Time Series • Introduction to Autoregressive Models and Forecasting
Time Series / Seasonal ARIMA Models

Seasonal ARIMA Models

Notes 4 Introduction to Autoregressive Models and Forecasting
Quick concept

A Seasonal ARIMA model, written SARIMA, extends ARIMA by adding seasonal AR, seasonal differencing and seasonal MA at lag m. It is used when the series has a repeating seasonal pattern as well as possible non-seasonal dependence.

Notes

Seasonal ARIMA Models

Definition

A Seasonal ARIMA model, written SARIMA, extends ARIMA by adding seasonal AR, seasonal differencing and seasonal MA at lag m. It is used when the series has a repeating seasonal pattern as well as possible non-seasonal dependence.

Example

Monthly packet sales with a December peak every year may use m = 12. A small teaching model such as SARIMA(0, 1, 1)(0, 1, 1)12 combines ordinary differencing with seasonal differencing at lag 12.

Notation

SARIMA(p, d, q)(P, D, Q)m

Symbol Role
p, d, q Non-seasonal AR, differencing and MA (same idea as ARIMA)
P, D, Q Seasonal AR, seasonal differencing and seasonal MA
m Seasonal period (for example 12 for months)

Seasonal Components in Words

  • Seasonal AR (P): the current value can depend on the value from m periods ago (last year, for monthly data), not only on last month.
  • Seasonal differencing (D): compare this period with the same period one season earlier, for example Yt − Yt−12 when m = 12. This is not the same as ordinary (non-seasonal) differencing Yt − Yt−1.
  • Seasonal MA (Q): the current value can depend on the shock from m periods ago.

Ordinary d and seasonal D answer different questions: nearby wandering versus a repeating seasonal level. Both can be 0 or 1 in simple teaching models. Large grids of P, D, Q are not the first step.

ARIMA vs SARIMA

ARIMA SARIMA
Notation (p, d, q) (p, d, q)(P, D, Q)m
Seasonal lag m Not built in Included
Typical use Non-seasonal or weakly seasonal series Clear repeating seasonal pattern
Differencing Non-seasonal d Non-seasonal d and/or seasonal D

SARIMA extends ARIMA; it does not replace the need to plot the series and to check residuals. ACF at lag m supports the choice of m, but does not by itself prove a unique (P, D, Q).

Exam-Oriented Key Points

  1. SARIMA = Seasonal ARIMA with extra orders (P, D, Q) and period m.
  2. p, d, q are non-seasonal; P, D, Q are seasonal.
  3. Seasonal differencing uses lag m; non-seasonal differencing uses lag 1.
  4. For monthly yearly seasonality, m = 12 is the usual teaching choice.