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# GASX models

## Introduction

GASX models extend GAS models by including exogenous factors X. For a conditional observation density p\left(y_{t}\mid{\theta_{t}}\right) with an observation y_{t} and a latent time-varying parameter \theta_{t}, we assume the parameter \theta_{t} follows the recursion:

\theta_{t} = \mu + \sum^{K}_{k=1}\beta_{k}X_{t,k} + \sum^{p}_{i=1}\phi_{i}\theta_{t-i} + \sum^{q}_{j=1}\alpha_{j}S\left(x_{j-1}\right)\frac{\partial\log p\left(y_{t-j}\mid{\theta_{t-j}}\right) }{\partial{\theta_{t-j}}}


For example, for the Poisson family, where the default scaling is \exp\left(\theta_{t}\right), the time-varying latent variable follows:

\theta_{t} = \mu + \sum^{K}_{k=1}\beta_{k}X_{t,k} + \sum^{p}_{i=1}\phi_{i}\theta_{t-i} + \sum^{q}_{j=1}\alpha_{j}\left(\frac{y_{t-j}}{\exp\left(\theta_{t-j}\right)} - 1\right)


The model can be viewed as an approximation to a non-linear ARIMAX model.

## Example

Below we estimate the \beta for a stock – the systematic (market) component of returns – using a heavy tailed distribution and some short-term autoregressive effects. First let’s load some data:

from pandas_datareader.data import DataReader
from datetime import datetime

a = DataReader('AMZN',  'yahoo', datetime(2012,1,1), datetime(2016,6,1))
a_returns.index = a.index.values[1:a.index.values.shape[0]]
a_returns.columns = ["Amazon Returns"]

spy = DataReader('SPY',  'yahoo', datetime(2012,1,1), datetime(2016,6,1))
spy_returns.index = spy.index.values[1:spy.index.values.shape[0]]
spy_returns.columns = ['S&P500 Returns']

one_day = np.log(1+one_mon)/365

returns = pd.concat([one_day,a_returns,spy_returns],axis=1).dropna()
excess_m = returns["Amazon Returns"].values - returns['DGS1MO'].values
excess_spy = returns["S&P500 Returns"].values - returns['DGS1MO'].values
final_returns = pd.DataFrame(np.transpose([excess_m,excess_spy, returns['DGS1MO'].values]))
final_returns.columns=["Amazon","SP500","Risk-free rate"]
final_returns.index = returns.index

plt.figure(figsize=(15,5))
plt.title("Excess Returns")
x = plt.plot(final_returns);
plt.legend(iter(x), final_returns.columns);

Below we estimate a point mass estimate z^{MLE} of the latent variables for a GASX(1,1) model:

model = pf.GASX(formula="Amazon~SP500",data=final_returns,ar=1,sc=1,family=pf.GASSkewt())
x = model.fit()
x.summary()

Skewt GASX(1,0,1)
======================================== =================================================
Dependent Variable: Amazon               Method: MLE
Start Date: 2012-01-05 00:00:00          Log Likelihood: 3165.9237
End Date: 2016-06-01 00:00:00            AIC: -6317.8474
Number of observations: 1100             BIC: -6282.8259
==========================================================================================
Latent Variable           Estimate   Std Error  z        P>|z|    95% C.I.
========================= ========== ========== ======== ======== ========================
AR(1)                     0.0807     0.0202     3.9956   0.0001   (0.0411 | 0.1203)
SC(1)                     -0.0       0.0187     -0.0001  0.9999   (-0.0367 | 0.0367)
Beta 1                    -0.0005    0.0249     -0.0184  0.9853   (-0.0493 | 0.0484)
Beta SP500                1.2683     0.0426     29.7473  0.0      (1.1848 | 1.3519)
Skewness                  1.017
Skewt Scale               0.0093
v                         2.7505
==========================================================================================
WARNING: Skew t distribution is not well-suited for MLE or MAP inference
Workaround 1: Use a t-distribution instead for MLE/MAP
Workaround 2: Use M-H or BBVI inference for Skew t distribution

