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I have recently fit a Fourier regression ARIMA error model to some time series data, which has weekly and yearly seasonality. The model is of this form (except there is another sum since I have multiple seasonality):

enter image description here

My ARIMA error term (N_t) is ARIMA[5,1,3]. I probably can't publically disclose this data. I have fit a Fourier model with 3 weekly sin/cos terms and 9 yearly ones (by choosing the K values which minimize the AIC).

When I plot the ACF of the residuals (ignore the graphs title) I get this:

enter image description here

There are significant spikes in the ACF every 14 terms. To me this would indicate the need to add a moving average term of period 14 for some Q (I'd have to see how far back the significant spikes go to determine Q),to give an error term of SARIMA[5,1,3][0,0,Q][14]. However Rob Hydnman says "Note that the ARIMA [term in this type of model] should be non-seasonal". I suppose because the Fourier terms are meant to capture all the seasonality.

So I guess my questions are:

A) Do I extend my Fourier model to capture this biweekly link? I.e. do I look for weekly, biweekly and yearly seasonality in my Fourier terms?

B) What is so wrong with having a SARIMA error term in these Fourier regression models?

Any help would be much appreciated. Thanks!

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