statsmodels.tsa.statespace.sarimax.SARIMAX

class statsmodels.tsa.statespace.sarimax.SARIMAX(endog, exog=None, order=1, 0, 0, seasonal_order=0, 0, 0, 0, trend=None, measurement_error=False, time_varying_regression=False, mle_regression=True, simple_differencing=False, enforce_stationarity=True, enforce_invertibility=True, hamilton_representation=False, concentrate_scale=False, trend_offset=1, use_exact_diffuse=False, dates=None, freq=None, missing='none', validate_specification=True, **kwargs)[source]

Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors model

Parameters
endogarray_like

The observed time-series process \(y\)

exogarray_like, optional

Array of exogenous regressors, shaped nobs x k.

orderiterable or iterable of iterables, optional

The (p,d,q) order of the model for the number of AR parameters, differences, and MA parameters. d must be an integer indicating the integration order of the process, while p and q may either be an integers indicating the AR and MA orders (so that all lags up to those orders are included) or else iterables giving specific AR and / or MA lags to include. Default is an AR(1) model: (1,0,0).

seasonal_orderiterable, optional

The (P,D,Q,s) order of the seasonal component of the model for the AR parameters, differences, MA parameters, and periodicity. D must be an integer indicating the integration order of the process, while P and Q may either be an integers indicating the AR and MA orders (so that all lags up to those orders are included) or else iterables giving specific AR and / or MA lags to include. s is an integer giving the periodicity (number of periods in season), often it is 4 for quarterly data or 12 for monthly data. Default is no seasonal effect.

trendstr{‘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 non-zero polynomial exponents to include, in increasing order. For example, [1,1,0,1] denotes \(a + bt + ct^3\). Default is to not include a trend component.

measurement_errorbool, optional

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

time_varying_regressionbool, optional

Used when an explanatory variables, exog, are provided provided to select whether or not coefficients on the exogenous regressors are allowed to vary over time. Default is False.

mle_regressionbool, optional

Whether or not to use estimate the regression coefficients for the exogenous variables as part of maximum likelihood estimation or through the Kalman filter (i.e. recursive least squares). If time_varying_regression is True, this must be set to False. Default is True.

simple_differencingbool, optional

Whether or not to use partially conditional maximum likelihood estimation. If True, differencing is performed prior to estimation, which discards the first \(s D + d\) initial rows but results in a smaller state-space formulation. See the Notes section for important details about interpreting results when this option is used. If False, the full SARIMAX model is put in state-space form so that all datapoints can be used in estimation. Default is False.

enforce_stationaritybool, optional

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

enforce_invertibilitybool, optional

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

hamilton_representationbool, optional

Whether or not to use the Hamilton representation of an ARMA process (if True) or the Harvey representation (if False). Default is False.

concentrate_scalebool, optional

Whether or not to concentrate the scale (variance of the error term) out of the likelihood. This reduces the number of parameters estimated by maximum likelihood by one, but standard errors will then not be available for the scale parameter.

trend_offsetint, optional

The offset at which to start time trend values. Default is 1, so that if trend=’t’ the trend is equal to 1, 2, …, nobs. Typically is only set when the model created by extending a previous dataset.

use_exact_diffusebool, optional

Whether or not to use exact diffuse initialization for non-stationary states. Default is False (in which case approximate diffuse initialization is used).

**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

The SARIMA model is specified \((p, d, q) \times (P, D, Q)_s\).

\[\phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D y_t = A(t) + \theta_q (L) \tilde \theta_Q (L^s) \zeta_t\]

In terms of a univariate structural model, this can be represented as

\[\begin{split}y_t & = u_t + \eta_t \\ \phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D u_t & = A(t) + \theta_q (L) \tilde \theta_Q (L^s) \zeta_t\end{split}\]

where \(\eta_t\) is only applicable in the case of measurement error (although it is also used in the case of a pure regression model, i.e. if p=q=0).

In terms of this model, regression with SARIMA errors can be represented easily as

\[\begin{split}y_t & = \beta_t x_t + u_t \\ \phi_p (L) \tilde \phi_P (L^s) \Delta^d \Delta_s^D u_t & = A(t) + \theta_q (L) \tilde \theta_Q (L^s) \zeta_t\end{split}\]

this model is the one used when exogenous regressors are provided.

