Differences in differences with fuzzy groups

Abstract

This paper considers inference on treatment effects using repeated cross sections. Contrary to the standard setting of difference-in-differences, we suppose that perfect control and treatment groups cannot be defined. Units may be treated in the "control group", while some units may remain untreated in the "treatment group". The composition of the groups is supposed to remain constant over time. In this setting and under two common trends conditions, the commonly used Wald-DID estimand receives a causal interpretation if the treatment rate remains stable or decreases in the control group. We also consider two alternate sets of assumptions. Under each of them, a local average treatment effect is point identified if the treatment rate is stable over time in the control group, and partially identified otherwise. We extend our initial setting to several groups, several periods of time and to include covariates. We show that simple plug-in estimators of treatment effects are asymptotically normal and that the bootstrap is valid. Finally, we use our results to reanalyze the findings of Duflo (2001). With Clément de Chaisemartin (Warwick)