Factorial Difference-in-Differences

Abstract

We formulate factorial difference-in-differences (FDID) as a research design that extends the canonical difference-in-differences (DID) to settings without clean controls. Such situations often arise when researchers exploit cross-sectional variation in a baseline factor and temporal variation in an event affecting all units. In these applications, the exact estimand is often unspecified, and justification for using the DID estimator is unclear. We formalize FDID by characterizing its data structure, target parameters, and identifying assumptions. Framing FDID as a factorial design with two factors -- the baseline factor G and the exposure level Z, we define effect modification and causal moderation as the associative and causal effects of G on the effect of Z. Under standard DID 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. We also show that the canonical DID is a special case of FDID under an exclusion restriction. We extend the framework to conditionally valid assumptions and clarify regression-based implementations. We then discuss extensions to repeated cross-sectional data and continuous G. We illustrate the approach with an empirical example on the role of social capital in famine relief in China.

Keywords: causal inference, panel data, parallel trends, factorial design, difference-in-differences