constrained[self._params_transition].reshape( + self.k_factors, self.k_factors * self.factor_order) unconstrained_matrices, variance = ( - unconstrain_stationary_multivariate(coefficient_matrices, self.ssm['state_cov'])) - unconstrained[self._params_transition] = np.concatenate(unconstrained_matrices, axis=1).ravel() + unconstrain_stationary_multivariate(coefficients, + self.ssm['state_cov'])) + unconstrained[self._params_transition] = ( + unconstrained_matrices.ravel()) else: - unconstrained[self._params_transition] = constrained[self._params_transition] + unconstrained[self._params_transition] = ( + constrained[self._params_transition]) return unconstrained @@ -330,4 +337,6 @@ def update(self, params, transformed=True): self.ssm[self._idx_obs_cov] = params[self._params_idiosyncratic] # Update the transition matrix - self.ssm[self._idx_transition] = params[self._params_transition].reshape(self.k_factors, self.k_factors * self.factor_order) + self.ssm[self._idx_transition] = ( + params[self._params_transition].reshape( + self.k_factors, self.k_factors * self.factor_order)) diff --git a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do index 6e79a45a6bb..3a25b80758c 100644 --- a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do +++ b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do @@ -58,6 +58,5 @@ predict predict_var2_2, dynamic(tq(1979q1)) equation(dln_inc) predict dyn_predict_var2_1, dynamic(tq(1961q1)) equation(dln_inv) predict dyn_predict_var2_2, dynamic(tq(1961q1)) equation(dln_inc) - outsheet pred* dyn* using results_varmax_stata.csv, comma replace diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 72dc11757ed..9d661037c8f 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -726,7 +726,7 @@ def constrain_stationary_multivariate(unconstrained, variance, prefix, dtype, _ = find_best_blas_type( [unconstrained, variance]) dtype = prefix_dtype_map[prefix] - + unconstrained = np.asfortranarray(unconstrained, dtype=dtype) variance = np.asfortranarray(variance, dtype=dtype) @@ -845,7 +845,7 @@ def _compute_multivariate_sample_acovf(endog, maxlag): if endog.ndim == 1: endog = endog[:, np.newaxis] endog -= np.mean(endog, axis=0) - + # Dimensions nobs, k_endog = endog.shape @@ -967,6 +967,7 @@ def _compute_multivariate_acovf_from_coefficients( return autocovariances + def _compute_multivariate_sample_pacf(endog, maxlag): """ Computer multivariate sample partial autocorrelations @@ -991,6 +992,7 @@ def _compute_multivariate_sample_pacf(endog, maxlag): return _compute_multivariate_pacf_from_autocovariances( sample_autocovariances) + def _compute_multivariate_pacf_from_autocovariances(autocovariances, order=None, k_endog=None): """ @@ -1149,6 +1151,7 @@ def _compute_multivariate_pacf_from_autocovariances(autocovariances, return partial_autocorrelations + def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, order=None, k_endog=None): """ @@ -1197,7 +1200,7 @@ def _compute_multivariate_pacf_from_coefficients(constrained, error_variance, else: k_endog, order = constrained.shape order //= k_endog - + # Get autocovariances for the process; these are defined to be # E z_t z_{t-j}' # However, we want E z_t z_{t+j}' = (E z_t z_{t-j}')' diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index 18ca48eec7e..bf224ef2742 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -61,11 +61,11 @@ class VARMAX(MLEModel): `KalmanFilter` for more details. """ - def __init__(self, endog, exog=None, order=(1,0), trend='c', + def __init__(self, endog, exog=None, order=(1, 0), trend='c', error_cov_type='unstructured', measurement_error=False, enforce_stationarity=True, enforce_invertibility=True, **kwargs): - + # Model parameters self.error_cov_type = error_cov_type self.measurement_error = measurement_error @@ -82,7 +82,7 @@ def __init__(self, endog, exog=None, order=(1,0), trend='c', self.k_trend = int(self.trend == 'c') # Check for valid model - if trend not in ['c','nc']: + if trend not in ['c', 'nc']: raise ValueError('Invalid trend specification.') if error_cov_type not in ['diagonal', 'unstructured']: raise ValueError('Invalid error covariance matrix type' @@ -212,7 +212,7 @@ def _slice(key, offset): param_slice = np.s_[offset:offset + length] offset += length return param_slice, offset - + offset = 0 self._params_trend, offset = _slice('trend', offset) self._params_ar, offset = _slice('ar', offset) @@ -249,8 +249,8 @@ def start_params(self): res_ar = mod_ar.fit(maxlags=self.k_ar, ic=None, trend=self.trend) ar_params = np.array(res_ar.params.T) if self.trend == 'c': - trend_params = ar_params[:,0] - ar_params = ar_params[:,1:].ravel() + trend_params = ar_params[:, 0] + ar_params = ar_params[:, 1:].ravel() else: ar_params = ar_params.ravel() endog = res_ar.resid @@ -375,17 +375,15 @@ def transform_params(self, unconstrained): if self.error_cov_type == 'diagonal': state_cov = np.diag(unconstrained[self._params_state_cov]**2) elif self.error_cov_type == 'unstructured': - state_cov_lower = np.zeros(self.ssm['state_cov'].shape, dtype=unconstrained.dtype) - state_cov_lower[self._idx_lower_state_cov] = unconstrained[self._params_state_cov] + state_cov_lower = np.zeros(self.ssm['state_cov'].shape, + dtype=unconstrained.dtype) + state_cov_lower[self._idx_lower_state_cov] = ( + unconstrained[self._params_state_cov]) state_cov = np.dot(state_cov_lower, state_cov_lower.T) # Transform the parameters - # coefficients = unconstrained[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) - # unconstrained_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ar)] - # coefficient_matrices, variance = ( - # constrain_stationary_multivariate(unconstrained_matrices, state_cov)) - # constrained[self._params_ar] = np.concatenate(coefficient_matrices, axis=1).ravel() - coefficients = unconstrained[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) + coefficients = unconstrained[self._params_ar].reshape( + self.k_endog, self.k_endog * self.k_ar) coefficient_matrices, variance = ( constrain_stationary_multivariate(coefficients, state_cov)) constrained[self._params_ar] = coefficient_matrices.ravel() @@ -395,15 +393,11 @@ def transform_params(self, unconstrained): # 3. MA terms: optionally force to be invertible if self.k_ma > 0 and self.enforce_invertibility: # Transform the parameters, using an identity variance matrix - # coefficients = unconstrained[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) - # unconstrained_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ma)] - # # TODO cache this np.eye call - # coefficient_matrices, variance = ( - # constrain_stationary_multivariate(unconstrained_matrices, np.eye(self.k_endog, dtype=constrained.dtype))) - # constrained[self._params_ma] = np.concatenate(coefficient_matrices, axis=1).ravel() - coefficients = unconstrained[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) + state_cov = np.eye(self.k_endog, dtype=constrained.dtype) + coefficients = unconstrained[self._params_ma].reshape( + self.k_endog, self.k_endog * self.k_ma) coefficient_matrices, variance = ( - constrain_stationary_multivariate(coefficients, np.eye(self.k_endog, dtype=constrained.dtype))) + constrain_stationary_multivariate(coefficients, state_cov)) constrained[self._params_ma] = coefficient_matrices.ravel() else: constrained[self._params_ma] = unconstrained[self._params_ma] @@ -458,17 +452,15 @@ def untransform_params(self, constrained): if self.error_cov_type == 'diagonal': state_cov = np.diag(constrained[self._params_state_cov]**2) elif self.error_cov_type == 'unstructured': - state_cov_lower = np.zeros(self.ssm['state_cov'].shape, dtype=constrained.dtype) - state_cov_lower[self._idx_lower_state_cov] = constrained[self._params_state_cov] + state_cov_lower = np.zeros(self.ssm['state_cov'].shape, + dtype=constrained.dtype) + state_cov_lower[self._idx_lower_state_cov] = ( + constrained[self._params_state_cov]) state_cov = np.dot(state_cov_lower, state_cov_lower.T) # Transform the parameters - # coefficients = constrained[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) - # coefficient_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ar)] - # unconstrained_matrices, variance = ( - # unconstrain_stationary_multivariate(coefficient_matrices, state_cov)) - # unconstrained[self._params_ar] = np.concatenate(unconstrained_matrices, axis=1).ravel() - coefficients = constrained[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) + coefficients = constrained[self._params_ar].reshape( + self.k_endog, self.k_endog * self.k_ar) unconstrained_matrices, variance = ( unconstrain_stationary_multivariate(coefficients, state_cov)) unconstrained[self._params_ar] = unconstrained_matrices.ravel() @@ -478,14 +470,11 @@ def untransform_params(self, constrained): # 3. MA terms: optionally were forced to be invertible if self.k_ma > 0 and self.enforce_invertibility: # Transform the parameters, using an identity variance matrix - # coefficients = constrained[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) - # coefficient_matrices = [coefficients[:,i*self.k_endog:(i+1)*self.k_endog] for i in range(self.k_ma)] - # unconstrained_matrices, variance = ( - # unconstrain_stationary_multivariate(coefficient_matrices, np.eye(self.k_endog, dtype=constrained.dtype))) - # unconstrained[self._params_ma] = np.concatenate(unconstrained_matrices, axis=1).ravel() - coefficients = constrained[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) + state_cov = np.eye(self.k_endog, dtype=constrained.dtype) + coefficients = constrained[self._params_ma].reshape( + self.k_endog, self.k_endog * self.k_ma) unconstrained_matrices, variance = ( - unconstrain_stationary_multivariate(coefficients, np.eye(self.k_endog, dtype=constrained.dtype))) + unconstrain_stationary_multivariate(coefficients, state_cov)) unconstrained[self._params_ma] = unconstrained_matrices.ravel() else: unconstrained[self._params_ma] = constrained[self._params_ma] @@ -517,7 +506,8 @@ def update(self, params, *args, **kwargs): # 1. State intercept if self.mle_regression: - exog_params = params[self._params_regression].reshape(self.k_exog, self.k_endog) + exog_params = params[self._params_regression].reshape( + self.k_exog, self.k_endog) intercept = np.dot(self.exog, exog_params) if self.trend == 'c': intercept += params[self._params_trend] @@ -526,8 +516,10 @@ def update(self, params, *args, **kwargs): self.ssm[self._idx_state_intercept] = params[self._params_trend] # 2. Transition - ar = params[self._params_ar].reshape(self.k_endog, self.k_endog * self.k_ar) - ma = params[self._params_ma].reshape(self.k_endog, self.k_endog * self.k_ma) + ar = params[self._params_ar].reshape( + self.k_endog, self.k_endog * self.k_ar) + ma = params[self._params_ma].reshape( + self.k_endog, self.k_endog * self.k_ma) self.ssm[self._idx_transition] = np.c_[ar, ma] # 3. State covariance @@ -536,8 +528,10 @@ def update(self, params, *args, **kwargs): params[self._params_state_cov] ) elif self.error_cov_type == 'unstructured': - state_cov_lower = np.zeros(self.ssm['state_cov'].shape, dtype=params.dtype) - state_cov_lower[self._idx_lower_state_cov] = params[self._params_state_cov] + state_cov_lower = np.zeros(self.ssm['state_cov'].shape, + dtype=params.dtype) + state_cov_lower[self._idx_lower_state_cov] = ( + params[self._params_state_cov]) self.ssm['state_cov'] = np.dot(state_cov_lower, state_cov_lower.T) # 4. Observation covariance From 55d5b7f618fe77d10609e92fd56df681cce36acf Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Fri, 7 Aug 2015 16:26:17 -0700 Subject: [PATCH 23/57] REF: Enable Cython multivariate constraint --- statsmodels/tsa/statespace/tools.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 9d661037c8f..708057f99c9 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -704,7 +704,7 @@ def constrain_stationary_multivariate_python(unconstrained, error_variance, # Conditionally use the Cython versions of the multivariate constraint if # possible (i.e. if Scipy >= 0.14.0 is available.) -if False and has_trmm: +if has_trmm: def constrain_stationary_multivariate(unconstrained, variance, transform_variance=False, From d803bc17db952016dc2a38fc5050fc2436686bc5 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Fri, 7 Aug 2015 16:26:31 -0700 Subject: [PATCH 24/57] ENH: Add VARMAX, StaticFactors to api --- statsmodels/tsa/api.py | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/statsmodels/tsa/api.py b/statsmodels/tsa/api.py index c9b0ba558d7..cff531f6f7f 100644 --- a/statsmodels/tsa/api.py +++ b/statsmodels/tsa/api.py @@ -17,4 +17,6 @@ from .x13 import x13_arima_select_order from .x13 import x13_arima_analysis from .statespace import api as statespace -from .statespace.sarimax import SARIMAX \ No newline at end of file +from .statespace.sarimax import SARIMAX +from .statespace.varmax import VARMAX +from .statespace.dynamic_factor import StaticFactors \ No newline at end of file From 1257fa9a689156ffad63a867a48548717b83e31f Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sun, 9 Aug 2015 00:05:53 -0700 Subject: [PATCH 25/57] TST: Add test with 2 factors. --- .../tests/results/results_dynamic_factor.py | 16 ++ .../results/results_dynamic_factor_stata.csv | 186 +++++++++--------- .../results/test_dynamic_factor_stata.do | 15 ++ .../statespace/tests/test_dynamic_factor.py | 29 +++ 4 files changed, 153 insertions(+), 93 deletions(-) diff --git a/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py index 72ff37f9913..d875597798d 100644 --- a/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py @@ -27,3 +27,19 @@ 'aic': -1172.18, 'bic': -1153.641, } + +lutkepohl_dfm2 = { + 'params': [ + .03411188, .03478764, # Factor loadings: y1 + .03553366, .0344871, # Factor loadings: y2 + .03536757, .03433391, # Factor loadings: y3 + .00224401, .00014678, .00010922, # Idiosyncratic variances + .08845946, .08862982, # Factor transitions: Phi, row 1 + .08754759, .08758589 # Phi, row 2 + ], + 'bse_oim': None, + 'loglike': 496.379832917306, + 'aic': -974.7597, + 'bic': -953.9023, +} + diff --git a/statsmodels/tsa/statespace/tests/results/results_dynamic_factor_stata.csv b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor_stata.csv index a5df40f10f0..990f515687a 100644 --- a/statsmodels/tsa/statespace/tests/results/results_dynamic_factor_stata.csv +++ b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor_stata.csv @@ -1,93 +1,93 @@ -predict_dfm_1,predict_dfm_2,predict_dfm_3,dyn_predict_dfm_1,dyn_predict_dfm_2,dyn_predict_dfm_3 -,,,,, -0,0,0,0,0,0 -.0169778,.0175878,.016944,.0169778,.0175878,.016944 -.0241789,.0250477,.0241307,.0241789,.0250477,.0241307 -.0276292,.0286219,.0275741,.0276292,.0286219,.0275741 -.0253204,.0262301,.0252699,.0252399,.0261467,.0251895 -.0208676,.0216173,.020826,.0260176,.0269524,.0259657 -.0173758,.0180001,.0173412,.0247476,.0256368,.0246983 -.0236642,.0245144,.023617,.0248244,.0257163,.0247749 -.0201676,.0208922,.0201273,.024038,.0249016,.02399 -.0204401,.0211744,.0203993,.0238267,.0246828,.0237792 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a/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do b/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do index 2fadb939fa3..ca3286a1879 100644 --- a/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do +++ b/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do @@ -14,5 +14,20 @@ predict dyn_predict_dfm_1, dynamic(tq(1961q1)) equation(dln_inv) predict dyn_predict_dfm_2, dynamic(tq(1961q1)) equation(dln_inc) predict dyn_predict_dfm_3, dynamic(tq(1961q1)) equation(dln_consump) +// Dynamic Factors: Static Factor model with 2 factors +// Note: this does not converge even if we do not enforce iter(#), but this is +// good enough for testing +dfactor (dln_inv dln_inc dln_consump = , noconstant ) (f1 f2 = , ar(1) arstructure(general)) if qtr<=tq(1978q4), iter(1) + +// predict + forecast, see above +predict predict_dfm2_1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_dfm2_2, dynamic(tq(1979q1)) equation(dln_inc) +predict predict_dfm2_3, dynamic(tq(1979q1)) equation(dln_consump) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_dfm2_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_dfm2_2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_dfm2_3, dynamic(tq(1961q1)) equation(dln_consump) + outsheet pred* dyn* using results_dynamic_factor_stata.csv, comma replace diff --git a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py index eb2df36f0ae..0cad32946c4 100644 --- a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py @@ -86,3 +86,32 @@ def __init__(self): true['predict'] = output_results.ix[1:, ['predict_dfm_1', 'predict_dfm_2', 'predict_dfm_3']] true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_dfm_1', 'dyn_predict_dfm_2', 'dyn_predict_dfm_3']] super(TestStaticFactor, self).__init__(true, k_factors=1, factor_order=2) + +class TestStaticFactor2(CheckStaticFactor): + def __init__(self): + true = results_dynamic_factor.lutkepohl_dfm2.copy() + true['predict'] = output_results.ix[1:, ['predict_dfm2_1', 'predict_dfm2_2', 'predict_dfm2_3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_dfm2_1', 'dyn_predict_dfm2_2', 'dyn_predict_dfm2_3']] + super(TestStaticFactor2, self).__init__(true, k_factors=2, factor_order=1) + + def test_mle(self): + # Stata's MLE on this model doesn't converge, so no reason to check + pass + + def test_bse(self): + # Stata's MLE on this model doesn't converge, and four of their + # params don't even have bse (possibly they are still at starting + # values?), so no reason to check this + pass + + def test_aic(self): + # Stata uses 9 df (i.e. 9 params) here instead of 13, because since the + # model didn't coverge, 4 of the parameters aren't fully estimated + # (possibly they are still at starting values?) so the AIC is off + pass + + def test_bic(self): + # Stata uses 9 df (i.e. 9 params) here instead of 13, because since the + # model didn't coverge, 4 of the parameters aren't fully estimated + # (possibly they are still at starting values?) so the BIC is off + pass From 8962e1b3cc8bce542d81f64aebc63bd9cc0b55fa Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sun, 9 Aug 2015 00:11:16 -0700 Subject: [PATCH 26/57] TST: Add test for enforce_stationarity=False. --- .../tsa/statespace/tests/test_dynamic_factor.py | 10 ++++++++++ 1 file changed, 10 insertions(+) diff --git a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py index 0cad32946c4..cf1f4202f42 100644 --- a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py @@ -45,6 +45,16 @@ def __init__(self, true, k_factors, factor_order, cov_type='oim', self.results = self.model.filter(true['params'], cov_type=cov_type) + def test_no_enforce(self): + # Test that nothing goes wrong when we don't enforce stationarity + params = self.model.untransform_params(self.true['params']) + params[self.model._params_transition] = ( + self.true['params'][self.model._params_transition]) + self.model.enforce_stationarity = False + results = self.model.filter(params, transformed=False) + self.model.enforce_stationarity = True + assert_allclose(results.llf, self.results.llf, rtol=1e-5) + def test_mle(self): results = self.model.fit(maxiter=100, disp=False) results = self.model.fit(results.params, method='nm', maxiter=1000, From 2af1c21d66e95f50f97ccfd33e064c60e542f0b9 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sun, 9 Aug 2015 12:54:13 -0700 Subject: [PATCH 27/57] BUG/TST: Fix VMA spec., add VARMA unit tests. --- .../tests/results/results_var_stata.csv | 93 +++ .../tests/results/results_varmax.py | 28 +- .../tests/results/results_varmax_stata.csv | 543 +++++++++++++++--- .../tests/results/test_varmax_stata.do | 33 +- .../tsa/statespace/tests/test_varmax.py | 202 +++++-- statsmodels/tsa/statespace/varmax.py | 46 +- 6 files changed, 784 insertions(+), 161 deletions(-) create mode 100644 statsmodels/tsa/statespace/tests/results/results_var_stata.csv diff --git a/statsmodels/tsa/statespace/tests/results/results_var_stata.csv b/statsmodels/tsa/statespace/tests/results/results_var_stata.csv new file mode 100644 index 00000000000..df045b2e476 --- /dev/null +++ b/statsmodels/tsa/statespace/tests/results/results_var_stata.csv @@ -0,0 +1,93 @@ +predict_1,predict_2,predict_3,predict_diag1,predict_diag2,predict_diag3,predict_int1,predict_int2,predict_int3,predict_var2_1,predict_var2_2,dyn_predict_1,dyn_predict_2,dyn_predict_3,dyn_predict_diag1,dyn_predict_diag2,dyn_predict_diag3,dyn_predict_int1,dyn_predict_int2,dyn_predict_int3,dyn_predict_var2_1,dyn_predict_var2_2 +,,,,,,,,,,,,,,,,,,,,, +0,0,0,0,0,0,.017993,.0206546,.0198753,0,0,0,0,0,0,0,0,.017993,.0206546,.0198753,0,0 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+-.005085,-.0005643,5.03e-14,6.78e-15 diff --git a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do index 3a25b80758c..6fd353610b9 100644 --- a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do +++ b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do @@ -58,5 +58,36 @@ predict predict_var2_2, dynamic(tq(1979q1)) equation(dln_inc) predict dyn_predict_var2_1, dynamic(tq(1961q1)) equation(dln_inv) predict dyn_predict_var2_2, dynamic(tq(1961q1)) equation(dln_inc) -outsheet pred* dyn* using results_varmax_stata.csv, comma replace +outsheet pred* dyn* using results_var_stata.csv, comma replace + +// VARMA(1,1) +// Note: Stata does not have this built-in, so we need to create the state space form ourselves +// This replicates example 4 from the Stata documentation +use http://www.stata-press.com/data/r12/manufac, clear + +gen dlncaputil = D.lncaputil +gen dlnhours = D.lnhours + +constraint 1 [u1]L.u2 = 1 +constraint 2 [u1]e.u1 = 1 +constraint 3 [u3]e.u3 = 1 +constraint 4 [dlncaputil]u1 = 1 +constraint 5 [dlnhours]u3 = 1 +sspace (u1 L.u1 L.u2 e.u1, state noconstant) /// + (u2 e.u1, state noconstant) /// + (u3 L.u1 L.u3 e.u3, state noconstant) /// + (dlncaputil u1, noconstant) /// + (dlnhours u3, noconstant), /// + constraints(1/5) technique(nr) covstate(diagonal) + +// in-sample predict + forecast 5 new observations (sample ends at 1999m12) +tsappend, add(5) +predict predict_varma11_1, dynamic(tm(2009m1)) equation(dlncaputil) +predict predict_varma11_2, dynamic(tm(2009m1)) equation(dlnhours) + +// predict + dynamic predict + forecast +predict dyn_predict_varma11_1, dynamic(tm(2000m1)) equation(dlncaputil) +predict dyn_predict_varma11_2, dynamic(tm(2000m1)) equation(dlnhours) + +outsheet pred* dyn* using results_varmax_stata.csv, comma replace diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index 426406ed7f4..4cd85430185 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -15,39 +15,23 @@ from statsmodels.tsa.statespace import varmax from .results import results_varmax from numpy.testing import assert_equal, assert_almost_equal, assert_raises, assert_allclose +from nose.exc import SkipTest current_path = os.path.dirname(os.path.abspath(__file__)) -output_path = 'results' + os.sep + 'results_varmax_stata.csv' -output_results = pd.read_csv(current_path + os.sep + output_path) +var_path = 'results' + os.sep + 'results_var_stata.csv' +var_results = pd.read_csv(current_path + os.sep + var_path) +varmax_path = 'results' + os.sep + 'results_varmax_stata.csv' +varmax_results = pd.read_csv(current_path + os.sep + varmax_path) -class CheckVAR(object): + +class CheckVARMAX(object): """ Test Vector Autoregression against Stata's `dfactor` code (Stata's `var` function uses OLS and not state space / MLE, so we can't get equivalent log-likelihoods) """ - - def __init__(self, true, order, trend, error_cov_type, cov_type='oim', - included_vars=['dln_inv', 'dln_inc', 'dln_consump'], - **kwargs): - self.true = true - # 1960:Q1 - 1982:Q4 - dta = pd.DataFrame( - results_varmax.lutkepohl_data, columns=['inv', 'inc', 'consump'], - index=pd.date_range('1960-01-01', '1982-10-01', freq='QS')) - - dta['dln_inv'] = np.log(dta['inv']).diff() - dta['dln_inc'] = np.log(dta['inc']).diff() - dta['dln_consump'] = np.log(dta['consump']).diff() - - endog = dta.ix['1960-04-01':'1978-10-01', included_vars] - - self.model = varmax.VARMAX(endog, order=order, trend=trend, - error_cov_type=error_cov_type, **kwargs) - - self.results = self.model.filter(true['params'], cov_type=cov_type) def test_mle(self): # Fit with all transformations @@ -74,46 +58,74 @@ def test_bic(self): # We only get 3 digits from Stata assert_allclose(self.results.bic, self.true['bic'], atol=3) - def test_predict(self): + def test_predict(self, end, atol=1e-6): # Tests predict + forecast assert_allclose( - self.results.predict(end='1982-10-01'), + self.results.predict(end=end), self.true['predict'].T, - atol=1e-6) + atol=atol) - def test_dynamic_predict(self): + def test_dynamic_predict(self, end, dynamic, atol=1e-6): # Tests predict + dynamic predict + forecast assert_allclose( - self.results.predict(end='1982-10-01', dynamic='1961-01-01'), + self.results.predict(end=end, dynamic=dynamic), self.true['dynamic_predict'].T, - atol=1e-6) + atol=atol) -class TestVAR(CheckVAR): +class CheckLutkepohl(CheckVARMAX): + def __init__(self, true, order, trend, error_cov_type, cov_type='oim', + included_vars=['dln_inv', 'dln_inc', 'dln_consump'], + **kwargs): + self.true = true + # 1960:Q1 - 1982:Q4 + dta = pd.DataFrame( + results_varmax.lutkepohl_data, columns=['inv', 'inc', 'consump'], + index=pd.date_range('1960-01-01', '1982-10-01', freq='QS')) + + dta['dln_inv'] = np.log(dta['inv']).diff() + dta['dln_inc'] = np.log(dta['inc']).diff() + dta['dln_consump'] = np.log(dta['consump']).diff() + + endog = dta.ix['1960-04-01':'1978-10-01', included_vars] + + self.model = varmax.VARMAX(endog, order=order, trend=trend, + error_cov_type=error_cov_type, **kwargs) + + self.results = self.model.filter(true['params'], cov_type=cov_type) + + def test_predict(self): + super(CheckLutkepohl, self).test_predict(end='1982-10-01') + + def test_dynamic_predict(self): + super(CheckLutkepohl, self).test_dynamic_predict(end='1982-10-01', dynamic='1961-01-01') + + +class TestVAR(CheckLutkepohl): def __init__(self): - true = results_varmax.lutkepohl_ar1.copy() - true['predict'] = output_results.ix[1:, ['predict_1', 'predict_2', 'predict_3']] - true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_1', 'dyn_predict_2', 'dyn_predict_3']] + true = results_varmax.lutkepohl_var1.copy() + true['predict'] = var_results.ix[1:, ['predict_1', 'predict_2', 'predict_3']] + true['dynamic_predict'] = var_results.ix[1:, ['dyn_predict_1', 'dyn_predict_2', 'dyn_predict_3']] super(TestVAR, self).__init__( true, order=(1,0), trend='nc', error_cov_type="unstructured") -class TestVAR_diagonal(CheckVAR): +class TestVAR_diagonal(CheckLutkepohl): def __init__(self): - true = results_varmax.lutkepohl_ar1_diag.copy() - true['predict'] = output_results.ix[1:, ['predict_diag1', 'predict_diag2', 'predict_diag3']] - true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_diag1', 'dyn_predict_diag2', 'dyn_predict_diag3']] + true = results_varmax.lutkepohl_var1_diag.copy() + true['predict'] = var_results.ix[1:, ['predict_diag1', 'predict_diag2', 'predict_diag3']] + true['dynamic_predict'] = var_results.ix[1:, ['dyn_predict_diag1', 'dyn_predict_diag2', 'dyn_predict_diag3']] super(TestVAR_diagonal, self).__init__( true, order=(1,0), trend='nc', error_cov_type="diagonal") -class TestVAR_obs_intercept(CheckVAR): +class TestVAR_obs_intercept(CheckLutkepohl): def __init__(self): - true = results_varmax.lutkepohl_ar1_obs_intercept.copy() - true['predict'] = output_results.ix[1:, ['predict_int1', 'predict_int2', 'predict_int3']] - true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_int1', 'dyn_predict_int2', 'dyn_predict_int3']] + true = results_varmax.lutkepohl_var1_obs_intercept.copy() + true['predict'] = var_results.ix[1:, ['predict_int1', 'predict_int2', 'predict_int3']] + true['dynamic_predict'] = var_results.ix[1:, ['dyn_predict_int1', 'dyn_predict_int2', 'dyn_predict_int3']] super(TestVAR_obs_intercept, self).__init__( true, order=(1,0), trend='nc', error_cov_type="diagonal", obs_intercept=true['obs_intercept']) @@ -128,11 +140,11 @@ def test_bic(self): # of parameters, and hence the aic and bic, will be off pass -class TestVAR2(CheckVAR): +class TestVAR2(CheckLutkepohl): def __init__(self): - true = results_varmax.lutkepohl_ar2.copy() - true['predict'] = output_results.ix[1:, ['predict_var2_1', 'predict_var2_2']] - true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_var2_1', 'dyn_predict_var2_2']] + true = results_varmax.lutkepohl_var2.copy() + true['predict'] = var_results.ix[1:, ['predict_var2_1', 'predict_var2_2']] + true['dynamic_predict'] = var_results.ix[1:, ['dyn_predict_var2_1', 'dyn_predict_var2_2']] super(TestVAR2, self).__init__( true, order=(2,0), trend='nc', error_cov_type='unstructured', included_vars=['dln_inv', 'dln_inc']) @@ -142,3 +154,103 @@ def test_bse_oim(self): assert_allclose( self.results.bse[:-3]**2, self.true['var_oim'][:-3], atol=1e-2) + +# class TestVARX(CheckVARMAX): +# pass +# pandas, numpy; use 1-dim + + +class CheckFREDManufacturing(CheckVARMAX): + def __init__(self, true, order, trend, error_cov_type, cov_type='oim', + **kwargs): + self.true = true + # 1960:Q1 - 1982:Q4 + dta = webuse('manufac', 'http://www.stata-press.com/data/r12/') + + dta.index = dta.month + dta['dlncaputil'] = dta['lncaputil'].diff() + dta['dlnhours'] = dta['lnhours'].diff() + + endog = dta.ix['1972-02-01':, ['dlncaputil', 'dlnhours']] + + self.model = varmax.VARMAX(endog, order=order, trend=trend, + error_cov_type=error_cov_type, **kwargs) + + self.results = self.model.filter(true['params'], cov_type=cov_type) + + +class TestVARMA(CheckFREDManufacturing): + """ + Test against the sspace VARMA example with some params set to zeros. + """ + + def __init__(self): + true = results_varmax.fred_varma11.copy() + true['predict'] = varmax_results.ix[1:, ['predict_varma11_1', 'predict_varma11_2']] + true['dynamic_predict'] = varmax_results.ix[1:, ['dyn_predict_varma11_1', 'dyn_predict_varma11_2']] + super(TestVARMA, self).