statsmodels.tsa.statespace.varmax.VARMAX

class statsmodels.tsa.statespace.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 (str{'n','c','t','ct'} or iterable, optional) – Parameter controlling the deterministic trend polynomial \(A(t)\). Can be specified as a string where ‘c’ indicates a constant (i.e. a degree zero component of the trend polynomial), ‘t’ indicates a linear trend with time, and ‘ct’ is both. Can also be specified as an iterable defining the polynomial as in numpy.poly1d, where [1,1,0,1] would denote \(a + bt + ct^3\). Default is a constant trend component.
  • 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.
order

The (p,q) order of the model for the number of AR and MA parameters to use.

Type:iterable
trend

Parameter controlling the deterministic trend polynomial \(A(t)\). Can be specified as a string where ‘c’ indicates a constant (i.e. a degree zero component of the trend polynomial), ‘t’ indicates a linear trend with time, and ‘ct’ is both. Can also be specified as an iterable defining the polynomial as in numpy.poly1d, where [1,1,0,1] would denote \(a + bt + ct^3\).

Type:str{‘n’,’c’,’t’,’ct’} or iterable
error_cov_type

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”.

Type:{‘diagonal’, ‘unstructured’}, optional
measurement_error

Whether or not to assume the endogenous observations endog were measured with error. Default is False.

Type:boolean, optional
enforce_stationarity

Whether or not to transform the AR parameters to enforce stationarity in the autoregressive component of the model. Default is True.

Type:boolean, optional
enforce_invertibility

Whether or not to transform the MA parameters to enforce invertibility in the moving average component of the model. Default is True.

Type:boolean, optional

Notes

Generically, the VARMAX model is specified (see for example chapter 18 of [1]):

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

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 [1] for more information. 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](1, 2) Lütkepohl, Helmut. 2007. New Introduction to Multiple Time Series Analysis. Berlin: Springer.

Methods

filter(params[, transformed, complex_step, …]) Kalman filtering
fit([start_params, transformed, cov_type, …]) Fits the model by maximum likelihood via Kalman filter.
from_formula(formula, data[, subset]) Not implemented for state space models
hessian(params, *args, **kwargs) Hessian matrix of the likelihood function, evaluated at the given parameters
impulse_responses(params[, steps, impulse, …]) Impulse response function
information(params) Fisher information matrix of model
initialize() Initialize (possibly re-initialize) a Model instance.
initialize_approximate_diffuse([variance])
initialize_known(initial_state, …)
initialize_statespace(**kwargs) Initialize the state space representation
initialize_stationary()
loglike(params, *args, **kwargs) Loglikelihood evaluation
loglikeobs(params[, transformed, complex_step]) Loglikelihood evaluation
observed_information_matrix(params[, …]) Observed information matrix
opg_information_matrix(params[, …]) Outer product of gradients information matrix
predict(params[, exog]) After a model has been fit predict returns the fitted values.
prepare_data() Prepare data for use in the state space representation
score(params, *args, **kwargs) Compute the score function at params.
score_obs(params[, method, transformed, …]) Compute the score per observation, evaluated at params
set_conserve_memory([conserve_memory]) Set the memory conservation method
set_filter_method([filter_method]) Set the filtering method
set_inversion_method([inversion_method]) Set the inversion method
set_smoother_output([smoother_output]) Set the smoother output
set_stability_method([stability_method]) Set the numerical stability method
simulate(params, nsimulations[, …]) Simulate a new time series following the state space model
simulation_smoother([simulation_output]) Retrieve a simulation smoother for the state space model.
smooth(params[, transformed, complex_step, …]) Kalman smoothing
transform_jacobian(unconstrained[, …]) Jacobian matrix for the parameter transformation function
transform_params(unconstrained) Transform unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation
untransform_params(constrained) Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer.
update(params, **kwargs) Update the parameters of the model

Attributes

endog_names Names of endogenous variables
exog_names
initial_variance
initialization
loglikelihood_burn
param_names (list of str) List of human readable parameter names (for parameters actually included in the model).
start_params (array) Starting parameters for maximum likelihood estimation.
tolerance