Factorial Difference-in-Differences

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.

💡 Research Summary

The paper introduces “Factorial Difference‑in‑Differences” (FDID), a methodological framework that extends the canonical difference‑in‑differences (DID) design to settings where a single event affects every observational unit. In many panel‑data applications, researchers exploit cross‑sectional variation in a pre‑treatment characteristic (denoted G) together with temporal variation in an event (denoted Z). Although the empirical practice often involves estimating an interaction term between G and a post‑event indicator in a two‑way fixed‑effects (TWFE) regression and calling the procedure a DID, the causal target of this estimator has been ambiguous.

FDID formalizes the setting as a factorial experiment with two factors: a baseline factor G (binary or continuous) and an exposure indicator Z. The key “universal exposure” assumption states that Z = 1 for all units; the event therefore has no untreated control group. Potential outcomes are defined with respect to both G and Z, yielding four counterfactual outcomes for each unit (pre‑ and post‑event outcomes under Z = 0 and Z = 1). Because Z = 0 is never observed, all four Z = 0 potential outcomes are missing.

Within this structure the authors define four causal quantities: (i) the unit‑level effect of the event conditional on G, τ_i,Z|G=g; (ii) the unit‑level effect of the baseline factor conditional on Z, τ_i,G|Z=z; (iii) effect modification, the difference in average event effects between the two G groups, E