Seasonal AutoRegressive Integrated Moving Average with eXogenous regressors model
Parameters: | endog : array_like
exog : array_like, optional
order : iterable or iterable of iterables, optional
seasonal_order : iterable, optional
trend : str{‘n’,’c’,’t’,’ct’} or iterable, optional
measurement_error : boolean, optional
time_varying_regression : boolean, optional
mle_regression : boolean, optional
simple_differencing : boolean, optional
enforce_stationarity : boolean, optional
enforce_invertibility : boolean, optional
hamilton_representation : boolean, optional
**kwargs :
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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
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
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
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
this model is the one used when exogenous regressors are provided.
Note that the reduced form lag polynomials will be written as:
\Phi (L) \equiv \phi_p (L) \tilde \phi_P (L^s) \\ \Theta (L) \equiv \theta_q (L) \tilde \theta_Q (L^s)
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 [R80]. Some specific references are:
References
[R80] | (1, 2) Durbin, James, and Siem Jan Koopman. 2012. Time Series Analysis by State Space Methods: Second Edition. Oxford University Press. |
Attributes
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, zero-indexed. |
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, zero-indexed. |
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, zero-indexed. |
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, zero-indexed. |
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. |
Methods
filter(params, **kwargs) | |
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 |
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() | |
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 |
smooth(params, **kwargs) | |
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 |
update(params[, transformed, complex_step]) | Update the parameters of the model |
Methods
filter(params, **kwargs) | |
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 |
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() | |
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 |
smooth(params, **kwargs) | |
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 |
update(params[, transformed, complex_step]) | Update the parameters of the model |
Attributes
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 |