Note that the reduced form lag polynomials will be written as:

\[\begin{split}\Phi (L) \equiv \phi_p (L) \tilde \phi_P (L^s) \\ \Theta (L) \equiv \theta_q (L) \tilde \theta_Q (L^s)\end{split}\]

If mle_regression is True, regression coefficients are treated as additional parameters to be estimated via maximum likelihood. Otherwise they are included as part of the state with a diffuse initialization. In this case, however, with approximate diffuse initialization, results can be sensitive to the initial variance.

This class allows two different underlying representations of ARMA models as state space models: that of Hamilton and that of Harvey. Both are equivalent in the sense that they are analytical representations of the ARMA model, but the state vectors of each have different meanings. For this reason, maximum likelihood does not result in identical parameter estimates and even the same set of parameters will result in different loglikelihoods.

The Harvey representation is convenient because it allows integrating differencing into the state vector to allow using all observations for estimation.

In this implementation of differenced models, the Hamilton representation is not able to accommodate differencing in the state vector, so simple_differencing (which performs differencing prior to estimation so that the first d + sD observations are lost) must be used.

Many other packages use the Hamilton representation, so that tests against Stata and R require using it along with simple differencing (as Stata does).

If filter_concentrated = True is used, then the scale of the model is concentrated out of the likelihood. A benefit of this is that there the dimension of the parameter vector is reduced so that numerical maximization of the log-likelihood function may be faster and more stable. If this option in a model with measurement error, it is important to note that the estimated measurement error parameter will be relative to the scale, and is named “snr.measurement_error” instead of “var.measurement_error”. To compute the variance of the measurement error in this case one would multiply snr.measurement_error parameter by the scale.

If simple_differencing = True is used, then the endog and exog data are differenced prior to putting the model in state-space form. This has the same effect as if the user differenced the data prior to constructing the model, which has implications for using the results:

  • Forecasts and predictions will be about the differenced data, not about the original data. (while if simple_differencing = False is used, then forecasts and predictions will be about the original data).

  • If the original data has an Int64Index, a new RangeIndex will be created for the differenced data that starts from one, and forecasts and predictions will use this new index.

Detailed information about state space models can be found in [1]. Some specific references are:

  • Chapter 3.4 describes ARMA and ARIMA models in state space form (using the Harvey representation), and gives references for basic seasonal models and models with a multiplicative form (for example the airline model). It also shows a state space model for a full ARIMA process (this is what is done here if simple_differencing=False).

  • Chapter 3.6 describes estimating regression effects via the Kalman filter (this is performed if mle_regression is False), regression with time-varying coefficients, and regression with ARMA errors (recall from above that if regression effects are present, the model estimated by this class is regression with SARIMA errors).

  • Chapter 8.4 describes the application of an ARMA model to an example dataset. A replication of this section is available in an example IPython notebook in the documentation.

References

1

Durbin, James, and Siem Jan Koopman. 2012. Time Series Analysis by State Space Methods: Second Edition. Oxford University Press.

Attributes
measurement_errorbool

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

state_errorbool

Whether or not the transition equation has an error component.

mle_regressionbool

Whether or not the regression coefficients for the exogenous variables were estimated via maximum likelihood estimation.

state_regressionbool

Whether or not the regression coefficients for the exogenous variables are included as elements of the state space and estimated via the Kalman filter.

time_varying_regressionbool

Whether or not coefficients on the exogenous regressors are allowed to vary over time.

simple_differencingbool

Whether or not to use partially conditional maximum likelihood estimation.

enforce_stationaritybool

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

enforce_invertibilitybool

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

hamilton_representationbool

Whether or not to use the Hamilton representation of an ARMA process.

trendstr{‘n’,’c’,’t’,’ct’} or iterable

Parameter controlling the deterministic trend polynomial \(A(t)\). See the class parameter documentation for more information.

polynomial_arndarray

Array containing autoregressive lag polynomial lags, ordered from lowest degree to highest. The polynomial begins with lag 0. Initialized with ones, unless a coefficient is constrained to be zero (in which case it is zero).

