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I am trying to forecast daily data with (S)ARIMA, having observations for the last 180 days.

enter image description here

STL decomposition clearly shows seasonality and ACF plot shows spikes at 7, 14, 21, etc. days so that I assume weekly seasonality, which meets our preliminary expectations as data comes from network traffic export which is significantly and 'reliably' lower on weekends:

enter image description here enter image description here

ACF after manual seasonal differencing:

 y_diff = diff(y, k_diff = 0, k_seasonal_diff = 1, seasonal_periods = 7)

enter image description here

Using pmdarima_AutoARIMA without differencing on the raw (not differenced) data it suggests ARIMA(1,0,0)(2,0,0)7

model = pm.auto_arima(y, seasonal = True, m = 7, trace = False)

Residuals ACF:

enter image description here

Using pmdarima_AutoARIMA with forced seasonal differencing (D=1) on the raw (not differenced) data it suggests ARIMA(2,0,2)(1,1,1)7

model = pm.auto_arima(y, seasonal = True, D = 1, m = 7, trace = False)

Residuals ACF:

enter image description here

With seasonal differencing, results are worse in terms of residuals ACF and actually in terms of RMSSE on test set as well.

My question is that how should one know that seasonal differencing should be used or not for SARIMA at all? If there is a clear seasonality in time series, how it is possible to get better results without seasonal differencing? Generally, what is the point of a SARIMA model without seasonal differencing?

Any tutorials I have found starts with seasonal differencing if there is an identified seasonality.

Thank you very much!

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  • $\begingroup$ Search the term overdifferencing to learn why unwarranted differencing can be detrimental. $\endgroup$ Commented Jan 5, 2022 at 15:33
  • $\begingroup$ Thank you for your answer. I understand (I think at least) why overdifferencing can be dangerous. However, I still not understand at which point it should be recognized (despite the clear signs of seasonality like ACF spikes) that no seasonal differencing is required, as it turned out to be the very first step based on several ARIMA manuals. So, reworded, my question is how SARIMA is able to cope with seasonality without seasonal differencing tag. Thanks. $\endgroup$
    – EEEE77
    Commented Jan 5, 2022 at 15:55
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    $\begingroup$ Seasonal differencing needs to be motivated rather than taken for granted. It is only relevant under special circumstances, namely, when the time is seasonally integrated. In all other cases, it leads to overdifferencing. SARMA is relevant when there is autocorrelation at the seasonal frequency and its multiples. $\endgroup$ Commented Jan 5, 2022 at 16:11
  • $\begingroup$ Thank you very much! Sorry for many questions, but I am quite new in this topic. Could you please enlight somehow, how can it be distinguished if "time is seasonally integrated" or "there is autocorrelation at the seasonal frequency". Until now I thought that two are quite the same. $\endgroup$
    – EEEE77
    Commented Jan 5, 2022 at 16:15

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Seasonal differencing needs to be motivated rather than taken for granted. It is only relevant under special circumstances, namely, when the time is seasonally integrated. (This leads to some fairly unintuitive behavior such as trajectories of the different seasons diverging from each other. Try simulating from a SARIMA(0,0,0)(0,1,0) process and you will see for yourself. This suggests seasonal integration may not be a very common phenomenon in reality, despite the suggestions of many time series manuals that should know better...) In all other cases, it leads to overdifferencing.

SARIMA without seasonal differencing can be relevant when there is autocorrelation at the seasonal frequency and its integer multiples, as then the process can be approximated by a SARIMA without seasonal differencing.


I still not understand at which point it should be recognized (despite the clear signs of seasonality like ACF spikes) that no seasonal differencing is required, as it turned out to be the very first step based on several ARIMA manuals. So, reworded, my question is how SARIMA is able to cope with seasonality without seasonal differencing tag.

and

how can it be distinguished if "time is seasonally integrated" or "there is autocorrelation at the seasonal frequency". Until now I thought that two are quite the same.

The ACF at seasonal frequencies declines very slowly for a seasonally integrated process. It declines faster for those that can be approximated by SAR and SMA without seasonal unit roots. Again, you can simulate from SARIMA(0,0,0)(0,1,0) vs. SARIMA(0,0,0)(p,0,0) for several values of $p$ vs. SARIMA(0,0,0)(0,0,1) for several values of $q$ to see the difference in the patterns.

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  • $\begingroup$ Thank you sir, get it now I think :) $\endgroup$
    – EEEE77
    Commented Jan 5, 2022 at 16:23
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    $\begingroup$ @EEEE77, great to hear that. My perception is that over the recent years, there has been a proliferation of time series manuals that encourage careless use of seasonal differencing. When I studied time series, I got no such advice (and I think it was for the better). $\endgroup$ Commented Jan 5, 2022 at 16:25

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