Patterns of Effects and Sensitivity Analysis for Differences-in-Differences

Luke J. Keele, Dylan S. Small, Jesse Y. Hsu, and Colin B. Fogarty

Applied analysts often use the differences-in-differences (DID) method to estimate the causal effect of policy interventions with observational data. The method is widely used, as the required before and after comparison of a treated and control group is commonly encountered in practice. DID removes bias from unobserved time-invariant confounders. While DID removes bias from time-invariant confounders, bias from time-varying confounders may be present. Hence, like any observational comparison, DID studies remain susceptible to bias from hidden confounders. Here, we develop a method of sensitivity analysis that allows investigators to quantify the amount of bias necessary to change a study's conclusions. Our method operates within a matched design that removes bias from observed baseline covariates. We develop methods for both binary and continuous outcomes. We then apply our methods to two different empirical examples from the social sciences. In the first application, we study the effect of changes to disability payments in Germany. In the second, we re-examine whether election day registration increased turnout in Wisconsin.