State Space is a Python package that permits the symbolic specification of linear state space models with normally distributed innovations and measurement errors. Coefficients are defined via SymPy matrices which are then compiled into a numerical statsmodels implementation. These unobserved state is inferred via Kalman filtering and model parameters are estimated via maximum likelihood using statsmodels as the numerical backend.
A linear state space model consists of a state evolution equation and an observation equation. The state is not directly observed, instead a linear transformation of the state with added Gaussian noise is observed. In a linear state space model, the state evolves according to
where the coefficients T, c, and R may depend on exogenous variables but not on the state itself. They may involve parameters that must be estimated from the data. The state innovation \eta_t has a multivariate normal distribution with zero mean and covariance matrix Q, which may depend on exogenous variables but not the state itself.
The observation equation maps the unobserved state according to
The observation noise \epsilon_t has a multivariate normal distribution with zero mean and covariance matrix H. The coefficients Z, d, as well as H may depend on exogeneous data and involve unknown parameters that are estimated via MLE.
State Space adopts the following terminology:
T: transition matrix
c: state intercept vector
R: selection matrix
Q: state covariance matrix
Z: design matrix
d: observation intercept
H: observation covariance matrix
The coefficients T,c, R, Q, Z, d, H are specified as SymPy matrices and may involve unknown parameters and exogenous data.
State Space includes two examples in Jupyter notebooks:
1.) A conditional linear factor model for returns of the Ford motor corporation with S&P 500 returns as the factor.
2.) A model of time-variation in the equity premium applied to S&P 500 index data.
pip install state_space
or
pip3 install state_space
if not using Anaconda.