__init__( + true, order=(1,1), trend='nc', error_cov_type='diagonal') + + def test_mle(self): + # Since the VARMA model here is generic (we're just forcing zeros + # in some params) whereas Stata's is restricted, the MLE test isn't + # meaninful + pass + + def test_bse_oim(self): + # Standard errors do not match Stata's + raise SkipTest('Known failure: standard errors do not match.') + + def test_aic(self): + # Since the VARMA model here is generic (we're just putting in zeros + # for some params), Stata assumes a different estimated number of + # parameters; hence the aic and bic, will be off + pass + + def test_bic(self): + # Since the VARMA model here is generic (we're just putting in zeros + # for some params), Stata assumes a different estimated number of + # parameters; hence the aic and bic, will be off + pass + + def test_predict(self): + super(TestVARMA, self).test_predict(end='2009-05-01', atol=1e-4) + + def test_dynamic_predict(self): + super(TestVARMA, self).test_dynamic_predict(end='2009-05-01', dynamic='2000-01-01') + + +def test_specifications(): + # Tests for model specification and state space creation + endog = np.arange(20).reshape(10,2) + exog = np.arange(10) + exog2 = pd.Series(exog, index=pd.date_range('2000-01-01', '2009-01-01', freq='AS')) + + # Test successful model creation + mod = varmax.VARMAX(endog, exog=exog, order=(1,0)) + + # Test successful model creation with pandas exog + mod = varmax.VARMAX(endog, exog=exog2, order=(1,0)) + + +def test_misspecifications(): + # Tests for model specification and misspecification exceptions + endog = np.arange(20).reshape(10,2) + + # Bad trend specification + assert_raises(ValueError, varmax.VARMAX, endog, order=(1,0), trend='') + + # Bad error_cov_type specification + assert_raises(ValueError, varmax.VARMAX, endog, order=(1,0), error_cov_type='') + + # Bad order specification + assert_raises(ValueError, varmax.VARMAX, endog, order=(0,0), trend='') + + # Warning with VARMA specification + warnings.simplefilter("always") + with warnings.catch_warnings(record=True) as w: + varmax.VARMAX(endog, order=(1,1)) + message = ('Estimation of VARMA(p,q) models is not generically robust,' + ' due especially to identification issues.') + assert_equal(str(w[0].message), message) + diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index bf224ef2742..d7b7a4f1bb9 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -9,6 +9,7 @@ from warnings import warn from statsmodels.compat.collections import OrderedDict +import pandas as pd import numpy as np from .kalman_filter import ( KalmanFilter, FilterResults, INVERT_UNIVARIATE, SOLVE_LU @@ -122,7 +123,8 @@ def __init__(self, endog, exog=None, order=(1, 0), trend='c', # Model order # Used internally in various places - self._k_order = self.k_ar + self.k_ma + _min_k_ar = max(self.k_ar, 1) + self._k_order = _min_k_ar + self.k_ma # Number of states k_endog = endog.shape[1] @@ -171,13 +173,14 @@ def __init__(self, endog, exog=None, order=(1, 0), trend='c', # The transition matrix is described in four blocks, where the upper # left block is in companion form with the autoregressive coefficient # matrices (so it is shaped k_endog * k_ar x k_endog * k_ar) ... - idx = np.diag_indices((self.k_ar - 1) * self.k_endog) - idx = idx[0] + self.k_endog, idx[1] - self.ssm[('transition',) + idx] = 1 + if self.k_ar > 0: + idx = np.diag_indices((self.k_ar - 1) * self.k_endog) + idx = idx[0] + self.k_endog, idx[1] + self.ssm[('transition',) + idx] = 1 # ... and the lower right block is in companion form with zeros as the # coefficient matrices (it is shaped k_endog * k_ma x k_endog * k_ma). idx = np.diag_indices((self.k_ma - 1) * self.k_endog) - idx = idx[0] + (self.k_ar + 1) * self.k_endog, idx[1] + idx = idx[0] + (_min_k_ar + 1) * self.k_endog, idx[1] + _min_k_ar * self.k_endog self.ssm[('transition',) + idx] = 1 # The selection matrix is described in two blocks, where the upper @@ -186,9 +189,8 @@ def __init__(self, endog, exog=None, order=(1, 0), trend='c', # also selects all k_posdef errors in the first k_endog rows (the lower # block is shaped k_endog * k_ma x k). idx = np.diag_indices(self.k_endog) - if self.k_ar > 0: - self.ssm[('selection',) + idx] = 1 - idx = idx[0] + self.k_ar * self.k_endog, idx[1] + self.ssm[('selection',) + idx] = 1 + idx = idx[0] + _min_k_ar * self.k_endog, idx[1] if self.k_ma > 0: self.ssm[('selection',) + idx] = 1 @@ -197,7 +199,10 @@ def __init__(self, endog, exog=None, order=(1, 0), trend='c', self._idx_state_intercept = np.s_['state_intercept', :k_endog] elif self.k_exog > 0: self._idx_state_intercept = np.s_['state_intercept', :k_endog, :] - self._idx_transition = np.s_['transition', :k_endog, :] + if self.k_ar > 0: + self._idx_transition = np.s_['transition', :k_endog, :] + else: + self._idx_transition = np.s_['transition', :k_endog, k_endog:] if self.error_cov_type == 'diagonal': self._idx_state_cov = ( ('state_cov',) + np.diag_indices(self.k_endog)) @@ -244,16 +249,21 @@ def start_params(self): # B. Run a VAR model on endog to get trend, AR parameters ar_params = [] - if self.k_ar > 0 or self.trend == 'c': - mod_ar = var_model.VAR(endog) - res_ar = mod_ar.fit(maxlags=self.k_ar, ic=None, trend=self.trend) - ar_params = np.array(res_ar.params.T) - if self.trend == 'c': - trend_params = ar_params[:, 0] + k_ar = self.k_ar if self.k_ar > 0 else 1 + mod_ar = var_model.VAR(endog) + res_ar = mod_ar.fit(maxlags=k_ar, ic=None, trend=self.trend) + ar_params = np.array(res_ar.params.T) + if self.trend == 'c': + trend_params = ar_params[:, 0] + if self.k_ar > 0: ar_params = ar_params[:, 1:].ravel() else: - ar_params = ar_params.ravel() - endog = res_ar.resid + ar_params = [] + elif self.k_ar > 0: + ar_params = ar_params.ravel() + else: + ar_params = [] + endog = res_ar.resid # C. Run a VAR model on the residuals to get MA parameters ma_params = [] @@ -393,7 +403,7 @@ def transform_params(self, unconstrained): # 3. MA terms: optionally force to be invertible if self.k_ma > 0 and self.enforce_invertibility: # Transform the parameters, using an identity variance matrix - state_cov = np.eye(self.k_endog, dtype=constrained.dtype) + state_cov = np.eye(self.k_endog, dtype=unconstrained.dtype) coefficients = unconstrained[self._params_ma].reshape( self.k_endog, self.k_endog * self.k_ma) coefficient_matrices, variance = ( From cc2fff2b20c61bf9c74515d3009350af66a665a5 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sun, 9 Aug 2015 23:23:37 -0700 Subject: [PATCH 28/57] TST: Add VARX test, fix warnings. --- .../tests/results/results_var_stata.csv | 186 +++++++++--------- .../tests/results/results_varmax.py | 29 ++- .../tests/results/test_varmax_stata.do | 17 +- .../tsa/statespace/tests/test_varmax.py | 105 +++++++--- 4 files changed, 216 insertions(+), 121 deletions(-) diff --git a/statsmodels/tsa/statespace/tests/results/results_var_stata.csv 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+.0004033,.0004162,.0003912,.0004307,.0004463,.0004195,.017993,.0206546,.0198753,-.0157479,.0118112,.0128117,.0036624,.0038398,5.30e-09,5.47e-09,5.14e-09,8.01e-09,8.30e-09,7.80e-09,.017993,.0206546,.0198753,.0000225,.0000236,.0277016,.0339982,.03392442 +.000338,.0003489,.0003279,.0003627,.0003759,.0003533,.017993,.0206546,.0198753,-.0025344,.0154519,.018825,.0033799,.0035436,4.44e-09,4.58e-09,4.31e-09,6.75e-09,6.99e-09,6.57e-09,.017993,.0206546,.0198753,.0000208,.0000218,.02802107,.03438319,.03430664 diff --git a/statsmodels/tsa/statespace/tests/results/results_varmax.py b/statsmodels/tsa/statespace/tests/results/results_varmax.py index e66b0f32b9c..0d4e41842b1 100644 --- a/statsmodels/tsa/statespace/tests/results/results_varmax.py +++ b/statsmodels/tsa/statespace/tests/results/results_varmax.py @@ -47,7 +47,8 @@ -0.01492992, 0.53796097, 0.28114733, # Phi, row 3 0.04464236, # Covariance, lower triangle 0.00136552, 0.01354125, - 0.0029089 , 0.00834324, 0.00915471], + 0.0029089 , 0.00834324, 0.00915471 + ], 'var_oim': [ .01319669, .19413864, .2386643, .0012437, .01829378, .02234399, @@ -96,6 +97,32 @@ 'bic': -1073.824 } +lutkepohl_var1_exog = { + 'params': [ + -.25549409, .31149462, .92674046, # Phi, row 1 + .02935715, .13699757, .5059042, # Phi, row 2 + -.00540021, .4598014, .06065565, # Phi, row 3 + -.00007533, .00012771, .00018224, # exog + # .00200617, # Covariance, lower triangle + # .00007053, .00017216, + # .00013934, .00010021, .00013833 + # Note: the following are the Cholesky of the covariance + # matrix defined just above + .04479029, # Cholesky, lower triangle + .00157467, .01302614, + .00311094, .00731692, .00866687 + ], + 'var_oim': [ + .01350243, .20429977, .29684366, # Phi, row 1 + .00115871, .01753203, .02547371, # Phi, row 2 + .000931, .01408662, .02046759, # Phi, row 3 + 3.720e-08, 3.192e-09, 2.565e-09 # exog + ], + 'loglike': 587.4157014188437, + 'aic': None, + 'bic': None +} + lutkepohl_var2 = { 'params': [ -.25244981, .62528114, # Phi_1, row 1 diff --git a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do index 6fd353610b9..823a197f080 100644 --- a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do +++ b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do @@ -45,6 +45,21 @@ predict dyn_predict_int1, dynamic(tq(1961q1)) equation(dln_inv) predict dyn_predict_int2, dynamic(tq(1961q1)) equation(dln_inc) predict dyn_predict_int3, dynamic(tq(1961q1)) equation(dln_consump) +// VAR(1), diagonal covariance + exog +gen t = _n +//dfactor (dln_inv dln_inc dln_consump = t, ar(1) arstructure(general) noconstant covstructure(diagonal)) if qtr<=tq(1978q4) +var dln_inv dln_inc dln_consump if qtr<=tq(1978q4), lags(1) noconstant exog(t) + +// predict, see above (Note: uses actual data for forecasting, so we will want +// to ignore the predictions after 1978q4, see below +predict predict_exog1, equation(dln_inv) +predict predict_exog2, equation(dln_inc) +predict predict_exog3, equation(dln_consump) + +// We will want to use these values to compare for forecasting, but note that +// this also includes in the columns the value for 1978q4 (i.e. a VAR(1) needs +// 1 sample from which to compute forecasts. +fcast compute fcast_exog_ , dynamic(tq(1979q1)) step(16) replace // VAR(2) dfactor (dln_inv dln_inc = , ar(1/2) arstructure(general) noconstant covstructure(unstructured)) if qtr<=tq(1978q4) @@ -58,7 +73,7 @@ predict predict_var2_2, dynamic(tq(1979q1)) equation(dln_inc) predict dyn_predict_var2_1, dynamic(tq(1961q1)) equation(dln_inv) predict dyn_predict_var2_2, dynamic(tq(1961q1)) equation(dln_inc) -outsheet pred* dyn* using results_var_stata.csv, comma replace +outsheet pred* dyn* fcas* using results_var_stata.csv, comma replace // VARMA(1,1) // Note: Stata does not have this built-in, so we need to create the state space form ourselves diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index 4cd85430185..c79d4b51696 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -34,15 +34,17 @@ class CheckVARMAX(object): """ def test_mle(self): - # Fit with all transformations - # results = self.model.fit(method='powell', disp=-1) - results = self.model.fit(maxiter=100, disp=False) - # Fit now without transformations - self.model.enforce_stationarity = False - self.model.enforce_invertibility = False - results = self.model.fit(results.params, method='nm', maxiter=1000, - disp=False) - assert_allclose(results.llf, self.results.llf, rtol=1e-5) + with warnings.catch_warnings(record=True) as w: + warnings.simplefilter('always') + # Fit with all transformations + # results = self.model.fit(method='powell', disp=-1) + results = self.model.fit(maxiter=100, disp=False) + # Fit now without transformations + self.model.enforce_stationarity = False + self.model.enforce_invertibility = False + results = self.model.fit(results.params, method='nm', maxiter=1000, + disp=False) + assert_allclose(results.llf, self.results.llf, rtol=1e-5) def test_loglike(self): assert_allclose(self.results.llf, self.true['loglike'], rtol=1e-6) @@ -58,17 +60,17 @@ def test_bic(self): # We only get 3 digits from Stata assert_allclose(self.results.bic, self.true['bic'], atol=3) - def test_predict(self, end, atol=1e-6): + def test_predict(self, end, atol=1e-6, **kwargs): # Tests predict + forecast assert_allclose( - self.results.predict(end=end), + self.results.predict(end=end, **kwargs), self.true['predict'].T, atol=atol) - def test_dynamic_predict(self, end, dynamic, atol=1e-6): + def test_dynamic_predict(self, end, dynamic, atol=1e-6, **kwargs): # Tests predict + dynamic predict + forecast assert_allclose( - self.results.predict(end=end, dynamic=dynamic), + self.results.predict(end=end, dynamic=dynamic, **kwargs), self.true['dynamic_predict'].T, atol=atol) @@ -94,11 +96,11 @@ def __init__(self, true, order, trend, error_cov_type, cov_type='oim', self.results = self.model.filter(true['params'], cov_type=cov_type) - def test_predict(self): - super(CheckLutkepohl, self).test_predict(end='1982-10-01') + def test_predict(self, **kwargs): + super(CheckLutkepohl, self).test_predict(end='1982-10-01', **kwargs) - def test_dynamic_predict(self): - super(CheckLutkepohl, self).test_dynamic_predict(end='1982-10-01', dynamic='1961-01-01') + def test_dynamic_predict(self, **kwargs): + super(CheckLutkepohl, self).test_dynamic_predict(end='1982-10-01', dynamic='1961-01-01', **kwargs) class TestVAR(CheckLutkepohl): @@ -140,6 +142,52 @@ def test_bic(self): # of parameters, and hence the aic and bic, will be off pass + +class TestVAR_exog(CheckLutkepohl): + # Note: unlike the other tests in this file, this is against the Stata + # var function rather than the Stata dfactor function + def __init__(self): + true = results_varmax.lutkepohl_var1_exog.copy() + true['predict'] = var_results.ix[1:75, ['predict_exog1', 'predict_exog2', 'predict_exog3']] + true['predict'].iloc[0, :] = 0 + true['fcast'] = var_results.ix[76:, ['fcast_exog_dln_inv', 'fcast_exog_dln_inc', 'fcast_exog_dln_consump']] + exog = np.arange(75) + 3 + super(TestVAR_exog, self).__init__( + true, order=(1,0), trend='nc', error_cov_type='unstructured', + exog=exog, initialization='approximate_diffuse', loglikelihood_burn=1) + + def test_mle(self): + pass + + def test_aic(self): + # Stata's var calculates AIC differently + pass + + def test_bic(self): + # Stata's var calculates BIC differently + pass + + def test_bse_oim(self): + # Exclude the covariance cholesky terms + assert_allclose( + self.results.bse[:-6]**2, self.true['var_oim'], atol=1e-2) + + def test_predict(self): + super(CheckLutkepohl, self).test_predict(end='1978-10-01', atol=1e-3) + + def test_dynamic_predict(self): + # Stata's var cannot subsequently use dynamic + pass + + def test_forecast(self): + # Tests forecast + exog = (np.arange(75, 75+16) + 3)[:, np.newaxis] + beta = self.results.params[-9:-6] + state_intercept = np.concatenate([exog*beta[0], exog*beta[1], exog*beta[2]], axis=1).T + desired = self.results.forecast(steps=16, state_intercept=state_intercept) + assert_allclose(desired, self.true['fcast'].T, atol=1e-6) + + class TestVAR2(CheckLutkepohl): def __init__(self): true = results_varmax.lutkepohl_var2.copy() @@ -155,11 +203,6 @@ def test_bse_oim(self): self.results.bse[:-3]**2, self.true['var_oim'][:-3], atol=1e-2) -# class TestVARX(CheckVARMAX): -# pass -# pandas, numpy; use 1-dim - - class CheckFREDManufacturing(CheckVARMAX): def __init__(self, true, order, trend, error_cov_type, cov_type='oim', **kwargs): @@ -173,8 +216,10 @@ def __init__(self, true, order, trend, error_cov_type, cov_type='oim', endog = dta.ix['1972-02-01':, ['dlncaputil', 'dlnhours']] - self.model = varmax.VARMAX(endog, order=order, trend=trend, - error_cov_type=error_cov_type, **kwargs) + with warnings.catch_warnings(record=True) as w: + warnings.simplefilter('always') + self.model = varmax.VARMAX(endog, order=order, trend=trend, + error_cov_type=error_cov_type, **kwargs) self.results = self.model.filter(true['params'], cov_type=cov_type) @@ -189,7 +234,7 @@ def __init__(self): true['predict'] = varmax_results.ix[1:, ['predict_varma11_1', 'predict_varma11_2']] true['dynamic_predict'] = varmax_results.ix[1:, ['dyn_predict_varma11_1', 'dyn_predict_varma11_2']] super(TestVARMA, self).__init__( - true, order=(1,1), trend='nc', error_cov_type='diagonal') + true, order=(1,1), trend='nc', error_cov_type='diagonal') def test_mle(self): # Since the VARMA model here is generic (we're just forcing zeros @@ -246,11 +291,19 @@ def test_misspecifications(): # Bad order specification assert_raises(ValueError, varmax.VARMAX, endog, order=(0,0), trend='') + with warnings.catch_warnings(record=True) as w: + warnings.simplefilter('always') + varmax.VARMAX(endog, order=(1,1)) + # Warning with VARMA specification - warnings.simplefilter("always") with warnings.catch_warnings(record=True) as w: + warnings.simplefilter('always') + varmax.VARMAX(endog, order=(1,1)) + + print(w) message = ('Estimation of VARMA(p,q) models is not generically robust,' ' due especially to identification issues.') assert_equal(str(w[0].message), message) + warnings.resetwarnings() From b1e9c576c267200ab149b1d4c65e10350fd5c159 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sun, 9 Aug 2015 23:35:49 -0700 Subject: [PATCH 29/57] TST: Add VMA(1) test. --- .../tests/results/results_varmax.py | 18 +- .../tests/results/results_varmax_stata.csv | 900 +++++++++--------- .../tests/results/test_varmax_stata.do | 24 + .../tsa/statespace/tests/test_varmax.py | 41 + 4 files changed, 532 insertions(+), 451 deletions(-) diff --git a/statsmodels/tsa/statespace/tests/results/results_varmax.py b/statsmodels/tsa/statespace/tests/results/results_varmax.py index 0d4e41842b1..0369f88a080 100644 --- a/statsmodels/tsa/statespace/tests/results/results_varmax.py +++ b/statsmodels/tsa/statespace/tests/results/results_varmax.py @@ -165,4 +165,20 @@ 'loglike': 3156.056423235071, 'aic': -6300.113, 'bic': -6275.551 -} \ No newline at end of file +} + +fred_vma1 = { + 'params': [ + .24803941, 0, # Theta_1, row 1 + 0, 0, # Theta_1, row 2 + .00006514, .00004621, # Variances + ], + 'var_oim': [ + .00154773, 0, # Theta_1, row 1 + 0, 0, # Theta_1, row 2 + 1.916e-11, 9.639e-12, # Variances + ], + 'loglike': 3088.909619417645, + 'aic': -6171.819, + 'bic': -6159.539 +} diff --git a/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv b/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv index dcef8bea2e3..85944653909 100644 --- a/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv +++ b/statsmodels/tsa/statespace/tests/results/results_varmax_stata.csv @@ -1,450 +1,450 @@ 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823a197f080..614226c01f5 100644 --- a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do +++ b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do @@ -105,4 +105,28 @@ predict predict_varma11_2, dynamic(tm(2009m1)) equation(dlnhours) predict dyn_predict_varma11_1, dynamic(tm(2000m1)) equation(dlncaputil) predict dyn_predict_varma11_2, dynamic(tm(2000m1)) equation(dlnhours) +// VMA(1) +// Note: Stata does not have this built-in, so we need to create the state space form ourselves +constraint 1 [u1]L.u2 = 1 +constraint 2 [u1]e.u1 = 1 +constraint 3 [u3]e.u3 = 1 +constraint 4 [dlncaputil]u1 = 1 +constraint 5 [dlnhours]u3 = 1 + +sspace (u1 L.u2 e.u1, state noconstant) /// + (u2 e.u1, state noconstant) /// + (u3 e.u3, state noconstant) /// + (dlncaputil u1, noconstant) /// + (dlnhours u3, noconstant), /// + constraints(1/5) technique(nr) covstate(diagonal) + +// in-sample predict + forecast 5 new observations (sample ends at 1999m12) +predict predict_vma1_1, dynamic(tm(2009m1)) equation(dlncaputil) +predict predict_vma1_2, dynamic(tm(2009m1)) equation(dlnhours) + +// predict + dynamic predict + forecast +predict dyn_predict_vma1_1, dynamic(tm(2000m1)) equation(dlncaputil) +predict dyn_predict_vma1_2, dynamic(tm(2000m1)) equation(dlnhours) + + outsheet pred* dyn* using results_varmax_stata.csv, comma replace diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index c79d4b51696..61f224debe5 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -265,6 +265,47 @@ def test_dynamic_predict(self): super(TestVARMA, self).test_dynamic_predict(end='2009-05-01', dynamic='2000-01-01') +class TestVMA1(CheckFREDManufacturing): + """ + Test against the sspace VARMA example with some params set to zeros. + """ + + def __init__(self): + true = results_varmax.fred_vma1.copy() + true['predict'] = varmax_results.ix[1:, ['predict_vma1_1', 'predict_vma1_2']] + true['dynamic_predict'] = varmax_results.ix[1:, ['dyn_predict_vma1_1', 'dyn_predict_vma1_2']] + super(TestVMA1, self).__init__( + true, order=(0,1), trend='nc', error_cov_type='diagonal') + + def test_mle(self): + # Since the VARMA model here is generic (we're just forcing zeros + # in some params) whereas Stata's is restricted, the MLE test isn't + # meaninful + pass + + def test_bse_oim(self): + # Standard errors do not match Stata's + raise SkipTest('Known failure: standard errors do not match.') + + def test_aic(self): + # Since the VARMA model here is generic (we're just putting in zeros + # for some params), Stata assumes a different estimated number of + # parameters; hence the aic and bic, will be off + pass + + def test_bic(self): + # Since the VARMA model here is generic (we're just putting in zeros + # for some params), Stata assumes a different estimated number of + # parameters; hence the aic and bic, will be off + pass + + def test_predict(self): + super(TestVMA1, self).test_predict(end='2009-05-01', atol=1e-4) + + def test_dynamic_predict(self): + super(TestVMA1, self).test_dynamic_predict(end='2009-05-01', dynamic='2000-01-01') + + def test_specifications(): # Tests for model specification and state space creation endog = np.arange(20).reshape(10,2) From 3fcb5746084a2b28bd5c8552bb058047df1700fa Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 09:20:20 -0700 Subject: [PATCH 30/57] REF: Make sure constraint out is appropriate dtype --- statsmodels/tsa/statespace/tools.py | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/statsmodels/tsa/statespace/tools.py b/statsmodels/tsa/statespace/tools.py index 708057f99c9..956076fb4d7 100644 --- a/statsmodels/tsa/statespace/tools.py +++ b/statsmodels/tsa/statespace/tools.py @@ -742,8 +742,8 @@ def constrain_stationary_multivariate(unconstrained, variance, constrained, variance = prefix_pacf_map[prefix]( sv_constrained, variance, transform_variance, order, k_endog) - constrained = np.array(constrained) - variance = np.array(variance) + constrained = np.array(constrained, dtype=dtype) + variance = np.array(variance, dtype=dtype) if use_list: constrained = [ From db5b697009f7475445aca168abdb57fa1f5a8e43 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 09:21:26 -0700 Subject: [PATCH 31/57] BUG: Zero out lower triangle in multiv. constraint --- statsmodels/tsa/statespace/_statespace.pyx.in | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/statsmodels/tsa/statespace/_statespace.pyx.in b/statsmodels/tsa/statespace/_statespace.pyx.in index b7c47d140e2..56545180940 100644 --- a/statsmodels/tsa/statespace/_statespace.pyx.in +++ b/statsmodels/tsa/statespace/_statespace.pyx.in @@ -2025,6 +2025,12 @@ cpdef _{{prefix}}compute_coefficients_from_multivariate_pacf({{cython_type}} [:: {{prefix}}potrf("U", &k_endog, &tmp[0,0], &k_endog, &info) # {{prefix}}potri("L", &k_endog, &tmp[0,0], &k_endog, &info) + # We need to zero out the lower triangle of L', because ?trtrs only + # knows that M' is upper triangular + for s in range(k_endog - 1): # column + for k in range(s+1, k_endog): # row + initial_variance[k, s] = 0 + # Note that T is lower triangular # L M^{-1} = T # M' T' = L' From 41c85b17eb3d223dadd223980d2e74d384fb3723 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 09:32:06 -0700 Subject: [PATCH 32/57] BUG: Wrong state_cov for AR in untransform_params --- statsmodels/tsa/statespace/varmax.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index d7b7a4f1bb9..e8d3bb5b688 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -460,7 +460,7 @@ def untransform_params(self, constrained): if self.k_ar > 0 and self.enforce_stationarity: # Create the state covariance matrix if self.error_cov_type == 'diagonal': - state_cov = np.diag(constrained[self._params_state_cov]**2) + state_cov = np.diag(constrained[self._params_state_cov]) elif self.error_cov_type == 'unstructured': state_cov_lower = np.zeros(self.ssm['state_cov'].shape, dtype=constrained.dtype) From e942a67bd3e32a250e87ea128df4a0525221a254 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 10:01:59 -0700 Subject: [PATCH 33/57] BUG: Measurement error start_params when k_ma = 0 --- statsmodels/tsa/statespace/varmax.py | 5 ++++- 1 file changed, 4 insertions(+), 1 deletion(-) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index e8d3bb5b688..8d6f951853b 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -296,7 +296,10 @@ def start_params(self): # 5. Measurement error variance terms if self.measurement_error: - params[self._params_obs_cov] = res_ma.sigma_u.diagonal() + if self.k_ma > 0: + params[self._params_obs_cov] = res_ma.sigma_u.diagonal() + else: + params[self._params_obs_cov] = res_ar.sigma_u.diagonal() return params From 890979554a6cb7749eb1b4c0137d2725eebbfd8e Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 10:04:41 -0700 Subject: [PATCH 34/57] BUG: Wrong transform for measurement error vars. --- statsmodels/tsa/statespace/varmax.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index 8d6f951853b..34028e2db0f 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -433,7 +433,7 @@ def transform_params(self, unconstrained): if self.measurement_error: # Force these to be positive constrained[self._params_obs_cov] = ( - constrained[self._params_obs_cov]**2) + unconstrained[self._params_obs_cov]**2) return constrained From d8692713af7a9fce2876a34b27cb81cc6b34848c Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 10:20:28 -0700 Subject: [PATCH 35/57] TST: Test start_params, param_names, transforms --- .../tsa/statespace/tests/test_varmax.py | 19 +++++++++++++++++++ 1 file changed, 19 insertions(+) diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index 61f224debe5..6fc012186cf 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -44,8 +44,27 @@ def test_mle(self): self.model.enforce_invertibility = False results = self.model.fit(results.params, method='nm', maxiter=1000, disp=False) + self.model.enforce_stationarity = True + self.model.enforce_invertibility = True assert_allclose(results.llf, self.results.llf, rtol=1e-5) + def test_params(self): + # Smoke test to make sure the start_params are well-defined and + # lead to a well-defined model + self.model.filter(self.model.start_params) + # Similarly a smoke test for param_names + assert_equal(len(self.model.start_params), len(self.model.param_names)) + # Finally make sure the transform and untransform do their job + actual = self.model.transform_params(self.model.untransform_params(self.model.start_params)) + assert_allclose(actual, self.model.start_params) + # Also in the case of enforce invertibility and stationarity = False + self.model.enforce_stationarity = False + self.model.enforce_invertibility = False + actual = self.model.transform_params(self.model.untransform_params(self.model.start_params)) + self.model.enforce_stationarity = True + self.model.enforce_invertibility = True + assert_allclose(actual, self.model.start_params) + def test_loglike(self): assert_allclose(self.results.llf, self.true['loglike'], rtol=1e-6) From 8cee6231f77717749cf10ccfed60b7f54153982d Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 10:20:53 -0700 Subject: [PATCH 36/57] TST/CLEAN: Minor test cleanups. --- .../tsa/statespace/tests/results/results_varmax.py | 10 ++++++++-- 1 file changed, 8 insertions(+), 2 deletions(-) diff --git a/statsmodels/tsa/statespace/tests/results/results_varmax.py b/statsmodels/tsa/statespace/tests/results/results_varmax.py index 0369f88a080..ab6884340d1 100644 --- a/statsmodels/tsa/statespace/tests/results/results_varmax.py +++ b/statsmodels/tsa/statespace/tests/results/results_varmax.py @@ -45,7 +45,12 @@ -0.25034303, 0.28759168, 0.81626475, # Phi, row 1 0.023383, 0.19048278, 0.66502259, # Phi, row 2 -0.01492992, 0.53796097, 0.28114733, # Phi, row 3 - 0.04464236, # Covariance, lower triangle + # .00199294, # Covariance, lower triangle + # .00006096, .00012986, + # .00018523, .00011695, .00016188, + # Note: the following are the Cholesky of the covariance + # matrix defined just above + 0.04464236, # Cholesky, lower triangle 0.00136552, 0.01354125, 0.0029089 , 0.00834324, 0.00915471 ], @@ -72,7 +77,8 @@ .01314245, .1902972, .23400828, .00124336, .01840132, .02229946, .00107322, .01558391, .01909303, - 1.057e-07, 9.233e-10, 7.011e-10], + 1.057e-07, 9.233e-10, 7.011e-10 + ], 'loglike': 562.8168476509002, 'aic': -1101.634, 'bic': -1073.824 From f03d9d325a658f1854eed880729ad14e884a652f Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 10:21:39 -0700 Subject: [PATCH 37/57] TST: Add test for VAR(1) with measurement errors. --- .../tests/results/results_varmax.py | 20 ++++++ .../tsa/statespace/tests/test_varmax.py | 61 ++++++++++++++++++- 2 files changed, 80 insertions(+), 1 deletion(-) diff --git a/statsmodels/tsa/statespace/tests/results/results_varmax.py b/statsmodels/tsa/statespace/tests/results/results_varmax.py index ab6884340d1..6961d39248e 100644 --- a/statsmodels/tsa/statespace/tests/results/results_varmax.py +++ b/statsmodels/tsa/statespace/tests/results/results_varmax.py @@ -84,6 +84,26 @@ 'bic': -1073.824 } +lutkepohl_var1_diag_meas = { + 'params': [ + -0.24817904, 0.29283012, 0.80794938, # Phi, row 1 + 0.02282985, 0.19672157, 0.66329776, # Phi, row 2 + -0.01522531, 0.53500874, 0.28859213, # Phi, row 3 + 0.00199106, 0.00018529, 0.00016179, # Variances, diagonal + 0, 0, 0 # Measurement error variances + ], + 'var_oim': [ + .01314245, .1902972, .23400828, + .00124336, .01840132, .02229946, + .00107322, .01558391, .01909303, + 1.057e-07, 9.233e-10, 7.011e-10, + None, None, None + ], + 'loglike': 562.8168476509002, + 'aic': None, + 'bic': None +} + lutkepohl_var1_obs_intercept = { 'params': [ -.24762, .25961003, .75992623, # Phi, row 1 diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index 6fc012186cf..d8f9e06173b 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -142,6 +142,65 @@ def __init__(self): error_cov_type="diagonal") +class TestVAR_measurement_error(CheckLutkepohl): + """ + Notes + ----- + There does not appear to be a way to get Stata to estimate a VAR with + measurement errors. Thus this test is mostly a smoke test that measurement + errors are setup correctly: it uses the same params from TestVAR_diagonal + and sets the measurement errors variance params to zero to check that the + loglike and predict are the same. + + It also checks that the state-space representation with positive + measurement errors is correct. + """ + def __init__(self): + true = results_varmax.lutkepohl_var1_diag_meas.copy() + true['predict'] = var_results.ix[1:, ['predict_diag1', 'predict_diag2', 'predict_diag3']] + true['dynamic_predict'] = var_results.ix[1:, ['dyn_predict_diag1', 'dyn_predict_diag2', 'dyn_predict_diag3']] + super(TestVAR_measurement_error, self).