SARIMA — Seasonal ARIMA

SARIMA extends ARIMA to handle seasonality — patterns that repeat over a fixed cycle (month, quarter). SARIMA adds seasonal AR/MA/differencing components alongside the non-seasonal ones.

When to use

Use SARIMA when the series has a clear seasonal cycle (e.g. monthly retail sales, quarterly tourism). ACF spikes at seasonal lags signal the need for SARIMA.

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Model notation

$$\text{SARIMA}(p,d,q)(P,D,Q)_s$$

• $(p,d,q)$: the non-seasonal part (as in ARIMA).
• $(P,D,Q)$: the seasonal part with cycle $s$ (e.g. $s=12$ for monthly, $s=4$ for quarterly data).

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Running in EcoLab

1. Modeling module → Univariate time series family → SARIMA.
2. Choose $Y$; declare $(p,d,q)(P,D,Q)_s$ and the seasonal period $s$ (or auto).
3. Run; view seasonal forecasts + diagnostics; export the replication code.

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Replication code

Stata

* --- SARIMA(1,1,1)(1,1,1)_12 ---
tsset time

* Estimate SARIMA with seasonal component (s = 12)
arima y, arima(1,1,1) sarima(1,1,1,12)

* Residual diagnostics
predict resid, residuals
corrgram resid, lags(24)

* Forecast
tsappend, add(12)
predict yhat, dynamic(.)

R

# --- SARIMA ---
library(forecast)

# Create monthly time series
ts_data <- ts(df$y, start = c(2010, 1), frequency = 12)

# Fit SARIMA(1,1,1)(1,1,1)[12]
fit <- Arima(ts_data, order = c(1, 1, 1),
             seasonal = c(1, 1, 1))
summary(fit)

# Diagnostics and forecast
checkresiduals(fit)
fc <- forecast(fit, h = 24)
plot(fc)

Python

from statsmodels.tsa.statespace.sarimax import SARIMAX
import matplotlib.pyplot as plt

# Estimate SARIMA(1,1,1)(1,1,1,12)
model = SARIMAX(df['y'],
                order=(1, 1, 1),
                seasonal_order=(1, 1, 1, 12))
result = model.fit(disp=False)
print(result.summary())

# Diagnostics
result.plot_diagnostics(figsize=(10, 6))
plt.tight_layout(); plt.show()

# Forecast 24 months
forecast = result.get_forecast(steps=24)
print(forecast.summary_frame())

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Limitations

• Many parameters ⇒ needs a long series spanning several seasonal cycles.
• Assumes the seasonal pattern is stable over time.

Video tutorial

Video Tutorial: Running SARIMA in EcoLab

See also

• ARIMA · GARCH · Catalog