Benefits and costs of matching prior to a Difference in Differences analysis when parallel trends does not hold

Abstract: The Difference in Difference (DiD) estimator is a popular estimator built on the "parallel trends" assumption, which is an assertion that the treatment group, absent treatment, would change "similarly" to the control group over time. To bolster such a claim, one might generate a comparison group, via matching, that is similar to the treated group with respect to pre-treatment outcomes and/or pre-treatment covariates. Unfortunately, as has been previously pointed out, this intuitively appealing approach also has a cost in terms of bias. To assess the trade-offs of matching in our application, we first characterize the bias of matching prior to a DiD analysis under a linear structural model that allows for time-invariant observed and unobserved confounders with time-varying effects on the outcome. Given our framework, we verify that matching on baseline covariates generally reduces bias. We further show how additionally matching on pre-treatment outcomes has both cost and benefit. First, matching on pre-treatment outcomes partially balances 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. Offsetting these gains, matching also injects bias into the final estimate by undermining the second difference in the DiD via a regression-to-the-mean effect. Consequently, we provide heuristic guidelines for determining to what degree the bias reduction of matching is likely to outweigh the bias cost. We illustrate our guidelines by reanalyzing a principal turnover study that used matching prior to a DiD analysis and find that matching on both the pre-treatment outcomes and observed covariates makes the estimated treatment effect more credible.