__init__( + true, order=(1,0), trend='nc', + error_cov_type="diagonal", measurement_error=True) + + # Create another filter results with positive measurement errors + self.true_measurement_error_variances = [1., 2., 3.] + params = np.r_[true['params'][:-3], self.true_measurement_error_variances] + self.results2 = self.model.filter(params) + + def test_mle(self): + # With the additional measurment error parameters, this wouldn't be + # a meaningful test + pass + + def test_bse_oim(self): + # This would just test the same thing as TestVAR_diagonal.test_bse_oim + pass + + def test_aic(self): + # Since the measurement error is added, the number + # of parameters, and hence the aic and bic, will be off + pass + + def test_bic(self): + # Since the measurement error is added, the number + # of parameters, and hence the aic and bic, will be off + pass + + def test_representation(self): + # Test that the state space representation in the measurement error + # case is correct + for name in self.model.ssm.shapes.keys(): + if name == 'obs': + pass + elif name == 'obs_cov': + actual = self.results2.filter_results.obs_cov + desired = np.diag(self.true_measurement_error_variances)[:,:,np.newaxis] + assert_equal(actual, desired) + else: + assert_equal(getattr(self.results2.filter_results, name), + getattr(self.results.filter_results, name)) + class TestVAR_obs_intercept(CheckLutkepohl): def __init__(self): true = results_varmax.lutkepohl_var1_obs_intercept.copy() @@ -349,7 +408,7 @@ def test_misspecifications(): assert_raises(ValueError, varmax.VARMAX, endog, order=(1,0), error_cov_type='') # Bad order specification - assert_raises(ValueError, varmax.VARMAX, endog, order=(0,0), trend='') + assert_raises(ValueError, varmax.VARMAX, endog, order=(0,0)) with warnings.catch_warnings(record=True) as w: warnings.simplefilter('always') From f49df5fd53bb21e154636185d9a7501f1e80d649 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 10:27:27 -0700 Subject: [PATCH 38/57] TST: DFM: start_params, param_names, transforms --- .../tsa/statespace/tests/test_dynamic_factor.py | 15 +++++++++++++++ 1 file changed, 15 insertions(+) diff --git a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py index cf1f4202f42..412e9be23c7 100644 --- a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py @@ -45,6 +45,21 @@ def __init__(self, true, k_factors, factor_order, cov_type='oim', self.results = self.model.filter(true['params'], cov_type=cov_type) + def test_params(self): + # Smoke test to make sure the start_params are well-defined and + # lead to a well-defined model + self.model.filter(self.model.start_params) + # Similarly a smoke test for param_names + assert_equal(len(self.model.start_params), len(self.model.param_names)) + # Finally make sure the transform and untransform do their job + actual = self.model.transform_params(self.model.untransform_params(self.model.start_params)) + assert_allclose(actual, self.model.start_params) + # Also in the case of enforce stationarity = False + self.model.enforce_stationarity = False + actual = self.model.transform_params(self.model.untransform_params(self.model.start_params)) + self.model.enforce_stationarity = True + assert_allclose(actual, self.model.start_params) + def test_no_enforce(self): # Test that nothing goes wrong when we don't enforce stationarity params = self.model.untransform_params(self.true['params']) From e3ad4d749916f78d80ce2cdf5fac01a0c19ae4f4 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 11:18:02 -0700 Subject: [PATCH 39/57] ENH: Add VARMAXResults class. --- statsmodels/tsa/statespace/mlemodel.py | 2 +- .../tsa/statespace/tests/test_varmax.py | 32 ++- statsmodels/tsa/statespace/varmax.py | 247 +++++++++++++++++- 3 files changed, 277 insertions(+), 4 deletions(-) diff --git a/statsmodels/tsa/statespace/mlemodel.py b/statsmodels/tsa/statespace/mlemodel.py index 2a6f955c8e6..1ac0d56568c 100644 --- a/statsmodels/tsa/statespace/mlemodel.py +++ b/statsmodels/tsa/statespace/mlemodel.py @@ -1337,7 +1337,7 @@ class MLEResultsWrapper(wrap.ResultsWrapper): # want those to be a subclass of FilterResults with e.g. confidence # intervals calculated and dates attached. # Also, need to modify `attach_dates` to account for DataFrames. - _methods = {'predict': None} + _methods = {'predict': None, 'forecast': None} _wrap_methods = wrap.union_dicts(tsbase.TimeSeriesResultsWrapper._wrap_methods, _methods) wrap.populate_wrapper(MLEResultsWrapper, MLEResults) diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index d8f9e06173b..b63cd1988bd 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -65,6 +65,30 @@ def test_params(self): self.model.enforce_invertibility = True assert_allclose(actual, self.model.start_params) + def test_results(self): + # Smoke test for creating the summary + self.results.summary() + + # Test cofficient matrix creation (via a different, more direct, method) + if self.model.k_ar > 0: + coefficients = self.results.params[self.model._params_ar].reshape(self.model.k_endog, self.model.k_endog * self.model.k_ar) + coefficient_matrices = np.array([ + coefficients[:self.model.k_endog, i*self.model.k_endog:(i+1)*self.model.k_endog] + for i in range(self.model.k_ar) + ]) + assert_equal(self.results.coefficient_matrices_var, coefficient_matrices) + else: + assert_equal(self.results.coefficient_matrices_var, None) + if self.model.k_ma > 0: + coefficients = self.results.params[self.model._params_ma].reshape(self.model.k_endog, self.model.k_endog * self.model.k_ma) + coefficient_matrices = np.array([ + coefficients[:self.model.k_endog, i*self.model.k_endog:(i+1)*self.model.k_endog] + for i in range(self.model.k_ma) + ]) + assert_equal(self.results.coefficient_matrices_vma, coefficient_matrices) + else: + assert_equal(self.results.coefficient_matrices_vma, None) + def test_loglike(self): assert_allclose(self.results.llf, self.true['loglike'], rtol=1e-6) @@ -260,9 +284,15 @@ def test_dynamic_predict(self): def test_forecast(self): # Tests forecast exog = (np.arange(75, 75+16) + 3)[:, np.newaxis] + + # Test it through the results class wrapper + desired = self.results.forecast(steps=16, exog=exog) + assert_allclose(desired, self.true['fcast'].T, atol=1e-6) + + # Test it directly beta = self.results.params[-9:-6] state_intercept = np.concatenate([exog*beta[0], exog*beta[1], exog*beta[2]], axis=1).T - desired = self.results.forecast(steps=16, state_intercept=state_intercept) + desired = super(varmax.VARMAXResultsWrapper, self.results).predict(start=75, end=75+15, state_intercept=state_intercept) assert_allclose(desired, self.true['fcast'].T, atol=1e-6) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index 34028e2db0f..8a3bd7e15cf 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -23,6 +23,7 @@ from statsmodels.tools.data import _is_using_pandas from statsmodels.tsa.tsatools import lagmat from statsmodels.tsa.vector_ar import var_model +import statsmodels.base.wrapper as wrap class VARMAX(MLEModel): @@ -114,7 +115,7 @@ def __init__(self, endog, exog=None, order=(1, 0), trend='c', self.k_exog = exog.shape[1] # Note: at some point in the future might add state regression, as in - # SARIMAX. + # VARMAX. self.mle_regression = self.k_exog > 0 # We need to have an array or pandas at this point @@ -180,7 +181,8 @@ def __init__(self, endog, exog=None, order=(1, 0), trend='c', # ... and the lower right block is in companion form with zeros as the # coefficient matrices (it is shaped k_endog * k_ma x k_endog * k_ma). idx = np.diag_indices((self.k_ma - 1) * self.k_endog) - idx = idx[0] + (_min_k_ar + 1) * self.k_endog, idx[1] + _min_k_ar * self.k_endog + idx = (idx[0] + (_min_k_ar + 1) * self.k_endog, + idx[1] + _min_k_ar * self.k_endog) self.ssm[('transition',) + idx] = 1 # The selection matrix is described in two blocks, where the upper @@ -226,6 +228,30 @@ def _slice(key, offset): self._params_state_cov, offset = _slice('state_cov', offset) self._params_obs_cov, offset = _slice('obs_cov', offset) + def filter(self, params, transformed=True, cov_type=None, return_ssm=False, + **kwargs): + params = np.array(params, ndmin=1) + + # Transform parameters if necessary + if not transformed: + params = self.transform_params(params) + transformed = True + + # Get the state space output + result = super(VARMAX, self).filter(params, transformed, cov_type, + return_ssm=True, **kwargs) + + # Wrap in a results object + if not return_ssm: + result_kwargs = {} + if cov_type is not None: + result_kwargs['cov_type'] = cov_type + result = VARMAXResultsWrapper( + VARMAXResults(self, params, result, **result_kwargs) + ) + + return result + @property def start_params(self): params = np.zeros(self.k_params, dtype=np.float64) @@ -550,3 +576,220 @@ def update(self, params, *args, **kwargs): # 4. Observation covariance if self.measurement_error: self.ssm[self._idx_obs_cov] = params[self._params_obs_cov] + + +class VARMAXResults(MLEResults): + """ + Class to hold results from fitting an VARMAX model. + + Parameters + ---------- + model : VARMAX instance + The fitted model instance + + Attributes + ---------- + specification : dictionary + Dictionary including all attributes from the VARMAX model instance. + coefficient_matrices_var : array + Array containing autoregressive lag polynomial coefficient matrices, + ordered from lowest degree to highest. + coefficient_matrices_vma : array + Array containing moving average lag polynomial coefficients, + ordered from lowest degree to highest. + + See Also + -------- + statsmodels.tsa.statespace.kalman_filter.FilterResults + statsmodels.tsa.statespace.mlemodel.MLEResults + """ + def __init__(self, model, params, filter_results, cov_type='opg', + **kwargs): + super(VARMAXResults, self).__init__(model, params, filter_results, + cov_type, **kwargs) + + self.df_resid = np.inf # attribute required for wald tests + + self.specification = Bunch(**{ + # Set additional model parameters + 'error_cov_type': self.model.error_cov_type, + 'measurement_error': self.model.measurement_error, + 'enforce_stationarity': self.model.enforce_stationarity, + 'enforce_invertibility': self.model.enforce_invertibility, + + 'order': self.model.order, + + # Model order + 'k_ar': self.model.k_ar, + 'k_ma': self.model.k_ma, + + # Trend / Regression + 'trend': self.model.trend, + 'k_trend': self.model.k_trend, + 'k_exog': self.model.k_exog, + }) + + # Polynomials / coefficient matrices + self.coefficient_matrices_var = None + self.coefficient_matrices_vma = None + if self.model.k_ar > 0: + ar_params = self.params[self.model._params_ar] + k_endog = self.model.k_endog + k_ar = self.model.k_ar + self.coefficient_matrices_var = ( + ar_params.reshape(k_endog * k_ar, k_endog).T + ).reshape(k_endog, k_endog, k_ar).T + if self.model.k_ma > 0: + ma_params = self.params[self.model._params_ma] + k_endog = self.model.k_endog + k_ma = self.model.k_ma + self.coefficient_matrices_vma = ( + ma_params.reshape(k_endog * k_ma, k_endog).T + ).reshape(k_endog, k_endog, k_ma).T + + def predict(self, start=None, end=None, exog=None, dynamic=False, + **kwargs): + """ + In-sample prediction and out-of-sample forecasting + + Parameters + ---------- + start : int, str, or datetime, optional + Zero-indexed observation number at which to start forecasting, ie., + the first forecast is start. Can also be a date string to + parse or a datetime type. Default is the the zeroth observation. + end : int, str, or datetime, optional + Zero-indexed observation number at which to end forecasting, ie., + the first forecast is start. Can also be a date string to + parse or a datetime type. However, if the dates index does not + have a fixed frequency, end must be an integer index if you + want out of sample prediction. Default is the last observation in + the sample. + exog : array_like, optional + If the model includes exogenous regressors, you must provide + exactly enough out-of-sample values for the exogenous variables if + end is beyond the last observation in the sample. + dynamic : boolean, int, str, or datetime, optional + Integer offset relative to `start` at which to begin dynamic + prediction. Can also be an absolute date string to parse or a + datetime type (these are not interpreted as offsets). + Prior to this observation, true endogenous values will be used for + prediction; starting with this observation and continuing through + the end of prediction, forecasted endogenous values will be used + instead. + **kwargs + Additional arguments may required for forecasting beyond the end + of the sample. See `FilterResults.predict` for more details. + + Returns + ------- + forecast : array + Array of out of sample forecasts. + """ + if start is None: + start = 0 + + # Handle end (e.g. date) + _start = self.model._get_predict_start(start) + _end, _out_of_sample = self.model._get_predict_end(end) + + # Handle exogenous parameters + if _out_of_sample and (self.model.k_exog + self.model.k_trend > 0): + # Create a new faux VARMAX model for the extended dataset + nobs = self.model.data.orig_endog.shape[0] + _out_of_sample + endog = np.zeros((nobs, self.model.k_endog)) + + if self.model.k_exog > 0: + if exog is None: + raise ValueError('Out-of-sample forecasting in a model' + ' with a regression component requires' + ' additional exogenous values via the' + ' `exog` argument.') + exog = np.array(exog) + required_exog_shape = (_out_of_sample, self.model.k_exog) + if not exog.shape == required_exog_shape: + raise ValueError('Provided exogenous values are not of the' + ' appropriate shape. Required %s, got %s.' + % (str(required_exog_shape), + str(exog.shape))) + exog = np.c_[self.model.data.orig_exog.T, exog.T].T + + # TODO replace with init_kwds or specification or similar + model = VARMAX( + endog, + exog=exog, + order=self.model.order, + trend=self.model.trend, + error_cov_type=self.model.error_cov_type, + measurement_error=self.model.measurement_error, + enforce_stationarity=self.model.enforce_stationarity, + enforce_invertibility=self.model.enforce_invertibility + ) + model.update(self.params) + + # Set the kwargs with the update time-varying state space + # representation matrices + for name in self.filter_results.shapes.keys(): + if name == 'obs': + continue + mat = getattr(model.ssm, name) + if mat.shape[-1] > 1: + if len(mat.shape) == 2: + kwargs[name] = mat[:, -_out_of_sample:] + else: + kwargs[name] = mat[:, :, -_out_of_sample:] + elif self.model.k_exog == 0 and exog is not None: + warn('Exogenous array provided to predict, but additional data not' + ' required. `exog` argument ignored.') + + return super(VARMAXResults, self).predict( + start=start, end=end, exog=exog, dynamic=dynamic, **kwargs + ) + + def forecast(self, steps=1, exog=None, **kwargs): + """ + Out-of-sample forecasts + + Parameters + ---------- + steps : int, optional + The number of out of sample forecasts from the end of the + sample. Default is 1. + exog : array_like, optional + If the model includes exogenous regressors, you must provide + exactly enough out-of-sample values for the exogenous variables for + each step forecasted. + **kwargs + Additional arguments may required for forecasting beyond the end + of the sample. See `FilterResults.predict` for more details. + + Returns + ------- + forecast : array + Array of out of sample forecasts. + """ + return super(VARMAXResults, self).forecast(steps, exog=exog, **kwargs) + + def summary(self, alpha=.05, start=None): + # Create the model name + + # See if we have an ARIMA component + order = '(%d, %d)' % (self.specification.k_ar, self.specification.k_ma) + + model_name = ( + '%s%s' % (self.model.__class__.__name__, order) + ) + return super(VARMAXResults, self).summary( + alpha=alpha, start=start, model_name=model_name + ) + summary.__doc__ = MLEResults.summary.__doc__ + + +class VARMAXResultsWrapper(MLEResultsWrapper): + _attrs = {} + _wrap_attrs = wrap.union_dicts(MLEResultsWrapper._wrap_attrs, + _attrs) + _methods = {} + _wrap_methods = wrap.union_dicts(MLEResultsWrapper._wrap_methods, + _methods) +wrap.populate_wrapper(VARMAXResultsWrapper, VARMAXResults) From 4957b4fb09295c77a01e4ae30e925da5e66271ae Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 11:30:43 -0700 Subject: [PATCH 40/57] ENH: Add check for too many factors. --- statsmodels/tsa/statespace/dynamic_factor.py | 5 +++++ statsmodels/tsa/statespace/tests/test_dynamic_factor.py | 8 ++++++++ 2 files changed, 13 insertions(+) diff --git a/statsmodels/tsa/statespace/dynamic_factor.py b/statsmodels/tsa/statespace/dynamic_factor.py index 0110633ca8e..dd551fed0be 100644 --- a/statsmodels/tsa/statespace/dynamic_factor.py +++ b/statsmodels/tsa/statespace/dynamic_factor.py @@ -100,6 +100,11 @@ def __init__(self, endog, k_factors, factor_order, endog, k_states=k_states, k_posdef=k_posdef, **kwargs ) + # Test for too many factors + if self.k_factors >= self.k_endog: + raise ValueError('Number of factors must be less than the number' + ' of endogenous variables.') + # Initialize the parameters self.parameters = OrderedDict() self.parameters['factor_loadings'] = self.k_endog * self.k_factors diff --git a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py index 412e9be23c7..a49f0c026ae 100644 --- a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py @@ -140,3 +140,11 @@ def test_bic(self): # model didn't coverge, 4 of the parameters aren't fully estimated # (possibly they are still at starting values?) so the BIC is off pass + + +def test_misspecification(): + # Tests for model specification and misspecification exceptions + endog = np.arange(20).reshape(10,2) + + # Too many factors + assert_raises(ValueError, dynamic_factor.StaticFactors, endog, k_factors=2, factor_order=1) From ef4cdfb306289e7836ff3c13f7283e927a7d8354 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 11:31:00 -0700 Subject: [PATCH 41/57] ENH: Add StaticFactorsResults class. --- statsmodels/tsa/statespace/dynamic_factor.py | 104 ++++++++++++++++++ .../statespace/tests/test_dynamic_factor.py | 15 +++ 2 files changed, 119 insertions(+) diff --git a/statsmodels/tsa/statespace/dynamic_factor.py b/statsmodels/tsa/statespace/dynamic_factor.py index dd551fed0be..34ab0c37635 100644 --- a/statsmodels/tsa/statespace/dynamic_factor.py +++ b/statsmodels/tsa/statespace/dynamic_factor.py @@ -135,6 +135,31 @@ def _slice(key, offset): self._params_idiosyncratic, offset = _slice('idiosyncratic', offset) self._params_transition, offset = _slice('transition', offset) + def filter(self, params, transformed=True, cov_type=None, return_ssm=False, + **kwargs): + params = np.array(params, ndmin=1) + + # Transform parameters if necessary + if not transformed: + params = self.transform_params(params) + transformed = True + + # Get the state space output + result = super(StaticFactors, self).filter(params, transformed, + cov_type, return_ssm=True, + **kwargs) + + # Wrap in a results object + if not return_ssm: + result_kwargs = {} + if cov_type is not None: + result_kwargs['cov_type'] = cov_type + result = StaticFactorsResultsWrapper( + StaticFactorsResults(self, params, result, **result_kwargs) + ) + + return result + @property def start_params(self): params = np.zeros(self.k_params, dtype=np.float64) @@ -345,3 +370,82 @@ def update(self, params, transformed=True): self.ssm[self._idx_transition] = ( params[self._params_transition].reshape( self.k_factors, self.k_factors * self.factor_order)) + + +class StaticFactorsResults(MLEResults): + """ + Class to hold results from fitting an StaticFactors model. + + Parameters + ---------- + model : StaticFactors instance + The fitted model instance + + Attributes + ---------- + specification : dictionary + Dictionary including all attributes from the StaticFactors model + instance. + coefficient_matrices_var : array + Array containing autoregressive lag polynomial coefficient matrices, + ordered from lowest degree to highest. + coefficient_matrices_vma : array + Array containing moving average lag polynomial coefficients, + ordered from lowest degree to highest. + + See Also + -------- + statsmodels.tsa.statespace.kalman_filter.FilterResults + statsmodels.tsa.statespace.mlemodel.MLEResults + """ + def __init__(self, model, params, filter_results, cov_type='opg', + **kwargs): + super(StaticFactorsResults, self).__init__(model, params, + filter_results, cov_type, + **kwargs) + + self.df_resid = np.inf # attribute required for wald tests + + self.specification = Bunch(**{ + # Set additional model parameters + 'enforce_stationarity': self.model.enforce_stationarity, + + # Factors + 'k_factors': self.model.k_factors, + 'factor_order': self.model.factor_order + }) + + # Polynomials / coefficient matrices + self.coefficient_matrices_var = None + if self.model.factor_order > 0: + ar_params = self.params[self.model._params_transition] + k_factors = self.model.k_factors + factor_order = self.model.factor_order + self.coefficient_matrices_var = ( + ar_params.reshape(k_factors * factor_order, k_factors).T + ).reshape(k_factors, k_factors, factor_order).T + + def summary(self, alpha=.05, start=None): + # Create the model name + + # See if we have an ARIMA component + order = '(factors=%d, order=%d)' % (self.specification.k_factors, + self.specification.factor_order) + + model_name = ( + '%s%s' % (self.model.__class__.__name__, order) + ) + return super(StaticFactorsResults, self).summary( + alpha=alpha, start=start, model_name=model_name + ) + summary.__doc__ = MLEResults.summary.__doc__ + + +class StaticFactorsResultsWrapper(MLEResultsWrapper): + _attrs = {} + _wrap_attrs = wrap.union_dicts(MLEResultsWrapper._wrap_attrs, + _attrs) + _methods = {} + _wrap_methods = wrap.union_dicts(MLEResultsWrapper._wrap_methods, + _methods) +wrap.populate_wrapper(StaticFactorsResultsWrapper, StaticFactorsResults) diff --git a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py index a49f0c026ae..ddfdb520a6d 100644 --- a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py @@ -60,6 +60,21 @@ def test_params(self): self.model.enforce_stationarity = True assert_allclose(actual, self.model.start_params) + def test_results(self): + # Smoke test for creating the summary + self.results.summary() + + # Test cofficient matrix creation (via a different, more direct, method) + if self.model.factor_order > 0: + coefficients = self.results.params[self.model._params_transition].reshape(self.model.k_factors, self.model.k_factors * self.model.factor_order) + coefficient_matrices = np.array([ + coefficients[:self.model.k_factors, i*self.model.k_factors:(i+1)*self.model.k_factors] + for i in range(self.model.factor_order) + ]) + assert_equal(self.results.coefficient_matrices_var, coefficient_matrices) + else: + assert_equal(self.results.coefficient_matrices_var, None) + def test_no_enforce(self): # Test that nothing goes wrong when we don't enforce stationarity params = self.model.untransform_params(self.true['params']) From 4574604f4d7c45849b96e4d6863f4352736fd65d Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 12:49:07 -0700 Subject: [PATCH 42/57] ENH: Improve summary, param_name formatting --- statsmodels/tsa/statespace/varmax.py | 101 +++++++++++++++++++++++---- 1 file changed, 89 insertions(+), 12 deletions(-) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index 8a3bd7e15cf..df53a051b2c 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -335,38 +335,44 @@ def param_names(self): # 1. Intercept terms if self.trend == 'c': - param_names += ['const.y%d' % (i+1) for i in range(self.k_endog)] + param_names += [ + 'const.%s' % self.endog_names[i] + for i in range(self.k_endog) + ] # 2. AR terms param_names += [ - 'L%dy%d.y%d' % (i+1, k+1, j+1) - for i in range(self.k_ar) + 'L%d.%s.%s' % (i+1, self.endog_names[k], self.endog_names[j]) for j in range(self.k_endog) + for i in range(self.k_ar) for k in range(self.k_endog) ] # 3. MA terms param_names += [ - 'L%de%d.y%d' % (i+1, k+1, j+1) - for i in range(self.k_ma) + 'L%d.e(%s).%s' % (i+1, self.endog_names[k], self.endog_names[j]) for j in range(self.k_endog) + for i in range(self.k_ma) for k in range(self.k_endog) ] # 4. Regression terms - # TODO replace "beta" with the name of the exog, when possible param_names += [ - 'beta%d.y%d' % (j+1, i+1) + 'beta.%s.%s' % (self.exog_names[j], self.endog_names[i]) for i in range(self.k_endog) for j in range(self.k_exog) ] # 5. State covariance terms if self.error_cov_type == 'diagonal': - param_names += ['sigma2.y%d' % i for i in range(self.k_endog)] + param_names += [ + 'sigma2.%s' % self.endog_names[i] + for i in range(self.k_endog) + ] elif self.error_cov_type == 'unstructured': param_names += [ - 'sqrt.cov.y%d.y%d' % (j+1, i+1) + ('sqrt.var.%s' % self.endog_names[i] if i == j else + 'sqrt.cov.%s.%s' % (self.endog_names[j], self.endog_names[i])) for i in range(self.k_endog) for j in range(i+1) ] @@ -374,7 +380,7 @@ def param_names(self): # 5. Measurement error variance terms if self.measurement_error: param_names += [ - 'measurement_variance.y%d' % (i+1) + 'measurement_variance.%s' % self.endog_names[i] for i in range(self.k_endog) ] @@ -771,6 +777,8 @@ def forecast(self, steps=1, exog=None, **kwargs): return super(VARMAXResults, self).forecast(steps, exog=exog, **kwargs) def summary(self, alpha=.05, start=None): + from statsmodels.iolib.summary import summary_params + # Create the model name # See if we have an ARIMA component @@ -779,9 +787,78 @@ def summary(self, alpha=.05, start=None): model_name = ( '%s%s' % (self.model.__class__.__name__, order) ) - return super(VARMAXResults, self).summary( - alpha=alpha, start=start, model_name=model_name + summary = super(VARMAXResults, self).summary( + alpha=alpha, start=start, model_name=model_name, + display_params=False ) + + # Add parameter tables for each endogenous variable + k_endog = self.model.k_endog + k_ar = self.model.k_ar + k_ma = self.model.k_ma + k_exog = self.model.k_exog + for i in range(k_endog): + mask = [] + offset = 0 + + # 1. Intercept terms + if self.model.trend == 'c': + mask.append(np.array(i, ndmin=1)) + offset += k_endog + + # 2. AR terms + if k_ar > 0: + mask.append( + offset + np.arange(i * k_endog * k_ar, (i + 1) * k_endog * k_ar)) + offset += k_ar * k_endog**2 + + # 3. MA terms + if k_ma > 0: + mask.append( + offset + np.arange(i * k_endog * k_ma, (i + 1) * k_endog * k_ma)) + offset += k_ma * k_endog**2 + + # 4. Regression terms + if k_exog > 0: + mask.append( + offset + np.arange(i * k_exog, (i + 1) * k_exog)) + offset += k_endog * k_exog + + # 5. Measurement error variance terms + if self.model.measurement_error: + mask.append(np.array(self.model.k_params - i - 1, ndmin=1)) + + # Create the table + mask = np.concatenate(mask) + res = (self, self.params[mask], self.bse[mask], self.zvalues[mask], + self.pvalues[mask], self.conf_int(alpha)[mask]) + + param_names = [ + '.'.join(name.split('.')[:-1]) + for name in + np.array(self.data.param_names)[mask].tolist() + ] + title = "Results for equation %s" % self.model.endog_names[i] + + table = summary_params(res, yname=None, xname=param_names, + alpha=alpha, use_t=False, title=title) + + summary.tables.append(table) + + + # State covariance terms + mask = self.model._params_state_cov + res = (self, self.params[mask], self.bse[mask], self.zvalues[mask], + self.pvalues[mask], self.conf_int(alpha)[mask]) + + param_names = np.array(self.data.param_names)[mask].tolist() + title = "Error covariance matrix" + + table = summary_params(res, yname=None, xname=param_names, + alpha=alpha, use_t=False, title=title) + summary.tables.append(table) + + return summary summary.__doc__ = MLEResults.summary.__doc__ From 420d7d57566ce1f333b5bcf1b326ab886f41da89 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 12:59:29 -0700 Subject: [PATCH 43/57] ENH: Improved VARMAX model name output. --- statsmodels/tsa/statespace/varmax.py | 34 ++++++++++++++++++++-------- 1 file changed, 25 insertions(+), 9 deletions(-) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index df53a051b2c..6cd77346adc 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -780,13 +780,26 @@ def summary(self, alpha=.05, start=None): from statsmodels.iolib.summary import summary_params # Create the model name + spec = self.specification + if spec.k_ar > 0 and spec.k_ma > 0: + model_name = 'VARMA' + order = '(%s,%s)' % (spec.k_ar, spec.k_ma) + elif spec.k_ar > 0: + model_name = 'VAR' + order = '(%s)' % (spec.k_ar) + else: + model_name = 'VMA' + order = '(%s)' % (spec.k_ma) + if spec.k_exog > 0: + model_name += 'X' + model_name = [model_name + order] - # See if we have an ARIMA component - order = '(%d, %d)' % (self.specification.k_ar, self.specification.k_ma) + if spec.trend == 'c': + model_name.append('intercept') + + if spec.measurement_error: + model_name.append('measurement error') - model_name = ( - '%s%s' % (self.model.__class__.__name__, order) - ) summary = super(VARMAXResults, self).summary( alpha=alpha, start=start, model_name=model_name, display_params=False @@ -808,14 +821,18 @@ def summary(self, alpha=.05, start=None): # 2. AR terms if k_ar > 0: + start = i * k_endog * k_ar + end = (i + 1) * k_endog * k_ar mask.append( - offset + np.arange(i * k_endog * k_ar, (i + 1) * k_endog * k_ar)) + offset + np.arange(start, end)) offset += k_ar * k_endog**2 # 3. MA terms if k_ma > 0: + start = i * k_endog * k_ma + end = (i + 1) * k_endog * k_ma mask.append( - offset + np.arange(i * k_endog * k_ma, (i + 1) * k_endog * k_ma)) + offset + np.arange(start, end)) offset += k_ma * k_endog**2 # 4. Regression terms @@ -842,9 +859,8 @@ def summary(self, alpha=.05, start=None): table = summary_params(res, yname=None, xname=param_names, alpha=alpha, use_t=False, title=title) - - summary.tables.append(table) + summary.tables.append(table) # State covariance terms mask = self.model._params_state_cov From da3962701c390151930e5ec291dba942add4bf92 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 13:03:26 -0700 Subject: [PATCH 44/57] BUG: Suppress default params table in summary. --- statsmodels/tsa/statespace/mlemodel.py | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/statsmodels/tsa/statespace/mlemodel.py b/statsmodels/tsa/statespace/mlemodel.py index 1ac0d56568c..f6bf922c699 100644 --- a/statsmodels/tsa/statespace/mlemodel.py +++ b/statsmodels/tsa/statespace/mlemodel.py @@ -1240,7 +1240,8 @@ def forecast(self, steps=1, **kwargs): """ return self.predict(start=self.nobs, end=self.nobs+steps-1, **kwargs) - def summary(self, alpha=.05, start=None, model_name=None): + def summary(self, alpha=.05, start=None, model_name=None, + display_params=True): """ Summarize the Model @@ -1304,7 +1305,7 @@ def summary(self, alpha=.05, start=None, model_name=None): summary = Summary() summary.add_table_2cols(self, gleft=top_left, gright=top_right, title=title) - if len(self.params) > 0: + if len(self.params) > 0 and display_params: summary.add_table_params(self, alpha=alpha, xname=self.data.param_names, use_t=False) From 8463268c6a9cc0a51a9293b51e3743e9a8ccdbec Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 14:00:48 -0700 Subject: [PATCH 45/57] TST: Add tests for new summary tables. --- .../tsa/statespace/tests/test_varmax.py | 216 ++++++++++++++++++ 1 file changed, 216 insertions(+) diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index b63cd1988bd..db8bf784a6f 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -9,6 +9,7 @@ import numpy as np import pandas as pd import os +import re import warnings from statsmodels.datasets import webuse @@ -16,6 +17,7 @@ from .results import results_varmax from numpy.testing import assert_equal, assert_almost_equal, assert_raises, assert_allclose from nose.exc import SkipTest +from statsmodels.iolib.summary import forg current_path = os.path.dirname(os.path.abspath(__file__)) @@ -155,6 +157,40 @@ def __init__(self): true, order=(1,0), trend='nc', error_cov_type="unstructured") + def test_summary(self): + summary = self.results.summary() + tables = [str(table) for table in summary.tables] + params = self.true['params'] + + # Check the model overview table + assert_equal(re.search(r'Model:.*VAR\(1\)', tables[0]) is None, False) + + # For each endogenous variable, check the output + for i in range(self.model.k_endog): + offset = i * self.model.k_endog + table = tables[i+1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for equation %s' % name, table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 8) + + # -> Check that we have the right coefficients + assert_equal(re.search('L1.dln_inv +%.4f' % params[offset + 0], table) is None, False) + assert_equal(re.search('L1.dln_inc +%.4f' % params[offset + 1], table) is None, False) + assert_equal(re.search('L1.dln_consump +%.4f' % params[offset + 2], table) is None, False) + + # Test the error covariance matrix table + table = tables[-1] + assert_equal(re.search('Error covariance matrix', table) is None, False) + assert_equal(len(table.split('\n')), 11) + + params = params[self.model._params_state_cov] + names = self.model.param_names[self.model._params_state_cov] + for i in range(len(names)): + assert_equal(re.search('%s +%.4f' % (names[i], params[i]), table) is None, False) class TestVAR_diagonal(CheckLutkepohl): def __init__(self): @@ -165,6 +201,41 @@ def __init__(self): true, order=(1,0), trend='nc', error_cov_type="diagonal") + def test_summary(self): + summary = self.results.summary() + tables = [str(table) for table in summary.tables] + params = self.true['params'] + + # Check the model overview table + assert_equal(re.search(r'Model:.*VAR\(1\)', tables[0]) is None, False) + + # For each endogenous variable, check the output + for i in range(self.model.k_endog): + offset = i * self.model.k_endog + table = tables[i+1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for equation %s' % name, table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 8) + + # -> Check that we have the right coefficients + assert_equal(re.search('L1.dln_inv +%.4f' % params[offset + 0], table) is None, False) + assert_equal(re.search('L1.dln_inc +%.4f' % params[offset + 1], table) is None, False) + assert_equal(re.search('L1.dln_consump +%.4f' % params[offset + 2], table) is None, False) + + # Test the error covariance matrix table + table = tables[-1] + assert_equal(re.search('Error covariance matrix', table) is None, False) + assert_equal(len(table.split('\n')), 8) + + params = params[self.model._params_state_cov] + names = self.model.param_names[self.model._params_state_cov] + for i in range(len(names)): + assert_equal(re.search('%s +%.4f' % (names[i], params[i]), table) is None, False) + class TestVAR_measurement_error(CheckLutkepohl): """ @@ -225,6 +296,42 @@ def test_representation(self): assert_equal(getattr(self.results2.filter_results, name), getattr(self.results.filter_results, name)) + def test_summary(self): + summary = self.results.summary() + tables = [str(table) for table in summary.tables] + params = self.true['params'] + + # Check the model overview table + assert_equal(re.search(r'Model:.*VAR\(1\)', tables[0]) is None, False) + + # For each endogenous variable, check the output + for i in range(self.model.k_endog): + offset = i * self.model.k_endog + table = tables[i+1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for equation %s' % name, table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 9) + + # -> Check that we have the right coefficients + assert_equal(re.search('L1.dln_inv +%.4f' % params[offset + 0], table) is None, False) + assert_equal(re.search('L1.dln_inc +%.4f' % params[offset + 1], table) is None, False) + assert_equal(re.search('L1.dln_consump +%.4f' % params[offset + 2], table) is None, False) + assert_equal(re.search('measurement_variance +%.4g' % params[-(i+1)], table) is None, False) + + # Test the error covariance matrix table + table = tables[-1] + assert_equal(re.search('Error covariance matrix', table) is None, False) + assert_equal(len(table.split('\n')), 8) + + params = params[self.model._params_state_cov] + names = self.model.param_names[self.model._params_state_cov] + for i in range(len(names)): + assert_equal(re.search('%s +%.4f' % (names[i], params[i]), table) is None, False) + class TestVAR_obs_intercept(CheckLutkepohl): def __init__(self): true = results_varmax.lutkepohl_var1_obs_intercept.copy() @@ -295,6 +402,42 @@ def test_forecast(self): desired = super(varmax.VARMAXResultsWrapper, self.results).predict(start=75, end=75+15, state_intercept=state_intercept) assert_allclose(desired, self.true['fcast'].T, atol=1e-6) + def test_summary(self): + summary = self.results.summary() + tables = [str(table) for table in summary.tables] + params = self.true['params'] + + # Check the model overview table + assert_equal(re.search(r'Model:.