Please use this identifier to cite or link to this item: http://hdl.handle.net/10419/130082
Authors: 
de Chaisemartin, Clément
D'Haultfoeuille, Xavier
Year of Publication: 
2015
Series/Report no.: 
cemmap working paper, Centre for Microdata Methods and Practice CWP69/15
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 treatment effects be constant over time. 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.
Subjects: 
differences-in-differences
changes-in-changes
quantile treatment effects
partial identification
returns to education
JEL: 
C21
C23
Persistent Identifier of the first edition: 
Document Type: 
Working Paper

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