Dae Woong Ham and Luke Miratrix present "A devil’s bargain? Repairing a Difference in Differences parallel trends assumption with an initial matching step"
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. In this paper, we characterize the bias of matching under a class of linear structural models with both observed and unobserved confounders that have time varying effects. Given this framework, we find that matching on baseline covariates generally reduces the bias associated with these covariates, when compared to the original DiD estimator. 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. On the other hand, we find that matching on pre-treatment outcomes also undermines the second "difference" in a DiD estimate by forcing the treated and control group's pre-treatment outcomes to be equal. This injects bias into the final estimate, creating a bias-bias tradeoff. We extend our bias results to multivariate confounders with multiple pre-treatment periods and find similar results. We summarize our findings with heuristic guidelines on whether to match prior to a DiD analysis, along with a method for roughly estimating the reduction in bias. We illustrate our guidelines by reanalyzing a recent empirical study that used matching prior to a DiD analysis to explore the impact of principal turnover on student achievement.
The Applied Statistics Workshop (Gov 3009) meets all academic year, Wednesdays, 12pm-1:30pm, in CGIS K354. This workshop is a forum for advanced graduate students, faculty, and visiting scholars to present and discuss methodological or empirical work in progress in an interdisciplinary setting. The workshop features a tour of Harvard's statistical innovations and applications with weekly stops in different fields and disciplines and includes occasional presentations by invited speakers.
More information is available at the Gov 3009 website: https://projects.iq.harvard.edu/applied.stats.workshop-gov3009
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