SARIMA and Seasonal Models

Retail sales spike every December, natural gas demand rises every winter, and quarterly earnings often show predictable patterns tied to the calendar. An ordinary ARIMA model, built to capture short-lag dependence (today relates to yesterday, or last week), completely misses this kind of structure — it has no notion that "12 months ago" is special in a way that "11 months ago" isn't. SARIMA (Seasonal ARIMA) fixes this by adding a second, parallel set of AR, differencing, and MA terms that operate specifically at the seasonal lag, so the model can capture both the ordinary short-run dynamics and the recurring seasonal pattern simultaneously.

An analogy: two clocks running at once

Think of a business with two overlapping rhythms: a fast one (this week versus last week — did we get a new client, was there a promotion) and a slow one (December versus every previous December). A model that only looks at the fast rhythm will keep getting fooled every holiday season, treating a predictable seasonal spike as if it were a surprising shock. SARIMA runs two clocks side by side: an ordinary ARIMA clock ticking at lag 1, 2, 3… for short-run dynamics, and a seasonal clock ticking at lag $s$, $2s$, $3s$… (where $s$ is the season length, like 12 for monthly data) for the recurring pattern — and combines both into one forecast.

The model, one piece at a time

SARIMA is written $\text{ARIMA}(p,d,q)(P,D,Q)_s$. The lowercase $(p,d,q)$ are the ordinary, non-seasonal AR order, differencing order, and MA order — exactly as in plain ARIMA. The uppercase $(P,D,Q)_s$ are the seasonal counterparts, applied at multiples of the seasonal period $s$:

$$\Phi_P(L^s)\,\phi_p(L)\,(1-L)^d(1-L^s)^D X_t = \Theta_Q(L^s)\,\theta_q(L)\,\varepsilon_t .$$

This looks dense, but each piece has a plain-English job: $\phi_p(L)$ and $\theta_q(L)$ are the familiar short-lag AR and MA polynomials; $\Phi_P(L^s)$ and $\Theta_Q(L^s)$ are the same kind of polynomials but built from lags at $s, 2s, 3s\ldots$, capturing "how does this January relate to last January, and the January before"; $(1-L)^d$ is ordinary differencing (removing a regular trend), and $(1-L^s)^D$ is seasonal differencing — subtracting the value from exactly one season ago, $X_t - X_{t-s}$, which removes a recurring seasonal level shift the same way ordinary differencing removes a regular trend. In plain English: SARIMA models today's value as a combination of recent past values, recent past shocks, the same-season past values, and the same-season past shocks — letting a December be explained partly by November and partly by last December, at the same time.

Worked example 1: seasonal differencing by hand

Monthly retail sales (in $ millions) run 100, 102, 98, 250 (December spike), 103, 101… repeating a similar spike each December. Ordinary first differencing, $X_t - X_{t-1}$, gives a huge +147 jump into December and a huge −147 drop right back out — the seasonal spike still dominates the differenced series, which is not stationary in any useful sense. Seasonal differencing at $s=12$, $X_t - X_{t-12}$, instead compares each December to last December: if this December is 250 and last December was 248, the seasonally differenced value is just +2 — small, because the predictable seasonal spike has been subtracted out entirely, leaving only the genuine year-over-year change.

Worked example 2: a SARIMA(1,0,0)(1,0,0)₁₂ forecast

Suppose a fitted model is $X_t = 0.3 X_{t-1} + 0.6 X_{t-12} - 0.18 X_{t-13} + \varepsilon_t$ (this specific combination is what a SARIMA(1,0,0)(1,0,0)₁₂ model expands to, since the seasonal and non-seasonal AR terms multiply together). Given $X_{t-1}=105$, $X_{t-12}=250$ (last December), and $X_{t-13}=103$ (last November), the forecast is $0.3(105) + 0.6(250) - 0.18(103) = 31.5 + 150 - 18.54 = 162.96$. Notice the seasonal term dominates the forecast (+150 from last December alone) — this is exactly the behavior an ordinary ARIMA model, lacking any $X_{t-12}$ term, could never produce, since it has no way to "remember" that December is special.

What this means in practice

SARIMA is the workhorse for forecasting anything with a calendar-driven cycle: retail and e-commerce demand, seasonal commodity demand (natural gas, agricultural futures), and quarterly-reported financial metrics. Choosing $s$ correctly (12 for monthly data with an annual cycle, 4 for quarterly, 5 for daily trading-week effects) and checking whether seasonal differencing is actually needed (via the same ADF/KPSS logic used for ordinary differencing, applied at the seasonal lag) both come before fitting the AR and MA orders.