# Difference-in-Difference (DID) Regression with Non-Stationary (but Cointegrated) Treatment and Control Groups

I would like to run a DID regression between two periods where each period spans multiple years. For example: Period 1: 1970Q1-1990Q4 Period 2: 1991Q1-2010Q4.

My treatment and control variables are non-stationary, but they are cointegrated. Thus, the error term is stationary and normally distributed. All assumptions of OLS also hold for DID. However, in a standard setting where you have two cointegrated variables one would have to use a DOLS or FMOLS to properly estimate the standard errors. Yet, the setup of a DID regression is different than the standard assumptions of regressing Y on X (with both being I(1) and cointegrated) which is what all of the theory on cointegration (in levels) is based on.

My question is if the variables are non-stationary, but cointegrated is the DID estimation valid? If it is, then the estimates would be superconsistent as is the case with an OLS between cointegrated variables, however, I am not sure if one can correct for the standard errors in this setup.

Any help would be much appreciated!

• How can your treatment be non-stationary? If you are using DiD the treatment will be a dummy variable. A dummy variable will be non-stationary only if probability of 1/0 is not constant in each time period. I cannot imagine any way how that would be possible with a treatment dummy, which is arguably not even stochastic if you apply treatment at some given time t and in other times its just always set to 0 or 1, and even if that would be the case how would you run DiD on it?
– 1muflon1
Aug 5 '20 at 22:45
• I am sorry I didn't mean that the treatment dummy is non-stationary (it is either 0 or 1 as you said). What I meant is that both my treatment and control groups ( the actual data process describing the two groups) are not stationary, but cointegrated. Aug 5 '20 at 23:55
• @1muflon1 Assuming my previous comment clears up the confusion does this regression make sense econometrically given the non-stationary nature of the treatment and control groups. If this was simply a regression of the type: y_t=alpha+beta_t+e and Y and X are cointegrated the esitmates will be superconsistent although one would have to correct for the standard errors. In my case, however, the X and the Y processes are my treatment and control respectively which are non-stationary, but cointegrated. Aug 13 '20 at 23:09