When Should Pre-trends Be Parallel?

We analyze pre-trends tests through the lens of how units select into treatment. We derive necessary and sufficient conditions for pre-trends and trends to be parallel with and without covariates. These conditions show that even in the absence of structural breaks, pre-trends tests can be uninformative about parallel trends, except under specific restrictions on selection. Thus, correctly interpreting pre-trends tests requires an understanding of the units’ selection behavior, and pre-trends tests cannot replace economic arguments for parallel trends. We document additional issues with pre-trends tests when researchers control for pre-treatment values of time-varying covariates.