Does a seasonal ARIMA model involve seasonality?

The question posed in the title may seem a tautology, but it's not. If an ARIMA model (chosen by any manual or automated procedure) contains lags of seasonal order, it does not necessarily mean that there is a relevant seasonal pattern in the data. Below I elaborate on this idea as a complement to the discussion in this post.

It is illuminating to plot the gain of the corresponding ARIMA filter. The gain reveals the frequencies of those cycles that are captured by the filter. A gain close to zero indicates that the cycle at that frequency is captured by the ARIMA filter, whereas a peak in the gain implies that the corresponding cycles are overlooked by the filter.

The left-hand-side plot in the figure below displays the gain of the following seasonal ARIMA(1,1,0)(1,0,0) model for a monthly series:

(1 - 0.252L)(1 + 0.446L^{12})(1 - L)y_t = \epsilon_t \,, \quad \epsilon_t \sim NID(0, \sigma^2) \,, where L is the lag operator, such that L^i y_t = y_{t-i}.

Vertical dotted lines point to the seasonal frequencies, which for monthly series are by definition: \omega_j = 2\pi j / 12\,, j=1,2,...,6. We can see that those cycles related to the seasonal frequencies are not touched by this filter. The right-hand-side plot displays the gain for the same model by with a negative seasonal AR coefficient. In this case, the model captures cycles related to the seasonal frequencies. If the second filter were applied to a monthly series, the seasonal cycles would be filtered and removed from the original series. In the first model, the seasonal cycles will escape to the filter.

Gain of ARIMA filter

One interpretation in the context of time series analysis is that if model selected for the data is the first one, then there is no major seasonal pattern as the selected model is not related to any seasonal cycle. The reverse applies for the model with negative seasonal AR coefficient.

When ARIMA models are used to obtain forecasts, we may not care much about the implications of the parameters of the fitted model. We can stick to choose and fit the model that gives better forecasts according to the mean squared error or to other measure of accurary. In general, we may expect that a seasonal ARIMA model will perform better with monthly or quarterly series, but that's all.

If the purpose of the analysis is to extract a seasonal component or to explore whether a seasonal patter is present in the data, then we must pay attention to the properties of the filter defined by the chosen ARIMA model.

Concluding that there is a seasonal pattern in the data because the seasonal AR coefficient is significant is not appropriate. We have seen two seasonal ARIMA models with very different features. Displaying the gain of the filter is helpful to study the properties of the filter and the frequency of the cycles that explain the variability in the observed data.

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