Hi @EdMarino ,
I am glad the previous information was helpful to you!
If your ultimate goal is to write out a forecasting equation from your fitted model in order to calculate the one-step-ahead predicted values and multistep-ahead forecasts outside of Forecast Studio, then I would caution you that the computations are quite involved--particularly for transfer function models. Various textbooks, such as Box, Jenkins and Reinsel (1994), and Brockwell and Davis (1991), provide details on how the forecasts for ARIMA and ARIMAX models are computed. Forecast Server computes the forecasts using the decomposition described in the following section of the documentation:
https://go.documentation.sas.com/?docsetId=hpfug&docsetTarget=hpfug_hpfdet_sect026.htm&docsetVersion=15.2&locale=en
An alternative approach to "hand calculating" the forecasts would be to use the forecasting model score files and DATA step functions provided by Forecast Server. Please see the following link for details:
https://go.documentation.sas.com/?docsetId=hpfug&docsetTarget=hpfug_macros_toc.htm&docsetVersion=15.2&locale=en
Regarding your question on how to apply the backshift operator after a model has been fit, I can provide a simple illustration. Let's say you fit an ARMA(1,1) model as follows:
(1 - θB) yt = μ + --------------- * at (1 - ɸB)
If you multiply both sides of the equation by the AR polynomial in the denominator, you get:
(1 - ɸB)yt = (1 - ɸB)μ + (1 - θB)at
Multiplying the polynomials and applying the backshift operator you get:
yt - ɸyt-1 = (1 - ɸ)μ - θat-1 + at
Note that because μ does not have a time subscript, the backshift operator drops out of that term. We can refer to that term as the constant, C. Now, you can isolate yt and write out your final model equation into which you can plug in your estimated parameters:
yt = C + ɸyt-1 - θat-1 + at
I hope this illustration helps!
DW
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