SARIMA stands for Seasonal AutoRegressive Integrated Moving Average. It extends the standard ARIMA model by adding seasonal AR, differencing, and MA components that operate at seasonal lags. Use SARIMA when your time series data shows a repeating seasonal pattern over a fixed period — such as monthly sales data that peaks every holiday season, quarterly GDP that dips every Q1, or weekly website traffic that spikes every Monday. The model is written as SARIMA(p,d,q)(P,D,Q)[s], where s is the seasonal period.

Each parameter controls a specific part of the model: p = number of past values (lags) used in the non-seasonal autoregressive component; d = number of times the series is differenced to remove a trend; q = number of past forecast errors in the non-seasonal moving average; P = seasonal AR order (how many seasonal lags); D = seasonal differencing (removes seasonal trend); Q = seasonal MA order; s = number of time steps per season (12 for monthly, 4 for quarterly, 7 for daily-within-weekly). Choosing appropriate values requires inspecting ACF and PACF plots and comparing AIC/BIC across candidate models.

Start by plotting the series and checking for trend (d) and seasonal non-stationarity (D). If the series drifts upward, set d=1. If the seasonal amplitude changes year over year, set D=1. After differencing, inspect the ACF: significant spikes at lag 1, 2, … suggest q; spikes at lag s, 2s, … suggest Q. Inspect the PACF: spikes at lag 1, 2, … suggest p; spikes at lag s, 2s, … suggest P. Fit several candidate models and compare their AIC and BIC — lower values indicate a better fit. Always verify with the Ljung-Box test on residuals: a non-significant result (p > 0.05) means the model has captured all autocorrelation.

AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) both balance model fit (log-likelihood) against model complexity (number of parameters). Lower values are better — they indicate a model that fits the data well without overfitting. BIC applies a stronger penalty for additional parameters, so it tends to select simpler models than AIC. In practice, if AIC and BIC disagree, prefer BIC for parsimony in short series (n < 100) and AIC for predictive accuracy in longer series. Differences of less than 2 units between models are not meaningful.

The Ljung-Box test checks whether the residuals from your fitted SARIMA model are uncorrelated (white noise). The null hypothesis is that all autocorrelations in the residuals up to lag h are zero. A non-significant result (p > 0.05) means the model has successfully removed all autocorrelation from the data — this is what you want. A significant result (p ≤ 0.05) means residual autocorrelation remains, and the model is inadequate. Try increasing q or Q, or adding one more AR term. Always test at lag h = min(2s, n/5) for seasonal models.

The point forecast is the model's single best estimate of each future value, generated by propagating the fitted model equations forward. The confidence interval (CI) is the range in which the true future value is expected to fall with the stated probability (e.g., 95%). The CI widens with the forecast horizon because uncertainty accumulates. A 95% CI means that if you repeated this process many times, 95% of the intervals would contain the true value. In practice, SARIMA forecasts are most reliable within 1–2 seasonal cycles; beyond that, uncertainty can be very large.

MAPE (Mean Absolute Percentage Error) expresses forecast error as a percentage of actual values — it is scale-free and easy to interpret. General benchmarks: below 10% = excellent accuracy; 10–20% = good; 20–50% = acceptable for complex data; above 50% = poor. However, MAPE is undefined or misleading when actual values are near zero. For such data, use RMSE (Root Mean Squared Error) instead, which is in the same units as the original series. A model with lower RMSE fits the historical data more closely than one with higher RMSE.

ARIMA(p,d,q) handles non-seasonal structure: trend removal via differencing (d), autoregressive lag dependence (p), and moving average smoothing (q). SARIMA adds a second layer of these components at the seasonal lag s: seasonal differencing (D), seasonal AR terms (P), and seasonal MA terms (Q). If your data shows significant autocorrelation at lags s, 2s, 3s in the ACF, ARIMA alone will leave this structure in the residuals — you need SARIMA. A quick check: if the ACF of the differenced series shows spikes at multiples of 12 (for monthly data), use SARIMA.

As a minimum, you need at least 2 complete seasonal cycles (e.g., 24 months for s=12) to identify seasonal parameters. In practice, 4–5 cycles (48–60 months) are recommended for stable parameter estimates. With fewer than 30 observations, parameter estimates become unreliable and forecasts very wide. For complex models with large P and Q, even more data is needed. If you have very short series, consider simpler ETS or Holt-Winters models instead.

Report the model specification, parameter estimates (with standard errors), log-likelihood, AIC, BIC, and Ljung-Box diagnostic. Example: "A SARIMA(1,1,1)(1,1,1)[12] model was fitted to 48 months of retail sales data. The model yielded AIC = 234.5, BIC = 248.7, and RMSE = 12.3 units. Ljung-Box test on residuals at lag 24 indicated no significant autocorrelation, Q(24) = 18.3, p = .79, confirming adequate model fit. The 12-step-ahead forecast produced a mean prediction of 105.2 units (95% CI [87.4, 123.0])." See the How to Write Your Results section for five full reporting templates.

The seasonal period s is determined by the frequency at which the pattern repeats: monthly data = s=12; quarterly data = s=4; daily data with weekly pattern = s=7; hourly data with daily pattern = s=24; weekly data with annual pattern = s=52. Look at your raw data plot — if you see a repeating cycle every 12 data points, then s=12. You can also check the ACF: significant spikes at lags 12, 24, 36 confirm s=12. If unsure, start with the most natural period for your data collection frequency.

This tool is designed for education, exploration, and preliminary analysis. For formal research submissions, verify your results using peer-reviewed software: R (forecast package, auto.arima, or statsmodels in Python), Python (statsmodels.tsa.statespace.sarimax), or SAS PROC ARIMA. You can cite this tool as: StatsUnlock. (2025). SARIMA model calculator. Retrieved from https://statsunlock.com/sarima-model-calculator. The interpretations and write-up templates are suitable as a starting point for your methods and results sections.