Rigorous FDID theory for repeated cross-sectional data

Develop a rigorous theoretical framework for factorial difference-in-differences (FDID) using repeated cross-sectional data rather than panels; specifically, formalize the repeated cross-sectional data structure, define potential outcomes indexed by a baseline factor G and exposure level Z under repeated sampling, and derive identification results for effect modification and average causal interaction under appropriate no-anticipation and parallel trends assumptions, together with corresponding estimation procedures.

Background

The paper formalizes FDID primarily in the two-group, two-period panel setting, where all units are exposed to a one-time event in the post period and the baseline factor G partitions units. Identification relies on no-anticipation and parallel trends assumptions, and additional factorial parallel trends for causal interaction. However, many empirical applications use repeated cross-sections rather than panels, necessitating a tailored formulation of potential outcomes and identifying assumptions.

The authors note that extensions to repeated cross-sections appear similar but involve more complicated notation, and explicitly indicate that a rigorous theoretical treatment is deferred. A complete theory would clarify how FDID estimands and assumptions translate when the same units are not observed over time, which is common in survey and administrative data.