Factorial Difference-in-Differences

Abstract

We formulate factorial difference-in-differences (FDID), a research design that extends canonical difference-in-differences (DID) to settings in which an event affects all units. In many panel data applications, researchers exploit cross-sectional variation in a baseline factor alongside temporal variation in the event, but the corresponding estimand is often implicit and the justification for applying the DID estimator remains unclear. We frame FDID as a factorial design with two factors, the baseline factor $G$ and the exposure level $Z$, and define effect modification and causal moderation as the associative and causal effects of $G$ on the effect of $Z$, respectively. Under standard DID assumptions of no anticipation and parallel trends, the DID estimator identifies effect modification but not causal moderation. Identifying the latter requires an additional \emph{factorial parallel trends} assumption, that is, mean independence between $G$ and potential outcome trends. We extend the framework to conditionally valid assumptions and regression-based implementations, and further to repeated cross-sectional data and continuous $G$. We demonstrate the framework with an empirical application on the role of social capital in famine relief in China.