I have perhaps a rookie question regarding the use of exogenous predictors in Forecasting. I'm using R's forecast package (auto.arima). I noticed that the exog. variables never have AR or MA terms associated with them. By contrast, SPSS will determine the appropriate PDQ structure of all Exog. variables (i.e. Transfer Function). So, now I'm wondering: if I manually lag the Exog. variables, is that essentially the same as identifying their AR structure? This would mean that I would need to lag all predictors if I suspect that there are delayed effects. I have seen a few similar posts, but nothing that addressed this specifically. Although Rob Hyndman briefly touches on it in his reference, I wasn't clear about the specific relationship between lags and AR terms in the independent variables. Thank you!
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$\begingroup$ The generally accepted approach is sometimes called a Transfer Function or a SARMAX model autobox.com/pdfs/SARMAX.pdf $\endgroup$– IrishStatCommented Apr 22, 2020 at 22:54
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Hi: Note that if you use a lagged dependent variable, then the effects from the exogenous variable become delayed.
For example : $Y_t = \rho \times Y_{t-1} + \beta \times X_t +\epsilon_t$ still captures delayed efffects from $X_t$.
This is because it's equivalent to $Y_t = \beta \sum_{i=0}^{\infty} \rho^{i} X_{t-i} + \sum_{i=0}^{\infty} \rho^{i} \epsilon_{t-i}$
Google for or look up "distributed lag models" in a text. Baltagi's econometrics has a pretty good explanation of them. At the same time, if you have multiple exogenous variables, then it can be difficult to disentangle the effects so it's best to start off simpler by using one exogenous variable and then seeing how that goes.
