Autoregressive Integrated Moving Average (ARIMA) model, and extensions

This model is the basic interface for ARIMA-type models, including those with exogenous regressors and those with seasonal components. The most general form of the model is SARIMAX(p, d, q)x(P, D, Q, s). It also allows all specialized cases, including

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

Array of exogenous regressors.

The (p,d,q) order of the model for the autoregressive, differences, and moving average components. d is always an integer, while p and q may either be integers or lists of integers specifying exactly which lag orders are included. The order of differences is to achieve stationnarity in the context of a sochastic trend or seasonality. The default is (0, 0, 0).

The (P,D,Q,s) order of the seasonal component of the model for the AR parameters, differences, MA parameters, and periodicity. D is a non- negative integer, and s is an integer strictly greater than one. P and Q may either be integers or lists of positive integers specifying exactly which lag orders are included. The default is (0, 0, 0, 0).

Parameter controlling the deterministic trend. Can be specified as a string where ‘c’ indicates a constant term, ‘t’ indicates a linear trend in time, and ‘ct’ includes both. Can also be specified as an iterable defining a polynomial, as in numpy.poly1d , where [1,1,0,1] would denote \(a + bt + ct^3\) . Default is ‘c’ for models without integration, and no trend for models with integration. Note that all trend terms are included in the model as exogenous regressors, which differs from how trends are included in SARIMAX models. See the Notes section for a precise definition of the treatment of trend terms.

Whether or not to require the autoregressive parameters to correspond to a stationarity process. Default is True.

Whether or not to require the moving average parameters to correspond to an invertible process. Default is True.

Whether or not to concentrate the scale (variance of the error term) out of the likelihood. This reduces the number of parameters by one. This is only applicable when considering estimation by numerical maximum likelihood. Default is False.

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.

If no index is given by endog or exog , an array-like object of datetime objects can be provided.

If no index is given by endog or exog , the frequency of the time-series may be specified here as a Pandas offset or offset string.

Available options are ‘none’, ‘drop’, and ‘raise’. If ‘none’, no nan checking is done. If ‘drop’, any observations with nans are dropped. If ‘raise’, an error is raised. Default is ‘none’.

Names of endogenous variables

The names of the exogenous variables.

Initial design matrix

Initial selection matrix

Initial state intercept vector

Initial transition matrix

The latex names of all possible model parameters.

The plain text names of all possible model parameters.

The orders of each of the polynomials in the model.

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

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

Starting parameters for maximum likelihood estimation

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

Notes

This model incorporates both exogenous regressors and trend components through “regression with ARIMA errors”. This differs from the specification estimated using SARIMAX which treats the trend components separately from any included exogenous regressors. The full specification of the model estimated here is:

where \(\eta_t \sim WN(0,\sigma^2)\) is a white noise process, L is the lag operator, and \(G(L)\) are lag polynomials corresponding to the autoregressive ( \(\Phi\) ), seasonal autoregressive ( \(\Phi_s\) ), moving average ( \(\Theta\) ), and seasonal moving average components ( \(\Theta_s\) ).

enforce_stationarity and enforce_invertibility are specified in the constructor because they affect loglikelihood computations, and so should not be changed on the fly. This is why they are not instead included as arguments to the fit method.

See the notebook ARMA: Sunspots Data and ARMA: Artificial Data for an overview.

Examples

>>> mod = sm.tsa.arima.ARIMA(endog, order=(1, 0, 0))
>>> res = mod.fit()
>>> print(res.summary())
Methods
clone(endog[, exog])
Clone state space model with new data and optionally new specification
filter(params[, transformed, ...])
Kalman filtering
fit([start_params, transformed, ...])
Fit (estimate) the parameters of the model.
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, ...])
Ensure model parameters satisfy shape and other requirements
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