Factorial Difference-in-Differences

TL;DR

FDID reframes the widely used difference-in-differences approach for settings where the event affects all units, by introducing a factorial 2-by-2 design with a baseline factor $G$ and exposure $Z$. It clarifies that under canonical DID assumptions, the estimand identified by the DID estimator is an effect modification $\tau_{em}$, while causal moderation $\tau_{cm}$ requires the stronger factorial parallel-trends assumption; with additional exclusion restrictions, FDID can recover $G$'s causal effect given exposure $\tau_{G|Z=1}$. The paper develops conditional extensions with covariates, demonstrates regression-based estimation via OLS and TWFE models, and extends to repeated cross-sections and general $G$, including a comprehensive empirical application on social capital and famine relief in China. These contributions advance causal panel analysis by formalizing the target estimands, identification conditions, and estimation strategies for observational FDID settings, with practical guidance for applied researchers.

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.

Figures (8)

• Figure 1: Average mortality rate across counties with high and low social capital. This figure resembles Figure 5(a) in cao2022clans, although we use a balanced panel, which represents a subset of the data used by the authors.
• Figure 2: Potential outcomes under FDID. The four potential outcomes with $z = 0$ in dotted boxes are unobservable for any unit in the FDID setting.
• Figure 3: Roadmap for key identification results
• Figure 4: Identification result under FDID in Proposition \ref{['prop:tau_em']}
• Figure 5: Estimated causal moderation over time. Each estimate is obtained from fitting OLS$_*$ using data in the year of interest and 1957. $G$ is a binary indicator of social capital.

Theorems & Definitions (29)

• Definition 1: FDID setting
• Remark 1
• Definition 2: Effect modification and causal moderation
• Remark 2
• Definition 3: Conditional causal effects of $G$
• Proposition 1
• Proposition 2
• Definition 4: Reframed canonical DID research design
• Proposition 3
• Remark 3