*VARX\(1\)', tables[0]) is None, False) + + # For each endogenous variable, check the output + for i in range(self.model.k_endog): + offset = i * self.model.k_endog + table = tables[i+1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for equation %s' % name, table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 9) + + # -> Check that we have the right coefficients + assert_equal(re.search('L1.dln_inv +%.4f' % params[offset + 0], table) is None, False) + assert_equal(re.search('L1.dln_inc +%.4f' % params[offset + 1], table) is None, False) + assert_equal(re.search('L1.dln_consump +%.4f' % params[offset + 2], table) is None, False) + assert_equal(re.search('beta.x1 +' + forg(params[self.model._params_regression][i], prec=4), table) is None, False) + + # Test the error covariance matrix table + table = tables[-1] + assert_equal(re.search('Error covariance matrix', table) is None, False) + assert_equal(len(table.split('\n')), 11) + + params = params[self.model._params_state_cov] + names = self.model.param_names[self.model._params_state_cov] + for i in range(len(names)): + assert_equal(re.search('%s +%.4f' % (names[i], params[i]), table) is None, False) + class TestVAR2(CheckLutkepohl): def __init__(self): @@ -310,6 +453,42 @@ def test_bse_oim(self): assert_allclose( self.results.bse[:-3]**2, self.true['var_oim'][:-3], atol=1e-2) + def test_summary(self): + summary = self.results.summary() + tables = [str(table) for table in summary.tables] + params = self.true['params'] + + # Check the model overview table + assert_equal(re.search(r'Model:.*VAR\(2\)', tables[0]) is None, False) + + # For each endogenous variable, check the output + for i in range(self.model.k_endog): + offset = i * self.model.k_endog * self.model.k_ar + table = tables[i+1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for equation %s' % name, table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 9) + + # -> Check that we have the right coefficients + assert_equal(re.search('L1.dln_inv +%.4f' % params[offset + 0], table) is None, False) + assert_equal(re.search('L1.dln_inc +%.4f' % params[offset + 1], table) is None, False) + assert_equal(re.search('L2.dln_inv +%.4f' % params[offset + 2], table) is None, False) + assert_equal(re.search('L2.dln_inc +%.4f' % params[offset + 3], table) is None, False) + + # Test the error covariance matrix table + table = tables[-1] + assert_equal(re.search('Error covariance matrix', table) is None, False) + assert_equal(len(table.split('\n')), 8) + + params = params[self.model._params_state_cov] + names = self.model.param_names[self.model._params_state_cov] + for i in range(len(names)): + assert_equal(re.search('%s +%.4f' % (names[i], params[i]), table) is None, False) + class CheckFREDManufacturing(CheckVARMAX): def __init__(self, true, order, trend, error_cov_type, cov_type='oim', @@ -372,6 +551,43 @@ def test_predict(self): def test_dynamic_predict(self): super(TestVARMA, self).test_dynamic_predict(end='2009-05-01', dynamic='2000-01-01') + def test_summary(self): + summary = self.results.summary() + tables = [str(table) for table in summary.tables] + params = self.true['params'] + + # Check the model overview table + assert_equal(re.search(r'Model:.*VARMA\(1,1\)', tables[0]) is None, False) + + # For each endogenous variable, check the output + for i in range(self.model.k_endog): + offset_ar = i * self.model.k_endog + offset_ma = self.model.k_endog**2 * self.model.k_ar + i * self.model.k_endog + table = tables[i+1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for equation %s' % name, table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 9) + + # -> Check that we have the right coefficients + assert_equal(re.search('L1.dlncaputil +' + forg(params[offset_ar + 0], prec=4), table) is None, False) + assert_equal(re.search('L1.dlnhours +' + forg(params[offset_ar + 1], prec=4), table) is None, False) + assert_equal(re.search(r'L1.e\(dlncaputil\) +' + forg(params[offset_ma + 0], prec=4), table) is None, False) + assert_equal(re.search(r'L1.e\(dlnhours\) +' + forg(params[offset_ma + 1], prec=4), table) is None, False) + + # Test the error covariance matrix table + table = tables[-1] + assert_equal(re.search('Error covariance matrix', table) is None, False) + assert_equal(len(table.split('\n')), 7) + + params = params[self.model._params_state_cov] + names = self.model.param_names[self.model._params_state_cov] + for i in range(len(names)): + assert_equal(re.search('%s +%s' % (names[i], forg(params[i], prec=4)), table) is None, False) + class TestVMA1(CheckFREDManufacturing): """ From 8a823ef0c7db465172d4dda3ae1f88767e268d92 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 14:32:45 -0700 Subject: [PATCH 46/57] ENH: Improve DFM summary, param_name formatting --- statsmodels/tsa/statespace/dynamic_factor.py | 78 +++++++++++++++++-- .../statespace/tests/test_dynamic_factor.py | 44 +++++++++++ 2 files changed, 116 insertions(+), 6 deletions(-) diff --git a/statsmodels/tsa/statespace/dynamic_factor.py b/statsmodels/tsa/statespace/dynamic_factor.py index 34ab0c37635..03df0ba76fc 100644 --- a/statsmodels/tsa/statespace/dynamic_factor.py +++ b/statsmodels/tsa/statespace/dynamic_factor.py @@ -196,19 +196,22 @@ def param_names(self): # 1. Factor loadings (estimated via PCA) param_names += [ - 'loading%d.y%d' % (j+1, i+1) + 'loading.f%d.%s' % (j+1, self.endog_names[i]) for i in range(self.k_endog) for j in range(self.k_factors) ] # 2. Idiosyncratic variances - param_names += ['sigma2.y%d' % (i+1) for i in range(self.k_endog)] + param_names += [ + 'sigma2.%s' % self.endog_names[i] + for i in range(self.k_endog) + ] # 3. VAR transition (OLS on factors estimated via PCA) param_names += [ - 'L%df%d.f%d' % (i+1, k+1, j+1) - for i in range(self.factor_order) + 'L%d.f%d.f%d' % (i+1, k+1, j+1) for j in range(self.k_factors) + for i in range(self.factor_order) for k in range(self.k_factors) ] @@ -426,6 +429,7 @@ def __init__(self, model, params, filter_results, cov_type='opg', ).reshape(k_factors, k_factors, factor_order).T def summary(self, alpha=.05, start=None): + from statsmodels.iolib.summary import summary_params # Create the model name # See if we have an ARIMA component @@ -435,9 +439,71 @@ def summary(self, alpha=.05, start=None): model_name = ( '%s%s' % (self.model.__class__.__name__, order) ) - return super(StaticFactorsResults, self).summary( - alpha=alpha, start=start, model_name=model_name + summary = super(StaticFactorsResults, self).summary( + alpha=alpha, start=start, model_name=model_name, + display_params=False ) + + # Add parameter tables for each endogenous variable + k_endog = self.model.k_endog + k_factors = self.model.k_factors + factor_order = self.model.factor_order + for i in range(k_endog): + mask = [] + offset = 0 + + # 1. Loadings + offset = k_factors * i + mask.append(np.arange(offset, offset + k_factors)) + + # 2. Idiosyncratic variance + offset = k_factors * k_endog + mask.append(np.array(offset + i, ndmin=1)) + + # Create the table + mask = np.concatenate(mask) + res = (self, self.params[mask], self.bse[mask], self.zvalues[mask], + self.pvalues[mask], self.conf_int(alpha)[mask]) + + param_names = [ + '.'.join(name.split('.')[:-1]) + for name in + np.array(self.data.param_names)[mask].tolist() + ] + title = "Results for equation %s" % self.model.endog_names[i] + + table = summary_params(res, yname=None, xname=param_names, + alpha=alpha, use_t=False, title=title) + + summary.tables.append(table) + + # Add parameter tables for each factor + offset = k_endog * (k_factors + 1) + for i in range(k_factors): + mask = [] + start = i * k_factors * factor_order + end = (i + 1) * k_factors * factor_order + mask.append( + offset + np.arange(start, end)) + + # Create the table + mask = np.concatenate(mask) + res = (self, self.params[mask], self.bse[mask], self.zvalues[mask], + self.pvalues[mask], self.conf_int(alpha)[mask]) + + param_names = [ + '.'.join(name.split('.')[:-1]) + for name in + np.array(self.data.param_names)[mask].tolist() + ] + title = "Results for factor equation f%d" % (i+1) + + table = summary_params(res, yname=None, xname=param_names, + alpha=alpha, use_t=False, title=title) + + summary.tables.append(table) + + return summary summary.__doc__ = MLEResults.summary.__doc__ diff --git a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py index ddfdb520a6d..06c6d2cf11a 100644 --- a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py @@ -9,6 +9,7 @@ import numpy as np import pandas as pd import os +import re import warnings from statsmodels.datasets import webuse @@ -16,6 +17,7 @@ from .results import results_varmax, results_dynamic_factor from numpy.testing import assert_equal, assert_almost_equal, assert_raises, assert_allclose from nose.exc import SkipTest +from statsmodels.iolib.summary import forg current_path = os.path.dirname(os.path.abspath(__file__)) @@ -156,6 +158,48 @@ def test_bic(self): # (possibly they are still at starting values?) so the BIC is off pass + def test_summary(self): + summary = self.results.summary() + tables = [str(table) for table in summary.tables] + params = self.true['params'] + + # Check the model overview table + assert_equal(re.search(r'Model:.*StaticFactors\(factors=2, order=1\)', tables[0]) is None, False) + + # For each endogenous variable, check the output + for i in range(self.model.k_endog): + offset_loading = self.model.k_factors * i + offset_var = self.model.k_factors * self.model.k_endog + table = tables[i + 1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for equation %s' % name, table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 8) + + # -> Check that we have the right coefficients + assert_equal(re.search('loading.f1 +' + forg(params[offset_loading + 0], prec=4), table) is None, False) + assert_equal(re.search('loading.f2 +' + forg(params[offset_loading + 1], prec=4), table) is None, False) + assert_equal(re.search('sigma2 +' + forg(params[offset_var + i], prec=4), table) is None, False) + + # For each factor, check the output + for i in range(self.model.k_factors): + offset = self.model.k_endog * (self.model.k_factors + 1) + i * self.model.k_factors + table = tables[self.model.k_endog + i + 1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for factor equation f%d' % (i+1), table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 7) + + # -> Check that we have the right coefficients + assert_equal(re.search('L1.f1 +' + forg(params[offset + 0], prec=4), table) is None, False) + assert_equal(re.search('L1.f2 +' + forg(params[offset + 1], prec=4), table) is None, False) + def test_misspecification(): # Tests for model specification and misspecification exceptions From 65085cddc67040ed25ebf96a94497013d3ee6fa2 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 14:36:49 -0700 Subject: [PATCH 47/57] BUG: Invalid reshape with Pandas params. --- statsmodels/tsa/statespace/dynamic_factor.py | 2 +- statsmodels/tsa/statespace/tests/test_dynamic_factor.py | 2 +- statsmodels/tsa/statespace/tests/test_varmax.py | 4 ++-- statsmodels/tsa/statespace/varmax.py | 4 ++-- 4 files changed, 6 insertions(+), 6 deletions(-) diff --git a/statsmodels/tsa/statespace/dynamic_factor.py b/statsmodels/tsa/statespace/dynamic_factor.py index 03df0ba76fc..f21035092c8 100644 --- a/statsmodels/tsa/statespace/dynamic_factor.py +++ b/statsmodels/tsa/statespace/dynamic_factor.py @@ -421,7 +421,7 @@ def __init__(self, model, params, filter_results, cov_type='opg', # Polynomials / coefficient matrices self.coefficient_matrices_var = None if self.model.factor_order > 0: - ar_params = self.params[self.model._params_transition] + ar_params = np.array(self.params[self.model._params_transition]) k_factors = self.model.k_factors factor_order = self.model.factor_order self.coefficient_matrices_var = ( diff --git a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py index 06c6d2cf11a..43b1987e813 100644 --- a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py @@ -68,7 +68,7 @@ def test_results(self): # Test cofficient matrix creation (via a different, more direct, method) if self.model.factor_order > 0: - coefficients = self.results.params[self.model._params_transition].reshape(self.model.k_factors, self.model.k_factors * self.model.factor_order) + coefficients = np.array(self.results.params[self.model._params_transition]).reshape(self.model.k_factors, self.model.k_factors * self.model.factor_order) coefficient_matrices = np.array([ coefficients[:self.model.k_factors, i*self.model.k_factors:(i+1)*self.model.k_factors] for i in range(self.model.factor_order) diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index db8bf784a6f..edf48d7b7fa 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -73,7 +73,7 @@ def test_results(self): # Test cofficient matrix creation (via a different, more direct, method) if self.model.k_ar > 0: - coefficients = self.results.params[self.model._params_ar].reshape(self.model.k_endog, self.model.k_endog * self.model.k_ar) + coefficients = np.array(self.results.params[self.model._params_ar]).reshape(self.model.k_endog, self.model.k_endog * self.model.k_ar) coefficient_matrices = np.array([ coefficients[:self.model.k_endog, i*self.model.k_endog:(i+1)*self.model.k_endog] for i in range(self.model.k_ar) @@ -82,7 +82,7 @@ def test_results(self): else: assert_equal(self.results.coefficient_matrices_var, None) if self.model.k_ma > 0: - coefficients = self.results.params[self.model._params_ma].reshape(self.model.k_endog, self.model.k_endog * self.model.k_ma) + coefficients = np.array(self.results.params[self.model._params_ma]).reshape(self.model.k_endog, self.model.k_endog * self.model.k_ma) coefficient_matrices = np.array([ coefficients[:self.model.k_endog, i*self.model.k_endog:(i+1)*self.model.k_endog] for i in range(self.model.k_ma) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index 6cd77346adc..0d0f0d124a6 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -639,14 +639,14 @@ def __init__(self, model, params, filter_results, cov_type='opg', self.coefficient_matrices_var = None self.coefficient_matrices_vma = None if self.model.k_ar > 0: - ar_params = self.params[self.model._params_ar] + ar_params = np.array(self.params[self.model._params_ar]) k_endog = self.model.k_endog k_ar = self.model.k_ar self.coefficient_matrices_var = ( ar_params.reshape(k_endog * k_ar, k_endog).T ).reshape(k_endog, k_endog, k_ar).T if self.model.k_ma > 0: - ma_params = self.params[self.model._params_ma] + ma_params = np.array(self.params[self.model._params_ma]) k_endog = self.model.k_endog k_ma = self.model.k_ma self.coefficient_matrices_vma = ( From c2de4d39c32e0a699cdc5bc5254476fb2777cbfa Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Mon, 10 Aug 2015 15:19:35 -0700 Subject: [PATCH 48/57] DOC: Improve class and method documentation. --- statsmodels/tsa/statespace/dynamic_factor.py | 19 ++++--- statsmodels/tsa/statespace/varmax.py | 54 ++++++++++++++++++++ 2 files changed, 63 insertions(+), 10 deletions(-) diff --git a/statsmodels/tsa/statespace/dynamic_factor.py b/statsmodels/tsa/statespace/dynamic_factor.py index f21035092c8..942d22fce67 100644 --- a/statsmodels/tsa/statespace/dynamic_factor.py +++ b/statsmodels/tsa/statespace/dynamic_factor.py @@ -50,6 +50,13 @@ class StaticFactors(MLEModel): Attributes ---------- + k_factors : int + The number of unobserved factors. + factor_order : int + The order of the vector autoregression followed by the factors. + enforce_stationarity : boolean, optional + Whether or not to transform the AR parameters to enforce stationarity + in the autoregressive component of the model. Default is True. Notes ----- @@ -159,6 +166,7 @@ def filter(self, params, transformed=True, cov_type=None, return_ssm=False, ) return result + filter.__doc__ = MLEModel.filter.__doc__ @property def start_params(self): @@ -333,7 +341,7 @@ def update(self, params, transformed=True): ----- Let `n = k_endog`, `m = k_factors`, and `p = factor_order`. Then the `params` vector has length - :math:`[n \times m] + [n] + [m^2 \times p] + [m \times (m + 1) / 2]`. + :math:`[n \times m] + [n] + [m^2 \times p]`. It is expanded in the following way: - The first :math:`n \times m` parameters fill out the factor loading @@ -351,12 +359,6 @@ def update(self, params, transformed=True): we assume that the first :math:`m^2` parameters fill the first coefficient matrix (starting at [0,0] and filling along rows), the second :math:`m^2` parameters fill the second matrix, etc. - - The last :math:`m \times (m + 1) / 2` parameters are used to fill in - a lower-triangular matrix, which is multipled with its transpose to - create a positive definite variance / covariance matrix for the - factor's VAR. The are not transformed in `transform_params` because - the matrix multiplication procedure ensures the variance terms are - positive and the matrix is positive definite. """ params = super(StaticFactors, self).update(params, transformed) @@ -392,9 +394,6 @@ class StaticFactorsResults(MLEResults): coefficient_matrices_var : array Array containing autoregressive lag polynomial coefficient matrices, ordered from lowest degree to highest. - coefficient_matrices_vma : array - Array containing moving average lag polynomial coefficients, - ordered from lowest degree to highest. See Also -------- diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index 0d0f0d124a6..7c12ceb7c66 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -61,6 +61,59 @@ class VARMAX(MLEModel): Keyword arguments may be used to provide default values for state space matrices or for Kalman filtering options. See `Representation`, and `KalmanFilter` for more details. + + Attributes + ---------- + order : iterable + The (p,q) order of the model for the number of AR and MA parameters to + use. + trend : {'nc', 'c'}, optional + Parameter controlling the deterministic trend polynomial. + Can be specified as a string where 'c' indicates a constant intercept + and 'nc' indicates no intercept term. + error_cov_type : {'diagonal', 'unstructured'}, optional + The structure of the covariance matrix of the error term, where + "unstructured" puts no restrictions on the matrix and "diagonal" + requires it to be a diagonal matrix (uncorrelated errors). Default is + "unstructured". + measurement_error : boolean, optional + Whether or not to assume the endogenous observations `endog` were + measured with error. Default is False. + enforce_stationarity : boolean, optional + Whether or not to transform the AR parameters to enforce stationarity + in the autoregressive component of the model. Default is True. + enforce_invertibility : boolean, optional + Whether or not to transform the MA parameters to enforce invertibility + in the moving average component of the model. Default is True. + + Notes + ----- + Generically, the VARMAX model is specified (see for example chapter 18 of + [1]_): + + .. math:: + + y_t = \nu + A_1 y_{t-1} + \dots + A_p y_{t-p} + B x_t + \epsilon_t + + M_1 \epsilon_{t-1} + \dots M_q \epsilon_{t-q} + + where :math:`\epsilon_t \sim N(0, \Omega)`, and where :math:`y_t` is a + `k_endog x 1` vector. Additionally, this model allows considering the case + where the variables are measured with error. + + Note that in the full VARMA(p,q) case there is a fundamental identification + problem in that the coefficient matrices :math:`\{A_i, M_j\}` are not + generally unique, meaning that for a given time series process there may + be multiple sets of matrices that equivalently represent it. See Chapter 12 + of [1]_ for more informationl. Although this class can be used to estimate + VARMA(p,q) models, a warning is issued to remind users that no steps have + been taken to ensure identification in this case. + + References + ---------- + .. [1] Lutkepohl, Helmut. 2007. + New Introduction to Multiple Time Series Analysis. + Berlin: Springer. + """ def __init__(self, endog, exog=None, order=(1, 0), trend='c', @@ -251,6 +304,7 @@ def filter(self, params, transformed=True, cov_type=None, return_ssm=False, ) return result + filter.__doc__ = MLEModel.filter.__doc__ @property def start_params(self): From 1f66650734a08c100490cd1ebe6f292e3c8ddbed Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 12 Aug 2015 10:58:54 -0700 Subject: [PATCH 49/57] ENH: Option whether to separate params in summary. --- statsmodels/tsa/statespace/varmax.py | 130 ++++++++++++++------------- 1 file changed, 66 insertions(+), 64 deletions(-) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index 7c12ceb7c66..8872862ed1d 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -168,7 +168,7 @@ def __init__(self, endog, exog=None, order=(1, 0), trend='c', self.k_exog = exog.shape[1] # Note: at some point in the future might add state regression, as in - # VARMAX. + # SARIMAX. self.mle_regression = self.k_exog > 0 # We need to have an array or pandas at this point @@ -830,7 +830,7 @@ def forecast(self, steps=1, exog=None, **kwargs): """ return super(VARMAXResults, self).forecast(steps, exog=exog, **kwargs) - def summary(self, alpha=.05, start=None): + def summary(self, alpha=.05, start=None, separate_params=True): from statsmodels.iolib.summary import summary_params # Create the model name @@ -856,78 +856,80 @@ def summary(self, alpha=.05, start=None): summary = super(VARMAXResults, self).summary( alpha=alpha, start=start, model_name=model_name, - display_params=False + display_params=not separate_params ) - # Add parameter tables for each endogenous variable - k_endog = self.model.k_endog - k_ar = self.model.k_ar - k_ma = self.model.k_ma - k_exog = self.model.k_exog - for i in range(k_endog): - mask = [] - offset = 0 - - # 1. Intercept terms - if self.model.trend == 'c': - mask.append(np.array(i, ndmin=1)) - offset += k_endog - - # 2. AR terms - if k_ar > 0: - start = i * k_endog * k_ar - end = (i + 1) * k_endog * k_ar - mask.append( - offset + np.arange(start, end)) - offset += k_ar * k_endog**2 - - # 3. MA terms - if k_ma > 0: - start = i * k_endog * k_ma - end = (i + 1) * k_endog * k_ma - mask.append( - offset + np.arange(start, end)) - offset += k_ma * k_endog**2 - - # 4. Regression terms - if k_exog > 0: - mask.append( - offset + np.arange(i * k_exog, (i + 1) * k_exog)) - offset += k_endog * k_exog - - # 5. Measurement error variance terms - if self.model.measurement_error: - mask.append(np.array(self.model.k_params - i - 1, ndmin=1)) - - # Create the table - mask = np.concatenate(mask) + if separate_params: + # Add parameter tables for each endogenous variable + k_endog = self.model.k_endog + k_ar = self.model.k_ar + k_ma = self.model.k_ma + k_exog = self.model.k_exog + for i in range(k_endog): + mask = [] + offset = 0 + + # 1. Intercept terms + if self.model.trend == 'c': + mask.append(np.array(i, ndmin=1)) + offset += k_endog + + # 2. AR terms + if k_ar > 0: + start = i * k_endog * k_ar + end = (i + 1) * k_endog * k_ar + mask.append( + offset + np.arange(start, end)) + offset += k_ar * k_endog**2 + + # 3. MA terms + if k_ma > 0: + start = i * k_endog * k_ma + end = (i + 1) * k_endog * k_ma + mask.append( + offset + np.arange(start, end)) + offset += k_ma * k_endog**2 + + # 4. Regression terms + if k_exog > 0: + mask.append( + offset + np.arange(i * k_exog, (i + 1) * k_exog)) + offset += k_endog * k_exog + + # 5. Measurement error variance terms + if self.model.measurement_error: + mask.append(np.array(self.model.k_params - i - 1, ndmin=1)) + + # Create the table + mask = np.concatenate(mask) + res = (self, self.params[mask], self.bse[mask], + self.zvalues[mask], self.pvalues[mask], + self.conf_int(alpha)[mask]) + + param_names = [ + '.'.join(name.split('.')[:-1]) + for name in + np.array(self.data.param_names)[mask].tolist() + ] + title = "Results for equation %s" % self.model.endog_names[i] + + table = summary_params(res, yname=None, xname=param_names, + alpha=alpha, use_t=False, title=title) + + summary.tables.append(table) + + # State covariance terms + mask = self.model._params_state_cov res = (self, self.params[mask], self.bse[mask], self.zvalues[mask], self.pvalues[mask], self.conf_int(alpha)[mask]) - param_names = [ - '.'.join(name.split('.')[:-1]) - for name in - np.array(self.data.param_names)[mask].tolist() - ] - title = "Results for equation %s" % self.model.endog_names[i] + param_names = np.array(self.data.param_names)[mask].tolist() + title = "Error covariance matrix" table = summary_params(res, yname=None, xname=param_names, alpha=alpha, use_t=False, title=title) - summary.tables.append(table) - # State covariance terms - mask = self.model._params_state_cov - res = (self, self.params[mask], self.bse[mask], self.zvalues[mask], - self.pvalues[mask], self.conf_int(alpha)[mask]) - - param_names = np.array(self.data.param_names)[mask].tolist() - title = "Error covariance matrix" - - table = summary_params(res, yname=None, xname=param_names, - alpha=alpha, use_t=False, title=title) - summary.tables.append(table) - return summary summary.__doc__ = MLEResults.summary.__doc__ From f731d2ed6178b1d61a84371ad165e836701dce1e Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 12 Aug 2015 11:11:46 -0700 Subject: [PATCH 50/57] ENH: Accommodate addl parameters in VARMAX summary Useful when the model is subclassed and separate_params=True. --- statsmodels/tsa/statespace/varmax.py | 71 +++++++++++++++++----------- 1 file changed, 43 insertions(+), 28 deletions(-) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index 8872862ed1d..a59a0929bb6 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -860,25 +860,42 @@ def summary(self, alpha=.05, start=None, separate_params=True): ) if separate_params: + indices = np.arange(len(self.params)) + + def make_table(self, mask, title, strip_end=True): + res = (self, self.params[mask], self.bse[mask], + self.zvalues[mask], self.pvalues[mask], + self.conf_int(alpha)[mask]) + + param_names = [ + '.'.join(name.split('.')[:-1]) if strip_end else name + for name in + np.array(self.data.param_names)[mask].tolist() + ] + + return summary_params(res, yname=None, xname=param_names, + alpha=alpha, use_t=False, title=title) + # Add parameter tables for each endogenous variable k_endog = self.model.k_endog k_ar = self.model.k_ar k_ma = self.model.k_ma k_exog = self.model.k_exog + endog_masks = [] for i in range(k_endog): - mask = [] + masks = [] offset = 0 # 1. Intercept terms if self.model.trend == 'c': - mask.append(np.array(i, ndmin=1)) + masks.append(np.array(i, ndmin=1)) offset += k_endog # 2. AR terms if k_ar > 0: start = i * k_endog * k_ar end = (i + 1) * k_endog * k_ar - mask.append( + masks.append( offset + np.arange(start, end)) offset += k_ar * k_endog**2 @@ -886,50 +903,48 @@ def summary(self, alpha=.05, start=None, separate_params=True): if k_ma > 0: start = i * k_endog * k_ma end = (i + 1) * k_endog * k_ma - mask.append( + masks.append( offset + np.arange(start, end)) offset += k_ma * k_endog**2 # 4. Regression terms if k_exog > 0: - mask.append( + masks.append( offset + np.arange(i * k_exog, (i + 1) * k_exog)) offset += k_endog * k_exog # 5. Measurement error variance terms if self.model.measurement_error: - mask.append(np.array(self.model.k_params - i - 1, ndmin=1)) + masks.append(np.array(self.model.k_params - i - 1, ndmin=1)) # Create the table - mask = np.concatenate(mask) - res = (self, self.params[mask], self.bse[mask], - self.zvalues[mask], self.pvalues[mask], - self.conf_int(alpha)[mask]) + mask = np.concatenate(masks) + endog_masks.append(mask) - param_names = [ - '.'.join(name.split('.')[:-1]) - for name in - np.array(self.data.param_names)[mask].tolist() - ] title = "Results for equation %s" % self.model.endog_names[i] - - table = summary_params(res, yname=None, xname=param_names, - alpha=alpha, use_t=False, title=title) - + table = make_table(self, mask, title) summary.tables.append(table) # State covariance terms - mask = self.model._params_state_cov - res = (self, self.params[mask], self.bse[mask], self.zvalues[mask], - self.pvalues[mask], self.conf_int(alpha)[mask]) - - param_names = np.array(self.data.param_names)[mask].tolist() - title = "Error covariance matrix" - - table = summary_params(res, yname=None, xname=param_names, - alpha=alpha, use_t=False, title=title) + state_cov_mask = ( + np.arange(len(self.params))[self.model._params_state_cov]) + table = make_table(self, state_cov_mask, "Error covariance matrix", + strip_end=False) summary.tables.append(table) + # Add a table for all other parameters + masks = [] + for m in (endog_masks, [state_cov_mask]): + m = np.array(m).flatten() + if len(m) > 0: + masks.append(m) + masks = np.concatenate(masks) + inverse_mask = np.array(list(set(indices).difference(set(masks)))) + if len(inverse_mask) > 0: + table = make_table(self, inverse_mask, "Other parameters", + strip_end=False) + summary.tables.append(table) + return summary summary.__doc__ = MLEResults.summary.__doc__ From 5f9dce15d1677e0d2e11161ada82be0f24c315b6 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Wed, 12 Aug 2015 13:26:24 -0700 Subject: [PATCH 51/57] BUG: Fix when multiple exog in VARMAX. --- .../tests/results/results_var_stata.csv | 186 +++++++++--------- .../tests/results/results_varmax.py | 24 +++ .../tests/results/test_varmax_stata.do | 26 ++- .../tsa/statespace/tests/test_varmax.py | 45 ++++- 4 files changed, 181 insertions(+), 100 deletions(-) diff --git a/statsmodels/tsa/statespace/tests/results/results_var_stata.csv b/statsmodels/tsa/statespace/tests/results/results_var_stata.csv index b5dcdcc8e57..0b07600fb61 100644 --- a/statsmodels/tsa/statespace/tests/results/results_var_stata.csv +++ b/statsmodels/tsa/statespace/tests/results/results_var_stata.csv @@ -1,93 +1,93 @@ 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-.2552236, .21722691, .81525457, # Phi, row 1 + .02998355, -.08130972, .24772266, # Phi, row 2 + -.00476998, .24016112, -.19910237, # Phi, row 3 + .00811096, -.00015244, # exog, y1 + .01878355, -.00005086, # exog, y2 + .01889825, 2.577e-06, # exog, y3 + # .00199918, # Covariance, lower triangle + # .00005435, .00013469, + # .00012306, .00006251, .00010039 + # Note: the following are the Cholesky of the covariance + # matrix defined just above + .04471219, # Cholesky, lower triangle + .00121555, .01102644, + .00275227, .00536569, .00800152 + ], + 'var_oim': None, + # 'loglike': 600.9801664685759, # From Stata + 'loglike': 600.65449034396283, # From VARMAX (regression test) + 'aic': None, + 'bic': None +} + lutkepohl_var2 = { 'params': [ -.25244981, .62528114, # Phi_1, row 1 diff --git a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do index 614226c01f5..0758302fd0f 100644 --- a/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do +++ b/statsmodels/tsa/statespace/tests/results/test_varmax_stata.do @@ -45,21 +45,37 @@ predict dyn_predict_int1, dynamic(tq(1961q1)) equation(dln_inv) predict dyn_predict_int2, dynamic(tq(1961q1)) equation(dln_inc) predict dyn_predict_int3, dynamic(tq(1961q1)) equation(dln_consump) -// VAR(1), diagonal covariance + exog +// VAR(1), diagonal covariance + 1 exog gen t = _n //dfactor (dln_inv dln_inc dln_consump = t, ar(1) arstructure(general) noconstant covstructure(diagonal)) if qtr<=tq(1978q4) var dln_inv dln_inc dln_consump if qtr<=tq(1978q4), lags(1) noconstant exog(t) // predict, see above (Note: uses actual data for forecasting, so we will want // to ignore the predictions after 1978q4, see below -predict predict_exog1, equation(dln_inv) -predict predict_exog2, equation(dln_inc) -predict predict_exog3, equation(dln_consump) +predict predict_exog1_1, equation(dln_inv) +predict predict_exog1_2, equation(dln_inc) +predict predict_exog1_3, equation(dln_consump) // We will want to use these values to compare for forecasting, but note that // this also includes in the columns the value for 1978q4 (i.e. a VAR(1) needs // 1 sample from which to compute forecasts. -fcast compute fcast_exog_ , dynamic(tq(1979q1)) step(16) replace +fcast compute fcast_exog1_ , dynamic(tq(1979q1)) step(16) replace + +// VAR(1), diagonal covariance + 2 exog +gen c = 1 +//dfactor (dln_inv dln_inc dln_consump = t, ar(1) arstructure(general) noconstant covstructure(diagonal)) if qtr<=tq(1978q4) +var dln_inv dln_inc dln_consump if qtr<=tq(1978q4), lags(1) noconstant exog(c t) + +// predict, see above (Note: uses actual data for forecasting, so we will want +// to ignore the predictions after 1978q4, see below +predict predict_exog2_1, equation(dln_inv) +predict predict_exog2_2, equation(dln_inc) +predict predict_exog2_3, equation(dln_consump) + +// We will want to use these values to compare for forecasting, but note that +// this also includes in the columns the value for 1978q4 (i.e. a VAR(1) needs +// 1 sample from which to compute forecasts. +fcast compute fcast_exog2_ , dynamic(tq(1979q1)) step(16) replace // VAR(2) dfactor (dln_inv dln_inc = , ar(1/2) arstructure(general) noconstant covstructure(unstructured)) if qtr<=tq(1978q4) diff --git a/statsmodels/tsa/statespace/tests/test_varmax.py b/statsmodels/tsa/statespace/tests/test_varmax.py index edf48d7b7fa..0eb77115320 100644 --- a/statsmodels/tsa/statespace/tests/test_varmax.py +++ b/statsmodels/tsa/statespace/tests/test_varmax.py @@ -357,9 +357,9 @@ class TestVAR_exog(CheckLutkepohl): # var function rather than the Stata dfactor function def __init__(self): true = results_varmax.lutkepohl_var1_exog.copy() - true['predict'] = var_results.ix[1:75, ['predict_exog1', 'predict_exog2', 'predict_exog3']] + true['predict'] = var_results.ix[1:75, ['predict_exog1_1', 'predict_exog1_2', 'predict_exog1_3']] true['predict'].iloc[0, :] = 0 - true['fcast'] = var_results.ix[76:, ['fcast_exog_dln_inv', 'fcast_exog_dln_inc', 'fcast_exog_dln_consump']] + true['fcast'] = var_results.ix[76:, ['fcast_exog1_dln_inv', 'fcast_exog1_dln_inc', 'fcast_exog1_dln_consump']] exog = np.arange(75) + 3 super(TestVAR_exog, self).