Factorial Difference-in-Differences

In many panel data settings, researchers apply the difference-in-differences (DID) estimator, exploiting cross-sectional variation in a baseline factor and temporal variation in exposure to an event affecting all units. However, the exact estimand is often unspecified and the justification for this method remains unclear. This paper formalizes this empirical approach, which we term factorial DID (FDID), as a research design including its data structure, estimands, and identifying assumptions. We frame it as a factorial design with two factors - the baseline factor G and exposure level Z - and define effect modification and causal moderation as the associative and causal effects of G on the effect of Z, respectively. We show that under standard assumptions, including no anticipation and parallel trends, the DID estimator identifies effect modification but not causal moderation. To identify the latter, we propose an additional factorial parallel trends assumption. Moreover, we reconcile canonical DID as a special case of FDID with an additional exclusion restriction and link causal moderation to G's conditional effect with another exclusion restriction. We extend our framework to conditionally valid assumptions, clarify regression-based approaches, and illustrate our findings with an empirical example. We offer practical recommendations for FDID applications.