The results table warns us about using the Skew t distribution. This choice of family can sometimes be unstable, so we may want to opt for a t-distribution instead. But in this case, we seem to have obtained sensible results. We can plot the constant and the GAS latent variables by referencing their indices with :py:func:plot_z:

model.plot_z(indices=[0,1,2])

Similarly we can plot \beta:

model.plot_z(indices=[3])

Our \beta_{AMZN} estimate is above 1.0 (fairly strong systematic risk). Let us plot the model fit and the systematic component of returns with :py:func:plot_fit:

model.plot_fit(figsize=(15,10))

## Class Description

.. py:class:: GASX(data, formula, ar, sc, integ, target, family)

**Generalized Autoregressive Score Exogenous Variable Models (GASX).**

==================   ===============================    ======================================
Parameter            Type                                Description
==================   ===============================    ======================================
data                 pd.DataFrame or np.ndarray         Contains the univariate time series
formula              string                             Patsy notation specifying the regression
ar                   int                                The number of autoregressive lags
sc                   int                                The number of score function lags
integ                int                                How many times to difference the data
(default: 0)
target               string or int                      Which column of DataFrame/array to use.
family               pf.Family instance                 The distribution for the time series,
e.g pf.Normal()
==================   ===============================    ======================================

**Attributes**

.. py:attribute:: latent_variables

A pf.LatentVariables() object containing information on the model latent variables,
prior settings. any fitted values, starting values, and other latent variable
information. When a model is fitted, this is where the latent variables are updated/stored.
Please see the documentation on Latent Variables for information on attributes within this
object, as well as methods for accessing the latent variable information.

**Methods**

Adjusts the priors for the model latent variables. The latent variables and their indices
can be viewed by printing the latent_variables attribute attached to the model instance.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
index                int                         Index of the latent variable to change
prior                pf.Family instance          Prior distribution, e.g. pf.Normal()
==================   ========================    ======================================

**Returns**: void - changes the model latent_variables attribute

.. py:method:: fit(method, **kwargs)

Estimates latent variables for the model. User chooses an inference option and the
method returns a results object, as well as updating the model's latent_variables
attribute.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
method               str                         Inference option: e.g. 'M-H' or 'MLE'
==================   ========================    ======================================

See Bayesian Inference and Classical Inference sections of the documentation for the
full list of inference options. Optional parameters can be entered that are relevant
to the particular mode of inference chosen.

**Returns**: pf.Results instance with information for the estimated latent variables

.. py:method:: plot_fit(**kwargs)

Plots the fit of the model against the data. Optional arguments include *figsize*,
the dimensions of the figure to plot.

**Returns** : void - shows a matplotlib plot

.. py:method:: plot_ppc(T, nsims)

Plots a histogram for a posterior predictive check with a discrepancy measure of the
user's choosing. This method only works if you have fitted using Bayesian inference.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
T                    function                    Discrepancy, e.g. np.mean or np.max
nsims                int                         How many simulations for the PPC
==================   ========================    ======================================

**Returns**: void - shows a matplotlib plot

.. py:method:: plot_predict(h, oos_data, past_values, intervals, **kwargs)

Plots predictions of the model, along with intervals.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
h                    int                         How many steps to forecast ahead
oos_data             pd.DataFrame                Exogenous variables in a frame for h steps
past_values          int                         How many past datapoints to plot
intervals            boolean                     Whether to plot intervals or not
==================   ========================    ======================================

To be clear, the *oos_data* argument should be a DataFrame in the same format as the initial
dataframe used to initialize the model instance. The reason is that to predict future values,
you need to specify assumptions about exogenous variables for the future. For example, if you
predict *h* steps ahead, the method will take the h first rows from *oos_data* and take the
values for the exogenous variables that you asked for in the patsy formula.

