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Seasonal nonstationarity

Just as a time series may have a trend in its regular component and must be differenced,a time series may also have a trend in its seasonal component.A time series with a seasonal trend only might appear to increase in regular,annual steps:

Of course,a time series may have both a regular trend and a seasonal trend, and in this case,the time series would have to differenced both regulary and seasonally before an ARIMA model can be identified.

If a time series nonstatonary in its seasonal component,the seasonal lags of the ACF will die out slowly.In figure 6.3(a) we show the ACF and PACF of a seasonally nonstationary time saeries.It would a have to be differenced seasonally to make it a stationary.The key to the identification of seasonal nonstationary is the existence of spikes at lags 24,36,and 48 that are comparable in magnitude to the spike at lag 24 of the ACF ,it is crucial to consider a minimum of 25 lags of the ACF , when analyzing monthly time-series data.In figure 6.3(a),we show 48 lags to demonstrate the principle underlying seasonal nonstationary.In practice, and in later axamples.we use ACFs and PACFs of only 30 lags since usually this is sufficient for estabilishing whether seasonal differencing is required.

Seasonal Autoregressive Processes

The most common seasonal autoregressive model is one where p=P=1,the ARIMA(1,0,0) (1.0.0)s model.The ACF for this model decasy exponentially but at the seasonal lags.In other words,for monthly data there is an exponential

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