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I have a time series on quarterly data with seasonality which I am trying to fit a SARIMA model to. The seasonal variation seems to be decreasing with time. I am wondering what the best way to model this with a SARIMA model is. Here is the series decomposed:

My confusion stems from when it is appropriate to take the first seasonal difference. In the literature, taking the first seasonal difference seems to be the correct way to handle seasonality. But if I do that, I model that the seasonal difference is constant over time, which it doesn't seem to be. If I then try to model without taking the first seasonal difference, all reasonable models violate the assumption of the AR constants being between -1 and 1. Here are the four quarters plotted separately:

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My confusion stems from when it is appropriate to take the first seasonal difference. In the literature, taking the first seasonal difference seems to be the correct way to handle seasonality.

Seasonal differencing should be applied when the series is seasonally integrated. That happens when the time series is generated from alternating random walks (one per period). As a consequence, any two consecutive observations of the time series diverge (!) under seasonal integration, which is often hard to justify. Thus I would think twice before applying seasonal differencing.

Regarding diminishing seasonality, I think seasonal AR or MA terms in the SARIMA model might work fine. But I am not quite sure. You could try auto.arima in R to see what SARIMA model is suggested and how the model residuals look; perhaps they will be adequate.

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  • $\begingroup$ SAR terms does seem to model to diminishing seasonality ok. If I should seasonally differate is still what I am unsure of. Canova-Hansen test tells me I should use one seasonal difference while the Osborn-Chui-Smith-Birchenhall and HEGY-test that I shouldn't. However, could this be due to lower power of the HEGY test? Because when adding the part of the time-series that I deleted to do the pseudo-out of sample forecast. I get the result that according to CH I should use one seasonal difference, OCSB that I shouldn't and HEGY that I should. $\endgroup$ – Dididi Mar 19 '17 at 12:54
  • $\begingroup$ Also, when using one seasonal difference, the best model achieved has a much lower insample RMSE than the the best model which doesn't have a one sample difference. $\endgroup$ – Dididi Mar 19 '17 at 12:54
  • $\begingroup$ @Dididi, What are the null hypotheses of these tests? If the more powerful test fails to reject a seasonal unit root, then you cannot say it is due to higher power, because a failure to reject can result from lower power, not higher power. Regarding RMSE, are you sure you are comparing it for the original data in both cases? Or are you comparing a fit for non-differenced data in the second case with a fit for differenced data in the first case (which would be wrong)? $\endgroup$ – Richard Hardy Mar 20 '17 at 20:36
  • $\begingroup$ I am using the RMSE reported from the arima() function in R. I guess I assumed that this is compared to the original data. Is this wrong? $\endgroup$ – Dididi Mar 21 '17 at 10:14
  • $\begingroup$ I don't know, it might be correct. One more test you can try is Canova-Hansen (available in the nsdiffs function in the "forecast" package in R). It is motivated by the low power of HEGY to reject the null of a seasonal unit root, so it considers a null of absence of a seasonal unit root instead. Just like HEGY generalizes the ADF test, so does Canova-Hansen generalize the KPSS test. The two can nicely complement each other. Also note that before using the Canova-Hansen test, non-seasonal unit roots and time trends should be removed as the test is not robust against these features. $\endgroup$ – Richard Hardy Mar 21 '17 at 10:38
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If you have strong feeling (and some theoretical foundation) that your time series have and will have decreasing seasonality, I would suggest trying two options:

1) Transform the time series (for instance by taking logarithm log(x))

2) Opt for second degree of seasonal differencing (D=2)

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    $\begingroup$ (1) can be a good idea. (2) sounds really radical. A seasonal unit root implies there are alternating random walks in the data so that any two consecutive observations of the time series diverge (!), which is often hard to justify. (IMHO, seasonal differentiation is overused.) If you take a second difference, you double that trouble. $\endgroup$ – Richard Hardy Mar 17 '17 at 13:19
  • $\begingroup$ Taking the log doesn't change much. The seasonality is still decreasing. I am guessing this due to the seasonality not being multiplicative. $\endgroup$ – Dididi Mar 19 '17 at 11:37

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