Estimate a Transfer Function from ARX Models vs. ARIMAX?
Hi: I don't know about BJ or OE but, as far as the other three go, it REALLY MATTERS which one you choose because ARX has no MA or differencing terms, ARMAX has no differencing terms and ARIMAX has both. But, even within any of the three models, there are major choices to be made as far as A) the number of lags of y, the number of lags of  X  ( and, in the ARDL case, the number of X's). This is an EE forum so  I'm only on it because I'm trying to learn some EE. My background is more econometrics so my suggestion is that you should probably look at a good econometrics textbook that discusses model selection given the question you asked. There is the top down method of hendry, the schwert criterion, AIC etc.  Luktepohl's text probably discusses model selection in these types of models but focuses on AR and ARI and not so much the MA terms. Also, it sounds like the response y_t is a scalar but, in case it isn't, you want to look at VARS which luktepohl covers in great detail. Good luck. Oh,  Also, you may want to look at the koyck distributed lag which is a special ARMAX model.