I am dealing with time series regression, where I have stationary and nonstationary variables. Can I regress nonstationary I(1) variable on stationary variable when controlling for the lag of the nonstationary I(1) variable?

I know regressing nonstationary variable on stationary variable would lead to bias. However, would it be okay if I control for the lag of the nonstationary variable in my regression.

My regression would be something like this in Stata (where y is I(1), x is I(0) and for illustration I use up to 3 lags of y)

reg  y_t+h    x    l(1/3).y_t

1 Answer 1


No. If you truly have i(1) variables then coefficient at lag will be one. It is a unit root process, so regress the differences on x properly

  • $\begingroup$ Thanks. I thought about first differencing before. But I will estimate impulse responses afterwards. Will first differencing provide normal impulse responses, or should I estimate cumulative impulse responses? Also, I do not get coefficient as 1 when regressing on lags. $\endgroup$
    – sverdi
    Apr 25, 2021 at 18:17
  • $\begingroup$ If the coeffiicient is not 1 then why do you think it is I(1) process? Run a unit root test on your y, then run with the lag and post what you get. The model you are estimating is called ARMAX(1,0,0). $\endgroup$
    – Aksakal
    Apr 25, 2021 at 20:38
  • $\begingroup$ You can do this with ARDL. See the bound testing approach that allows some variables to be I[1] and others I[0] $\endgroup$
    – user54285
    Apr 26, 2021 at 1:32
  • $\begingroup$ @Aksakal, recall that the distribution of the OLS estimator of AR(1) is peculiar when the process has a unit root. So no wonder the OP is not getting the estimated coefficient at 1 even if the true coefficient is 1. $\endgroup$ Apr 26, 2021 at 5:27
  • $\begingroup$ @Richard Hardy? Can you even run OLS with a unit root (non-stationarity). $\endgroup$
    – user54285
    Apr 26, 2021 at 22:40

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