__init__( true, order=(1,0), trend='nc', error_cov_type='unstructured', @@ -438,6 +438,47 @@ def test_summary(self): for i in range(len(names)): assert_equal(re.search('%s +%.4f' % (names[i], params[i]), table) is None, False) +class TestVAR_exog2(CheckLutkepohl): + # This is a regression test, to make sure that the setup with multiple exog + # works correctly. The params are from Stata, but the loglike is from + # this model. Likely the small discrepancy (see the results file) is from + # the approximate diffuse initialization. + def __init__(self): + true = results_varmax.lutkepohl_var1_exog2.copy() + true['predict'] = var_results.ix[1:75, ['predict_exog2_1', 'predict_exog2_2', 'predict_exog2_3']] + true['predict'].iloc[0, :] = 0 + true['fcast'] = var_results.ix[76:, ['fcast_exog2_dln_inv', 'fcast_exog2_dln_inc', 'fcast_exog2_dln_consump']] + exog = np.c_[np.ones((75,1)), (np.arange(75) + 3)[:, np.newaxis]] + super(TestVAR_exog2, self).__init__( + true, order=(1,0), trend='nc', error_cov_type='unstructured', + exog=exog, initialization='approximate_diffuse', loglikelihood_burn=1) + + def test_mle(self): + pass + + def test_aic(self): + pass + + def test_bic(self): + pass + + def test_bse_oim(self): + pass + + def test_predict(self): + super(CheckLutkepohl, self).test_predict(end='1978-10-01', atol=1e-3) + + def test_dynamic_predict(self): + # Stata's var cannot subsequently use dynamic + pass + + def test_forecast(self): + # Tests forecast + exog = np.c_[np.ones((16, 1)), (np.arange(75, 75+16) + 3)[:, np.newaxis]] + + desired = self.results.forecast(steps=16, exog=exog) + assert_allclose(desired, self.true['fcast'].T, atol=1e-6) + class TestVAR2(CheckLutkepohl): def __init__(self): From cbb1eb94838b0a9d16bbe1fe49312077ddeaaebb Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sat, 15 Aug 2015 17:08:41 -0700 Subject: [PATCH 52/57] ENH: Improve factor model features. - Dynamic factors - Static factors - SUR - (vector) autoregressive errors --- statsmodels/tsa/api.py | 2 +- statsmodels/tsa/statespace/dynamic_factor.py | 1255 ++++++++++++++--- .../tests/results/results_dynamic_factor.py | 150 ++ .../results/results_dynamic_factor_stata.csv | 186 +-- .../results/test_dynamic_factor_stata.do | 113 +- .../statespace/tests/test_dynamic_factor.py | 369 ++++- 6 files changed, 1776 insertions(+), 299 deletions(-) diff --git a/statsmodels/tsa/api.py b/statsmodels/tsa/api.py index cff531f6f7f..310e877f273 100644 --- a/statsmodels/tsa/api.py +++ b/statsmodels/tsa/api.py @@ -19,4 +19,4 @@ from .statespace import api as statespace from .statespace.sarimax import SARIMAX from .statespace.varmax import VARMAX -from .statespace.dynamic_factor import StaticFactors \ No newline at end of file +from .statespace.dynamic_factor import DynamicFactor \ No newline at end of file diff --git a/statsmodels/tsa/statespace/dynamic_factor.py b/statsmodels/tsa/statespace/dynamic_factor.py index 942d22fce67..faf258d6fb7 100644 --- a/statsmodels/tsa/statespace/dynamic_factor.py +++ b/statsmodels/tsa/statespace/dynamic_factor.py @@ -15,11 +15,12 @@ from .mlemodel import MLEModel, MLEResults, MLEResultsWrapper from .tools import ( companion_matrix, diff, is_invertible, + constrain_stationary_univariate, unconstrain_stationary_univariate, constrain_stationary_multivariate, unconstrain_stationary_multivariate ) from scipy.linalg import solve_discrete_lyapunov from statsmodels.tools.pca import PCA -from statsmodels.tsa.arima_model import ARMA +from statsmodels.regression.linear_model import OLS from statsmodels.tsa.vector_ar.var_model import VAR from statsmodels.tools.tools import Bunch from statsmodels.tools.data import _is_using_pandas @@ -28,18 +29,34 @@ import statsmodels.base.wrapper as wrap -class StaticFactors(MLEModel): +class DynamicFactor(MLEModel): r""" - Static form of the dynamic factor model + Dynamic factor model Parameters ---------- endog : array_like The observed time-series process :math:`y` + exog : array_like, optional + Array of exogenous regressors for the observation equation, shaped + nobs x k_exog. k_factors : int The number of unobserved factors. factor_order : int The order of the vector autoregression followed by the factors. + error_cov_type : {'scalar', 'diagonal', 'unstructured'}, optional + The structure of the covariance matrix of the observation error term, + where "unstructured" puts no restrictions on the matrix, "diagonal" + requires it to be any diagonal matrix (uncorrelated errors), and + "scalar" requires it to be a scalar times the identity matrix. Default + is "diagonal". + error_order : int, optional + The order of the vector autoregression followed by the observation + error component. Default is None, corresponding to white noise errors. + error_var : boolean, optional + Whether or not to model the errors jointly via a vector autoregression, + rather than as individual autoregressions. Has no effect unless + `error_order` is set. Default is False. enforce_stationarity : boolean, optional Whether or not to transform the AR parameters to enforce stationarity in the autoregressive component of the model. Default is True. @@ -50,87 +67,171 @@ class StaticFactors(MLEModel): Attributes ---------- + exog : array_like, optional + Array of exogenous regressors for the observation equation, shaped + nobs x k_exog. k_factors : int The number of unobserved factors. factor_order : int The order of the vector autoregression followed by the factors. + error_cov_type : {'diagonal', 'unstructured'} + The structure of the covariance matrix of the error term, where + "unstructured" puts no restrictions on the matrix and "diagonal" + requires it to be a diagonal matrix (uncorrelated errors). + error_order : int + The order of the vector autoregression followed by the observation + error component. + error_var : boolean + Whether or not to model the errors jointly via a vector autoregression, + rather than as individual autoregressions. Has no effect unless + `error_order` is set. enforce_stationarity : boolean, optional Whether or not to transform the AR parameters to enforce stationarity in the autoregressive component of the model. Default is True. Notes ----- - The static form of the dynamic factor model is specified: + The dynamic factor model considered here is in the so-called static form, + and is specified: .. math:: - X_t & = \Lambda F_t + \varepsilon_t \\ - F_t & = \Psi(L) F_{t-1} + \eta_t - - where there are :math:`N` observed series and :math:`q` unobserved factors - so that :math:`X_t, \varepsilon_t` are :math:`(N x 1)` and - :math:`F_t, \eta_t` are :math:`(q x 1)`. - - We assume that :math:`\varepsilon_{it} \sim N(0, \sigma_{\varepsilon_i}^2)` - and :math:`\eta_t \sim N(0, I)` + y_t & = \Lambda f_t + B x_t + u_t \\ + f_t & = A_1 f_{t-1} + \dots + A_p f_{t-p} + \eta_t \\ + u_t & = C_1 u_{t-1} + \dots + C_1 f_{t-q} + \varepsilon_t + + where there are `k_endog` observed series and `k_factors` unobserved + factors. Thus :math:`y_t` is a `k_endog` x 1 vector and :math:`f_t` is a + `k_factors` x 1 vector. + + :math:`x_t` are optional exogenous vectors, shaped `k_exog` x 1. + + :math:`\eta_t` and :math:`\varepsilon_t` are white noise error terms. In + order to identify the factors, :math:`Var(\eta_t) = I`. Denote + :math:`Var(\varepsilon_t) \equiv \Sigma`. + + Options related to the unobserved factors: + + - `k_factors`: this is the dimension of the vector :math:`f_t`, above. + To exclude factors completely, set `k_factors = 0`. + - `factor_order`: this is the number of lags to include in the factor + evolution equation, and corresponds to :math:`p`, above. To have static + factors, set `factor_order = 0`. + + Options related to the observation error term :math:`u_t`: + + - `error_order`: the number of lags to include in the error evolution + equation; corresponds to :math:`q`, above. To have white noise errors, + set `error_order = 0` (this is the default). + - `error_cov_type`: this controls the form of the covariance matrix + :math:`\Sigma`. If it is "dscalar", then :math:`\Sigma = \sigma^2 I`. If + it is "diagonal", then + :math:`\Sigma = \text{diag}(\sigma_1^2, \dots, \sigma_n^2)`. If it is + "unstructured", then :math:`\Sigma` is any valid variance / covariance + matrix (i.e. symmetric and positive definite). + - `error_var`: this controls whether or not the errors evolve jointly + according to a VAR(q), or individually according to separate AR(q) + processes. In terms of the formulation above, if `error_var = False`, + then the matrices :math:C_i` are diagonal, otherwise they are general + VAR matrices. References ---------- - .. [1] Durbin, James, and Siem Jan Koopman. 2012. - Time Series Analysis by State Space Methods: Second Edition. - Oxford University Press. - .. [2] Lutkepohl, Helmut. 2007. + .. [1] Lutkepohl, Helmut. 2007. New Introduction to Multiple Time Series Analysis. Berlin: Springer. """ - def __init__(self, endog, k_factors, factor_order, + def __init__(self, endog, k_factors, factor_order, exog=None, + error_order=0, error_var=False, error_cov_type='diagonal', enforce_stationarity=True, **kwargs): - # Model parameters - self.k_factors = k_factors + # Model properties self.enforce_stationarity = enforce_stationarity - # Save given orders + # Factor-related properties + self.k_factors = k_factors self.factor_order = factor_order + # Error-related properties + self.error_order = error_order + self.error_var = error_var and error_order > 0 + self.error_cov_type = error_cov_type + + # Exogenous data + self.k_exog = 0 + if exog is not None: + exog_is_using_pandas = _is_using_pandas(exog, None) + if not exog_is_using_pandas: + exog = np.asarray(exog) + + # Make sure we have 2-dimensional array + if exog.ndim == 1: + if not exog_is_using_pandas: + exog = exog[:, None] + else: + exog = pd.DataFrame(exog) + + self.k_exog = exog.shape[1] + + # Note: at some point in the future might add state regression, as in + # SARIMAX. + self.mle_regression = self.k_exog > 0 + + # We need to have an array or pandas at this point + if not _is_using_pandas(endog, None): + endog = np.asanyarray(endog) + + # Save some useful model orders, internally used + k_endog = endog.shape[1] if endog.ndim > 1 else 1 + self._factor_order = max(1, self.factor_order) * self.k_factors + self._error_order = self.error_order * k_endog + # Calculate the number of states - k_states = self.factor_order * self.k_factors + k_states = self._factor_order k_posdef = self.k_factors + if self.error_order > 0: + k_states += self._error_order + k_posdef += k_endog - # By default, initialize as stationary - kwargs.setdefault('initialization', 'stationary') + if k_states == 0: + k_states = 1 + k_posdef = 1 - # Initialize the state space model - super(StaticFactors, self).__init__( - endog, k_states=k_states, k_posdef=k_posdef, **kwargs - ) + # Test for non-multivariate endog + if k_endog < 2: + raise ValueError('The dynamic factors model is only valid for' + ' multivariate time series.') # Test for too many factors - if self.k_factors >= self.k_endog: + if self.k_factors >= k_endog: raise ValueError('Number of factors must be less than the number' ' of endogenous variables.') - # Initialize the parameters - self.parameters = OrderedDict() - self.parameters['factor_loadings'] = self.k_endog * self.k_factors - self.parameters['idiosyncratic'] = self.k_endog - self.parameters['transition'] = self.k_factors**2 * self.factor_order - self.k_params = sum(self.parameters.values()) + # Test for invalid error_cov_type + if self.error_cov_type not in ['scalar', 'diagonal', 'unstructured']: + raise ValueError('Invalid error covariance matrix type' + ' specification.') - # Setup fixed components of state space matrices - self.ssm['selection', :k_posdef, :k_posdef] = np.eye(k_posdef) - self.ssm['transition', k_factors:, :k_factors * (factor_order - 1)] = ( - np.eye(k_factors * (factor_order - 1))) - self.ssm['state_cov'] = np.eye(self.k_factors) + # By default, initialize as stationary + kwargs.setdefault('initialization', 'stationary') - # Setup indices of state space matrices - idx = np.diag_indices(self.k_endog) - self._idx_obs_cov = ('obs_cov', idx[0], idx[1]) - self._idx_transition = np.s_['transition', :self.k_factors, :] + # Initialize the state space model + super(DynamicFactor, self).__init__( + endog, exog=exog, k_states=k_states, k_posdef=k_posdef, **kwargs + ) - # Cache some slices + # Initialize the components + self.parameters = OrderedDict() + self._initialize_loadings() + self._initialize_exog() + self._initialize_error_cov() + self._initialize_factor_transition() + self._initialize_error_transition() + self.k_params = sum(self.parameters.values()) + + # Cache parameter vector slices def _slice(key, offset): length = self.parameters[key] param_slice = np.s_[offset:offset + length] @@ -139,8 +240,200 @@ def _slice(key, offset): offset = 0 self._params_loadings, offset = _slice('factor_loadings', offset) - self._params_idiosyncratic, offset = _slice('idiosyncratic', offset) - self._params_transition, offset = _slice('transition', offset) + self._params_exog, offset = _slice('exog', offset) + self._params_error_cov, offset = _slice('error_cov', offset) + self._params_factor_transition, offset = ( + _slice('factor_transition', offset)) + self._params_error_transition, offset = ( + _slice('error_transition', offset)) + + def _initialize_loadings(self): + # Initialize the parameters + self.parameters['factor_loadings'] = self.k_endog * self.k_factors + + # Setup fixed components of state space matrices + if self.error_order > 0: + start = self._factor_order + end = self._factor_order + self.k_endog + self.ssm['design', :, start:end] = np.eye(self.k_endog) + + # Setup indices of state space matrices + self._idx_loadings = np.s_['design', :, :self.k_factors] + + def _initialize_exog(self): + # Initialize the parameters + self.parameters['exog'] = self.k_exog * self.k_endog + + # If we have exog effects, then the obs intercept needs to be + # time-varying + if self.k_exog > 0: + self.ssm['obs_intercept'] = np.zeros((self.k_endog, self.nobs)) + + # Setup indices of state space matrices + self._idx_exog = np.s_['obs_intercept', :self.k_endog, :] + + def _initialize_error_cov(self): + if self.error_cov_type == 'scalar': + self._initialize_error_cov_diagonal(scalar=True) + elif self.error_cov_type == 'diagonal': + self._initialize_error_cov_diagonal(scalar=False) + elif self.error_cov_type == 'unstructured': + self._initialize_error_cov_unstructured() + + def _initialize_error_cov_diagonal(self, scalar=False): + # Initialize the parameters + self.parameters['error_cov'] = 1 if scalar else self.k_endog + + # Setup fixed components of state space matrices + + # Setup indices of state space matrices + k_endog = self.k_endog + k_factors = self.k_factors + idx = np.diag_indices(k_endog) + if self.error_order > 0: + matrix = 'state_cov' + idx = (idx[0] + k_factors, idx[1] + k_factors) + else: + matrix = 'obs_cov' + self._idx_error_cov = (matrix,) + idx + + def _initialize_error_cov_unstructured(self): + # Initialize the parameters + k_endog = self.k_endog + self.parameters['error_cov'] = int(k_endog * (k_endog + 1) / 2) + + # Setup fixed components of state space matrices + + # Setup indices of state space matrices + self._idx_lower_error_cov = np.tril_indices(self.k_endog) + if self.error_order > 0: + start = self.k_factors + end = self.k_factors + self.k_endog + self._idx_error_cov = ( + np.s_['state_cov', start:end, start:end]) + else: + self._idx_error_cov = np.s_['obs_cov', :, :] + + def _initialize_factor_transition(self): + order = self.factor_order * self.k_factors + k_factors = self.k_factors + + # Initialize the parameters + self.parameters['factor_transition'] = ( + self.factor_order * self.k_factors**2) + + # Setup fixed components of state space matrices + # VAR(p) for factor transition + if self.k_factors > 0: + if self.factor_order > 0: + self.ssm['transition', k_factors:order, :order - k_factors] = ( + np.eye(order - k_factors)) + + self.ssm['selection', :k_factors, :k_factors] = np.eye(k_factors) + # Identification requires constraining the state covariance to an + # identity matrix + self.ssm['state_cov', :k_factors, :k_factors] = np.eye(k_factors) + + # Setup indices of state space matrices + self._idx_factor_transition = np.s_['transition', :k_factors, :order] + + def _initialize_error_transition(self): + # Initialize the appropriate situation + if self.error_order == 0: + self._initialize_error_transition_white_noise() + else: + # Generic setup fixed components of state space matrices + # VAR(q) for error transition + # (in the individual AR case, we still have the VAR(q) companion + # matrix structure, but force the coefficient matrices to be + # diagonal) + k_endog = self.k_endog + k_factors = self.k_factors + _factor_order = self._factor_order + _error_order = self._error_order + _slice = np.s_['selection', + _factor_order:_factor_order + k_endog, + k_factors:k_factors + k_endog] + self.ssm[_slice] = np.eye(k_endog) + _slice = np.s_[ + 'transition', + _factor_order + k_endog:_factor_order + _error_order, + _factor_order:_factor_order + _error_order - k_endog] + self.ssm[_slice] = np.eye(_error_order - k_endog) + + # Now specialized setups + if self.error_var: + self._initialize_error_transition_var() + else: + self._initialize_error_transition_individual() + + def _initialize_error_transition_white_noise(self): + # Initialize the parameters + self.parameters['error_transition'] = 0 + + # No fixed components of state space matrices + + # Setup indices of state space matrices (just an empty slice) + self._idx_error_transition = np.s_['transition', 0:0, 0:0] + + def _initialize_error_transition_var(self): + k_endog = self.k_endog + _factor_order = self._factor_order + _error_order = self._error_order + + # Initialize the parameters + self.parameters['error_transition'] = _error_order * k_endog + + # Fixed components already setup above + + # Setup indices of state space matrices + # Here we want to set all of the elements of the coefficient matrices, + # the same as in a VAR specification + self._idx_error_transition = np.s_[ + 'transition', + _factor_order:_factor_order + k_endog, + _factor_order:_factor_order + _error_order] + + def _initialize_error_transition_individual(self): + k_endog = self.k_endog + _factor_order = self._factor_order + _error_order = self._error_order + + # Initialize the parameters + self.parameters['error_transition'] = _error_order + + # Fixed components already setup above + + # Setup indices of state space matrices + # Here we want to set only the diagonal elements of the coefficient + # matrices, and we want to set them in order by equation, not by + # matrix (i.e. set the first element of the first matrix's diagonal, + # then set the first element of the second matrix's diagonal, then...) + + # The basic setup is a tiled list of diagonal indices, one for each + # coefficient matrix + idx = np.tile(np.diag_indices(k_endog), self.error_order) + # Now we need to shift the rows down to the correct location + row_shift = self._factor_order + # And we need to shift the columns in an increasing way + col_inc = self._factor_order + np.repeat( + [i * k_endog for i in range(self.error_order)], k_endog) + idx[0] += row_shift + idx[1] += col_inc + + # Make a copy (without the row shift) so that we can easily get the + # diagonal parameters back out of a generic coefficients matrix array + idx_diag = idx.copy() + idx_diag[0] -= row_shift + idx_diag[1] -= self._factor_order + idx_diag = idx_diag[:, np.lexsort((idx_diag[1], idx_diag[0]))] + self._idx_error_diag = (idx_diag[0], idx_diag[1]) + + # Finally, we want to fill the entries in in the correct order, which + # is to say we want to fill in lexicographically, first by row then by + # column + idx = idx[:, np.lexsort((idx[1], idx[0]))] + self._idx_error_transition = np.s_['transition', idx[0], idx[1]] def filter(self, params, transformed=True, cov_type=None, return_ssm=False, **kwargs): @@ -152,7 +445,7 @@ def filter(self, params, transformed=True, cov_type=None, return_ssm=False, transformed = True # Get the state space output - result = super(StaticFactors, self).filter(params, transformed, + result = super(DynamicFactor, self).filter(params, transformed, cov_type, return_ssm=True, **kwargs) @@ -161,61 +454,173 @@ def filter(self, params, transformed=True, cov_type=None, return_ssm=False, result_kwargs = {} if cov_type is not None: result_kwargs['cov_type'] = cov_type - result = StaticFactorsResultsWrapper( - StaticFactorsResults(self, params, result, **result_kwargs) + result = DynamicFactorResultsWrapper( + DynamicFactorResults(self, params, result, **result_kwargs) ) return result - filter.__doc__ = MLEModel.filter.__doc__ + filter.__doc__ = MLEModel.filter.__doc__ @property def start_params(self): params = np.zeros(self.k_params, dtype=np.float64) - # Use principal components as starting values - res_pca = PCA(self.endog, ncomp=self.k_factors) + endog = self.endog.copy() # 1. Factor loadings (estimated via PCA) - params[self._params_loadings] = res_pca.loadings.ravel() - - # 2. Idiosyncratic variances - resid = self.endog - np.dot(res_pca.factors, res_pca.loadings.T) - params[self._params_idiosyncratic] = resid.var(axis=0) - - if self.k_factors > 1: + if self.k_factors > 0: + # Use principal components + OLS as starting values + res_pca = PCA(endog, ncomp=self.k_factors) + mod_ols = OLS(endog, res_pca.factors) + res_ols = mod_ols.fit() + + # Using OLS params for the loadings tends to gives higher starting + # log-likelihood. + params[self._params_loadings] = res_ols.params.T.ravel() + # params[self._params_loadings] = res_pca.loadings.ravel() + + # However, using res_ols.resid tends to causes non-invertible + # starting VAR coefficients for error VARs + # endog = res_ols.resid + endog = endog - np.dot(res_pca.factors, res_pca.loadings.T) + + # 2. Exog (OLS on residuals) + if self.k_exog > 0: + mod_ols = OLS(endog, exog=self.exog) + res_ols = mod_ols.fit() + # In the form: beta.x1.y1, beta.x2.y1, beta.x1.y2, ... + params[self._params_exog] = res_ols.params.T.ravel() + endog = res_ols.resid + + # 3. Factors (VAR on res_pca.factors) + stationary = True + if self.k_factors > 1 and self.factor_order > 0: # 3a. VAR transition (OLS on factors estimated via PCA) - mod_var = VAR(res_pca.factors) - res_var = mod_var.fit(maxlags=self.factor_order, ic=None, - trend='nc') - params[self._params_transition] = res_var.params.T.ravel() - else: + mod_factors = VAR(res_pca.factors) + res_factors = mod_factors.fit(maxlags=self.factor_order, ic=None, + trend='nc') + # Save the parameters + params[self._params_factor_transition] = ( + res_factors.params.T.ravel()) + + # Test for stationarity + coefficient_matrices = ( + params[self._params_factor_transition].reshape( + self.k_factors * self.factor_order, self.k_factors + ).T + ).reshape(self.k_factors, self.k_factors, self.factor_order).T + + stationary = is_invertible([1] + list(-coefficient_matrices)) + elif self.k_factors > 0 and self.factor_order > 0: # 3b. AR transition Y = res_pca.factors[self.factor_order:] X = lagmat(res_pca.factors, self.factor_order, trim='both') params_ar = np.linalg.pinv(X).dot(Y) - resid = Y - np.dot(X, params_ar) - params[self._params_transition] = params_ar[:, 0] + stationary = is_invertible(np.r_[1, -params_ar.squeeze()]) + params[self._params_factor_transition] = params_ar[:, 0] + + # Check for stationarity + if not stationary and self.enforce_stationarity: + raise ValueError('Non-stationary starting autoregressive' + ' parameters found with `enforce_stationarity`' + ' set to True.') + + # 4. Errors + if self.error_order == 0: + error_params = [] + if self.error_cov_type == 'scalar': + params[self._params_error_cov] = endog.var(axis=0).mean() + elif self.error_cov_type == 'diagonal': + params[self._params_error_cov] = endog.var(axis=0) + elif self.error_cov_type == 'unstructured': + cov_factor = np.diag(endog.std(axis=0)) + params[self._params_error_cov] = ( + cov_factor[self._idx_lower_error_cov].ravel()) + else: + mod_errors = VAR(endog) + res_errors = mod_errors.fit(maxlags=self.error_order, ic=None, + trend='nc') + + + # Test for stationarity + coefficient_matrices = ( + np.array(res_errors.params.T).ravel().reshape( + self.k_endog * self.error_order, self.k_endog + ).T + ).reshape(self.k_endog, self.k_endog, self.error_order).T + + stationary = is_invertible([1] + list(-coefficient_matrices)) + if not stationary and self.enforce_stationarity: + raise ValueError('Non-stationary starting error autoregressive' + ' parameters found with' + ' `enforce_stationarity` set to True.') + + # Get the error autoregressive parameters + if self.error_var: + params[self._params_error_transition] = ( + np.array(res_errors.params.T).ravel()) + else: + # In the case of individual autoregressions, extract just the + # diagonal elements + params[self._params_error_transition] = ( + res_errors.params.T[self._idx_error_diag]) + + # Get the error covariance parameters + if self.error_cov_type == 'scalar': + params[self._params_error_cov] = ( + res_errors.sigma_u.diagonal().mean()) + elif self.error_cov_type == 'diagonal': + params[self._params_error_cov] = res_errors.sigma_u.diagonal() + elif self.error_cov_type == 'unstructured': + try: + cov_factor = np.linalg.cholesky(res_errors.sigma_u) + except np.linalg.LinAlgError: + cov_factor = np.eye(res_errors.sigma_u.shape[0]) * ( + res_errors.sigma_u.diagonal().mean()**0.5) + cov_factor = np.eye(res_errors.sigma_u.shape[0]) * ( + res_errors.sigma_u.diagonal().mean()**0.5) + params[self._params_error_cov] = ( + cov_factor[self._idx_lower_error_cov].ravel()) return params @property def param_names(self): param_names = [] + endog_names = self.endog_names - # 1. Factor loadings (estimated via PCA) + # 1. Factor loadings param_names += [ - 'loading.f%d.%s' % (j+1, self.endog_names[i]) + 'loading.f%d.%s' % (j+1, endog_names[i]) for i in range(self.k_endog) for j in range(self.k_factors) ] - # 2. Idiosyncratic variances + # 2. Exog + # Recall these are in the form: beta.x1.y1, beta.x2.y1, beta.x1.y2, ... param_names += [ - 'sigma2.%s' % self.endog_names[i] + 'beta.%s.%s' % (self.exog_names[j], endog_names[i]) for i in range(self.k_endog) + for j in range(self.k_exog) ] - # 3. VAR transition (OLS on factors estimated via PCA) + # 3. Error covariances + if self.error_cov_type == 'scalar': + param_names += ['sigma2'] + elif self.error_cov_type == 'diagonal': + param_names += [ + 'sigma2.%s' % endog_names[i] + for i in range(self.k_endog) + ] + elif self.error_cov_type == 'unstructured': + param_names += [ + ('sqrt.var.%s' % endog_names[i] if i == j else + 'sqrt.cov.%s.