Extend FDID to continuous baseline factors

Extend the factorial difference-in-differences (FDID) framework to allow a continuous baseline factor G, including precise definitions of estimands (effect modification and average causal interaction) for continuous G, identification conditions and assumptions analogous to the binary case, and practical estimation strategies (e.g., regression/TWFE formulations) under this generalization.

Background

Throughout the paper, the baseline factor G is treated as binary to simplify the exposition and identification arguments. In many empirical applications, baseline measures (e.g., intensities or indices) are continuous, making the binary framework restrictive.

The authors explicitly state that extending FDID beyond the binary baseline factor and to repeated cross-sections is left for future work. A formal extension to continuous G would specify suitable potential outcomes and interaction estimands, clarify identification via generalized parallel trends assumptions, and delineate estimation approaches that accommodate continuous moderators.