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Title: A devils bargain? Repairing a difference in differences parallel trends assumption with an initial matching step Authors:  Luke Miratrix - Harvard University (United States) [presenting]
Dae Woong Ham - Harvard University (United States)
Abstract: The Difference in Difference (DiD) estimator is a popular estimator built on the ``parallel trends'' assumption that the treatment group, absent treatment, would change ``similarly'' to the control group over time. To increase the plausibility of this assumption, a natural idea is to match treated and control units prior to a DiD analysis. We characterize the bias of such matching under a class of linear structural models with both observed and unobserved confounders with time-varying effects. We find matching on baseline covariates generally reduces the bias associated with these covariates. We further find that additionally matching on pre-treatment outcomes has both cost and benefit. First, matching on pre-treatment outcomes will partially balance unobserved confounders, which mitigates some bias. This reduction is proportional to the outcome's reliability, a measure of how coupled the outcomes are with the latent covariates. Matching on pre-treatment outcomes also undermines the second ``difference'' in a DiD estimate. This injects bias into the final estimate, creating a bias-bias tradeoff. We extend our results to multivariate confounders with multiple pre-treatment periods and find similar results. We summarize with heuristic guidelines on whether to match prior to a DiD analysis, along with a rough bias reduction estimator. We illustrate our guidelines by reanalyzing a recent empirical study of principal turnover on student achievement.