Difference-in-Differences Estimators of Intertemporal Treatment Effects
We study treatment-effect estimation, with a panel where groups may experience multiple changes of their treatment dose. We make parallel trends assumptions, but do not restrict treatment effect heterogeneity, unlike the linear regressions that have been used in such designs. We extend the event-study approach for binary-and-staggered treatments, by redefining the event as the first time a group’s treatment changes. This yields an event-study graph, with reduced-form estimates of the effect of having been exposed to a weakly higher amount of treatment for ℓ periods. We show that the reduced-form estimates can be combined into an economically interpretable cost-benefit ratio.
We are very grateful to Timothy Armstrong, Michal Kolesár, Isabelle Méjean, Ulrich Müller, Aureo de Paula, Ashesh Rambachan, Jonathan Roth, Pedro Sant’Anna, Jesse Shapiro and seminar participants at Berkeley, CREST, the Harris School of Public Policy, Mannheim, McGill, Oxford, Princeton, Paris School of Economics, Santa Clara, Sciences Po, UC Dublin, Universidad Javeriana, Université Paris Dauphine, and Yale for their helpful comments. Xavier D’Haultfoeuille gratefully acknowledges financial support from the research grants Otelo (ANR-17-CE26-0015-041). Part of this research was conducted while Xavier D’Haultfoeuille was at PSE, which he thanks for its hospitality. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.