There is an interesting post about the connection of lagged exogenous variables and the autoregressive time series model: Forecasting - Lags vs. AR terms for Exogenous Variables
Consequently, by using an ARIMA model (I assume that this not only holds for AR models) including a non-lagged exogenous variable you are able to catch the decaying effects of all lagged versions. That sounds quite nice in my opinion.
My question now is whether this also holds for a Regression with ARIMA errors?
I would say no because in those models there is only an autoregressive part for the error term and no for the dependent time series variable modeled by the help of exogenous variables.
Thank you in advance.