Fuzzy Differences-in-Differences

TL;DR

This paper examines the limitations of the standard fuzzy DID estimator, proposes two new estimands that do not rely on strong assumptions, and demonstrates their asymptotic properties, enhancing causal inference in partial treatment scenarios.

Contribution

It introduces two alternative estimands for fuzzy DID that do not depend on homogeneous treatment effect assumptions and are applicable when control group treatment rates are stable.

Findings

Standard fuzzy DID ratio requires strong homogeneity assumptions.

Proposed estimands are valid without treatment effect homogeneity.

New estimators are asymptotically normal and applicable in broader settings.

Abstract

Difference-in-differences (DID) is a method to evaluate the effect of a treatment. In its basic version, a "control group" is untreated at two dates, whereas a "treatment group" becomes fully treated at the second date. However, in many applications of the DID method, the treatment rate only increases more in the treatment group. In such fuzzy designs, a popular estimator of treatment effects is the DID of the outcome divided by the DID of the treatment. We show that this ratio identifies a local average treatment effect only if two homogeneous treatment effect assumptions are satisfied. We then propose two alternative estimands that do not rely on any assumption on treatment effects, and that can be used when the treatment rate does not change over time in the control group. We prove that the corresponding estimators are asymptotically normal. Finally, we use our results to revisit Duflo (2001).