Optional arguments include *figsize* - the dimensions of the figure to plot. Please note
that if you use Maximum Likelihood or Variational Inference, the intervals shown will not
reflect latent variable uncertainty. Only Metropolis-Hastings will give you fully Bayesian
prediction intervals. Bayesian intervals with variational inference are not shown because
of the limitation of mean-field inference in not accounting for posterior correlations.

**Returns** : void - shows a matplotlib plot

.. py:method:: plot_predict_is(h, fit_once, fit_method, **kwargs)

Plots in-sample rolling predictions for the model. This means that the user pretends a
last subsection of data is out-of-sample, and forecasts after each period and assesses
how well they did. The user can choose whether to fit parameters once at the beginning
or every time step.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
h                    int                         How many previous timesteps to use
fit_once             boolean                     Whether to fit once, or every timestep
fit_method           str                         Which inference option, e.g. 'MLE'
==================   ========================    ======================================

Optional arguments include *figsize* - the dimensions of the figure to plot. **h** is an int of how many previous steps to simulate performance on.

**Returns** : void - shows a matplotlib plot

.. py:method:: plot_sample(nsims, plot_data=True)

Plots samples from the posterior predictive density of the model. This method only works
if you fitted the model using Bayesian inference.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
nsims                int                         How many samples to draw
plot_data            boolean                     Whether to plot the real data as well
==================   ========================    ======================================

**Returns** : void - shows a matplotlib plot

.. py:method:: plot_z(indices, figsize)

Returns a plot of the latent variables and their associated uncertainty.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
indices              int or list                 Which latent variable indices to plot
figsize              tuple                       Size of the matplotlib figure
==================   ========================    ======================================

**Returns** : void - shows a matplotlib plot

.. py:method:: ppc(T, nsims)

Returns a p-value for a posterior predictive check. This method only works if you have
fitted using Bayesian inference.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
T                    function                    Discrepancy, e.g. np.mean or np.max
nsims                int                         How many simulations for the PPC
==================   ========================    ======================================

**Returns**: int - the p-value for the discrepancy test

.. py:method:: predict(h, oos_data, intervals=False)

Returns a DataFrame of model predictions.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
h                    int                         How many steps to forecast ahead
oos_data             pd.DataFrame                Exogenous variables in a frame for h steps
intervals            boolean                     Whether to return prediction intervals
==================   ========================    ======================================

To be clear, the *oos_data* argument should be a DataFrame in the same format as the initial
dataframe used to initialize the model instance. The reason is that to predict future values,
you need to specify assumptions about exogenous variables for the future. For example, if you
predict *h* steps ahead, the method will take the 5 first rows from *oos_data* and take the
values for the exogenous variables that you specified as exogenous variables in the patsy formula.

Please note that if you use Maximum Likelihood or Variational Inference, the intervals shown
will not reflect latent variable uncertainty. Only Metropolis-Hastings will give you fully
Bayesian prediction intervals. Bayesian intervals with variational inference are not shown
because of the limitation of mean-field inference in not accounting for posterior correlations.

**Returns** : pd.DataFrame - the model predictions

.. py:method:: predict_is(h, fit_once, fit_method)

Returns DataFrame of in-sample rolling predictions for the model.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
h                    int                         How many previous timesteps to use
fit_once             boolean                     Whether to fit once, or every timestep
fit_method           str                         Which inference option, e.g. 'MLE'
==================   ========================    ======================================

**Returns** : pd.DataFrame - the model predictions

.. py:method:: sample(nsims)

Returns np.ndarray of draws of the data from the posterior predictive density. This
method only works if you have fitted the model using Bayesian inference.

==================   ========================    ======================================
Parameter            Type                        Description
==================   ========================    ======================================
nsims                int                         How many posterior draws to take
==================   ========================    ======================================

**Returns** : np.ndarray - samples from the posterior predictive density.



## References

Creal, D; Koopman, S.J.; Lucas, A. (2013). Generalized Autoregressive Score Models with Applications. Journal of Applied Econometrics, 28(5), 777–795. doi:10.1002/jae.1279.

Harvey, A.C. (2013). Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series. Cambridge University Press.

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