ARIMA Model (Autoregressive Integrated Moving Average)

First determine d by running an ADF or PP unit root test and differencing until the series is stationary. Then plot the ACF and PACF of the differenced series: a sharp cutoff in the PACF suggests the AR order p; a sharp cutoff in the ACF suggests the MA order q. Confirm with AIC/BIC across competing specifications, or use auto.arima in R to automate the search.

ARMA(p,q) models a stationary series directly. ARIMA(p,d,q) adds d rounds of differencing before fitting the ARMA part, making it applicable to non-stationary series with stochastic trends. An ARIMA(p,0,q) is identical to an ARMA(p,q).

When the series exhibits regular seasonal spikes at fixed intervals (e.g., monthly retail sales peaking every December), plain ARIMA cannot capture the pattern adequately. SARIMA adds seasonal AR and MA terms and seasonal differencing to handle periodic structure explicitly.

Standard ARIMA assumes constant parameters throughout the sample. A major structural break — such as a financial crisis or policy regime change — can distort the estimated orders and produce poor forecasts. Dummy variables, the Zivot-Andrews test, or regime-switching models should be considered when breaks are suspected.

Forecast accuracy degrades quickly with the horizon because each step compounds uncertainty from the estimated parameters and the innovations. As a rule of thumb, ARIMA forecasts are most reliable within a horizon equal to roughly one quarter of the sample length. Beyond that, prediction intervals widen substantially and the point forecast gravitates toward the sample mean.