Difference-in-Difference (DID) Regression with Non-Stationary (but Cointegrated) Treatment and Control Groups
If your question is the broad: &quot;How do I estimate the differences between these two groups under these problematic contexts?&quot; I recommend taking a look at event study literature, which has a lot of similarities to the DiD literature, but is frequently applied in the context of stocks (which can be non-stationary).
MacKinlay, A. Craig. &quot;Event studies in economics and finance.&quot; Journal of economic literature 35.1 (1997): 13-39.