Fuzzy Differences-in-Differences

Causal Question / Estimand

Treatment effects in fuzzy DiD designs, where the treatment rate rises more in the treatment group than the control group (rather than going cleanly from 0 to 1). The popular “Wald-DiD” ratio (DiD of outcome ÷ DiD of treatment) targets a local average treatment effect (LATE, Causal-Estimand) — but only under strong conditions.

Identification Strategy

Shows the Wald-DiD ratio identifies a LATE only if treatment effects are stable over time and equal across the treatment and control groups. Relaxing these, proposes alternative estimands (Wald-TC and Wald-CIC, the latter using a changes-in-changes logic) that require no assumption on treatment-effect heterogeneity, valid when the treatment rate is stable over time in the control group. Establishes partial identification where point identification fails.

Key Assumptions

Parallel-Trends for potential outcomes, a fuzzy first stage (differential change in treatment rates), and for the alternative estimands a stable control-group treatment rate. The changes-in-changes variant invokes Monotonicity of the outcome in a latent index; SUTVA throughout. Estimand interpreted as a LATE.

Threats to Validity

The standard Wald-DiD is biased if effects change over time or differ between groups — exactly the Treatment-Effect-Heterogeneity problem in a fuzzy setting. The robust estimands need a control group whose treatment rate is stable across periods.

Setting / Data

n/a — econometric methodology. Empirical reassessment of the returns to schooling in Indonesia (the Duflo/INPRES school-construction setting).

Key Claims

• Wald-DiD ≠ a clean LATE unless effects are time-stable and group-homogeneous.
• Offers heterogeneity-robust fuzzy-DiD estimands (Wald-TC, Wald-CIC) and partial- identification bounds.
• Bridges DiD, changes-in-changes, and the IV/LATE tradition.

Connections

• By the same authors: DeChaisemartinDHaultfoeuille2020-IntertemporalTE, surveyed in DeChaisemartinDHaultfoeuille2023-TWFESurvey
• Uses changes-in-changes from: AtheyImbens2006-NonlinearDiD
• Shares the LATE estimand with: Angrist2022-EmpiricalStrategiesIlluminatingPath
• See also: DiD, IV

Citation

de Chaisemartin, C., & D’Haultfœuille, X. (2018). Fuzzy Differences-in-Differences. The Review of Economic Studies, 85(2), 999–1028. https://doi.org/10.1093/restud/rdx049