0.1 SARIMA(p,d,q)(P,D,Q)\(_s\) Processes
Many time series (e.g. monthly sales, quarterly GDP) exhibit both ordinary (short-lag) dependence and seasonal dependence at a fixed periodicity \(s\) (e.g. \(s=12\) for monthly data). The seasonal ARIMA model extends ARIMA(p,d,q) by adding a seasonal ARMA polynomial acting on lag-\(s\) multiples.
\[ \Phi(B^s)\phi(B)(1-B^s)^D(1-B)^d X_t = \Theta(B^s)\theta(B)Z_t, \] where: - \(\phi(B), \theta(B)\) are the ordinary AR/MA polynomials of orders \(p,q\), - \(\Phi(B^s), \Theta(B^s)\) are the seasonal AR/MA polynomials of orders \(P,Q\) in \(B^s\), - \((1-B)^d\) is ordinary differencing, \((1-B^s)^D\) is seasonal differencing of order \(D\) at period \(s\).
Seasonal differencing \((1-B^s)X_t = X_t - X_{t-s}\) removes deterministic (or slowly evolving) seasonal patterns, analogous to how ordinary differencing removes trend.
Model identification proceeds as for \(ARMA\), but examining the ACF/PACF at the seasonal lags \(s, 2s, 3s,\dots\) in addition to the low lags, since seasonal AR/MA orders manifest there.
An alternative to differencing out seasonality is explicit decomposition into trend + seasonal + remainder components — see Time Series Decomposition — which is often preferred for interpretability or when the seasonal pattern is closer to deterministic than stochastic.