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I have the monthly number of patients in a psychiatric facility from Jan 2010 to Dec 2018 - the data shows a seasonal pattern. I want to forecast the number of patients in the facility from Jan 2019 to Dec 2020 using seasonal ARIMAX, with the number of people receiving a mental health diagnosis as an exogenous variable. I have the monthly number of people receiving a mental health diagnosis from Jan 2010 to Dec 2018 also, but obviously I don't know what the future numbers will be out to Dec 2020. So first I ran a seasonal ARIMA on the diagnosis data to get predictions from Jan 2019 to Dec 2020, and then I used these predictions for the exogenous variable component of the seasonal ARIMAX model (0,1,1)*(0,1,1)12 to forecast patient numbers at the psychiatric facility over the same time period.

Is this a reasonable approach to take?

What is the ARIMAX model actually doing with the exogenous variable predictions I feed into it? Is it treating them as known values (which they are not), or does it incorporate some measure of uncertainty for these in its predictions.

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In general your approach is correct. In terms of specifics when you build the arima model for your predictor it is wise to consider possible anomalies ala https://pdfs.semanticscholar.org/09c4/ba8dd3cc88289caf18d71e8985bdd11ad21c.pdf . Secondly when you go to form the way X enters the model , you have to use pre-whitening to identify the contemporaneous and lag structure ala https://onlinecourses.science.psu.edu/stat510/node/75 and Transfer function in forecasting models - interpretation . Finally you have touched on a third rail issue regarding the encoding/incorporating the uncertainty in the predictor INTO the uncertainty of the dependent variable. This is accomplished via boot-strapping the forecast probability distribution for the predictor AND the forecast probability distribution for the model errors. See https://autobox.com/cms/images/dllupdate/AutoboxUsersguide.pdf page 56 for a discussion for why and what you need to do. In terms of transparency , I have been involoved in implementing this very important feature that most analysts ignore .. but apparently not you and I and increasingly many others !

Unable to remove seasonal flucatuations from NDVI data might help you further in understanding and explaining the ARIMAX model.

In specific when you feed the predictor into your model your software is most likely assuming a purely contemporaneous relationship. Not a good idea in general to assume this but some software places the analytical burden is on you to pre-specify this.

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