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')[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

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.

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.


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’.


This model incorporates both exogenous regressors and trend components through “regression with ARIMA errors”.

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.

TODO: should we use concentrate_scale=True by default?


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


clone(endog[, exog])

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 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


Fisher information matrix of model.


Initialize the SARIMAX model.


Initialize approximate diffuse


Initialize default

initialize_known(initial_state, …)

Initialize known


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 the memory conservation method


Set the filtering method


Set the inversion method


Set the smoother output


Set the numerical stability method

simulate(params, nsimulations[, …])

Simulate a new time series following the state space model


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 unconstrained parameters used by the optimizer to constrained parameters used in likelihood evaluation.


Transform constrained parameters used in likelihood evaluation to unconstrained parameters used by the optimizer

update(params[, transformed, …])

Update the parameters of the model



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.