Seasonal dummies significance issues

Question

I have a question concerning the significance of the seasonal dummies in my ARIMA-model (I do not use seasonal differencing or seasonal AR/MA as I have quite regular seasonality and I get better performance results):

What to do, if say 1 or 2 are significant but the rest are not? This happens to me particularly when I use monthly dummies. Shall I exclude the insignificant dummy variables?

If I do so (say I exclude from 11 monthly dummies the dummies for February, March), this will lead to a change again in the significance and coefficients of the remaining monthly dummy variables significantly!

I could not find any information on that in my standard forecasting textbooks.

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Finally I found a reference which I wanted to share to conclude the post:

The exclusion of some seasonal dummies because their estimated coefficients have low t-scores is not recommended. Preferably, testing seasonal dummy coefficients should be done with the F-Test instead of with the t-test because seasonality is usually a single compound hypothesis rather than 3 (or 11 with monthly data) individual hypothesis having to do with each quarter (or month). To the extent that a hypothesis is a joint one, it should be tested with the F-Test. If the hypothesis of seasonal variation can be summarized into a single dummy variable, then the use of the t-test will cause no problems. Often, where seasonal dummies are unambiguously called for, no hypothesis testing at all is undertaking.

Studenmund, A.H. and Cassidy, H.J. (1997). Using econometrics: A practical guide. The Addison-Wesley series in economics, 3rd ed., p. 257, Addison-Wesley, Reading, Mass.

Answer 1

This is partly a matter of taste and style, but my attitude here is that the dummies (indicators) are a team and should be used as such. The appropriate significance test thus considers the seasonal indicators jointly.

Once you start fiddling around and making ad hoc decisions you run risks of arbitrary analyses, accusations of data dredging, and so forth.

It is consistent even with well-defined seasonality that month-to-month shifts can be quite subtle or that two months have very similar coefficients.

A natural reservation is that you need to have enough data here to make adding 11 coefficients to a model reasonable in terms of degrees of freedom.

Answer 2

It has been my experience that deleting non-significant seasonal dummies is preferable as otherwise unwarranted blips are injected into the prediction that make no statistical sense. ARIMA models of order 12 can often be both more powerful and more parsimonious. AUTOBOX (a piece of software that I have contributed to) evaluates both approaches, thus allowing the data to have a "clear voice".

We often find that there are significant changes in the seasonal dummies suggesting "seasonal pulses". This can arise if the June effect over 6 years goes from 4,4,4 for the first 3 years to 10,10,10 for the last 3 years. Simple seasonal dummies as suggested by the OP and simple SARIMA do not correctly step up to the plate.