ARIMASARIMA

ARIMASARIMA is a term used to describe a comprehensive time series model that combines the features of ARIMA with both seasonality and exogenous predictors. In practice, it is equivalent to what is commonly called SARIMAX: a seasonal ARIMA model that allows external variables to influence the series.

Model structure

The model incorporates nonseasonal autoregressive terms, seasonal autoregressive terms, nonseasonal moving average terms, seasonal moving average terms, differencing to achieve stationarity, and a linear effect from exogenous variables. Let y_t denote the target series, X_t a vector of exogenous predictors, and s the seasonal period. With p and d for nonseasonal AR order and differencing, P and D for seasonal AR and differencing, and q and Q for nonseasonal and seasonal MA orders, a typical representation is:

Phi_p(B) Phi_P(B^s) (1 - B)^d (1 - B^s)^D y_t = Theta_q(B) Theta_Q(B^s) a_t + beta' X_t

where a_t is white noise, and beta is the vector of coefficients for the exogenous variables. This formulation encompasses a wide range of patterns, including nonseasonal and seasonal persistence, shocks, and external drivers.

Estimation and use

Parameters are usually estimated by maximum likelihood or conditional likelihood methods. Diagnostic checks on residuals and stability are standard steps. ARIMASARIMA is used for forecasting when past values, seasonal patterns, and external factors jointly influence the series, such as economic indicators guided by policy variables, energy demand with weather covariates, or sales data with marketing activity.

Relation to related models

ARIMASARIMA generalizes ARIMA, ARIMAX, SARIMA, and SARIMAX. In software, similar specifications are often implemented as SARIMAX, sometimes with terminology highlighting the combined exogenous and seasonal components.