Just as the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots helped us identify potential orders for the non-seasonal components ( $p, q$ ) of ARIMA models in Chapter 3, they are also instrumental in identifying the seasonal components ( $P, Q$ ) for SARIMA models. The main difference lies in which lags we pay close attention to.

Focusing on Seasonal Lags

When dealing with seasonality, we are interested in correlations at lags corresponding to the seasonal frequency. If a time series has a seasonal period $m$ , it means observations $m$ time steps apart are related. For instance:

Monthly data often has $m=12$ (e.g., January sales relate to previous January sales).
Quarterly data often has $m=4$ .
Daily data might have $m=7$ (weekday patterns).

The seasonal period $m$ is usually determined based on the nature of the data and visual inspection of the time series plot, often showing repeating patterns at regular intervals.

To identify the seasonal orders $P$ and $Q$ , we examine the ACF and PACF plots specifically at lags that are multiples of the seasonal period: $m, 2m, 3m, \dots$

Interpreting Seasonal Patterns in ACF/PACF

The interpretation rules are analogous to those for non-seasonal components, but applied to the seasonal lags:

Seasonal Autoregressive (AR) Order (P):
- Look at the PACF plot at seasonal lags ( $m, 2m, \dots$ ).
- A sharp cut-off after lag $P \times m$ in the PACF, combined with a slower, often sinusoidal or exponential decay across seasonal lags in the ACF, suggests a seasonal AR process of order $P$ . For example, if $m=12$ and the PACF has a significant spike only at lag 12 but not at 24, 36, etc., it suggests $P=1$ .
Seasonal Moving Average (MA) Order (Q):
- Look at the ACF plot at seasonal lags ( $m, 2m, \dots$ ).
- A sharp cut-off after lag $Q \times m$ in the ACF, combined with a slower decay across seasonal lags in the PACF, suggests a seasonal MA process of order $Q$ . For example, if $m=12$ and the ACF has a significant spike only at lag 12 but cuts off afterwards, it suggests $Q=1$ .
Seasonal Differencing (D):
- Strong, persistent positive correlations at seasonal lags in the ACF (e.g., high values at lags $m, 2m, 3m$ ) often indicate that seasonal differencing is needed ( $D > 0$ ). If you suspect seasonal non-stationarity, you should apply seasonal differencing (subtracting the observation from $m$ periods ago, $y'_t = y_t - y_{t-m}$ ) and then re-examine the ACF/PACF plots of the differenced series to determine $P$ and $Q$ . The need for seasonal differencing is often apparent even before looking at ACF/PACF plots if the seasonality clearly changes in level or amplitude over time.

Example: ACF/PACF Plots with Seasonality

Imagine we have monthly data ( $m=12$ ) that we suspect requires seasonal differencing ( $D=1$ ). After applying this differencing, we generate the ACF and PACF plots of the differenced series.

ACF plot showing a significant negative spike at lag 12 ( $m=12$ ) and smaller, insignificant values at lag 24. Non-seasonal lags (e.g., lag 1) might also be significant. The confidence interval is shown by the dashed lines.

PACF plot showing a significant negative spike at lag 12 and decaying values at further seasonal lags (lag 24 is smaller). Non-seasonal lags might also show significance.

Interpretation of Example Plots:

ACF: The significant spike at lag 12 that cuts off immediately (lag 24 is insignificant) strongly suggests a seasonal MA component of order $Q=1$ .
PACF: The spike at lag 12 is significant, but the spike at lag 24 is much smaller and potentially insignificant, suggesting decay. This pattern is consistent with a seasonal MA(1) process.
Non-Seasonal: We also observe significance at non-seasonal lags (e.g., lag 1 in both ACF and PACF). These would be used to inform the non-seasonal orders $p$ and $q$ , just as in Chapter 3.

Based on these plots for the seasonally differenced data (meaning $D=1$ ), a potential seasonal order could be $(P=0, D=1, Q=1)_{12}$ . The non-seasonal orders $(p, d, q)$ would be determined by examining the spikes at the initial lags (1, 2, 3, ...).

Remember, these plots provide guidance, not definitive answers. Often, you might need to try a few candidate models based on slightly different interpretations of the ACF/PACF plots, especially when the patterns aren't perfectly clear. The next section discusses strategies for combining these observations to select the full $SARIMA(p, d, q)(P, D, Q)_m$ order.