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Fuzzy Differences-in-Differences

Author

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  • de Chaisemartin, Clement

    (Department of Economics University of Warwick)

  • D'Haultfoeuille, Xavier

    (CREST)

Abstract

In many applications of the differences-in-differences (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 effects is the DID of the outcome divided by the DID of the treatment, or OLS and 2SLS regressions with time and group fixed effects 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 effect only if treatment effects are homogenous in the two groups. Even when the distribution of treatment is stable, it requires that the effect of time be the same on all counterfactual outcomes. As this assumption is not always applicable, we propose two alternative estimators. The first 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 effects are partially identified. Finally, we prove that our estimators are asymptotically normal and use them to revisit applied papers using fuzzy designs.

Suggested Citation

  • de Chaisemartin, Clement & D'Haultfoeuille, Xavier, 2015. "Fuzzy Differences-in-Differences," The Warwick Economics Research Paper Series (TWERPS) 1065, University of Warwick, Department of Economics.
    • Handle: RePEc:wrk:warwec:1065
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    More about this item

    Keywords

    differences-in-differences ; changes-in-changes ; quantile treatment effects ; partitial identification ; returns to education;
    All these keywords.

    JEL classification:

    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models

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