Factorial Difference-in-Differences

TL;DR

This paper introduces factorial difference-in-differences (FDID), a new research design extending DID to settings with all units affected by an event, clarifying estimands and assumptions for effect modification and causal moderation.

Contribution

It formulates FDID as a factorial design, defines effect modification and causal moderation, and extends DID assumptions and methods to more complex data settings.

Findings

FDID clarifies effect modification and causal moderation identification.

Additional assumptions are needed for causal moderation.

Framework applied to social capital's role in famine relief in China.

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.