I'm very familiar with the theoretical underpinnings of ARIMA/SARIMA models but I've been struggling to understand the theory behind fitting an ARIMAX model. I'm not looking for a practical application on R or Python since I know how to do that I'm rather looking for how exactly is the procedure done. I've been digging around the web for a while and found papers about Transfer Functions & Impulse Response but not at any depth or elaboration.
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$\begingroup$ So do you need theory (like statistical properties of estimators) or a recipe (description of procedure, pseudocode)? $\endgroup$– Richard HardyMay 28, 2019 at 8:12
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$\begingroup$ If you've found the answer helpful, don't to forget to upvote and accept it - it seems @IrishStat put a lot of effort in it. $\endgroup$– Martin ModrákMay 29, 2019 at 11:55
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1$\begingroup$ @MartinModrák tks for the nice words ...I enjoy these opportunities to teach and learn $\endgroup$– IrishStatMay 30, 2019 at 19:19
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1 Answer
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Please see my response to How to use Dynamic Regression models in R to forecast future sales . The whole idea about transfer function model identification is that we filter the stationary X to make it white noise (x) and apply that filter to the stationary Y to make y and then use the cross-correlation of x & y or it's proportional equivalent the Impulse Response Weights (regression coefficients) to identify a minimally sufficient set of lags (0,1,2,??).
The following should be studied closely ( follow the algebra ) to do this http://www.math.cts.nthu.edu.tw/download.php?filename=569_fe0ff1a2.pdf&dir=publish&title=Ruey+S.+Tsay-Lec1 ... particularly the bottom of page 4.
The final model errors need to be free of not only auto-correlation BUT cross-correlation AND need to be free of pulses,level/step shifts, seasonal pulses & Local time trends. Furthermore the parameters of the final model and the error variance need to be homogeneous over time. 
using something similar to https://autobox.com/pdfs/A.pdf
EDITED AFTER OP'S QUESTION:
b ( the delay) is the the # of periods before the first significant cross-correlation. s speaks to the denominator structure (output lag) and can be identified by examining the cross-correlation for possible "decay" . (this is similar to examining the acf for decay in univariate analysis) and r is the # of numerator coefficients (input lag structure) that are needed. AUTOBOX solves this problem via a heuristic search process similar to auto.arima in style that yields the answers to r,s, and b https://autobox.com/cms/index.php/blog/entry/watson-its-not-elementary
See http://viewer.zmags.com/publication/9d4dc62a#/9d4dc62a/66 for a very aggressive test of the AUTOBOX heuristic when the reviewer injected structure to test the viability of AUTOBOX.
This is the area of " Pankratz's subjectivity" which is dealt with via search procedures not so easily programmed which is why one uses "smart software" for help rather than spending a lifetime at the keyboard.
Various alternatives such as the corner method often fail to uncover the the correct combination of s and b . As a novice you might start simply by setting s=0 and r large enough to encompass the significant cross-correlations.
Finally the coefficents for the polynomial can be estimated by starting with the Impulse Response Weights.
If you are satisfied with my answer ...upcheck it and accept it to bring attention to the clarity it brings .
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$\begingroup$ Thanks. Your answer cleared up a lot of things for me. I have a couple questions: 1) How are the lags $(r, s, b)$ determined? 2) How are the weights for $V(b)$ estimated? are they found via equating coefficients of the rational polynomial such in a Koyck rational transfer model? In which case, how are the coefficients of those polynomials estimated? $\endgroup$ May 29, 2019 at 5:35
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$\begingroup$ stats.stackexchange.com/questions/410082/… might be relevant to you as it is a good exposure to the mechanics $\endgroup$ May 30, 2019 at 10:12
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$\begingroup$ Wish I could upvote this. Thanks a lot you've given a wealth of information to keep me busy for a while. I also liked Alan Pankratz's "Forecasting with Dynamic Regression Models" especially chapter 4, chapter 5 and chapter 9 which also adress these questions. I'll just leave this reference here in case it helps someone in the future. $\endgroup$ May 30, 2019 at 19:01
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$\begingroup$ ok ... then two things ... feel free to reach out me any way you can as I know that you know ! . Bye the way I had worked with Alan P. on some things ..he being an author/writer and I being a statistician/programmer . Nice guy ! $\endgroup$ May 30, 2019 at 19:17
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$\begingroup$ That's impressive. That's so nice of you thanks a lot ! $\endgroup$ May 30, 2019 at 20:36