In the previous chapter, you learned how to build Autoregressive Integrated Moving Average (ARIMA) models, denoted as $ARIMA(p, d, q)$ . These models are powerful tools for capturing serial correlation and trends in time series data. They work by relating the current value of a series to its own past values (the AR part) and to past forecast errors (the MA part), often after applying differencing to achieve stationarity (the I part).

However, many real-world time series exhibit seasonality, a pattern that repeats over a fixed, known period. Think of monthly retail sales data often peaking before holidays, quarterly company earnings reports, or daily website traffic showing weekday/weekend variations. These cycles occur at regular intervals (e.g., every 12 months, 4 quarters, or 7 days).

Standard $ARIMA(p, d, q)$ models face challenges when dealing with strong seasonality. Here's why:

Focus on Short-Term Lags: The $p$ and $q$ parameters in ARIMA primarily capture dependencies between an observation and those immediately preceding it (lag 1, lag 2, etc.). While differencing ( $d$ ) handles trends, it doesn't inherently account for the relationship between an observation and another observation exactly one season ago (e.g., this December vs. last December).
Ignoring Seasonal Dependencies: A strong seasonal pattern implies that the value at time $t$ is often highly correlated with the value at time $t-m$ , where $m$ is the seasonal period (e.g., $m=12$ for monthly data). An $ARIMA(p, d, q)$ model doesn't have a built-in mechanism to directly incorporate this lag- $m$ dependency without potentially making $p$ or $q$ very large.
Model Complexity: To force a standard ARIMA model to capture a seasonal effect at lag $m$ , you might need to set $p$ or $q$ to be at least $m$ . For monthly data ( $m=12$ ), this could mean including 12 or more AR or MA parameters. Such models become cumbersome, difficult to estimate reliably, and prone to overfitting the non-seasonal fluctuations in the data. The large number of parameters doesn't efficiently target the specific seasonal structure.
Inefficient Differencing: While the differencing order $d$ helps remove trends, it doesn't address seasonal patterns effectively. A different kind of differencing, seasonal differencing, which calculates the difference between an observation and the corresponding observation from the previous season ( $y_t - y_{t-m}$ ), is often required. This isn't part of the standard ARIMA framework.

Consider a time series representing monthly sales of air conditioners, which likely peaks in the summer months each year.

Illustrative monthly sales data showing a clear yearly seasonal pattern. Sales peak in months 6-8 (summer) each year.

An ARIMA model trying to forecast sales for next July would primarily look at sales in June, May, April, etc. (lags 1, 2, 3...). While these recent values provide some information, the sales figure from the previous July (lag 12) is often a much stronger predictor due to the seasonal nature of demand. A standard ARIMA model doesn't explicitly use this lag-12 information unless $p$ or $q$ is set to 12 or higher, which, as noted, creates an inefficient model structure.

Furthermore, if we look at the Autocorrelation Function (ACF) plot for such seasonal data (after potentially making it stationary), we often see significant correlations not just at short lags but also at lags corresponding to the seasonal frequency (e.g., 12, 24, 36 for monthly data).

Example ACF plot for data with monthly seasonality. Note the significant spikes at lag 12 and lag 24, indicating strong correlation at the seasonal frequency, in addition to shorter-lag correlations. The dashed lines represent approximate confidence bounds.

Standard ARIMA models are not structured to efficiently capture these distinct seasonal spikes in the ACF. They try to model the decaying pattern from lag 1 onwards but don't have specific terms dedicated to lags $m, 2m, 3m$ , etc.

Because of these limitations, relying solely on $ARIMA(p, d, q)$ for strongly seasonal data often leads to suboptimal forecasts. The models might fail to capture the recurring peaks and troughs accurately or require an excessive number of parameters. This necessitates an extension that explicitly incorporates seasonal components, leading us to the Seasonal ARIMA (SARIMA) model.