VARMAX¶

class dismalpy.ssm.varmax.VARMAX(endog, exog=None, order=(1, 0), trend='c', error_cov_type='unstructured', measurement_error=False, enforce_stationarity=True, enforce_invertibility=True, **kwargs)[source]¶

Vector Autoregressive Moving Average with eXogenous regressors model

Parameters:

endog : array_like

The observed time-series process \(y\), , 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.

Notes

Generically, the VARMAX model is specified (see for example chapter 18 of [R13]):

\[y_t = \nu + A_1 y_{t-1} + \dots + A_p y_{t-p} + B x_t + \epsilon_t + M_1 \epsilon_{t-1} + \dots M_q \epsilon_{t-q}\]

where \(\epsilon_t \sim N(0, \Omega)\), and where \(y_t\) 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 \(\{A_i, M_j\}\) 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 [R13] 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

[R13]

(1, 2, 3) Lutkepohl, Helmut. 2007. New Introduction to Multiple Time Series Analysis. Berlin: Springer.

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.

class dismalpy.ssm.varmax.VARMAXResults(model, params, smoother_results, cov_type='opg', cov_kwds=None, **kwargs)[source]¶

Class to hold results from fitting an VARMAX model.

Parameters:

model : VARMAX instance

The fitted model instance

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.