Abstract: Exchangeable arrays are natural tools to model common forms of dependence between units of a sample. Jointly exchangeable arrays are well suited to dyadic data, where observed random variables are indexed by two units from the same population. Examples include trade flows between countries or relationships in a network. Separately exchangeable arrays are well suited to multiway clustering, where units sharing the same cluster (e.g. geographical areas or sectors of activity when considering individual wages) may be dependent in an unrestricted way. We prove uniform laws of large numbers and central limit theorems for such exchangeable arrays. We obtain these results under the same moment restrictions and conditions on the class of functions as those typically assumed with i.i.d. data. We also show the convergence of bootstrap processes adapted to such arrays.

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Abstract: Differences-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 this method, the treatment rate only increases more in the treatment group. In such fuzzy designs, de Chaisemartin and D’Haultfœuille (2018b) propose various estimands that identify local average and quantile treatment effects under different assumptions. They also propose estimands that can be used in applications with a non-binary treatment, multiple periods and groups and covariates. This paper presents the Stata command fuzzydid, which computes the various corresponding estimators. We illustrate the use of the command by revisiting Gentzkow et al. (2011).

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