Is it possible to include lags of exogenous variables in an ARIMAX-model?

Question

Suppose I have two time series {x} and {y} and want to examine how the current realization of {y} is dependent on current and past realizations of {x} and past realizations of {y}. For this purpose, I could run a VAR(p)-model including p lags of {x} and {y}. However, I want to assume that {x} is exogenous, so I guess it would be better to run an ARIMAX-model, i.e. an ARIMA-model with one or several exogenous variables.

In order to estimate an ARIMAX-model in R, I can either use the function arimax() (https://www.rdocumentation.org/packages/TSA/versions/1.01/topics/arimax) from the package TSA or auto.arima() (https://www.rdocumentation.org/packages/forecast/versions/8.1/topics/auto.arima) from forecast that both allow to include exogenous variables (it's specified with the argument xreg)

I’m however wondering why it is not possible to define lags of the exogenous variables? Or are there any (theoretical) reasons why we shouldn’t estimate an ARIMAX model including lags of exogenous variables? I know that it is possible with an Autoregressive Distributed Lag model, so basically an AR-model with exogenous variables.

Answer 1

I stumbled in the same problem today. I did not try my idea yet, but I think we can pass the lagged exogenous series as if it were a common exogenous variable in the model.

Example:

When you pass the exogenous variable, instead of passing just exogenous in time t, you could also pass exogenous at t-1, t-2 and so on.