The difference-of-differences (DiD) estimator is a common estimator built on the assumption of “parallel trends”. To increase the validity of this assumption, a natural idea is to match the processing and control units prior to DiD analysis. In this paper, we characterize the matching bias prior to his DiD analysis under a linear structural model. Our framework takes into account both observed and unobserved confounders with time-varying effects. Given this framework, we find that matching baseline covariates reduces the bias associated with these covariates when compared to the original DiD estimator. Moreover, we find that there are both costs and benefits in further collating pre-treatment results. First, it reduces the bias associated with unobserved confounders. This is because collating pretreatment results partially reconciles the balance of these unobserved confounders. This reduction is proportional to the reliability of the results. This is a measure of how the results are combined with these potential covariates. On the other hand, we found that matching pretreatment results compromised the second ‘difference’ in DiD estimates, as it forced the treated and control groups to have equal pretreatment results. This introduces a bias in the final estimate, as if parallel trends were maintained. We extend the biased results to multivariate confounders with multiple pretreatment periods to find similar results. Finally, we provide practitioners with heuristic guidelines on whether to match prior to DiD analysis and how to roughly estimate bias reduction. We illustrate our guidelines by reanalyzing recent empirical studies that investigated the impact of principal turnover on student performance using matching prior to DiD analysis. We found that the authors’ decision to match pretreatment results was important in making the estimated treatment effects more reliable.
