Fuzzy Differences-in-Differences

Abstract

In many applications of the dierences-in-dierences (DID) method, the treatment increases more in the treatment group, but some units are also treated in the control group. In such fuzzy designs, a popular estimator of treatment eects is the DID of the outcome divided by the DID of the treatment, or OLS and 2SLS regressions with time and group xed eects estimating weighted averages of this ratio across groups. We start by showing that when the treatment also increases in the control group, this ratio estimates a causal eect only if treatment eects are homogenous in the two groups. Even when the distribution of treatment is stable, it requires that the eect of time be the same on all counterfactual outcomes. As this assumption is not always applicable, we propose two alternative estimators. The rst estimator relies on a generalization of common trends assumptions to fuzzy designs, while the second extends the changes-in-changes estimator of Athey & Imbens (2006). When the distribution of treatment changes in the control group, treatment eects are partially identied. Finally, we prove that our estimators are asymptotically normal and use them to revisit applied papers using fuzzy designs.