Identifying Seasonal Components (ACF/PACF)

The Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) plots are valuable for identifying potential orders for the seasonal components ( $P, Q$ ) of SARIMA models. For ARIMA models, these plots help determine non-seasonal components ( $p, q$ ). When analyzing seasonal components, a primary distinction lies in which lags require particular attention.

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.

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References

Time Series Analysis: Forecasting and Control, George E. P. Box, Gwilym M. Jenkins, Gregory C. Reinsel, Greta M. Ljung, 2015 (John Wiley & Sons) - This is the definitive reference for the Box-Jenkins methodology, offering a foundational understanding of identifying ARIMA and seasonal ARIMA (SARIMA) model components, including the interpretation of ACF and PACF plots for seasonal patterns.
Forecasting: Principles and Practice, Rob J Hyndman and George Athanasopoulos, 2021 (OTexts) - Provides a practical and accessible guide to time series forecasting, with clear explanations and examples of using ACF and PACF plots to identify seasonal and non-seasonal components for SARIMA models.
Introduction to Time Series and Forecasting, Peter J. Brockwell, Richard A. Davis, 2016 (Springer) DOI: 10.1007/978-3-319-29318-8 - A rigorous yet accessible textbook that covers the theoretical underpinnings and practical aspects of time series analysis, including detailed discussions on ARIMA and seasonal models and their identification via ACF/PACF.