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I am trying to build a model where both dependent and independent variables are time series some of which may not be stationary (this can apply to both types of variables).

My first port of call was (S)ARIMAX. But I have a few questions:

(1) Can ARIMAX capture the lagged impact of the independent time series?

I get the impression that in ARIMAX the independent time series is expected to have a simultaneous impact on the dependent one - which is not necessarily true in my case. How do I handle such a lagged impact?

Also, I am not really sure what are the lags. So my other question is:

(2) How can I estimate the order of such lagged impact?

Also, I am not sure what to do about the stationarity. It seems that either I need to treat (e.g. by differencing, power-transforming) all variables (even those which are stationary) or none at all. So my other question is:

(3) Do I need to treat the non-stationarity of some of the variables?

And my last question:

(4) Can you point to publicly accessible worked examples of such problems?

I would like to see some publicly available worked example of such problems (i.e. independent variables are time series with lagged impact on the dependent time series) with python or R code.

References to books on Amazon (or to some other non-open sources) won't be that helpful.

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(1) Can ARIMAX capture the lagged impact of the independent time series?

Yes it can ......

By identifying the nature of the X from the pre-whitened cross-correlations one can identify the current and lag effects of each predictor series .

See Why is prewhitening important? and http://www.math.cts.nthu.edu.tw/download.php?filename=569_fe0ff1a2.pdf&dir=publish&title=Ruey+S.+Tsay-Lec1 and How to use Dynamic Regression models in R to forecast future sales

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