The PCDID Approach: Difference-in-Differences when Trends are Potentially Unparallel and Stochastic

We develop a class of regression-based estimators, called Principal Components Differ-ence-in-Differences estimators (PCDID), for treatment effect estimation. Analogous to a control function approach, PCDID uses factor proxies constructed from control units to control for unobserved trends, assuming that the unobservables follow an interactive effects structure. We clarify the conditions under which the estimands in this regression-based approach represent useful causal parameters of interest. We establish consistency and asymptotic normality results of PCDID estimators under minimal assumptions on the specification of time trends. The PCDID approach is illustrated in an empirical exercise that examines the effects of welfare waiver programs on welfare caseloads in the US.