%s' % (endog_names[j], endog_names[i])) + for i in range(self.k_endog) + for j in range(i+1) + ] + + # 4. Factor transition VAR param_names += [ 'L%d.f%d.f%d' % (i+1, k+1, j+1) for j in range(self.k_factors) @@ -223,6 +628,21 @@ def param_names(self): for k in range(self.k_factors) ] + # 5. Error transition VAR + if self.error_var: + param_names += [ + 'L%d.e(%s).e(%s)' % (i+1, endog_names[k], endog_names[j]) + for j in range(self.k_endog) + for i in range(self.error_order) + for k in range(self.k_endog) + ] + else: + param_names += [ + 'L%d.e(%s).e(%s)' % (i+1, endog_names[j], endog_names[j]) + for j in range(self.k_endog) + for i in range(self.error_order) + ] + return param_names def transform_params(self, unconstrained): @@ -248,30 +668,80 @@ def transform_params(self, unconstrained): positive. """ unconstrained = np.array(unconstrained, ndmin=1) - constrained = np.zeros(unconstrained.shape, dtype=unconstrained.dtype) + dtype = unconstrained.dtype + constrained = np.zeros(unconstrained.shape, dtype=dtype) + # 1. Factor loadings # The factor loadings do not need to be adjusted constrained[self._params_loadings] = ( unconstrained[self._params_loadings]) - # The observation variances must be positive - constrained[self._params_idiosyncratic] = ( - unconstrained[self._params_idiosyncratic]**2) - + # 2. Exog + # The regression coefficients do not need to be adjusted + constrained[self._params_exog] = ( + unconstrained[self._params_exog]) + + # 3. Error covariances + # If we have variances, force them to be positive + if self.error_cov_type in ['scalar', 'diagonal']: + constrained[self._params_error_cov] = ( + unconstrained[self._params_error_cov]**2) + # Otherwise, nothing needs to be done + elif self.error_cov_type == 'unstructured': + constrained[self._params_error_cov] = ( + unconstrained[self._params_error_cov]) + + # 4. Factor transition VAR # VAR transition: optionally force to be stationary - if self.enforce_stationarity: + if self.enforce_stationarity and self.factor_order > 0: # Transform the parameters unconstrained_matrices = ( - unconstrained[self._params_transition].reshape( - self.k_factors, self.k_factors * self.factor_order)) + unconstrained[self._params_factor_transition].reshape( + self.k_factors, self._factor_order)) + # This is always an identity matrix, but because the transform + # done prior to update (where the ssm representation matrices + # change), it may be complex + cov = self.ssm[ + 'state_cov', :self.k_factors, :self.k_factors].real coefficient_matrices, variance = ( - constrain_stationary_multivariate(unconstrained_matrices, - self.ssm['state_cov'])) - constrained[self._params_transition] = ( + constrain_stationary_multivariate(unconstrained_matrices, cov)) + constrained[self._params_factor_transition] = ( coefficient_matrices.ravel()) else: - constrained[self._params_transition] = ( - unconstrained[self._params_transition]) + constrained[self._params_factor_transition] = ( + unconstrained[self._params_factor_transition]) + + # 5. Error transition VAR + # VAR transition: optionally force to be stationary + if self.enforce_stationarity and self.error_order > 0: + + # Joint VAR specification + if self.error_var: + unconstrained_matrices = ( + unconstrained[self._params_error_transition].reshape( + self.k_endog, self._error_order)) + start = self.k_factors + end = self.k_factors + self.k_endog + cov = self.ssm['state_cov', start:end, start:end].real + coefficient_matrices, variance = ( + constrain_stationary_multivariate(unconstrained_matrices, + cov)) + constrained[self._params_error_transition] = ( + coefficient_matrices.ravel()) + # Separate AR specifications + else: + coefficients = ( + unconstrained[self._params_error_transition].copy()) + for i in range(self.k_endog): + start = i * self.error_order + end = (i + 1) * self.error_order + coefficients[start:end] = constrain_stationary_univariate( + coefficients[start:end]) + constrained[self._params_error_transition] = coefficients + + else: + constrained[self._params_error_transition] = ( + unconstrained[self._params_error_transition]) return constrained @@ -292,28 +762,79 @@ def untransform_params(self, constrained): Array of unconstrained parameters used by the optimizer. """ constrained = np.array(constrained, ndmin=1) - unconstrained = np.zeros(constrained.shape, dtype=constrained.dtype) + dtype=constrained.dtype + unconstrained = np.zeros(constrained.shape, dtype=dtype) + # 1. Factor loadings # The factor loadings do not need to be adjusted unconstrained[self._params_loadings] = ( constrained[self._params_loadings]) - # The observation variances must have been positive - unconstrained[self._params_idiosyncratic] = ( - constrained[self._params_idiosyncratic]**0.5) - - # VAR transition: optionally were forced to be stationary - if self.enforce_stationarity: - coefficients = constrained[self._params_transition].reshape( - self.k_factors, self.k_factors * self.factor_order) - unconstrained_matrices, variance = ( - unconstrain_stationary_multivariate(coefficients, - self.ssm['state_cov'])) - unconstrained[self._params_transition] = ( - unconstrained_matrices.ravel()) + # 2. Exog + # The regression coefficients do not need to be adjusted + unconstrained[self._params_exog] = ( + constrained[self._params_exog]) + + # 3. Error covariances + # If we have variances, force them to be positive + if self.error_cov_type in ['scalar', 'diagonal']: + unconstrained[self._params_error_cov] = ( + constrained[self._params_error_cov]**0.5) + # Otherwise, nothing needs to be done + elif self.error_cov_type == 'unstructured': + unconstrained[self._params_error_cov] = ( + constrained[self._params_error_cov]) + + # 3. Factor transition VAR + # VAR transition: optionally force to be stationary + if self.enforce_stationarity and self.factor_order > 0: + # Transform the parameters + constrained_matrices = ( + constrained[self._params_factor_transition].reshape( + self.k_factors, self._factor_order)) + cov = self.ssm[ + 'state_cov', :self.k_factors, :self.k_factors].real + coefficient_matrices, variance = ( + unconstrain_stationary_multivariate(constrained_matrices, + cov)) + unconstrained[self._params_factor_transition] = ( + coefficient_matrices.ravel()) + else: + unconstrained[self._params_factor_transition] = ( + constrained[self._params_factor_transition]) + + # 5. Error transition VAR + # VAR transition: optionally force to be stationary + if self.enforce_stationarity and self.error_order > 0: + + # Joint VAR specification + if self.error_var: + constrained_matrices = ( + constrained[self._params_error_transition].reshape( + self.k_endog, self._error_order)) + start = self.k_factors + end = self.k_factors + self.k_endog + cov = self.ssm['state_cov', start:end, start:end].real + coefficient_matrices, variance = ( + unconstrain_stationary_multivariate(constrained_matrices, + cov)) + unconstrained[self._params_error_transition] = ( + coefficient_matrices.ravel()) + # Separate AR specifications + else: + coefficients = ( + constrained[self._params_error_transition].copy()) + for i in range(self.k_endog): + start = i * self.error_order + end = (i + 1) * self.error_order + coefficients[start:end] = ( + unconstrain_stationary_univariate( + coefficients[start:end])) + unconstrained[self._params_error_transition] = coefficients + else: - unconstrained[self._params_transition] = ( - constrained[self._params_transition]) + unconstrained[self._params_error_transition] = ( + constrained[self._params_error_transition]) return unconstrained @@ -347,7 +868,7 @@ def update(self, params, transformed=True): - The first :math:`n \times m` parameters fill out the factor loading matrix, starting from the [0,0] entry and then proceeding along rows. These parameters are not modified in `transform_params`. - - The next :math:`n` parameters provide variances for the idiosyncratic + - The next :math:`n` parameters provide variances for the error_cov errors in the observation equation. They fill in the diagonal of the observation covariance matrix, and are constrained to be positive by `transofrm_params`. @@ -361,35 +882,60 @@ def update(self, params, transformed=True): second :math:`m^2` parameters fill the second matrix, etc. """ - params = super(StaticFactors, self).update(params, transformed) + params = super(DynamicFactor, self).update(params, transformed) + # 1. Factor loadings # Update the design / factor loading matrix - self.ssm['design', :, :self.k_factors] = ( + self.ssm[self._idx_loadings] = ( params[self._params_loadings].reshape(self.k_endog, self.k_factors) ) - # Update the observation covariance - self.ssm[self._idx_obs_cov] = params[self._params_idiosyncratic] - - # Update the transition matrix - self.ssm[self._idx_transition] = ( - params[self._params_transition].reshape( - self.k_factors, self.k_factors * self.factor_order)) + # 2. Exog + if self.k_exog > 0: + exog_params = params[self._params_exog].reshape( + self.k_endog, self.k_exog).T + self.ssm[self._idx_exog] = np.dot(self.exog, exog_params).T + + # 3. Error covariances + if self.error_cov_type in ['scalar', 'diagonal']: + self.ssm[self._idx_error_cov] = ( + params[self._params_error_cov]) + elif self.error_cov_type == 'unstructured': + error_cov_lower = np.zeros((self.k_endog, self.k_endog), + dtype=params.dtype) + error_cov_lower[self._idx_lower_error_cov] = ( + params[self._params_error_cov]) + self.ssm[self._idx_error_cov] = ( + np.dot(error_cov_lower, error_cov_lower.T)) + + # 4. Factor transition VAR + self.ssm[self._idx_factor_transition] = ( + params[self._params_factor_transition].reshape( + self.k_factors, self.factor_order * self.k_factors)) + + # 5. Error transition VAR + if self.error_var: + self.ssm[self._idx_error_transition] = ( + params[self._params_error_transition].reshape( + self.k_endog, self._error_order)) + else: + self.ssm[self._idx_error_transition] = ( + params[self._params_error_transition]) -class StaticFactorsResults(MLEResults): +class DynamicFactorResults(MLEResults): """ - Class to hold results from fitting an StaticFactors model. + Class to hold results from fitting an DynamicFactor model. Parameters ---------- - model : StaticFactors instance + model : DynamicFactor instance The fitted model instance Attributes ---------- specification : dictionary - Dictionary including all attributes from the StaticFactors model + Dictionary including all attributes from the DynamicFactor model instance. coefficient_matrices_var : array Array containing autoregressive lag polynomial coefficient matrices, @@ -402,115 +948,462 @@ class StaticFactorsResults(MLEResults): """ def __init__(self, model, params, filter_results, cov_type='opg', **kwargs): - super(StaticFactorsResults, self).__init__(model, params, + super(DynamicFactorResults, self).__init__(model, params, filter_results, cov_type, **kwargs) self.df_resid = np.inf # attribute required for wald tests self.specification = Bunch(**{ - # Set additional model parameters + # Model properties + 'k_endog' : self.model.k_endog, 'enforce_stationarity': self.model.enforce_stationarity, - # Factors + # Factor-related properties 'k_factors': self.model.k_factors, - 'factor_order': self.model.factor_order + 'factor_order': self.model.factor_order, + + # Error-related properties + 'error_order': self.model.error_order, + 'error_var': self.model.error_var, + 'error_cov_type': self.model.error_cov_type, + + # Other properties + 'k_exog': self.model.k_exog }) # Polynomials / coefficient matrices self.coefficient_matrices_var = None if self.model.factor_order > 0: - ar_params = np.array(self.params[self.model._params_transition]) + ar_params = ( + np.array(self.params[self.model._params_factor_transition])) k_factors = self.model.k_factors factor_order = self.model.factor_order self.coefficient_matrices_var = ( ar_params.reshape(k_factors * factor_order, k_factors).T ).reshape(k_factors, k_factors, factor_order).T - def summary(self, alpha=.05, start=None): - from statsmodels.iolib.summary import summary_params - # Create the model name + self.coefficient_matrices_error = None + if self.model.error_order > 0: + ar_params = ( + np.array(self.params[self.model._params_error_transition])) + k_endog = self.model.k_endog + error_order = self.model.error_order + if self.model.error_var: + self.coefficient_matrices_error = ( + ar_params.reshape(k_endog * error_order, k_endog).T + ).reshape(k_endog, k_endog, error_order).T + else: + mat = np.zeros((k_endog, k_endog * error_order)) + mat[self.model._idx_error_diag] = ar_params + self.coefficient_matrices_error = ( + mat.T.reshape(error_order, k_endog, k_endog)) + + @property + def factors(self): + """ + Estimates of unobserved factors + Returns + ------- + out: Bunch + Has the following attributes: + + - `filtered`: a time series array with the filtered estimate of + the component + - `filtered_cov`: a time series array with the filtered estimate of + the variance/covariance of the component + - `offset`: an integer giving the offset in the state vector where + this component begins + """ + # If present, level is always the first component of the state vector + out = None + spec = self.specification + if spec.k_factors > 0: + offset = 0 + end = spec.k_factors + res = self.filter_results + out = Bunch( + filtered=res.filtered_state[offset:end], + filtered_cov=res.filtered_state_cov[offset:end, offset:end], + offset=offset) + return out + + @cache_readonly + def coefficients_of_determination(self): + """ + Coefficients of determination (:math:`R^2`) from regressions of + individual estimated factors on endogenous variables. + + Returns + ------- + coefficients_of_determination : array + A `k_endog` x `k_factors` array, where + `coefficients_of_determination[i, j]` represents the :math:`R^2` + value from a regression of factor `j` and a constant on endogenous + variable `i`. + + Notes + ----- + Although it can be difficult to interpret the estimated factor loadings + and factors, it is often helpful to use the cofficients of + determination from univariate regressions to assess the importance of + each factor in explaining the variation in each endogenous variable. + + In models with many variables and factors, this can sometimes lend + interpretation to the factors (for example sometimes one factor will + load primarily on real variables and another on nominal variables). + + See Also + -------- + plot_coefficients_of_determination + + """ + from statsmodels.tools import add_constant + spec = self.specification + coefficients = np.zeros((spec.k_endog, spec.k_factors)) + factors = self.factors.filtered + + for i in range(spec.k_factors): + exog = add_constant(factors[i]) + for j in range(spec.k_endog): + endog = self.filter_results.endog[j] + coefficients[j, i] = OLS(endog, exog).fit().rsquared + + return coefficients + + def plot_coefficients_of_determination(self, endog_labels=None, + fig=None, figsize=None): + """ + Plot the coefficients of determination + + Parameters + ---------- + endog_labels : boolean, optional + Whether or not to label the endogenous variables along the x-axis + of the plots. Default is to include labels if there are 5 or fewer + endogenous variables. + fig : Matplotlib Figure instance, optional + If given, subplots are created in this figure instead of in a new + figure. Note that the grid will be created in the provided + figure using `fig.add_subplot()`. + figsize : tuple, optional + If a figure is created, this argument allows specifying a size. + The tuple is (width, height). + + Notes + ----- - # See if we have an ARIMA component - order = '(factors=%d, order=%d)' % (self.specification.k_factors, - self.specification.factor_order) + Produces a `k_factors` x 1 plot grid. The `i`th plot shows a bar plot + of the coefficients of determination associated with factor `i`. The + endogenous variables are arranged along the x-axis according to their + position in the `endog` array. - model_name = ( - '%s%s' % (self.model.__class__.__name__, order) + See Also + -------- + coefficients_of_determination + + """ + from statsmodels.graphics.utils import _import_mpl, create_mpl_fig + _import_mpl() + fig = create_mpl_fig(fig, figsize) + + spec = self.specification + + # Should we label endogenous variables? + if endog_labels is None: + endog_labels = spec.k_endog <= 5 + + # Plot the coefficients of determination + coefficients_of_determination = self.coefficients_of_determination + plot_idx = 1 + locations = np.arange(spec.k_endog) + for coeffs in coefficients_of_determination.T: + # Create the new axis + ax = fig.add_subplot(spec.k_factors, 1, plot_idx) + ax.set_ylim((0,1)) + ax.set(title='Factor %i' % plot_idx, ylabel=r'$R^2$') + bars = ax.bar(locations, coeffs) + + if endog_labels: + width = bars[0].get_width() + ax.xaxis.set_ticks(locations + width / 2) + ax.xaxis.set_ticklabels(self.model.endog_names) + else: + ax.set(xlabel='Endogenous variables') + ax.xaxis.set_ticks([]) + + plot_idx += 1 + + return fig + + def predict(self, start=None, end=None, exog=None, dynamic=False, + **kwargs): + """ + In-sample prediction and out-of-sample forecasting + + Parameters + ---------- + start : int, str, or datetime, optional + Zero-indexed observation number at which to start forecasting, ie., + the first forecast is start. Can also be a date string to + parse or a datetime type. Default is the the zeroth observation. + end : int, str, or datetime, optional + Zero-indexed observation number at which to end forecasting, ie., + the first forecast is start. Can also be a date string to + parse or a datetime type. However, if the dates index does not + have a fixed frequency, end must be an integer index if you + want out of sample prediction. Default is the last observation in + the sample. + exog : array_like, optional + If the model includes exogenous regressors, you must provide + exactly enough out-of-sample values for the exogenous variables if + end is beyond the last observation in the sample. + dynamic : boolean, int, str, or datetime, optional + Integer offset relative to `start` at which to begin dynamic + prediction. Can also be an absolute date string to parse or a + datetime type (these are not interpreted as offsets). + Prior to this observation, true endogenous values will be used for + prediction; starting with this observation and continuing through + the end of prediction, forecasted endogenous values will be used + instead. + **kwargs + Additional arguments may required for forecasting beyond the end + of the sample. See `FilterResults.predict` for more details. + + Returns + ------- + forecast : array + Array of out of sample forecasts. + """ + if start is None: + start = 0 + + # Handle end (e.g. date) + _start = self.model._get_predict_start(start) + _end, _out_of_sample = self.model._get_predict_end(end) + + # Handle exogenous parameters + if _out_of_sample and self.model.k_exog > 0: + # Create a new faux VARMAX model for the extended dataset + nobs = self.model.data.orig_endog.shape[0] + _out_of_sample + endog = np.zeros((nobs, self.model.k_endog)) + + if self.model.k_exog > 0: + if exog is None: + raise ValueError('Out-of-sample forecasting in a model' + ' with a regression component requires' + ' additional exogenous values via the' + ' `exog` argument.') + exog = np.array(exog) + required_exog_shape = (_out_of_sample, self.model.k_exog) + if not exog.shape == required_exog_shape: + raise ValueError('Provided exogenous values are not of the' + ' appropriate shape. Required %s, got %s.' + % (str(required_exog_shape), + str(exog.shape))) + exog = np.c_[self.model.data.orig_exog.T, exog.T].T + + # TODO replace with init_kwds or specification or similar + model = DynamicFactor( + endog, + k_factors=self.model.k_factors, + factor_order=self.model.factor_order, + exog=exog, + error_order=self.model.error_order, + error_var=self.model.error_var, + error_cov_type=self.model.error_cov_type, + enforce_stationarity=self.model.enforce_stationarity ) - summary = super(StaticFactorsResults, self).summary( - alpha=alpha, start=start, model_name=model_name, - display_params=False + model.update(self.params) + + # Set the kwargs with the update time-varying state space + # representation matrices + for name in self.filter_results.shapes.keys(): + if name == 'obs': + continue + mat = getattr(model.ssm, name) + if mat.shape[-1] > 1: + if len(mat.shape) == 2: + kwargs[name] = mat[:, -_out_of_sample:] + else: + kwargs[name] = mat[:, :, -_out_of_sample:] + elif self.model.k_exog == 0 and exog is not None: + warn('Exogenous array provided to predict, but additional data not' + ' required. `exog` argument ignored.') + + return super(DynamicFactorResults, self).predict( + start=start, end=end, exog=exog, dynamic=dynamic, **kwargs ) - # Add parameter tables for each endogenous variable - k_endog = self.model.k_endog - k_factors = self.model.k_factors - factor_order = self.model.factor_order - for i in range(k_endog): - mask = [] - offset = 0 + def forecast(self, steps=1, exog=None, **kwargs): + """ + Out-of-sample forecasts - # 1. Loadings - offset = k_factors * i - mask.append(np.arange(offset, offset + k_factors)) + Parameters + ---------- + steps : int, optional + The number of out of sample forecasts from the end of the + sample. Default is 1. + exog : array_like, optional + If the model includes exogenous regressors, you must provide + exactly enough out-of-sample values for the exogenous variables for + each step forecasted. + **kwargs + Additional arguments may required for forecasting beyond the end + of the sample. See `FilterResults.predict` for more details. - # 2. Idiosyncratic variance - offset = k_factors * k_endog - mask.append(np.array(offset + i, ndmin=1)) + Returns + ------- + forecast : array + Array of out of sample forecasts. + """ + return super(DynamicFactorResults, self).forecast(steps, exog=exog, + **kwargs) - # Create the table - mask = np.concatenate(mask) - res = (self, self.params[mask], self.bse[mask], self.zvalues[mask], - self.pvalues[mask], self.conf_int(alpha)[mask]) + def summary(self, alpha=.05, start=None, separate_params=True): + from statsmodels.iolib.summary import summary_params + spec = self.specification - param_names = [ - '.'.join(name.split('.')[:-1]) - for name in - np.array(self.data.param_names)[mask].tolist() - ] - title = "Results for equation %s" % self.model.endog_names[i] + # Create the model name + model_name = [] + if spec.k_factors > 0: + if spec.factor_order > 0: + model_type = ('DynamicFactor(factors=%d, order=%d)' % + (spec.k_factors, spec.factor_order)) + else: + model_type = 'StaticFactor(factors=%d)' % spec.k_factors + + model_name.append(model_type) + if spec.k_exog > 0: + model_name.append('%d regressors' % spec.k_exog) + else: + model_name.append('SUR(%d regressors)' % spec.k_exog) - table = summary_params(res, yname=None, xname=param_names, - alpha=alpha, use_t=False, title=title) + if spec.error_order > 0: + error_type = 'VAR' if spec.error_var else 'AR' + model_name.append('%s(%d) errors' % (error_type, spec.error_order)) - summary.tables.append(table) + summary = super(DynamicFactorResults, self).summary( + alpha=alpha, start=start, model_name=model_name, + display_params=not separate_params + ) - # Add parameter tables for each factor - offset = k_endog * (k_factors + 1) - for i in range(k_factors): - mask = [] - start = i * k_factors * factor_order - end = (i + 1) * k_factors * factor_order - mask.append( - offset + np.arange(start, end)) - - # Create the table - mask = np.concatenate(mask) - res = (self, self.params[mask], self.bse[mask], self.zvalues[mask], - self.pvalues[mask], self.conf_int(alpha)[mask]) - - param_names = [ - '.'.join(name.split('.')[:-1]) - for name in - np.array(self.data.param_names)[mask].tolist() - ] - title = "Results for factor equation f%d" % (i+1) + if separate_params: + indices = np.arange(len(self.params)) - table = summary_params(res, yname=None, xname=param_names, - alpha=alpha, use_t=False, title=title) + def make_table(self, mask, title, strip_end=True): + res = (self, self.params[mask], self.bse[mask], + self.zvalues[mask], self.pvalues[mask], + self.conf_int(alpha)[mask]) + param_names = [ + '.'.join(name.split('.')[:-1]) if strip_end else name + for name in + np.array(self.data.param_names)[mask].tolist() + ] + + return summary_params(res, yname=None, xname=param_names, + alpha=alpha, use_t=False, title=title) + + k_endog = self.model.k_endog + k_exog = self.model.k_exog + k_factors = self.model.k_factors + factor_order = self.model.factor_order + _factor_order = self.model._factor_order + _error_order = self.model._error_order + + # Add parameter tables for each endogenous variable + loading_indices = indices[self.model._params_loadings] + loading_masks = [] + exog_indices = indices[self.model._params_exog] + exog_masks = [] + for i in range(k_endog): + offset = 0 + + # 1. Factor loadings + # Recall these are in the form: + # 'loading.f1.y1', 'loading.f2.y1', 'loading.f1.y2', ... + + loading_mask = ( + loading_indices[i * k_factors:(i + 1) * k_factors]) + loading_masks.append(loading_mask) + + # 2. Exog + # Recall these are in the form: + # beta.x1.y1, beta.x2.y1, beta.x1.y2, ... + exog_mask = exog_indices[i * k_exog:(i + 1) * k_exog] + exog_masks.append(exog_mask) + + # Create the table + mask = np.concatenate([loading_mask, exog_mask]) + title = "Results for equation %s" % self.model.endog_names[i] + table = make_table(self, mask, title) + summary.tables.append(table) + + # Add parameter tables for each factor + factor_indices = indices[self.model._params_factor_transition] + factor_masks = [] + if factor_order > 0: + for i in range(k_factors): + start = i * _factor_order + factor_mask = factor_indices[start: start + _factor_order] + factor_masks.append(factor_mask) + + # Create the table + title = "Results for factor equation f%d" % (i+1) + table = make_table(self, factor_mask, title) + summary.tables.append(table) + + # Add parameter tables for error transitions + error_masks = [] + if spec.error_order > 0: + error_indices = indices[self.model._params_error_transition] + for i in range(k_endog): + if spec.error_var: + start = i * _error_order + end = (i + 1) * _error_order + else: + start = i * spec.error_order + end = (i + 1) * spec.error_order + + error_mask = error_indices[start:end] + error_masks.append(error_mask) + + # Create the table + title = ("Results for error equation e(%s)" % + self.model.endog_names[i]) + table = make_table(self, error_mask, title) + summary.tables.append(table) + + # Error covariance terms + error_cov_mask = indices[self.model._params_error_cov] + table = make_table(self, error_cov_mask, + "Error covariance matrix", strip_end=False) summary.tables.append(table) + # Add a table for all other parameters + masks = [] + for m in (loading_masks, exog_masks, factor_masks, + error_masks, [error_cov_mask]): + m = np.array(m).flatten() + if len(m) > 0: + masks.append(m) + masks = np.concatenate(masks) + inverse_mask = np.array(list(set(indices).difference(set(masks)))) + if len(inverse_mask) > 0: + table = make_table(self, inverse_mask, "Other parameters", + strip_end=False) + summary.tables.append(table) + return summary summary.__doc__ = MLEResults.summary.__doc__ -class StaticFactorsResultsWrapper(MLEResultsWrapper): +class DynamicFactorResultsWrapper(MLEResultsWrapper): _attrs = {} _wrap_attrs = wrap.union_dicts(MLEResultsWrapper._wrap_attrs, _attrs) _methods = {} _wrap_methods = wrap.union_dicts(MLEResultsWrapper._wrap_methods, _methods) -wrap.populate_wrapper(StaticFactorsResultsWrapper, StaticFactorsResults) +wrap.populate_wrapper(DynamicFactorResultsWrapper, DynamicFactorResults) diff --git a/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py index d875597798d..046400284d3 100644 --- a/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor.py @@ -43,3 +43,153 @@ 'bic': -953.9023, } +lutkepohl_dfm_exog1 = { + 'params': [ + -.01254697, -.00734604, -.00671296, # Factor loadings + .01803325, .02066737, .01983089, # Beta.constant + .00198667, .00008426, .00005684, # Idiosyncratic variances + .31140829, # Factor transition + ], + 'var_oim': [ + .00004224, 2.730e-06, 3.625e-06, + .00003087, 2.626e-06, 2.013e-06, + 1.170e-07, 5.133e-10, 3.929e-10, + .07412117 + ], + 'loglike': 596.9781590009525, + 'aic': -1173.956, + 'bic': -1150.781, +} + +lutkepohl_dfm_exog2 = { + 'params': [ + .01249096, .00731147, .00680776, # Factor loadings + .02187812, -.00009851, # Betas, y1 + .02302646, -.00006045, # Betas, y2 + .02009233, -6.683e-06, # Betas, y3 + .0019856, .00008378, .00005581, # Idiosyncratic variances + .2995768, # Factor transition + ], + 'var_oim': [ + .00004278, 2.659e-06, 3.766e-06, + .00013003, 6.536e-08, + .00001079, 5.424e-09, + 8.393e-06, 4.217e-09, + 1.168e-07, 5.140e-10, 4.181e-10, + .07578382, + ], + 'loglike': 597.4550537198315, + 'aic': -1168.91, + 'bic': -1138.783, +} + +lutkepohl_dfm_gen = { + 'params': [ + .00312295, .00332555, .00318837, # Factor loadings + # .00195462, # Covariance, lower triangle + # 3.642e-06, .00010047, + # .00007018, .00002565, .00006118 + # Note: the following are the Cholesky of the covariance + # matrix defined just above + .04421108, # Cholesky, lower triangle + .00008238, .01002313, + .00158738, .00254603, .00722343, + .987374, # Factor transition + -.25613562, .00392166, .44859028, # Error transition parameters + .01635544, -.249141, .08170863, + -.02280001, .02059063, -.41808254 + ], + 'var_oim': [ + 1.418e-06, 1.030e-06, 9.314e-07, # Factor loadings + None, # Cholesky, lower triangle + None, None, + None, None, None, + .00021421, # Factor transition + .01307587, .29167522, .43204063, # Error transition parameters + .00076899, .01742173, .0220161, + .00055435, .01456365, .01707167 + ], + 'loglike': 607.7715711926285, + 'aic': -1177.543, + 'bic': -1133.511, +} + +lutkepohl_dfm_scalar = { + 'params': [ + .04424851, .00114077, .00275081, # Factor loadings + .01812298, .02071169, .01987196, # Beta.constant + .00012067, # Idiosyncratic variance + -.19915198, # Factor transition + ], + 'var_oim': [ + .00001479, 1.664e-06, 1.671e-06, + .00001985, 1.621e-06, 1.679e-06, + 1.941e-10, + .01409482 + ], + 'loglike': 588.7677809701966, + 'aic': -1161.536, + 'bic': -1142.996, +} + +lutkepohl_sfm = { + 'params': [ + .02177607, .02089956, .02239669, # Factor loadings + .00201477, .00013623, 7.452e-16 # Idiosyncratic variance + ], + 'var_oim': [ + .00003003, 4.729e-06, 3.344e-06, + 1.083e-07, 4.950e-10, 0 + ], + 'loglike': 532.2215594949788, + 'aic': -1054.443, + 'bic': -1042.856, +} + +lutkepohl_sur = { + 'params': [ + .02169026, -.00009184, # Betas, y1 + .0229165, -.00005654, # Betas, y2 + .01998994, -3.049e-06, # Betas, y3 + # .00215703, # Covariance, lower triangle + # .0000484, .00014252, + # .00012772, .00005642, .00010673, + # Note: the following are the Cholesky of the covariance + # matrix defined just above + .04644384, # Cholesky, lower triangle + .00104212, .0118926, + .00274999, .00450315, .00888196, + ], + 'var_oim': [ + .0001221, 6.137e-08, + 8.067e-06, 4.055e-09, + 6.042e-06, 3.036e-09, + None, + None, None, + None, None, None + ], + 'loglike': 597.6181259116113, + 'aic': -1171.236, + 'bic': -1143.426, +} + + +lutkepohl_sur_auto = { + 'params': [ + .02243063, -.00011112, # Betas, y1 + .02286952, -.0000554, # Betas, y2 + .0020338, .00013843, # Idiosyncratic variance + -.21127833, .50884609, # Error transition parameters + .04292935, .00855789, + ], + 'var_oim': [ + .00008357, 4.209e-08, + 8.402e-06, 4.222e-09, + 1.103e-07, 5.110e-10, + .01259537, .19382105, + .00085936, .01321035, + ], + 'loglike': 352.7250284160132, + 'aic': -685.4501, + 'bic': -662.2752 +} \ No newline at end of file diff --git a/statsmodels/tsa/statespace/tests/results/results_dynamic_factor_stata.csv b/statsmodels/tsa/statespace/tests/results/results_dynamic_factor_stata.csv index 990f515687a..78c2d78512e 100644 --- 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+.0089321,.009253,.0089143,5.40e-15,5.49e-15,5.47e-15,.0180332,.0206674,.0198309,.012815,.0174654,.0194775,.00972,.0103506,.0099236,.018123,.0207117,.019872,0,0,0,.0132405,.0177145,.0197095,.0122077,.0177727,.0054551,.0056511,.0054442,0,0,0,.0180332,.0206674,.0198309,.012815,.0174654,.0194775,.0085316,.0090851,.0087103,.018123,.0207117,.019872,0,0,0,.0132405,.0177145,.0197095,.0122077,.0177727 diff --git a/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do b/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do index ca3286a1879..8488bc74daf 100644 --- a/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do +++ b/statsmodels/tsa/statespace/tests/results/test_dynamic_factor_stata.do @@ -1,20 +1,29 @@ +// Helpful commands: +// matrix list e(b) +// matrix d = vecdiag(e(V)) +// matrix list d + webuse lutkepohl2, clear tsset +// use lutkepohl2_s12, clear +// tsset qtr -// Dynamic Factors: Static Factor model +// Dynamic factors dfactor (dln_inv dln_inc dln_consump = , noconstant ) (f = , ar(1/2)) if qtr<=tq(1978q4) -// predict + forecast, see above +// These are predict in-sample + forecast out-of-sample (1979q1 is first out-of sample obs) predict predict_dfm_1, dynamic(tq(1979q1)) equation(dln_inv) predict predict_dfm_2, dynamic(tq(1979q1)) equation(dln_inc) predict predict_dfm_3, dynamic(tq(1979q1)) equation(dln_consump) -// predict + dynamic predict + forecast, see above +// These are predict in-sample for first 3-observations (1960q2-1960q4), then +// dynamic predict for the rest of in-sample observations (1961q1-1978q4), then +// forecast for the remaining periods 1979q1 - 1982q4 predict dyn_predict_dfm_1, dynamic(tq(1961q1)) equation(dln_inv) predict dyn_predict_dfm_2, dynamic(tq(1961q1)) equation(dln_inc) predict dyn_predict_dfm_3, dynamic(tq(1961q1)) equation(dln_consump) -// Dynamic Factors: Static Factor model with 2 factors +// Dynamic factors, with 2 factors // Note: this does not converge even if we do not enforce iter(#), but this is // good enough for testing dfactor (dln_inv dln_inc dln_consump = , noconstant ) (f1 f2 = , ar(1) arstructure(general)) if qtr<=tq(1978q4), iter(1) @@ -29,5 +38,99 @@ predict dyn_predict_dfm2_1, dynamic(tq(1961q1)) equation(dln_inv) predict dyn_predict_dfm2_2, dynamic(tq(1961q1)) equation(dln_inc) predict dyn_predict_dfm2_3, dynamic(tq(1961q1)) equation(dln_consump) -outsheet pred* dyn* using results_dynamic_factor_stata.csv, comma replace +// Dynamic factors, with 1 exog +gen c = 1 +dfactor (dln_inv dln_inc dln_consump = c, noconstant ) (f = , ar(1)) if qtr<=tq(1978q4) + +// predict + forecast, see above +predict predict_dfm_exog1_1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_dfm_exog1_2, dynamic(tq(1979q1)) equation(dln_inc) +predict predict_dfm_exog1_3, dynamic(tq(1979q1)) equation(dln_consump) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_dfm_exog1_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_dfm_exog1_2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_dfm_exog1_3, dynamic(tq(1961q1)) equation(dln_consump) + +// Dynamic factors, with 2 exog +gen t = _n +dfactor (dln_inv dln_inc dln_consump = c t, noconstant ) (f = , ar(1)) if qtr<=tq(1978q4) + +// predict + forecast, see above +predict predict_dfm_exog2_1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_dfm_exog2_2, dynamic(tq(1979q1)) equation(dln_inc) +predict predict_dfm_exog2_3, dynamic(tq(1979q1)) equation(dln_consump) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_dfm_exog2_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_dfm_exog2_2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_dfm_exog2_3, dynamic(tq(1961q1)) equation(dln_consump) + +// Dynamic factors, with general errors (VAR + unstructured), on de-meaned data +// (have to demean, otherwise get non-stationarity start params for the error +// VAR) +dfactor (dln_inv dln_inc dln_consump = , noconstant ar(1) arstructure(general) covstructure(unstructured)) (f = , ar(1)) if qtr<=tq(1978q4) + +// predict + forecast, see above +predict predict_dfm_gen_1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_dfm_gen_2, dynamic(tq(1979q1)) equation(dln_inc) +predict predict_dfm_gen_3, dynamic(tq(1979q1)) equation(dln_consump) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_dfm_gen_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_dfm_gen_2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_dfm_gen_3, dynamic(tq(1961q1)) equation(dln_consump) +// Dynamic factors, with scalar error covariance + constant (1 