An Averaging Alternative to Pre-Trend Testing

Abstract

We study difference-in-differences (DID) estimation of average treatment effects on the treated when the researcher is uncertain about which pre-treatment periods satisfy the parallel trends assumption. Roth (2022) shows that pre-trend testing can induce bias, complementing the statistical literature on post-selection inference. We propose the model averaged difference-in-differences (MADID) estimator, a weighted average of the candidate $2\times2$ DID estimators. The weights are normalized exponential functions of the candidate residual sums of squares, so implementation requires no specification test. We derive MADID's asymptotic properties under conditions that make invalid comparisons detectably more variable than at least one valid comparison. Under these conditions, MADID is consistent when the parallel trends assumption holds for at least one pre-treatment period. Although its joint limiting distribution across post-treatment periods is generally non-Gaussian, inference can be conducted by subsampling or simulation.