State Space

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.

Stochastic Process

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.

Examples

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.

Installation

    pip install state_space

or

    pip3 install state_space

if not using Anaconda.