Seasonal ARIMA (SARIMA)
Seasonal Autoregressive Integrated Moving Average · Also known as: seasonal ARIMA, Box-Jenkins seasonal model, SARIMA — Mevsimsel ARIMA
SARIMA is a seasonal extension of the Box-Jenkins ARIMA model that adds seasonal differencing and seasonal autoregressive and moving-average terms. Developed within the Box, Jenkins, Reinsel and Ljung framework (5th edition, 2015), it forecasts series whose pattern repeats on a yearly, monthly, or weekly period.
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When to use it
Use SARIMA for a single continuous time series that shows a clear seasonal cycle of known period (for example 12 for monthly data with yearly seasonality, or 7 for daily data with weekly seasonality), and when the goal is forecasting or prediction. It needs a reasonably long history — at least about 48 observations — so that the seasonal structure can be estimated. The series must be made stationary, which seasonal and ordinary differencing usually achieve, the seasonal period must be known in advance, and the model residuals should behave like independent noise. It is not the right tool when there is no seasonality, when external predictors drive the series (use SARIMAX), or when the history is too short to reveal a stable seasonal pattern.
Strengths & limitations
- Explicitly models seasonality on top of trend, capturing repeating yearly, monthly, or weekly patterns that plain ARIMA misses.
- Built on the well-established Box-Jenkins methodology, with mature diagnostics and information criteria for order selection.
- Produces principled forecasts with prediction intervals for a single series without needing external regressors.
- Requires a fairly long history (at least about 48 observations) to estimate the seasonal terms reliably.
- The seasonal period must be known in advance and assumed roughly constant.
- Order selection across (p, d, q)(P, D, Q)_s can be intricate, and a misspecified model leaves autocorrelated residuals that bias the forecasts.
Frequently asked
How is SARIMA different from ARIMA?
ARIMA models trend and short-term dependence but not recurring seasonal patterns. SARIMA adds seasonal differencing and seasonal AR/MA terms at the seasonal lag s, so it can capture cycles that repeat every year, month, or week on top of the ordinary dynamics.
What is the seasonal period s and how do I choose it?
s is the number of steps in one full seasonal cycle: 12 for monthly data with yearly seasonality, 4 for quarterly data, or 7 for daily data with a weekly cycle. It must be known and specified in advance from the nature of the data.
How much data does SARIMA need?
At least about 48 observations. Because the seasonal terms are estimated from how the series behaves across whole seasons, several complete cycles of history are required for stable estimates.
When should I use SARIMAX instead?
Use SARIMAX when external (exogenous) variables — holidays, economic indicators, or policy changes — help explain the series. SARIMAX adds those regressors to the SARIMA structure; plain SARIMA models the series from its own past alone.
Sources
- Box, G.E.P., Jenkins, G.M., Reinsel, G.C. & Ljung, G.M. (2015). Time Series Analysis: Forecasting and Control (5th ed.). Wiley. ISBN: 978-1118675021
- Hyndman, R.J. & Athanasopoulos, G. (2021). Forecasting: Principles and Practice (3rd ed.). OTexts. ISBN: 978-0987507136
How to cite this page
ScholarGate. (2026, June 1). Seasonal Autoregressive Integrated Moving Average. ScholarGate. https://scholargate.app/en/econometrics/sarima
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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