ARIMAX how to forecast past exogenous lags?

Question

I have an ARIMAX model that has AR & MA terms along with exogeneous variables at prior lags. I am predicting on a continuous process and attempting to forecast to 1 and 10 units ahead. However, the important lags for the exogeneous variables are within the 1-10 units. Thus, it fails to produce 10 units ahead forecast on the most recent data.

For example, I have a x variable with a lag of 3 (x_3) in the model. I just received the most recent result (0 units ahead).

If I forecast to 10 units ahead, that forecast requires the x_3 to be non-missing at 7 units, which isn't possible since it hasn't occurred yet. Therefore, no prediction is made at 10 units ahead.

How do I resolve this issue? Do I attempt to fit an ARIMA model to the x variables? Use the last value for x and propagate it forward? Am I using this method incorrectly? Create a new model that only uses exogeneous lags that exceed the forecast step size (x_11 for the 10 step forecast)?

Answer 1

You will need to provide "future values" of the exogeneous variables. These can be "known" values, e.g., if they are promotions in a retail context. Or they could be "forecasted" values, e.g., if they are weather data. You could also try multiple different values, e.g., for scenario analysis.

Yes, especially the latter means that ARIMAX or regressions with ARIMA errors (there is a difference) are dubious the farther out we forecast.

This holds regardless of whether you use lagged or non-lagged values of your predictors. What you don't know you will need to forecast or assume.