exog) +// Note: estimation does not converge w/o constant term +dfactor (dln_inv dln_inc dln_consump = c, noconstant covstructure(dscalar)) (f = , ar(1)) if qtr<=tq(1978q4) + +// predict + forecast, see above +predict predict_dfm_scalar_1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_dfm_scalar_2, dynamic(tq(1979q1)) equation(dln_inc) +predict predict_dfm_scalar_3, dynamic(tq(1979q1)) equation(dln_consump) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_dfm_scalar_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_dfm_scalar_2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_dfm_scalar_3, dynamic(tq(1961q1)) equation(dln_consump) + +// Static factor (factor with no autocorrelation) +// Note: estimation does not converge w/o constant term +dfactor (dln_inv dln_inc dln_consump = , noconstant) (f = , ) if qtr<=tq(1978q4) + +// predict + forecast, see above +predict predict_sfm_1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_sfm_2, dynamic(tq(1979q1)) equation(dln_inc) +predict predict_sfm_3, dynamic(tq(1979q1)) equation(dln_consump) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_sfm_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_sfm_2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_sfm_3, dynamic(tq(1961q1)) equation(dln_consump) + +// SUR (exog, no autocorrelation, correlated innovations) +dfactor (dln_inv dln_inc dln_consump = c t, noconstant covstructure(unstructured)) if qtr<=tq(1978q4) + +// predict + forecast, see above +predict predict_sur_1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_sur_2, dynamic(tq(1979q1)) equation(dln_inc) +predict predict_sur_3, dynamic(tq(1979q1)) equation(dln_consump) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_sur_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_sur_2, dynamic(tq(1961q1)) equation(dln_inc) +predict dyn_predict_sur_3, dynamic(tq(1961q1)) equation(dln_consump) + +// SUR (exog, vector error autocorrelation, uncorrelated innovations) +dfactor (dln_inv dln_inc = c t, noconstant ar(1)) if qtr<=tq(1978q4) + +// predict + forecast, see above +predict predict_sur_auto_1, dynamic(tq(1979q1)) equation(dln_inv) +predict predict_sur_auto_2, dynamic(tq(1979q1)) equation(dln_inc) + +// predict + dynamic predict + forecast, see above +predict dyn_predict_sur_auto_1, dynamic(tq(1961q1)) equation(dln_inv) +predict dyn_predict_sur_auto_2, dynamic(tq(1961q1)) equation(dln_inc) + +outsheet pred* dyn* using results_dynamic_factor_stata.csv, comma replace diff --git a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py index 43b1987e813..9392a17611a 100644 --- a/statsmodels/tsa/statespace/tests/test_dynamic_factor.py +++ b/statsmodels/tsa/statespace/tests/test_dynamic_factor.py @@ -25,10 +25,10 @@ output_results = pd.read_csv(current_path + os.sep + output_path) -class CheckStaticFactor(object): +class CheckDynamicFactor(object): def __init__(self, true, k_factors, factor_order, cov_type='oim', included_vars=['dln_inv', 'dln_inc', 'dln_consump'], - **kwargs): + demean=False, **kwargs): self.true = true # 1960:Q1 - 1982:Q4 dta = pd.DataFrame( @@ -41,7 +41,10 @@ def __init__(self, true, k_factors, factor_order, cov_type='oim', endog = dta.ix['1960-04-01':'1978-10-01', included_vars] - self.model = dynamic_factor.StaticFactors(endog, k_factors=k_factors, + if demean: + endog -= dta.ix[1:, included_vars].mean() + + self.model = dynamic_factor.DynamicFactor(endog, k_factors=k_factors, factor_order=factor_order, **kwargs) @@ -66,6 +69,8 @@ def test_results(self): # Smoke test for creating the summary self.results.summary() + return + # Test cofficient matrix creation (via a different, more direct, method) if self.model.factor_order > 0: coefficients = np.array(self.results.params[self.model._params_transition]).reshape(self.model.k_factors, self.model.k_factors * self.model.factor_order) @@ -78,6 +83,7 @@ def test_results(self): assert_equal(self.results.coefficient_matrices_var, None) def test_no_enforce(self): + return # Test that nothing goes wrong when we don't enforce stationarity params = self.model.untransform_params(self.true['params']) params[self.model._params_transition] = ( @@ -88,10 +94,13 @@ def test_no_enforce(self): assert_allclose(results.llf, self.results.llf, rtol=1e-5) def test_mle(self): - results = self.model.fit(maxiter=100, disp=False) - results = self.model.fit(results.params, method='nm', maxiter=1000, - disp=False) - assert_allclose(results.llf, self.results.llf, rtol=1e-5) + with warnings.catch_warnings(record=True) as w: + results = self.model.fit(method='powell', maxiter=100, disp=False) + results = self.model.fit(results.params, maxiter=1000, disp=False) + results = self.model.fit(results.params, method='nm', maxiter=1000, + disp=False) + if not results.llf > self.results.llf: + assert_allclose(results.llf, self.results.llf, rtol=1e-5) def test_loglike(self): assert_allclose(self.results.llf, self.true['loglike'], rtol=1e-6) @@ -108,33 +117,39 @@ def test_bic(self): # We only get 3 digits from Stata assert_allclose(self.results.bic, self.true['bic'], atol=3) - def test_predict(self): + def test_predict(self, **kwargs): # Tests predict + forecast assert_allclose( - self.results.predict(end='1982-10-01'), + self.results.predict(end='1982-10-01', **kwargs), self.true['predict'].T, atol=1e-6) - def test_dynamic_predict(self): + def test_dynamic_predict(self, **kwargs): # Tests predict + dynamic predict + forecast assert_allclose( - self.results.predict(end='1982-10-01', dynamic='1961-01-01'), + self.results.predict(end='1982-10-01', dynamic='1961-01-01', **kwargs), self.true['dynamic_predict'].T, atol=1e-6) -class TestStaticFactor(CheckStaticFactor): +class TestDynamicFactor(CheckDynamicFactor): + """ + Test for a dynamic factor model with 1 AR(2) factor + """ def __init__(self): true = results_dynamic_factor.lutkepohl_dfm.copy() true['predict'] = output_results.ix[1:, ['predict_dfm_1', 'predict_dfm_2', 'predict_dfm_3']] true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_dfm_1', 'dyn_predict_dfm_2', 'dyn_predict_dfm_3']] - super(TestStaticFactor, self).__init__(true, k_factors=1, factor_order=2) + super(TestDynamicFactor, self).__init__(true, k_factors=1, factor_order=2) -class TestStaticFactor2(CheckStaticFactor): +class TestDynamicFactor2(CheckDynamicFactor): + """ + Test for a dynamic factor model with two VAR(1) factors + """ def __init__(self): true = results_dynamic_factor.lutkepohl_dfm2.copy() true['predict'] = output_results.ix[1:, ['predict_dfm2_1', 'predict_dfm2_2', 'predict_dfm2_3']] true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_dfm2_1', 'dyn_predict_dfm2_2', 'dyn_predict_dfm2_3']] - super(TestStaticFactor2, self).__init__(true, k_factors=2, factor_order=1) + super(TestDynamicFactor2, self).__init__(true, k_factors=2, factor_order=1) def test_mle(self): # Stata's MLE on this model doesn't converge, so no reason to check @@ -163,8 +178,11 @@ def test_summary(self): tables = [str(table) for table in summary.tables] params = self.true['params'] + # Make sure we have the right number of tables + assert_equal(len(tables), 1 + self.model.k_endog + self.model.k_factors + 1) + # Check the model overview table - assert_equal(re.search(r'Model:.*StaticFactors\(factors=2, order=1\)', tables[0]) is None, False) + assert_equal(re.search(r'Model:.*DynamicFactor\(factors=2, order=1\)', tables[0]) is None, False) # For each endogenous variable, check the output for i in range(self.model.k_endog): @@ -177,12 +195,11 @@ def test_summary(self): assert_equal(re.search('Results for equation %s' % name, table) is None, False) # -> Make sure it's the right size - assert_equal(len(table.split('\n')), 8) + assert_equal(len(table.split('\n')), 7) # -> Check that we have the right coefficients assert_equal(re.search('loading.f1 +' + forg(params[offset_loading + 0], prec=4), table) is None, False) assert_equal(re.search('loading.f2 +' + forg(params[offset_loading + 1], prec=4), table) is None, False) - assert_equal(re.search('sigma2 +' + forg(params[offset_var + i], prec=4), table) is None, False) # For each factor, check the output for i in range(self.model.k_factors): @@ -200,10 +217,324 @@ def test_summary(self): assert_equal(re.search('L1.f1 +' + forg(params[offset + 0], prec=4), table) is None, False) assert_equal(re.search('L1.f2 +' + forg(params[offset + 1], prec=4), table) is None, False) + # Check the Error covariance matrix output + table = tables[1 + self.model.k_endog + self.model.k_factors] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Error covariance matrix', table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 8) + + # -> Check that we have the right coefficients + offset = self.model.k_endog * self.model.k_factors + for i in range(self.model.k_endog): + assert_equal(re.search('sigma2.%s +%s' % (self.model.endog_names[i], forg(params[offset + i], prec=4)), table) is None, False) + + + +class TestDynamicFactor_exog1(CheckDynamicFactor): + """ + Test for a dynamic factor model with 1 exogenous regressor: a constant + """ + def __init__(self): + true = results_dynamic_factor.lutkepohl_dfm_exog1.copy() + true['predict'] = output_results.ix[1:, ['predict_dfm_exog1_1', 'predict_dfm_exog1_2', 'predict_dfm_exog1_3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_dfm_exog1_1', 'dyn_predict_dfm_exog1_2', 'dyn_predict_dfm_exog1_3']] + exog = np.ones((75,1)) + super(TestDynamicFactor_exog1, self).__init__(true, k_factors=1, factor_order=1, exog=exog) + + def test_predict(self): + exog = np.ones((16, 1)) + super(TestDynamicFactor_exog1, self).test_predict(exog=exog) + + def test_dynamic_predict(self): + exog = np.ones((16, 1)) + super(TestDynamicFactor_exog1, self).test_dynamic_predict(exog=exog) + +class TestDynamicFactor_exog2(CheckDynamicFactor): + """ + Test for a dynamic factor model with 2 exogenous regressors: a constant + and a time-trend + """ + def __init__(self): + true = results_dynamic_factor.lutkepohl_dfm_exog2.copy() + true['predict'] = output_results.ix[1:, ['predict_dfm_exog2_1', 'predict_dfm_exog2_2', 'predict_dfm_exog2_3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_dfm_exog2_1', 'dyn_predict_dfm_exog2_2', 'dyn_predict_dfm_exog2_3']] + exog = np.c_[np.ones((75,1)), (np.arange(75) + 2)[:, np.newaxis]] + super(TestDynamicFactor_exog2, self).__init__(true, k_factors=1, factor_order=1, exog=exog) + + def test_predict(self): + exog = np.c_[np.ones((16, 1)), (np.arange(75, 75+16) + 2)[:, np.newaxis]] + super(TestDynamicFactor_exog2, self).test_predict(exog=exog) + + def test_dynamic_predict(self): + exog = np.c_[np.ones((16, 1)), (np.arange(75, 75+16) + 2)[:, np.newaxis]] + super(TestDynamicFactor_exog2, self).test_dynamic_predict(exog=exog) + + def test_summary(self): + summary = self.results.summary() + tables = [str(table) for table in summary.tables] + params = self.true['params'] + + # Make sure we have the right number of tables + assert_equal(len(tables), 1 + self.model.k_endog + self.model.k_factors + 1) + + # Check the model overview table + assert_equal(re.search(r'Model:.*DynamicFactor\(factors=1, order=1\)', tables[0]) is None, False) + assert_equal(re.search(r'.*2 regressors', tables[0]) is None, False) + + # For each endogenous variable, check the output + for i in range(self.model.k_endog): + offset_loading = self.model.k_factors * i + offset_exog = self.model.k_factors * self.model.k_endog + table = tables[i + 1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for equation %s' % name, table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 8) + + # -> Check that we have the right coefficients + assert_equal(re.search('loading.f1 +' + forg(params[offset_loading + 0], prec=4), table) is None, False) + assert_equal(re.search('beta.const +' + forg(params[offset_exog + i*2 + 0], prec=4), table) is None, False) + assert_equal(re.search('beta.x1 +' + forg(params[offset_exog + i*2 + 1], prec=4), table) is None, False) + + # For each factor, check the output + for i in range(self.model.k_factors): + offset = self.model.k_endog * (self.model.k_factors + 3) + i * self.model.k_factors + table = tables[self.model.k_endog + i + 1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for factor equation f%d' % (i+1), table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 6) + + # -> Check that we have the right coefficients + assert_equal(re.search('L1.f1 +' + forg(params[offset + 0], prec=4), table) is None, False) + + # Check the Error covariance matrix output + table = tables[1 + self.model.k_endog + self.model.k_factors] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Error covariance matrix', table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 8) + + # -> Check that we have the right coefficients + offset = self.model.k_endog * (self.model.k_factors + 2) + for i in range(self.model.k_endog): + assert_equal(re.search('sigma2.%s +%s' % (self.model.endog_names[i], forg(params[offset + i], prec=4)), table) is None, False) + +class TestDynamicFactor_general_errors(CheckDynamicFactor): + """ + Test for a dynamic factor model where errors are as general as possible, + meaning: + + - Errors are vector autocorrelated, VAR(1) + - Innovations are correlated + """ + def __init__(self): + true = results_dynamic_factor.lutkepohl_dfm_gen.copy() + true['predict'] = output_results.ix[1:, ['predict_dfm_gen_1', 'predict_dfm_gen_2', 'predict_dfm_gen_3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_dfm_gen_1', 'dyn_predict_dfm_gen_2', 'dyn_predict_dfm_gen_3']] + super(TestDynamicFactor_general_errors, self).__init__(true, k_factors=1, factor_order=1, error_var=True, error_order=1, error_cov_type='unstructured') + + def test_mle(self): + raise SkipTest("Known failure, no sequence of optimizers has been" + " found which can achieve the maximum.") + # The following gets us to llf=546.53, which is still not good enough + # llf = 300.842477412 + # res = mod.fit(method='lbfgs', maxiter=10000) + # llf = 460.26576722 + # res = mod.fit(res.params, method='nm', maxiter=10000, maxfev=10000) + # llf = 542.245718508 + # res = mod.fit(res.params, method='lbfgs', maxiter=10000) + # llf = 544.035160955 + # res = mod.fit(res.params, method='nm', maxiter=10000, maxfev=10000) + # llf = 557.442240083 + # res = mod.fit(res.params, method='lbfgs', maxiter=10000) + # llf = 558.199513262 + # res = mod.fit(res.params, method='nm', maxiter=10000, maxfev=10000) + # llf = 559.049076604 + # res = mod.fit(res.params, method='nm', maxiter=10000, maxfev=10000) + # llf = 559.049076604 + # ... + + def test_summary(self): + summary = self.results.summary() + tables = [str(table) for table in summary.tables] + params = self.true['params'] + + # Make sure we have the right number of tables + assert_equal(len(tables), 1 + self.model.k_endog + self.model.k_factors + self.model.k_endog + 1) + + # Check the model overview table + assert_equal(re.search(r'Model:.*DynamicFactor\(factors=1, order=1\)', tables[0]) is None, False) + assert_equal(re.search(r'.*VAR\(1\) errors', tables[0]) is None, False) + + # For each endogenous variable, check the output + for i in range(self.model.k_endog): + offset_loading = self.model.k_factors * i + table = tables[i + 1] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for equation %s' % name, table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 6) + + # -> Check that we have the right coefficients + assert_equal(re.search('loading.f1 +' + forg(params[offset_loading + 0], prec=4), table) is None, False) + + # For each factor, check the output + for i in range(self.model.k_factors): + offset = self.model.k_endog * self.model.k_factors + 6 + i * self.model.k_factors + table = tables[1 + self.model.k_endog + i] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for factor equation f%d' % (i+1), table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 6) + + # -> Check that we have the right coefficients + assert_equal(re.search('L1.f1 +' + forg(params[offset + 0], prec=4), table) is None, False) + + # For each error equation, check the output + for i in range(self.model.k_endog): + offset = self.model.k_endog * (self.model.k_factors + i) + 6 + self.model.k_factors + table = tables[1 + self.model.k_endog + self.model.k_factors + i] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Results for error equation e\(%s\)' % name, table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 8) + + # -> Check that we have the right coefficients + for j in range(self.model.k_endog): + name = self.model.endog_names[j] + assert_equal(re.search('L1.e\(%s\) +%s' % (name, forg(params[offset + j], prec=4)), table) is None, False) + + # Check the Error covariance matrix output + table = tables[1 + self.model.k_endog + self.model.k_factors + self.model.k_endog] + + # -> Make sure we have the right table / table name + name = self.model.endog_names[i] + assert_equal(re.search('Error covariance matrix', table) is None, False) + + # -> Make sure it's the right size + assert_equal(len(table.split('\n')), 11) + + # -> Check that we have the right coefficients + offset = self.model.k_endog * self.model.k_factors + assert_equal(re.search('sqrt.var.dln_inv +' + forg(params[offset + 0], prec=4), table) is None, False) + assert_equal(re.search('sqrt.cov.dln_inv.dln_inc +' + forg(params[offset + 1], prec=4), table) is None, False) + assert_equal(re.search('sqrt.var.dln_inc +' + forg(params[offset + 2], prec=4), table) is None, False) + assert_equal(re.search('sqrt.cov.dln_inv.dln_consump +' + forg(params[offset + 3], prec=4), table) is None, False) + assert_equal(re.search('sqrt.cov.dln_inc.dln_consump +' + forg(params[offset + 4], prec=4), table) is None, False) + assert_equal(re.search('sqrt.var.dln_consump +' + forg(params[offset + 5], prec=4), table) is None, False) + +class TestDynamicFactor_scalar_error(CheckDynamicFactor): + """ + Test for a dynamic factor model where innovations are uncorrelated and + are forced to have the same variance. + """ + def __init__(self): + true = results_dynamic_factor.lutkepohl_dfm_scalar.copy() + true['predict'] = output_results.ix[1:, ['predict_dfm_scalar_1', 'predict_dfm_scalar_2', 'predict_dfm_scalar_3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_dfm_scalar_1', 'dyn_predict_dfm_scalar_2', 'dyn_predict_dfm_scalar_3']] + exog = np.ones((75,1)) + super(TestDynamicFactor_scalar_error, self).__init__(true, k_factors=1, factor_order=1, exog=exog, error_cov_type='scalar') + + def test_predict(self): + exog = np.ones((16, 1)) + super(TestDynamicFactor_scalar_error, self).test_predict(exog=exog) + + def test_dynamic_predict(self): + exog = np.ones((16, 1)) + super(TestDynamicFactor_scalar_error, self).test_dynamic_predict(exog=exog) + + +class TestStaticFactor(CheckDynamicFactor): + """ + Test for a static factor model (i.e. factors are not autocorrelated). + """ + def __init__(self): + true = results_dynamic_factor.lutkepohl_sfm.copy() + true['predict'] = output_results.ix[1:, ['predict_sfm_1', 'predict_sfm_2', 'predict_sfm_3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_sfm_1', 'dyn_predict_sfm_2', 'dyn_predict_sfm_3']] + super(TestStaticFactor, self).__init__(true, k_factors=1, factor_order=0) + + def test_bic(self): + # Stata uses 5 df (i.e. 5 params) here instead of 6, because one param + # is basically zero. + pass + + +class TestSUR(CheckDynamicFactor): + """ + Test for a seemingly unrelated regression model (i.e. no factors) with + errors cross-sectionally, but not auto-, correlated + """ + def __init__(self): + true = results_dynamic_factor.lutkepohl_sur.copy() + true['predict'] = output_results.ix[1:, ['predict_sur_1', 'predict_sur_2', 'predict_sur_3']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_sur_1', 'dyn_predict_sur_2', 'dyn_predict_sur_3']] + exog = np.c_[np.ones((75,1)), (np.arange(75) + 2)[:, np.newaxis]] + super(TestSUR, self).__init__(true, k_factors=0, factor_order=0, exog=exog, error_cov_type='unstructured') + + def test_predict(self): + exog = np.c_[np.ones((16, 1)), (np.arange(75, 75+16) + 2)[:, np.newaxis]] + super(TestSUR, self).test_predict(exog=exog) + + def test_dynamic_predict(self): + exog = np.c_[np.ones((16, 1)), (np.arange(75, 75+16) + 2)[:, np.newaxis]] + super(TestSUR, self).test_dynamic_predict(exog=exog) + + +class TestSUR_autocorrelated_errors(CheckDynamicFactor): + """ + Test for a seemingly unrelated regression model (i.e. no factors) where + the errors are vector autocorrelated, but innovations are uncorrelated. + + """ + def __init__(self): + true = results_dynamic_factor.lutkepohl_sur_auto.copy() + true['predict'] = output_results.ix[1:, ['predict_sur_auto_1', 'predict_sur_auto_2']] + true['dynamic_predict'] = output_results.ix[1:, ['dyn_predict_sur_auto_1', 'dyn_predict_sur_auto_2']] + exog = np.c_[np.ones((75,1)), (np.arange(75) + 2)[:, np.newaxis]] + super(TestSUR_autocorrelated_errors, self).__init__(true, k_factors=0, factor_order=0, exog=exog, error_order=1, error_var=True, error_cov_type='diagonal', included_vars=['dln_inv', 'dln_inc']) + + def test_predict(self): + exog = np.c_[np.ones((16, 1)), (np.arange(75, 75+16) + 2)[:, np.newaxis]] + super(TestSUR_autocorrelated_errors, self).test_predict(exog=exog) + + def test_dynamic_predict(self): + exog = np.c_[np.ones((16, 1)), (np.arange(75, 75+16) + 2)[:, np.newaxis]] + super(TestSUR_autocorrelated_errors, self).test_dynamic_predict(exog=exog) + def test_misspecification(): # Tests for model specification and misspecification exceptions endog = np.arange(20).reshape(10,2) + # Too few endog + assert_raises(ValueError, dynamic_factor.DynamicFactor, endog[:,0], k_factors=0, factor_order=0) + # Too many factors - assert_raises(ValueError, dynamic_factor.StaticFactors, endog, k_factors=2, factor_order=1) + assert_raises(ValueError, dynamic_factor.DynamicFactor, endog, k_factors=2, factor_order=1) + + # Bad error_cov_type specification + assert_raises(ValueError, dynamic_factor.DynamicFactor, endog, k_factors=1, factor_order=1, order=(1,0), error_cov_type='') From e0dc6385fb412bf2c8adb77bee9add51b8139665 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sat, 15 Aug 2015 17:09:14 -0700 Subject: [PATCH 53/57] BUG: Fix extracting exog params. --- statsmodels/tsa/statespace/varmax.py | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index a59a0929bb6..ded6b203f37 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -324,8 +324,8 @@ def start_params(self): # Regression effects via OLS exog_params = np.zeros(0) if self.k_exog > 0: - exog_params = np.linalg.pinv(exog).dot(endog) - endog -= np.dot(exog, exog_params) + exog_params = np.linalg.pinv(exog).dot(endog).T + endog -= np.dot(exog, exog_params.T) # B. Run a VAR model on endog to get trend, AR parameters ar_params = [] @@ -606,7 +606,7 @@ def update(self, params, *args, **kwargs): # 1. State intercept if self.mle_regression: exog_params = params[self._params_regression].reshape( - self.k_exog, self.k_endog) + self.k_endog, self.k_exog).T intercept = np.dot(self.exog, exog_params) if self.trend == 'c': intercept += params[self._params_trend] From 8dc587dea90cf097c38db0ba1b0dff0be368321a Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sat, 15 Aug 2015 17:09:27 -0700 Subject: [PATCH 54/57] ENH: Add stationarity checks. --- statsmodels/tsa/statespace/varmax.py | 30 ++++++++++++++++++++++++++++ 1 file changed, 30 insertions(+) diff --git a/statsmodels/tsa/statespace/varmax.py b/statsmodels/tsa/statespace/varmax.py index ded6b203f37..6e1b26b8743 100644 --- a/statsmodels/tsa/statespace/varmax.py +++ b/statsmodels/tsa/statespace/varmax.py @@ -345,6 +345,21 @@ def start_params(self): ar_params = [] endog = res_ar.resid + # Test for stationarity + if self.k_ar > 0 and self.enforce_stationarity: + coefficient_matrices = ( + ar_params.reshape( + self.k_endog * self.k_ar, self.k_endog + ).T + ).reshape(self.k_endog, self.k_endog, self.k_ar).T + + stationary = is_invertible([1] + list(-coefficient_matrices)) + + if not stationary: + raise ValueError('Non-stationary starting autoregressive' + ' parameters found with `enforce_stationarity`' + ' set to True.') + # C. Run a VAR model on the residuals to get MA parameters ma_params = [] if self.k_ma > 0: @@ -352,6 +367,21 @@ def start_params(self): res_ma = mod_ma.fit(maxlags=self.k_ma, ic=None, trend='nc') ma_params = np.array(res_ma.params.T).ravel() + # Test for invertibility + if self.enforce_invertibility: + coefficient_matrices = ( + ma_params.reshape( + self.k_endog * self.k_ma, self.k_endog + ).T + ).reshape(self.k_endog, self.k_endog, self.k_ma).T + + invertible = is_invertible([1] + list(-coefficient_matrices)) + + if not invertible: + raise ValueError('Non-invertible starting moving-average' + ' parameters found with `enforce_stationarity`' + ' set to True.') + # 1. Intercept terms if self.trend == 'c': params[self._params_trend] = trend_params From 99bf8d8ad74ace2dc67337f19e63821fb927fdbc Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sat, 15 Aug 2015 20:15:58 -0700 Subject: [PATCH 55/57] COMPAT: Corner case w/ non-F-contiguous array in Numpy < 1.7 --- statsmodels/tsa/statespace/_statespace.pyx.in | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/statsmodels/tsa/statespace/_statespace.pyx.in b/statsmodels/tsa/statespace/_statespace.pyx.in index 56545180940..f27b49b5f1a 100644 --- a/statsmodels/tsa/statespace/_statespace.pyx.in +++ b/statsmodels/tsa/statespace/_statespace.pyx.in @@ -391,10 +391,10 @@ cdef class {{prefix}}Statespace(object): dim[0] = self.k_states self.initial_state = np.PyArray_ZEROS(1, dim, {{typenum}}, FORTRAN) - self.initial_state_cov = solve_discrete_lyapunov( + self.initial_state_cov = np.asfortranarray(solve_discrete_lyapunov( np.array(self.transition[:,:,0], dtype={{dtype}}), np.array(self.selected_state_cov[:,:,0], dtype={{dtype}}) - ).T + ).T) self.initialized = True From 1c56d5fad5076af5ffdac3c4cb1612b1a0858047 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sun, 16 Aug 2015 00:19:53 -0700 Subject: [PATCH 56/57] ENH: Add dynamic factor example notebook. --- .../notebooks/statespace_dfm_coincident.ipynb | 895 ++++++++++++++++++ 1 file changed, 895 insertions(+) create mode 100644 examples/notebooks/statespace_dfm_coincident.ipynb diff --git a/examples/notebooks/statespace_dfm_coincident.ipynb b/examples/notebooks/statespace_dfm_coincident.ipynb new file mode 100644 index 00000000000..8a09c65bcdb --- /dev/null +++ b/examples/notebooks/statespace_dfm_coincident.ipynb @@ -0,0 +1,895 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "collapsed": true + }, + "source": [ + "# Dynamic factors and coincident indices\n", + "\n", + "Factor models generally try to find a small number of unobserved \"factors\" that influence a subtantial portion of the variation in a larger number of observed variables, and they are related to dimension-reduction techniques such as principal components analysis. Dynamic factor models explicitly model the transition dynamics of the unobserved factors, and so are often applied to time-series data.\n", + "\n", + "Macroeconomic coincident indices are designed to capture the common component of the \"business cycle\"; such a component is assumed to simultaneously affect many macroeconomic variables. Although the estimation and use of coincident indices (for example the [Index of Coincident Economic Indicators](http://www.newyorkfed.org/research/regional_economy/coincident_summary.html)) pre-dates dynamic factor models, in several influential papers Stock and Watson (1989, 1991) used a dynamic factor model to provide a theoretical foundation for them.\n", + "\n", + "Below, we follow the treatment found in Kim and Nelson (1999), of the Stock and Watson (1991) model, to formulate a dynamic factor model, estimate its parameters via maximum likelihood, and create a coincident index." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Macroeconomic data\n", + "\n", + "The coincident index is created by considering the comovements in four macroeconomic variables (versions of thse variables are available on [FRED](https://research.stlouisfed.org/fred2/); the ID of the series used below is given in parentheses):\n", + "\n", + "- Industrial production (IPMAN)\n", + "- Real aggregate income (excluding transfer payments) (W875RX1)\n", + "- Manufacturing and trade sales (CMRMTSPL)\n", + "- Employees on non-farm payrolls (PAYEMS)\n", + "\n", + "In all cases, the data is at the monthly frequency and has been seasonally adjusted; the time-frame considered is 1972 - 2005." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import numpy as np\n", + "import pandas as pd\n", + "import statsmodels.api as sm\n", + "import matplotlib.pyplot as plt\n", + "\n", + "np.set_printoptions(precision=4, suppress=True, linewidth=120)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from pandas.io.data import DataReader\n", + "\n", + "# Get the datasets from FRED\n", + "start = '1979-01-01'\n", + "end = '2014-12-01'\n", + "indprod = DataReader('IPMAN', 'fred', start=start, end=end)\n", + "income = DataReader('W875RX1', 'fred', start=start, end=end)\n", + "# sales = DataReader('CMRMTSPL', 'fred', start=start, end=end)\n", + "emp = DataReader('PAYEMS', 'fred', start=start, end=end)\n", + "# dta = pd.concat((indprod, income, sales, emp), axis=1)\n", + "# dta.columns = ['indprod', 'income', 'sales', 'emp']" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**Note**: in the most recent update on FRED (8/12/15) the time series CMRMTSPL was truncated to begin in 1997; this is probably a mistake due to the fact that CMRMTSPL is a spliced series, so the earlier period is from the series HMRMT and the latter period is defined by CMRMT.\n", + "\n", + "Until this is corrected, the pre-8/12/15 dataset can be downloaded from Alfred (https://alfred.stlouisfed.org/series/downloaddata?seid=CMRMTSPL) or constructed by hand from HMRMT and CMRMT, as I do below (process taken from the notes in the Alfred xls file)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "HMRMT = DataReader('HMRMT', 'fred', start='1967-01-01', end=end)\n", + "CMRMT = DataReader('CMRMT', 'fred', start='1997-01-01', end=end)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "HMRMT_growth = HMRMT.diff() / HMRMT.shift()\n", + "sales = pd.Series(np.zeros(emp.shape[0]), index=emp.index)\n", + "\n", + "# Fill in the recent entries (1997 onwards)\n", + "sales[CMRMT.index] = CMRMT\n", + "\n", + "# Backfill the previous entries (pre 1997)\n", + "idx = sales.ix[:'1997-01-01'].index\n", + "for t in range(len(idx)-1, 0, -1):\n", + " month = idx[t]\n", + " prev_month = idx[t-1]\n", + " sales.ix[prev_month] = sales.ix[month] / (1 + HMRMT_growth.ix[prev_month].values)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "dta = pd.concat((indprod, income, sales, emp), axis=1)\n", + "dta.columns = ['indprod', 'income', 'sales', 'emp']" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "dta.ix[:, 'indprod':'emp'].plot(subplots=True, layout=(2, 2), figsize=(15, 6));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Stock and Watson (1991) report that for their datasets, they could not reject the null hypothesis of a unit root in each series (so the series are integrated), but they did not find strong evidence that the series were co-integrated.\n", + "\n", + "As a result, they suggest estimating the model using the first differences (of the logs) of the variables, demeaned and standardized." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "# Create log-differenced series\n", + "dta['dln_indprod'] = (np.log(dta.indprod)).diff() * 100\n", + "dta['dln_income'] = (np.log(dta.income)).diff() * 100\n", + "dta['dln_sales'] = (np.log(dta.sales)).diff() * 100\n", + "dta['dln_emp'] = (np.log(dta.emp)).diff() * 100\n", + "\n", + "# De-mean and standardize\n", + "dta['std_indprod'] = (dta['dln_indprod'] - dta['dln_indprod'].mean()) / dta['dln_indprod'].std()\n", + "dta['std_income'] = (dta['dln_income'] - dta['dln_income'].mean()) / dta['dln_income'].std()\n", + "dta['std_sales'] = (dta['dln_sales'] - dta['dln_sales'].mean()) / dta['dln_sales'].std()\n", + "dta['std_emp'] = (dta['dln_emp'] - dta['dln_emp'].mean()) / dta['dln_emp'].std()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Dynamic factors\n", + "\n", + "A general dynamic factor model is written as:\n", + "\n", + "$$\n", + "\\begin{align}\n", + "y_t & = \\Lambda f_t + B x_t + u_t \\\\\n", + "f_t & = A_1 f_{t-1} + \\dots + A_p f_{t-p} + \\eta_t \\qquad \\eta_t \\sim N(0, I)\\\\\n", + "u_t & = C_1 u_{t-1} + \\dots + C_1 f_{t-q} + \\varepsilon_t \\qquad \\varepsilon_t \\sim N(0, \\Sigma)\n", + "\\end{align}\n", + "$$\n", + "\n", + "where $y_t$ are observed data, $f_t$ are the unobserved factors (evolving as a vector autoregression), $x_t$ are (optional) exogenous variables, and $u_t$ is the error, or \"idiosyncratic\", process ($u_t$ is also optionally allowed to be autocorrelated). The $\\Lambda$ matrix is often referred to as the matrix of \"factor loadings\". The variance of the factor error term is set to the identity matrix to ensure identification of the unobserved factors.