(Empirical) Bayes Approaches to Parallel Trends

Abstract

We consider Bayes and Empirical Bayes (EB) approaches for dealing with violations of parallel trends. In the Bayes approach, the researcher specifies a prior over both the pretreatment violations of parallel trends delta-pre and the posttreatment violations delta-post. The researcher then updates their posterior about the posttreatment bias delta-post given an estimate of the pre-trends delta-pre. This allows them to form posterior means and credible sets for the treatment effect of interest, Delta-post. In the EB approach, the prior on the violations of parallel trends is learned from the pretreatment observations. We illustrate these approaches in two empirical applications.