Characterize the estimand of extended two-way fixed effects (ETWFE) cohort-time coefficients

Determine the precise statistical estimand associated with the extended two-way fixed effects (ETWFE) cohort-by-time coefficient β^W_{g,t} in staggered-adoption designs, including the exact weighting scheme across pre-treatment periods and not-yet-treated comparison groups and a clear decomposition that shows how β^W_{g,t} aggregates the underlying group-time average treatment effects ATT(g,t) under the “parallel trends for every period and group” assumption.

Background

The paper discusses several estimators for group-time average treatment effects ATT(g,t) in staggered adoption settings. Wooldridge (2021) proposes an extended TWFE specification that, under a strong parallel trends assumption holding across all periods and groups, produces coefficients β^W_{g,t} that are consistent for ATT(g,t).

Despite consistency, the authors note that the exact statistical estimand for β^W_{g,t}—that is, how it combines information from different pre-treatment periods and not-yet-treated comparison groups—is unknown. Clarifying this estimand would improve interpretability, align the ETWFE approach with the 2×2 building-block framework, and help practitioners understand the weighting implicit in these regression-based estimates.