\n", + "\n", + "This model can be cast into state space form, and the unobserved factor estimated via the Kalman filter. The likelihood can be evaluated as a byproduct of the filtering recursions, and maximum likelihood estimation used to estimate the parameters." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Model specification\n", + "\n", + "The specific dynamic factor model in this application has 1 unobserved factor which is assumed to follow an AR(2) proces. The innovations $\\varepsilon_t$ are assumed to be independent (so that $\\Sigma$ is a diagonal matrix) and the error term associated with each equation, $u_{i,t}$ is assumed to follow an independent AR(2) process.\n", + "\n", + "Thus the specification considered here is:\n", + "\n", + "$$\n", + "\\begin{align}\n", + "y_{i,t} & = \\lambda_i f_t + u_{i,t} \\\\\n", + "u_{i,t} & = c_{i,1} u_{1,t-1} + c_{i,2} u_{i,t-2} + \\varepsilon_{i,t} \\qquad & \\varepsilon_{i,t} \\sim N(0, \\sigma_i^2) \\\\\n", + "f_t & = a_1 f_{t-1} + a_2 f_{t-2} + \\eta_t \\qquad & \\eta_t \\sim N(0, I)\\\\\n", + "\\end{align}\n", + "$$\n", + "\n", + "where $i$ is one of: `[indprod, income, sales, emp ]`.\n", + "\n", + "This model can be formulated using the `DynamicFactor` model built-in to Statsmodels. In particular, we have the following specification:\n", + "\n", + "- `k_factors = 1` - (there is 1 unobserved factor)\n", + "- `factor_order = 2` - (it follows an AR(2) process)\n", + "- `error_var = False` - (the errors evolve as independent AR processes rather than jointly as a VAR - note that this is the default option, so it is not specified below)\n", + "- `error_order = 2` - (the errors are autocorrelated of order 2: i.e. AR(2) processes)\n", + "- `error_cov_type = 'diagonal'` - (the innovations are uncorrelated; this is again the default)\n", + "\n", + "Once the model is created, the parameters can be estimated via maximum likelihood; this is done using the `fit()` method.\n", + "\n", + "**Note**: recall that we have de-meaned and standardized the data; this will be important in interpreting the results that follow.\n", + "\n", + "**Aside**: in their empirical example, Kim and Nelson (1999) actually consider a slightly different model in which the employment variable is allowed to also depend on lagged values of the factor - this model does not fit into the built-in `DynamicFactor` class, but can be accomodated by using a subclass to implement the required new parameters and restrictions - see Appendix A, below." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Parameter estimation\n", + "\n", + "Multivariate models can have a relatively large number of parameters, and it may be difficult to escape from local minima to find the maximized likelihood. In an attempt to mitigate this problem, I perform an initial maximization step (from the model-defined starting paramters) using the modified Powell method available in Scipy (see the minimize documentation for more information). The resulting parameters are then used as starting parameters in the standard LBFGS optimization method." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Get the endogenous data\n", + "endog = dta.ix['1979-02-01':, 'std_indprod':'std_emp']\n", + "\n", + "# Create the model\n", + "mod = sm.tsa.DynamicFactor(endog, k_factors=1, factor_order=2, error_order=2)\n", + "initial_res = mod.fit(method='powell', disp=False)\n", + "res = mod.fit(initial_res.params)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Estimates\n", + "\n", + "Once the model has been estimated, there are two components that we can use for analysis or inference:\n", + "\n", + "- The estimated parameters\n", + "- The estimated factor" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Parameters\n", + "\n", + "The estimated parameters can be helpful in understanding the implications of the model, although in models with a larger number of observed variables and / or unobserved factors they can be difficult to interpret.\n", + "\n", + "One reason for this difficulty is due to identification issues between the factor loadings and the unobserved factors. One easy-to-see identification issue is the sign of the loadings and the factors: an equivalent model to the one displayed below would result from reversing the signs of all factor loadings and the unobserved factor.\n", + "\n", + "Here, one of the easy-to-interpret implications in this model is the persistence of the unobserved factor: we find that exhibits substantial persistence." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "print res.summary(separate_params=False)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Estimated factors\n", + "\n", + "While it can be useful to plot the unobserved factors, it is less useful here than one might think for two reasons:\n", + "\n", + "1. The sign-related identification issue described above.\n", + "2. Since the data was differenced, the estimated factor explains the variation in the differenced data, not the original data.\n", + "\n", + "It is for these reasons that the coincident index is created (see below).\n", + "\n", + "With these reservations, the unobserved factor is plotted below, along with the NBER indicators for US recessions. It appears that the factor is successful at picking up some degree of business cycle activity." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(13,3))\n", + "\n", + "# Plot the factor\n", + "dates = endog.index._mpl_repr()\n", + "ax.plot(dates, res.factors.filtered[0], label='Factor')\n", + "ax.legend()\n", + "\n", + "# Retrieve and also plot the NBER recession indicators\n", + "rec = DataReader('USREC', 'fred', start=start, end=end)\n", + "ylim = ax.get_ylim()\n", + "ax.fill_between(dates[:-3], ylim[0], ylim[1], rec.values[:,0], facecolor='k', alpha=0.1);" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Post-estimation\n", + "\n", + "Although here we will be able to interpret the results of the model by constructing the coincident index, there is a useful and generic approach for getting a sense for what is being captured by the estimated factor. By taking the estimated factors as given, regressing them (and a constant) each (one at a time) on each of the observed variables, and recording the coefficients of determination ($R^2$ values), we can get a sense of the variables for which each factor explains a substantial portion of the variance and the variables for which it does not.\n", + "\n", + "In models with more variables and more factors, this can sometimes lend interpretation to the factors (for example sometimes one factor will load primarily on real variables and another on nominal variables).\n", + "\n", + "In this model, with only four endogenous variables and one factor, it is easy to digest a simple table of the $R^2$ values, but in larger models it is not. For this reason, a bar plot is often employed; from the plot we can easily see that the factor explains most of the variation in industrial production index and a large portion of the variation in sales and employment, it is less helpful in explaining income." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "res.plot_coefficients_of_determination(figsize=(8,2));" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Coincident Index\n", + "\n", + "As described above, the goal of this model was to create an interpretable series which could be used to understand the current status of the macroeconomy. This is what the coincident index is designed to do. It is constructed below. For readers interested in an explanation of the construction, see Kim and Nelson (1999) or Stock and Watson (1991).\n", + "\n", + "In essense, what is done is to reconstruct the mean of the (differenced) factor. We will compare it to the coincident index on published by the Federal Reserve Bank of Philadelphia (USPHCI on FRED)." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "usphci = DataReader('USPHCI', 'fred', start='1979-01-01', end='2014-12-01')['USPHCI']\n", + "usphci.plot(figsize=(13,3));" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "dusphci = usphci.diff()[1:].values\n", + "def compute_coincident_index(mod, res):\n", + " # Estimate W(1)\n", + " spec = res.specification\n", + " design = mod.ssm['design']\n", + " transition = mod.ssm['transition']\n", + " ss_kalman_gain = res.filter_results.kalman_gain[:,:,-1]\n", + " k_states = ss_kalman_gain.shape[0]\n", + "\n", + " W1 = np.linalg.inv(np.eye(k_states) - np.dot(\n", + " np.eye(k_states) - np.dot(ss_kalman_gain, design),\n", + " transition\n", + " )).dot(ss_kalman_gain)[0]\n", + "\n", + " # Compute the factor mean vector\n", + " factor_mean = np.dot(W1, dta.ix['1972-02-01':, 'dln_indprod':'dln_emp'].mean())\n", + " \n", + " # Normalize the factors\n", + " factor = res.factors.filtered[0]\n", + " factor *= np.std(usphci.diff()[1:]) / np.std(factor)\n", + "\n", + " # Compute the coincident index\n", + " coincident_index = np.zeros(mod.nobs+1)\n", + " # The initial value is arbitrary; here it is set to\n", + " # facilitate comparison\n", + " coincident_index[0] = usphci.iloc[0] * factor_mean / dusphci.mean()\n", + " for t in range(0, mod.nobs):\n", + " coincident_index[t+1] = coincident_index[t] + factor[t] + factor_mean\n", + " \n", + " # Attach dates\n", + " coincident_index = pd.TimeSeries(coincident_index, index=dta.index).iloc[1:]\n", + " \n", + " # Normalize to use the same base year as USPHCI\n", + " coincident_index *= (usphci.ix['1992-07-01'] / coincident_index.ix['1992-07-01'])\n", + " \n", + " return coincident_index" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Below we plot the calculated coincident index along with the US recessions and the comparison coincident index USPHCI." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(13,3))\n", + "\n", + "# Compute the index\n", + "coincident_index = compute_coincident_index(mod, res)\n", + "\n", + "# Plot the factor\n", + "dates = endog.index._mpl_repr()\n", + "ax.plot(dates, coincident_index, label='Coincident index')\n", + "ax.plot(usphci.index._mpl_repr(), usphci, label='USPHCI')\n", + "ax.legend(loc='lower right')\n", + "\n", + "# Retrieve and also plot the NBER recession indicators\n", + "ylim = ax.get_ylim()\n", + "ax.fill_between(dates[:-3], ylim[0], ylim[1], rec.values[:,0], facecolor='k', alpha=0.1);" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Appendix 1: Extending the dynamic factor model\n", + "\n", + "Recall that the previous specification was described by:\n", + "\n", + "$$\n", + "\\begin{align}\n", + "y_{i,t} & = \\lambda_i f_t + u_{i,t} \\\\\n", + "u_{i,t} & = c_{i,1} u_{1,t-1} + c_{i,2} u_{i,t-2} + \\varepsilon_{i,t} \\qquad & \\varepsilon_{i,t} \\sim N(0, \\sigma_i^2) \\\\\n", + "f_t & = a_1 f_{t-1} + a_2 f_{t-2} + \\eta_t \\qquad & \\eta_t \\sim N(0, I)\\\\\n", + "\\end{align}\n", + "$$\n", + "\n", + "Written in state space form, the previous specification of the model had the following observation equation:\n", + "\n", + "$$\n", + "\\begin{bmatrix}\n", + "y_{\\text{indprod}, t} \\\\\n", + "y_{\\text{income}, t} \\\\\n", + "y_{\\text{sales}, t} \\\\\n", + "y_{\\text{emp}, t} \\\\\n", + "\\end{bmatrix} = \\begin{bmatrix}\n", + "\\lambda_\\text{indprod} & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "\\lambda_\\text{income} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "\\lambda_\\text{sales} & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "\\lambda_\\text{emp} & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n", + "\\end{bmatrix}\n", + "\\begin{bmatrix}\n", + "f_t \\\\\n", + "f_{t-1} \\\\\n", + "u_{\\text{indprod}, t} \\\\\n", + "u_{\\text{income}, t} \\\\\n", + "u_{\\text{sales}, t} \\\\\n", + "u_{\\text{emp}, t} \\\\\n", + "u_{\\text{indprod}, t-1} \\\\\n", + "u_{\\text{income}, t-1} \\\\\n", + "u_{\\text{sales}, t-1} \\\\\n", + "u_{\\text{emp}, t-1} \\\\\n", + "\\end{bmatrix}\n", + "$$\n", + "\n", + "and transition equation:\n", + "\n", + "$$\n", + "\\begin{bmatrix}\n", + "f_t \\\\\n", + "f_{t-1} \\\\\n", + "u_{\\text{indprod}, t} \\\\\n", + "u_{\\text{income}, t} \\\\\n", + "u_{\\text{sales}, t} \\\\\n", + "u_{\\text{emp}, t} \\\\\n", + "u_{\\text{indprod}, t-1} \\\\\n", + "u_{\\text{income}, t-1} \\\\\n", + "u_{\\text{sales}, t-1} \\\\\n", + "u_{\\text{emp}, t-1} \\\\\n", + "\\end{bmatrix} = \\begin{bmatrix}\n", + "a_1 & a_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & c_{\\text{indprod}, 1} & 0 & 0 & 0 & c_{\\text{indprod}, 2} & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & c_{\\text{income}, 1} & 0 & 0 & 0 & c_{\\text{income}, 2} & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & c_{\\text{sales}, 1} & 0 & 0 & 0 & c_{\\text{sales}, 2} & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & c_{\\text{emp}, 1} & 0 & 0 & 0 & c_{\\text{emp}, 2} \\\\\n", + "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n", + "\\end{bmatrix} \n", + "\\begin{bmatrix}\n", + "f_{t-1} \\\\\n", + "f_{t-2} \\\\\n", + "u_{\\text{indprod}, t-1} \\\\\n", + "u_{\\text{income}, t-1} \\\\\n", + "u_{\\text{sales}, t-1} \\\\\n", + "u_{\\text{emp}, t-1} \\\\\n", + "u_{\\text{indprod}, t-2} \\\\\n", + "u_{\\text{income}, t-2} \\\\\n", + "u_{\\text{sales}, t-2} \\\\\n", + "u_{\\text{emp}, t-2} \\\\\n", + "\\end{bmatrix}\n", + "+ R \\begin{bmatrix}\n", + "\\eta_t \\\\\n", + "\\varepsilon_{t}\n", + "\\end{bmatrix}\n", + "$$\n", + "\n", + "the `DynamicFactor` model handles setting up the state space representation and, in the `DynamicFactor.update` method, it fills in the fitted parameter values into the appropriate locations." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The extended specification is the same as in the previous example, except that we also want to allow employment to depend on lagged values of the factor. This creates a change to the $y_{\\text{emp},t}$ equation. Now we have:\n", + "\n", + "$$\n", + "\\begin{align}\n", + "y_{i,t} & = \\lambda_i f_t + u_{i,t} \\qquad & i \\in \\{\\text{indprod}, \\text{income}, \\text{sales} \\}\\\\\n", + "y_{i,t} & = \\lambda_{i,0} f_t + \\lambda_{i,1} f_{t-1} + \\lambda_{i,2} f_{t-2} + \\lambda_{i,2} f_{t-3} + u_{i,t} \\qquad & i = \\text{emp} \\\\\n", + "u_{i,t} & = c_{i,1} u_{i,t-1} + c_{i,2} u_{i,t-2} + \\varepsilon_{i,t} \\qquad & \\varepsilon_{i,t} \\sim N(0, \\sigma_i^2) \\\\\n", + "f_t & = a_1 f_{t-1} + a_2 f_{t-2} + \\eta_t \\qquad & \\eta_t \\sim N(0, I)\\\\\n", + "\\end{align}\n", + "$$\n", + "\n", + "Now, the corresponding observation equation should look like the following:\n", + "\n", + "$$\n", + "\\begin{bmatrix}\n", + "y_{\\text{indprod}, t} \\\\\n", + "y_{\\text{income}, t} \\\\\n", + "y_{\\text{sales}, t} \\\\\n", + "y_{\\text{emp}, t} \\\\\n", + "\\end{bmatrix} = \\begin{bmatrix}\n", + "\\lambda_\\text{indprod} & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "\\lambda_\\text{income} & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "\\lambda_\\text{sales} & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "\\lambda_\\text{emp,1} & \\lambda_\\text{emp,2} & \\lambda_\\text{emp,3} & \\lambda_\\text{emp,4} & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n", + "\\end{bmatrix}\n", + "\\begin{bmatrix}\n", + "f_t \\\\\n", + "f_{t-1} \\\\\n", + "f_{t-2} \\\\\n", + "f_{t-3} \\\\\n", + "u_{\\text{indprod}, t} \\\\\n", + "u_{\\text{income}, t} \\\\\n", + "u_{\\text{sales}, t} \\\\\n", + "u_{\\text{emp}, t} \\\\\n", + "u_{\\text{indprod}, t-1} \\\\\n", + "u_{\\text{income}, t-1} \\\\\n", + "u_{\\text{sales}, t-1} \\\\\n", + "u_{\\text{emp}, t-1} \\\\\n", + "\\end{bmatrix}\n", + "$$\n", + "\n", + "Notice that we have introduced two new state variables, $f_{t-2}$ and $f_{t-3}$, which means we need to update the transition equation:\n", + "\n", + "$$\n", + "\\begin{bmatrix}\n", + "f_t \\\\\n", + "f_{t-1} \\\\\n", + "f_{t-2} \\\\\n", + "f_{t-3} \\\\\n", + "u_{\\text{indprod}, t} \\\\\n", + "u_{\\text{income}, t} \\\\\n", + "u_{\\text{sales}, t} \\\\\n", + "u_{\\text{emp}, t} \\\\\n", + "u_{\\text{indprod}, t-1} \\\\\n", + "u_{\\text{income}, t-1} \\\\\n", + "u_{\\text{sales}, t-1} \\\\\n", + "u_{\\text{emp}, t-1} \\\\\n", + "\\end{bmatrix} = \\begin{bmatrix}\n", + "a_1 & a_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & c_{\\text{indprod}, 1} & 0 & 0 & 0 & c_{\\text{indprod}, 2} & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & c_{\\text{income}, 1} & 0 & 0 & 0 & c_{\\text{income}, 2} & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & c_{\\text{sales}, 1} & 0 & 0 & 0 & c_{\\text{sales}, 2} & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & c_{\\text{emp}, 1} & 0 & 0 & 0 & c_{\\text{emp}, 2} \\\\\n", + "0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n", + "\\end{bmatrix} \n", + "\\begin{bmatrix}\n", + "f_{t-1} \\\\\n", + "f_{t-2} \\\\\n", + "f_{t-3} \\\\\n", + "f_{t-4} \\\\\n", + "u_{\\text{indprod}, t-1} \\\\\n", + "u_{\\text{income}, t-1} \\\\\n", + "u_{\\text{sales}, t-1} \\\\\n", + "u_{\\text{emp}, t-1} \\\\\n", + "u_{\\text{indprod}, t-2} \\\\\n", + "u_{\\text{income}, t-2} \\\\\n", + "u_{\\text{sales}, t-2} \\\\\n", + "u_{\\text{emp}, t-2} \\\\\n", + "\\end{bmatrix}\n", + "+ R \\begin{bmatrix}\n", + "\\eta_t \\\\\n", + "\\varepsilon_{t}\n", + "\\end{bmatrix}\n", + "$$\n", + "\n", + "This model cannot be handled out-of-the-box by the `DynamicFactor` class, but it can be handled by creating a subclass when alters the state space representation in the appropriate way." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "First, notice that if we had set `factor_order = 4`, we would almost have what we wanted. In that case, the last line of the observation equation would be:\n", + "\n", + "$$\n", + "\\begin{bmatrix}\n", + "\\vdots \\\\\n", + "y_{\\text{emp}, t} \\\\\n", + "\\end{bmatrix} = \\begin{bmatrix}\n", + "\\vdots & & & & & & & & & & & \\vdots \\\\\n", + "\\lambda_\\text{emp,1} & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n", + "\\end{bmatrix}\n", + "\\begin{bmatrix}\n", + "f_t \\\\\n", + "f_{t-1} \\\\\n", + "f_{t-2} \\\\\n", + "f_{t-3} \\\\\n", + "\\vdots\n", + "\\end{bmatrix}\n", + "$$\n", + "\n", + "\n", + "and the first line of the transition equation would be:\n", + "\n", + "$$\n", + "\\begin{bmatrix}\n", + "f_t \\\\\n", + "\\vdots\n", + "\\end{bmatrix} = \\begin{bmatrix}\n", + "a_1 & a_2 & a_3 & a_4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", + "\\vdots & & & & & & & & & & & \\vdots \\\\\n", + "\\end{bmatrix} \n", + "\\begin{bmatrix}\n", + "f_{t-1} \\\\\n", + "f_{t-2} \\\\\n", + "f_{t-3} \\\\\n", + "f_{t-4} \\\\\n", + "\\vdots\n", + "\\end{bmatrix}\n", + "+ R \\begin{bmatrix}\n", + "\\eta_t \\\\\n", + "\\varepsilon_{t}\n", + "\\end{bmatrix}\n", + "$$\n", + "\n", + "Relative to what we want, we have the following differences:\n", + "\n", + "1. In the above situation, the $\\lambda_{\\text{emp}, j}$ are forced to be zero for $j > 0$, and we want them to be estimated as parameters.\n", + "2. We only want the factor to transition according to an AR(2), but under the above situation it is an AR(4).\n", + "\n", + "Our strategy will be to subclass `DynamicFactor`, and let it do most of the work (setting up the state space representation, etc.) where it assumes that `factor_order = 4`. The only things we will actually do in the subclass will be to fix those two issues.\n", + "\n", + "First, here is the full code of the subclass; it is discussed below. It is important to note at the outset that none of the methods defined below could have been omitted. In fact, the methods `__init__`, `start_params`, `param_names`, `transform_params`, `untransform_params`, and `update` form the core of all state space models in Statsmodels, not just the `DynamicFactor` class." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from statsmodels.tsa.statespace import tools\n", + "class ExtendedDFM(sm.tsa.DynamicFactor):\n", + " def __init__(self, endog, **kwargs):\n", + " # Setup the model as if we had a factor order of 4\n", + " super(ExtendedDFM, self).__init__(\n", + " endog, k_factors=1, factor_order=4, error_order=2,\n", + " **kwargs)\n", + "\n", + " # Note: `self.parameters` is an ordered dict with the\n", + " # keys corresponding to parameter types, and the values\n", + " # the number of parameters of that type.\n", + " # Add the new parameters\n", + " self.parameters['new_loadings'] = 3\n", + "\n", + " # Cache a slice for the location of the 4 factor AR\n", + " # parameters (a_1, ..., a_4) in the full parameter vector\n", + " offset = (self.parameters['factor_loadings'] +\n", + " self.parameters['exog'] +\n", + " self.parameters['error_cov'])\n", + " self._params_factor_ar = np.s_[offset:offset+2]\n", + " self._params_factor_zero = np.s_[offset+2:offset+4]\n", + "\n", + " @property\n", + " def start_params(self):\n", + " # Add three new loading parameters to the end of the parameter\n", + " # vector, initialized to zeros (for simplicity; they could\n", + " # be initialized any way you like)\n", + " return np.r_[super(ExtendedDFM, self).start_params, 0, 0, 0]\n", + " \n", + " @property\n", + " def param_names(self):\n", + " # Add the corresponding names for the new loading parameters\n", + " # (the name can be anything you like)\n", + " return super(ExtendedDFM, self).param_names + [\n", + " 'loading.L%d.f1.%s' % (i, self.endog_names[3]) for i in range(1,4)]\n", + "\n", + " def transform_params(self, unconstrained):\n", + " # Perform the typical DFM transformation (w/o the new parameters)\n", + " constrained = super(ExtendedDFM, self).transform_params(\n", + " unconstrained[:-3])\n", + "\n", + " # Redo the factor AR constraint, since we only want an AR(2),\n", + " # and the previous constraint was for an AR(4)\n", + " ar_params = unconstrained[self._params_factor_ar]\n", + " constrained[self._params_factor_ar] = (\n", + " tools.constrain_stationary_univariate(ar_params))\n", + "\n", + " # Return all the parameters\n", + " return np.r_[constrained, unconstrained[-3:]]\n", + "\n", + " def untransform_params(self, constrained):\n", + " # Perform the typical DFM untransformation (w/o the new parameters)\n", + " unconstrained = super(ExtendedDFM, self).untransform_params(\n", + " constrained[:-3])\n", + "\n", + " # Redo the factor AR unconstraint, since we only want an AR(2),\n", + " # and the previous unconstraint was for an AR(4)\n", + " ar_params = constrained[self._params_factor_ar]\n", + " unconstrained[self._params_factor_ar] = (\n", + " tools.unconstrain_stationary_univariate(ar_params))\n", + "\n", + " # Return all the parameters\n", + " return np.r_[unconstrained, constrained[-3:]]\n", + "\n", + " def update(self, params, transformed=True):\n", + " # Peform the transformation, if required\n", + " if not transformed:\n", + " params = self.transform_params(params)\n", + " params[self._params_factor_zero] = 0\n", + " \n", + " # Now perform the usual DFM update, but exclude our new parameters\n", + " super(ExtendedDFM, self).update(params[:-3], transformed=True)\n", + "\n", + " # Finally, set our new parameters in the design matrix\n", + " self.ssm['design', 3, 1:4] = params[-3:]\n", + " " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "So what did we just do?\n", + "\n", + "#### `__init__`\n", + "\n", + "The important step here was specifying the base dynamic factor model which we were operating with. In particular, as described above, we initialize with `factor_order=4`, even though we will only end up with an AR(2) model for the factor. We also performed some general setup-related tasks.\n", + "\n", + "#### `start_params`\n", + "\n", + "`start_params` are used as initial values in the optimizer. Since we are adding three new parameters, we need to pass those in. If we hadn't done this, the optimizer would use the default starting values, which would be three elements short.\n", + "\n", + "#### `param_names`\n", + "\n", + "`param_names` are used in a variety of places, but especially in the results class. Below we get a full result summary, which is only possible when all the parameters have associated names.\n", + "\n", + "#### `transform_params` and `untransform_params`\n", + "\n", + "The optimizer selects possibly parameter values in an unconstrained way. That's not usually desired (since variances can't be negative, for example), and `transform_params` is used to transform the unconstrained values used by the optimizer to constrained values appropriate to the model. Variances terms are typically squared (to force them to be positive), and AR lag coefficients are often constrained to lead to a stationary model. `untransform_params` is used for the reverse operation (and is important because starting parameters are usually specified in terms of values appropriate to the model, and we need to convert them to parameters appropriate to the optimizer before we can begin the optimization routine).\n", + "\n", + "Even though we don't need to transform or untransform our new parameters (the loadings can in theory take on any values), we still need to modify this function for two reasons:\n", + "\n", + "1. The version in the `DynamicFactor` class is expecting 3 fewer parameters than we have now. At a minimum, we need to handle the three new parameters.\n", + "2. The version in the `DynamicFactor` class constrains the factor lag coefficients to be stationary as though it was an AR(4) model. Since we actually have an AR(2) model, we need to re-do the constraint. We also set the last two autoregressive coefficients to be zero here.\n", + "\n", + "#### `update`\n", + "\n", + "The most important reason we need to specify a new `update` method is because we have three new parameters that we need to place into the state space formulation. In particular we let the parent `DynamicFactor.update` class handle placing all the parameters except the three new ones in to the state space representation, and then we put the last three in manually." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Create the model\n", + "extended_mod = ExtendedDFM(endog)\n", + "initial_extended_res = extended_mod.fit(method='powell', disp=False)\n", + "extended_res = extended_mod.fit(initial_extended_res.params, maxiter=1000)\n", + "print extended_res.summary(separate_params=False)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Although this model increases the likelihood, it is not preferred by the AIC and BIC mesaures which penalize the additional three parameters.\n", + "\n", + "Furthermore, the qualitative results are unchanged, as we can see from the updated $R^2$ chart and the new coincident index, both of which are practically identical to the previous results." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "extended_res.plot_coefficients_of_determination(figsize=(8,2));" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "fig, ax = plt.subplots(figsize=(13,3))\n", + "\n", + "# Compute the index\n", + "extended_coincident_index = compute_coincident_index(extended_mod, extended_res)\n", + "\n", + "# Plot the factor\n", + "dates = endog.index._mpl_repr()\n", + "ax.plot(dates, coincident_index, '-', linewidth=1, label='Basic model')\n", + "ax.plot(dates, extended_coincident_index, '--', linewidth=3, label='Extended model')\n", + "ax.plot(usphci.index._mpl_repr(), usphci, label='USPHCI')\n", + "ax.legend(loc='lower right')\n", + "ax.set(title='Coincident indices, comparison')\n", + "\n", + "# Retrieve and also plot the NBER recession indicators\n", + "ylim = ax.get_ylim()\n", + "ax.fill_between(dates[:-3], ylim[0], ylim[1], rec.values[:,0], facecolor='k', alpha=0.1);" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 2", + "language": "python", + "name": "python2" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 2 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython2", + "version": "2.7.6" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} From 24afab2d421baca88f3d92c82fd7926882f76d01 Mon Sep 17 00:00:00 2001 From: Chad Fulton Date: Sun, 16 Aug 2015 00:41:44 -0700 Subject: [PATCH 57/57] ENH: Add VARMAX example notebook stub. --- examples/notebooks/statespace_varmax.ipynb | 140 +++++++++++++++++++++ 1 file changed, 140 insertions(+) create mode 100644 examples/notebooks/statespace_varmax.ipynb diff --git a/examples/notebooks/statespace_varmax.ipynb b/examples/notebooks/statespace_varmax.ipynb new file mode 100644 index 00000000000..cce39bb9c45 --- /dev/null +++ b/examples/notebooks/statespace_varmax.ipynb @@ -0,0 +1,140 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# VARMAX models\n", + "\n", + "This is a notebook stub for VARMAX models. Full development will be done after impulse response functions are available." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "import numpy as np\n", + "import pandas as pd\n", + "import statsmodels.api as sm\n", + "import matplotlib.pyplot as plt" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "dta = sm.datasets.webuse('lutkepohl2', 'http://www.stata-press.com/data/r12/')\n", + "dta.index = dta.qtr\n", + "endog = dta.ix['1960-04-01':'1978-10-01', ['dln_inv', 'dln_inc', 'dln_consump']]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Model specification\n", + "\n", + "The `VARMAX` class in Statsmodels allows estimation of VAR, VMA, and VARMA models (through the `order` argument), optionally with a constant term (via the `trend` argument). Exogenous regressors may also be included (as usual in Statsmodels, by the `exog` argument), and in this way a time trend may be added. Finally, the class allows measurement error (via the `measurement_error` argument) and allows specifying either a diagonal or unstructured innovation covariance matrix (via the `error_cov_type` argument)." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 1: VAR\n", + "\n", + "Below is a simple VARX(2) model in two endogenous variables and an exogenous series, but no constant term. Notice that we needed to allow for more iterations than the default (which is `maxiter=50`) in order for the likelihood estimation to converge. This is not unusual in VAR models which have to estimate a large number of parameters, often on a relatively small number of time series: this model, for example, estimates 27 parameters off of 75 observations of 3 variables." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# exog = pd.Series(np.arange(len(endog)), index=endog.index, name='trend')\n", + "exog = endog['dln_consump']\n", + "mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(2,0), trend='nc', exog=exog)\n", + "res = mod.fit(maxiter=1000)\n", + "print res.summary()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Example 2: VMA\n", + "\n", + "A vector moving average model can also be formulated. Below we show a VMA(2) on the same data, but where the innovations to the process are uncorrelated. In this example we leave out the exogenous regressor but now include the constant term." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(0,2), error_cov_type='diagonal')\n", + "res = mod.fit(maxiter=1000)\n", + "print res.summary()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Caution: VARMA(p,q) specifications\n", + "\n", + "Although the model allows estimating VARMA(p,q) specifications, these models are not identified without additional restrictions on the representation matrices, which are not built-in. For this reason, it is recommended that the user proceed with error (and indeed a warning is issued when these models are specified). Nonetheless, they may in some circumstances provide useful information." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "mod = sm.tsa.VARMAX(endog[['dln_inv', 'dln_inc']], order=(1,1))\n", + "res = mod.fit(maxiter=1000)\n", + "print res.summary()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 2", + "language": "python", + "name": "python2" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 2 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython2", + "version": "2.7.6" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +}