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, **kwargs)
[source]
Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors model
Parameters: 


measurement_error
boolean – Whether or not to assume the endogenous observations endog
were measured with error.
state_error
boolean – Whether or not the transition equation has an error component.
mle_regression
boolean – Whether or not the regression coefficients for the exogenous variables were estimated via maximum likelihood estimation.
state_regression
boolean – 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_regression
boolean – Whether or not coefficients on the exogenous regressors are allowed to vary over time.
simple_differencing
boolean – Whether or not to use partially conditional maximum likelihood estimation.
enforce_stationarity
boolean – Whether or not to transform the AR parameters to enforce stationarity in the autoregressive component of the model.
enforce_invertibility
boolean – Whether or not to transform the MA parameters to enforce invertibility in the moving average component of the model.
hamilton_representation
boolean – Whether or not to use the Hamilton representation of an ARMA process.
trend
str{‘n’,’c’,’t’,’ct’} or iterable – Parameter controlling the deterministic trend polynomial \(A(t)\). See the class parameter documentation for more information.
polynomial_ar
array – Array containing autoregressive lag 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).
polynomial_ma
array – Array containing moving average lag 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).
polynomial_seasonal_ar
array – Array containing seasonal moving average lag 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).
polynomial_seasonal_ma
array – Array containing seasonal moving average lag 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).
polynomial_trend
array – 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_ar
int – Highest autoregressive order in the model, zeroindexed.
k_ar_params
int – Number of autoregressive parameters to be estimated.
k_diff
int – Order of intergration.
k_ma
int – Highest moving average order in the model, zeroindexed.
k_ma_params
int – Number of moving average parameters to be estimated.
seasonal_periods
int – Number of periods in a season.
k_seasonal_ar
int – Highest seasonal autoregressive order in the model, zeroindexed.
k_seasonal_ar_params
int – Number of seasonal autoregressive parameters to be estimated.
k_seasonal_diff
int – Order of seasonal intergration.
k_seasonal_ma
int – Highest seasonal moving average order in the model, zeroindexed.
k_seasonal_ma_params
int – Number of seasonal moving average parameters to be estimated.
k_trend
int – Order of the trend polynomial plus one (i.e. the constant polynomial would have k_trend=1
).
k_exog
int – Number of exogenous regressors.
The SARIMA model is specified \((p, d, q) \times (P, D, Q)_s\).
In terms of a univariate structural model, this can be represented as
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
this model is the one used when exogenous regressors are provided.
Note that the reduced form lag polynomials will be written as:
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 accomodate 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).
Detailed information about state space models can be found in [1]. Some specific references are:
simple_differencing=False
).mle_regression
is False), regression with timevarying 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).[1]  Durbin, James, and Siem Jan Koopman. 2012. Time Series Analysis by State Space Methods: Second Edition. Oxford University Press. 
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 the SARIMAX model. 
initialize_approximate_diffuse ([variance])  Initialize the statespace model with approximate diffuse values. 
initialize_known (initial_state, …)  Initialize the statespace model with known distribution for initial state. 
initialize_state ([variance, complex_step])  Initialize state and state covariance arrays in preparation for the Kalman filter. 
initialize_statespace (**kwargs)  Initialize the state space representation 
initialize_stationary ()  Initialize the statespace model as 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[, transformed, complex_step])  Update the parameters of the model 
endog_names  Names of endogenous variables 
exog_names  
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 
tolerance 
© 2009–2012 Statsmodels Developers
© 2006–2008 Scipy Developers
© 2006 Jonathan E. Taylor
Licensed under the 3clause BSD License.
http://www.statsmodels.org/stable/generated/statsmodels.tsa.statespace.sarimax.SARIMAX.html