Cohort-Anchored Robust Inference for Event-Study with Staggered Adoption

Abstract: I propose a cohort-anchored framework for robust inference in event studies with staggered adoption. Robust inference based on aggregated event-study coefficients, as in Rambachan and Roth (2023), can be misleading because pre- and post-treatment coefficients are identified from different cohort compositions and the not-yet-treated control group changes over time. To address these issues, I work at the cohort-period level and introduce the \textit{block bias}-the parallel-trends violation for a cohort relative to its anchored initial control group-whose interpretation is consistent across pre- and post-treatment periods. For both the imputation estimator and the estimator in Callaway and Sant'Anna (2021) that uses not-yet-treated units as controls, I show an invertible decomposition linking these estimators' biases in post-treatment periods to block biases. This allows researchers to place transparent restrictions on block biases (e.g., Relative Magnitudes and Second Differences) and conduct robust inference using the algorithm from Rambachan and Roth (2023). In simulations, when parallel-trends violations differ across cohorts, my framework yields better-centered (and sometimes narrower) confidence sets than the aggregated approach. In a reanalysis of the effect of minimum-wage changes on teen employment in the Callaway and Sant'Anna (2021) application, my inference framework with the Second Differences restriction yields confidence sets centered well below zero, indicating robust negative effects, whereas inference based on aggregated coefficients yields sets centered near zero. The proposed framework is most useful when there are several cohorts, adequate within-cohort precision, and substantial cross-cohort heterogeneity.