How does SARIMA handle seasonal patterns in time series?

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AcadiFi · Accepted Answer

Seasonal ARIMA, denoted SARIMA(p,d,q)(P,D,Q)s, adds seasonal AR, MA, and differencing terms at seasonal lag s. For monthly data with annual seasonality, s = 12; for quarterly data, s = 4. The seasonal differencing operator is (1 - B^s) where B is the backshift: for monthly data this computes year-over-year differences. Applied after regular differencing, SARIMA handles series with both trend and seasonal patterns. Example SARIMA(1,1,1)(0,1,1)12 on monthly retail sales: the regular part captures month-to-month dynamics with AR(1) and MA(1); seasonal MA(1) at lag 12 absorbs residual annual pattern; seasonal differencing removes the cycle. Fitted model: (1 - φB)(1 - B)(1 - B^12) yt = (1 + θB)(1 + ΘB^12) et. ```mermaid flowchart LR A[Raw Series] --> B[First Diff
remove trend] B --> C[Seasonal Diff
remove cycle] C --> D[Fit ARMA] D --> E[Forecast] ``` Model identification examines ACF and PACF at both low lags (for regular orders) and seasonal lags (12, 24, 36 for monthly). Significant spikes at seasonal lags indicate seasonal terms. Information criteria guide final selection. Forecasts incorporate both short-term persistence and seasonal structure, producing patterns that preserve annual cycles. SARIMA struggles with complex seasonality (multiple overlapping cycles, evolving patterns)—state-space models or Prophet handle those better, but SARIMA remains the workhorse for classical seasonal forecasting in finance and economics.

How does SARIMA handle seasonal patterns in time series?

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