(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 pre-treatment violations of parallel trends $\hat{I}{´}_{pre}$ and the post-treatment violations $\hat{I}{´}_{post}$. The researcher then updates their posterior about the post-treatment bias $\hat{I}{´}_{post}$ given an estimate of the pre-trends $\hat{I}{´}_{pre}$. This allows them to form posterior means and credible sets for the treatment effect of interest, $\ddot{I}{„}_{post}$. 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.