# statsmodels.tsa.arima.model.ARIMA¶

class statsmodels.tsa.arima.model.ARIMA(endog, exog=None, order=(0, 0, 0), seasonal_order=(0, 0, 0, 0), trend=None, enforce_stationarity=True, enforce_invertibility=True, concentrate_scale=False, trend_offset=1, dates=None, freq=None, missing='none', validate_specification=True)[source]

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

• autoregressive models: AR(p)

• moving average models: MA(q)

• mixed autoregressive moving average models: ARMA(p, q)

• integration models: ARIMA(p, d, q)

• seasonal models: SARIMA(P, D, Q, s)

• regression with errors that follow one of the above ARIMA-type models

Parameters
endogarray_like, optional

The observed time-series process $$y$$.

exogarray_like, optional

Array of exogenous regressors.

ordertuple, optional

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.

seasonal_ordertuple, optional

The (P,D,Q,s) order of the seasonal component of the model for the AR parameters, differences, MA parameters, and periodicity. Default is (0, 0, 0, 0). D and s are always integers, while P and Q may either be integers or lists of positive integers.

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

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.

enforce_stationaritybool, optional

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

enforce_invertibilitybool, optional

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

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 by one. This is only applicable when considering estimation by numerical maximum likelihood.

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.

datesarray_like of datetime, optional

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

freqstr, optional

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.

missingstr

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’.

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:

$\begin{split}Y_{t}-\delta_{0}-\delta_{1}t-\ldots-\delta_{k}t^{k}-X_{t}\beta & =\epsilon_{t} \\ \left(1-L\right)^{d}\left(1-L^{s}\right)^{D}\Phi\left(L\right) \Phi_{s}\left(L\right)\epsilon_{t} & =\Theta\left(L\right)\Theta_{s}\left(L\right)\eta_{t}\end{split}$

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.

Examples

>>> mod = sm.tsa.arima.ARIMA(endog, order=(1, 0, 0))
>>> res = mod.fit()
>>> print(res.summary())

Attributes
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.

start_params

Starting parameters for maximum likelihood estimation

state_names

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

tolerance

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 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 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 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