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I have built a SARIMA model for forecasting monthly income data in python using the pmdarima library. It gives a confidence interval for each monthly prediction. I want to combine the data til the end of the year to get a yearly prediction and confidence interval which is simple assuming data within the same 12 month period is independant and follows a normal distribution.

If all non-seasonal variables of the SARIMA are zero for example SARIMA(0,0,0)(1,1,0)[12], does this imply each prediction within the same 12 month period is independent?

For example would the prediction for March 2020 be independant of the prediction for February 2020?

If not why is this so when all non seasonal variables are zero?

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  • $\begingroup$ Could you be more specific about what you mean by each prediction within the same season is independent? Do you e.g. mean the prediction for January of 2021 is independent of prediction for January of 2020? What are you conditioning on? $\endgroup$ – Richard Hardy Oct 20 '20 at 9:54
  • $\begingroup$ Thanks for the comment, I have edited the question to make it more clear. I actually asking if each data point within the 12 month period would be independent. e.g. the prediction for January 2020 is independent of the prediction for Febuary 2020 is independent of the prediction for December 2020. I trained on historical income figures. $\endgroup$ – David King Oct 20 '20 at 10:19
  • $\begingroup$ I think you might be using the term season in an awkward way. If we have monthly data, January is a season, and it is a different one from February. The year 2020 consists of 12 different seasons. The season of January 2020 and January 2021 is the same, however, as is February 2020 and February 2021. $\endgroup$ – Richard Hardy Oct 20 '20 at 13:35
  • $\begingroup$ Yes you are correct, I thought that the season was the length of time before a repetition (in this case a year or 12 months). Is period the correct term to use to describe this insteaed? $\endgroup$ – David King Oct 20 '20 at 14:09
  • $\begingroup$ Probably. You may wish to edit your question accordingly. $\endgroup$ – Richard Hardy Oct 20 '20 at 14:16
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After quite a bit of thinking I am quite certain that if all the non-Seasonal variables are 0 it implies that each prediction is calculated from historical data from the same season only (the prediction for January would be calculated from the previous values for January only). This implied that the values for that season are independent from every other season (in this example February would be indepentent from January).

If we take SARIMA(0,0,0)(1,1,0)[12] i belive the formula for a prediction would look like the folowing:

$$\Delta X_i = c + \phi_1\Delta X_{i-12} + \epsilon$$ Where $\Delta X$ is the differenced value. This presentation avoids the backshift operator which i found confusing. As we can see in this case the prediction only depends on the values from the same season $X_{i-12}$ and $X_{i-24}$ (because $\Delta X_{i-12} = X_{i-12} - X_{i-24}$)

I hope my reasoning is correct. I would appreciate if someone could verify this is correct (or show me why it isn't)

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