SARIMA Model — Seasonal Autoregressive Integrated Moving Average
Seasonal Autoregressive Integrated Moving Average Model · Also known as: SARIMA, seasonal ARIMA, Box-Jenkins seasonal model, ARIMA with seasonal component
SARIMA extends ARIMA by adding seasonal autoregressive and moving-average operators to capture repeating patterns at fixed intervals — such as monthly, quarterly, or annual cycles. Denoted SARIMA(p,d,q)(P,D,Q)s, it is the standard workhorse for univariate seasonal time series forecasting in econometrics, economics, and official statistics.
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When to use it
Use SARIMA when you have a univariate time series that displays both trend and a stable, periodic seasonal pattern at a known frequency (e.g., monthly, quarterly). It is especially valuable in macroeconomics, finance, and official statistics for series such as GDP, retail sales, unemployment, or electricity demand. Prefer SARIMA over plain ARIMA whenever ACF/PACF plots reveal significant autocorrelation at seasonal lags. Do not use SARIMA if the seasonal pattern is irregular or changes amplitude over time — consider TBATS or ETS/state-space models instead. It is also unsuitable for multivariate problems (use VAR or VARMA) or very short series (fewer than two or three complete seasonal cycles).
Strengths & limitations
- Handles both trend and seasonal non-stationarity in a single, unified framework.
- Parsimonious: a small number of integer orders (p,d,q,P,D,Q) can capture complex dynamics.
- Well-understood theoretical properties and diagnostic toolkit (ACF, PACF, Ljung-Box).
- Produces prediction intervals, enabling probabilistic forecasting rather than point estimates only.
- Widely implemented in R (forecast, fable), Python (statsmodels, pmdarima), EViews, and Stata.
- Assumes a linear, constant-parameter data-generating process — cannot model structural breaks or regime changes without extension.
- Requires at least two to three full seasonal cycles of data for reliable estimation; short series yield unstable parameter estimates.
- Seasonal length s must be pre-specified; misspecifying s leads to poor fit and invalid forecasts.
- Forecasting accuracy degrades rapidly at longer horizons because uncertainty compounds with each step ahead.
- Cannot handle multiple seasonalities (e.g., daily and weekly) in a single model — use TBATS or Facebook Prophet for that.
Frequently asked
How do I choose the seasonal period s?
The seasonal period s is determined by the data frequency: s = 12 for monthly, s = 4 for quarterly, s = 7 for daily series with a weekly cycle, and so on. It is not estimated from the data but set by the analyst based on domain knowledge. If you are uncertain, plot the series and its ACF — peaks at regular intervals reveal the dominant season length.
What is the difference between d and D?
d is the number of regular (non-seasonal) differences needed to remove a trend, and D is the number of seasonal differences needed to remove a seasonal unit root. In practice D = 1 is almost always sufficient for one seasonal cycle, and d = 1 handles most linear trends. Applying D = 1 before selecting d is a common strategy.
How do I decide between SARIMA and ETS (exponential smoothing)?
Both can handle trend and seasonality, but they make different structural assumptions. SARIMA is best when the ACF/PACF show interpretable patterns and you want explicit lag structure. ETS is simpler to specify and often performs comparably in practice. Running both and comparing out-of-sample error metrics (RMSE, MAPE on a hold-out set) is the standard approach.
Can SARIMA handle multiple seasonal cycles?
Standard SARIMA handles only one seasonal period. For data with multiple seasonalities — such as hourly data with daily and weekly patterns — use TBATS, BATS, or harmonic regression combined with ARIMA instead.
What should I do if Ljung-Box tests still show autocorrelation after fitting?
Remaining autocorrelation at seasonal lags usually means D or Q is too small. Remaining autocorrelation at short lags usually means p or q needs to be increased. Check the ACF of residuals, compare candidate models by AIC/BIC, and rerun diagnostics after refitting.
Sources
- Box, G. E. P., Jenkins, G. M., & Reinsel, G. C. (1976). Time Series Analysis: Forecasting and Control (revised ed.). Holden-Day. ISBN: 978-0130607744
- Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and Practice (3rd ed.). OTexts. link ↗
How to cite this page
ScholarGate. (2026, June 3). Seasonal Autoregressive Integrated Moving Average Model. ScholarGate. https://scholargate.app/en/econometrics/sarima-model
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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