# Regression with ARIMA Errors for non-stationary timeseries: Mixing of stationary/non-stationary covariates?

Given I want to forecast e.g. monthly sales (dependent variable, likely non-stationary) with regression and ARIMA errors (ARIMA in R with xreg) I have e.g. two independent variables/covariates:

• Holidays per month (stationary)
• GDP or similar (non-stationary)

In R an ARIMA(p,1,q) will automatically difference the covariates automatically as well - could that be an issue, given one variable is already stationary? If yes, how could it be avoided?

Edit: I would like to avoid to check for stationarity and do differencing manually - as it gets ugly/tedious especially with seasonality. So far I'm using mostly the auto.arima from the forecast package with some restrictions.

• It is definitely necessary to difference all independent variables if you have to difference dependent variable. See the link under "non stationary". robjhyndman.com/hyndsight/arimax Commented Aug 5, 2017 at 11:35
• Yes, I read everything from Mr. Hyndman who as far as I know even filed a bugreport (accepted) for the basic R arima to difference all independent variables when d > 0. But I never saw any explicit example or mention of a stationary/non-stationary mix. As far as I understand you its not hurting, when the stationary variable gets differenced as well? Commented Aug 5, 2017 at 11:44
• That is correct, you don't have a choice you HAVE to difference all the variables if your dependent variable is non stationary. If you use auto.arima then it is regression with arima errors. No need to difference beforehand everything is handled automatically Commented Aug 5, 2017 at 12:22
• I think it might hurt if you difference stationary variables. There is a well-known notion of overdifferencing. I would think twice before mechanically differencing anything that is not integrated in the first place. Commented Aug 5, 2017 at 12:37
• As @forecaster said It is necessary to IDENTIFY the relationship by suitably differencing (possibly different) all series in the model. If Y and X are themselves non-stationary the final model can still be Y= a + bX as compared to a potentially bloated model containing differencing factors. This is why AUTOBOX ( a piece of software that I have helped to develop) allows an option in this regard. Some very bad implementations e.g. SAS force unnecessary differencing into the TF MODEL.. A careful look at the TF schemes in the back of the B-J text will help. Commented Aug 6, 2017 at 16:20