SARIMA Models: Seasonal ARIMA#

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The seasonal problem#

Plain ARIMA handles trend but not seasonality — a pattern that repeats every \(s\) steps (12 for monthly data, 4 for quarterly). SARIMA extends it by adding a second, seasonal ARIMA operating at the seasonal lag, so one model captures both short-term and season-to-season structure.

The notation#

A SARIMA is written SARIMA(p, d, q)(P, D, Q) with seasonal period s. The first triple is the ordinary non-seasonal part; the second is its seasonal mirror — \(P\) seasonal AR terms, \(D\) seasonal differences, \(Q\) seasonal MA terms — each acting at multiples of \(s\). In operator form the two multiply:

\[\Phi_P(B^s)\,\phi_p(B)\,(1-B)^d\,(1-B^s)^D\, x_t = \Theta_Q(B^s)\,\theta_q(B)\, w_t.\]

The seasonal difference#

The workhorse is the seasonal difference \((1 - B^s)\), which subtracts the value from one full season ago:

\[\nabla_s x_t = x_t - x_{t-s}.\]

Just as ordinary differencing removes a trend, this removes a repeating seasonal pattern. Data with both trend and seasonality may need both a regular difference (\(d\)) and a seasonal one (\(D\)). Spikes in the ACF / PACF at lags \(s, 2s, \dots\) point to the seasonal orders.

Fitting and pitfalls#

In statsmodels this is SARIMAX(order=(p,d,q), seasonal_order=(P,D,Q,s)). Two cautions: seasonal terms cost parameters, so imposing them on a non-seasonal series adds noise and can degrade forecasts; and for multiple overlapping seasonalities (say daily and weekly), specialised tools like TBATS or Prophet fit better than a single seasonal period.

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