(Empirical) Bayes Approaches to Parallel Trends

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 pre-treatment violations of parallel trends δₚᵣₑ and the post-treatment violations δₚₒₛₜ. The researcher then updates their posterior about the post-treatment bias δₚₒₛₜ given an estimate of the pre-trends δₚᵣₑ. This allows them to form posterior means and credible sets for the treatment effect of interest, τₚₒₛₜ. In the EB approach, the prior on the violations of parallel trends is learned from the pre-treatment observations. We illustrate these approaches in two empirical applications.