Estimate the best ARMAX model with one lagged independent variable (time series)?

I have two time series to work with, let's say $X_1$ and $X_2$.

First I have to estimate the best pure ARMA model for $X_1$; which is no problem. For that I perform the following steps:

1. Stationarize (if needed) the time series by differencing
2. Determine whether AR/MA terms are needed to correct any autocorrelation in the differenced series -> tentatively identify the maximum number of AR and/or MA terms using ACF and PACF plots
3. Then estimate different models, store BIC values, construct a matrix of BIC values, and select the ARMA model with the lowest BIC value.

Now comes the part that confuses me. I have to estimate an ARMAX model for $X_1$, where I need to put $X_2$ as a lagged explanatory variable in the ARMAX model.

• I don't think that including the exogenous variable will change the number of AR and MA terms for the best model relative to model estimated before. If e.g. ARMA(2,1) has the lowest BIC relative to the other ARMA models, ARMAX(2,1) will also have the lowest BIC relative to the other ARMAX models. Is that true?
• What is meant by including the 'lagged' explanatory variable? Does that mean the one-period lagged values of $X_2$? Or do I have to find the optimum number of lags for $X_2$ using a statistical technique?

I don't think that including the exogenous variable will change the number of AR and MA terms for the best model relative to model estimated before.

I highly doubt that. I do not remember any result regarding under which conditions this would hold. This would be a special case.

What is meant by including the 'lagged' explanatory variable? Does that mean the one-period lagged values of X2? Or do I have to find the optimum number of lags for X2 using a statistical technique?

It is difficult to know what someone meant without seeing the context, but generally you would either rely on subject-matter knowledge or use statistics (or both).

For a reference, you may find Rob J. Hyndman's blog post about ARIMAX and related models useful. It highlights the differences between the different versions of the model (or should I say, the different but related models), mentions software implementations and has a take on their interpretations, too.

Hi: An ARIMAX can be viewed as a transfer response model so you could look that up. If there are no MA terms in the model, then it can also be viewed as an ARDL(p,1) so I would look up how to estimate ARDL's also. This material is involved so I wouldn't try to give an estimation algorithm here. Hendry's approach is somewhat popular for ARDL's but you already know that q = 1. Generally speaking, if you are going to limit yourself to q=1, then the latest lag that's available should be used. But you should check if it helps ( does the regressor help explain the response ) using say a granger causality type of test.

As far as your claim about the ARIMA part staying the same when one adds the regressor, that is not the case.

The text by Pankratz contains a lot of examples of dynamic regression models and yours is a special case of those so you might want to check that book out. It's not amazing and it's pretty old but there are a lot of examples and he gives steps on the estimation approach. All the best.

You can start by reviewing a very basic tutorial here https://onlinecourses.science.psu.edu/stat510/node/75/ Also How to include control variables in an Intervention analysis with ARIMA? might be of help.