Abstract. Abadie and Imbens (2008) showed that the naive bootstrap is not asymptotically valid for a matching estimator of the average treatment effect with a fixed number of matches. In this article, we propose asymptotically valid inference methods for matching estimators based on the weighted bootstrap. The key is to construct bootstrap counterparts by resampling based on certain linear forms of the estimators.Our weighted bootstrap is applicable for the matching estimators of both the average treatment effect and its counterpart for the treated population. Also, by incorporating a bias correction method in Abadie and Imbens (2011), our method can be asymptotically valid even for matching based on a vector of covariates. A simulation study indicates that the weighted bootstrap method is favorably comparable with the asymptotic normal approximation by Abadie and Imbens (2006). As an empirical illustration, we apply the proposed method to the National Supported Work data.
Abstract. This paper revisits testability of complementarity in economic models with multiple equilibria studied by Echenique and Komunjer (2009) where Y ∈ R is a dependent variable, X ∈ X ⊆ R is an explanatory variable, U ∈ R is a disturbance term, and r : R×X → R is a function implied by economic theory. We observe X and Y but do not observe U . EK studied testability of this model when there are complementarities between X and Y without assuming a parametric functional form of r, dependence structure between X and U , and specific equilibrium selection rule. Let E xu = {y ∈ R| |r (y, x) = u} be the equilibrium set for given x and u, F U |X=x be the conditional distribution of U given X = x (which is assumed to have a strictly positive density), andF Y |X=x (y) = 1 − F Y |X=x (y), whereBy an innovative argument to focus on the largest or smallest equilibrium and to apply a change of variable technique, EK obtained the following result (the comments in the parentheses are added by the authors).Assumption S1. (i) The function r : R × X → R is continuous (on y ∈ R for each given X ∈ X ); (ii) for any x ∈ X , lim y→−∞ r (y, x) = +∞ and lim y→+∞ r (y, x) = −∞; (iii) for any (x, u) ∈ X × R, E xu is a finite set. We write E xu = {ξ 1xu , . . . , ξ nxxu } (ξ 1xu ≤ · · · ≤ ξ nxxu ) with n x = Card (E xu ) (which is finite and does not depend on u). (Also, the selection rule P xu , a probability distribution over E xu , assigns probabilities {π 1x , . . . , π nxx } to {ξ 1xu , . . . , ξ nxxu } for all u ∈ R with π 1x > 0 and π nxx > 0.)
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