Post-Matching Two-Way Fixed Effects Estimation

When estimating treatment effects with two-way fixed effects (2WFE) models, researchers often use matching as a pre-processing step when the parallel trends assumption is thought to hold conditionally on covariates. Specifically, in a first step, each treated unit is matched to one or more untreated units based on observed time-invariant covariates. In the second step, treatment effects are estimated with a 2WFE regression in the matched sample, reweighting the untreated units by the number of times they are matched. We formally analyze this common practice and highlight two problems. First, when different treatment cohorts enter treatment in different time periods, the post-matching 2WFE estimator that pools all treated cohorts has an asymptotic bias, even when the treatment effect is constant across units and over time. Second, failing to account for the variability introduced by the matching procedure yields invalid standard error estimators, which can be biased upwards or downwards depending on the data generating process. We propose simple post-matching difference-in-differences estimators that compare each treated cohort to the never-treated separately, instead of pooling all treated cohorts. We provide conditions under which these estimators are consistent for well-defined causal parameters, and derive valid standard errors that account for the matching step. We illustrate our results with simulations and with an empirical application.