polynomial_mandarray

Array containing moving average lag polynomial lags, ordered from lowest degree to highest. Initialized with ones, unless a coefficient is constrained to be zero (in which case it is zero).

polynomial_seasonal_arndarray

Array containing seasonal moving average lag polynomial lags, ordered from lowest degree to highest. Initialized with ones, unless a coefficient is constrained to be zero (in which case it is zero).

polynomial_seasonal_mandarray

Array containing seasonal moving average lag polynomial lags, ordered from lowest degree to highest. Initialized with ones, unless a coefficient is constrained to be zero (in which case it is zero).

polynomial_trendndarray

Array containing trend polynomial coefficients, ordered from lowest degree to highest. Initialized with ones, unless a coefficient is constrained to be zero (in which case it is zero).

k_arint

Highest autoregressive order in the model, zero-indexed.

k_ar_paramsint

Number of autoregressive parameters to be estimated.

k_diffint

Order of integration.

k_maint

Highest moving average order in the model, zero-indexed.

k_ma_paramsint

Number of moving average parameters to be estimated.

seasonal_periodsint

Number of periods in a season.

k_seasonal_arint

Highest seasonal autoregressive order in the model, zero-indexed.

k_seasonal_ar_paramsint

Number of seasonal autoregressive parameters to be estimated.

k_seasonal_diffint

Order of seasonal integration.

k_seasonal_maint

Highest seasonal moving average order in the model, zero-indexed.

k_seasonal_ma_paramsint

Number of seasonal moving average parameters to be estimated.

k_trendint

Order of the trend polynomial plus one (i.e. the constant polynomial would have k_trend=1).

k_exogint

Number of exogenous regressors.

Methods

clone(endog[, exog])

Clone state space model with new data and optionally new specification

filter(params[, transformed, …])

Kalman filtering

fit([start_params, transformed, …])

Fits the model by maximum likelihood via Kalman filter.

fit_constrained(constraints[, start_params])

Fit the model with some parameters subject to equality constraints.

fix_params(params)

Fix parameters to specific values (context manager)

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 the SARIMAX model.

initialize_approximate_diffuse([variance])

Initialize approximate diffuse

initialize_default([…])

Initialize default

initialize_known(initial_state, …)

Initialize known

initialize_statespace(**kwargs)

Initialize the state space representation

initialize_stationary()

Initialize stationary

loglike(params, *args, **kwargs)

Loglikelihood evaluation

loglikeobs(params[, transformed, …])

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, …])

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[, transformed, …])

Update the parameters of the model

handle_params

Methods

clone(endog[, exog])

Clone state space model with new data and optionally new specification

filter(params[, transformed, …])

Kalman filtering

fit([start_params, transformed, …])

Fits the model by maximum likelihood via Kalman filter.

fit_constrained(constraints[, start_params])

Fit the model with some parameters subject to equality constraints.

fix_params(params)

Fix parameters to specific values (context manager)

from_formula(formula, data[, subset])

Not implemented for state space models

handle_params(params[, transformed, …])

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 the SARIMAX model.

initialize_approximate_diffuse([variance])

Initialize approximate diffuse

initialize_default([…])

Initialize default

initialize_known(initial_state, …)

Initialize known

initialize_statespace(**kwargs)

Initialize the state space representation

initialize_stationary()

Initialize stationary

loglike(params, *args, **kwargs)

Loglikelihood evaluation

loglikeobs(params[, transformed, …])

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, …])

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[, transformed, …])

Update the parameters of the model

Properties

endog_names

Names of endogenous variables

exog_names

The names of the exogenous variables.

initial_design

Initial design matrix

initial_selection

Initial selection matrix

initial_state_intercept

Initial state intercept vector

initial_transition

Initial transition matrix

initial_variance

initialization

loglikelihood_burn

model_latex_names

The latex names of all possible model parameters.

model_names

The plain text names of all possible model parameters.

model_orders

The orders of each of the polynomials in the model.

param_names

List of human readable parameter names (for parameters actually included in the model).

param_terms

List of parameters actually included in the model, in sorted order.

params_complete

start_params

Starting parameters for maximum likelihood estimation

state_names

(list of str) List of human readable names for unobserved states.

tolerance