Large Sample Properties of Matching Estimators for Average Treatment Effects


  • Alberto Abadie,

    1. John F. Kennedy School of Government, Harvard University, 79 John F. Kennedy Street, Cambridge, MA 02138, U.S.A.; and NBER;;
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  • Guido W. Imbens

    1. Dept. of Economics and Dept. of Agricultural and Resource Economics, University of California at Berkeley, 661 Evans Hall #3880, Berkeley, CA 94720-3880, U.S.A.; and NBER;;
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    • We wish to thank Donald Andrews, Joshua Angrist, Gary Chamberlain, Geert Dhaene, Jinyong Hahn, James Heckman, Keisuke Hirano, Hidehiko Ichimura, Whitney Newey, Jack Porter, James Powell, Geert Ridder, Paul Rosenbaum, Edward Vytlacil, a co-editor and two anonymous referees, and seminar participants at various universities for comments, and Don Rubin for many discussions on the topic of this article. Financial support for this research was generously provided through National Science Foundation Grants SES-0350645 (Abadie), SBR-9818644, and SES-0136789 (Imbens). Imbens also acknowledges financial support from the Giannini Foundation and the Agricultural Experimental Station at UC Berkeley.


Matching estimators for average treatment effects are widely used in evaluation research despite the fact that their large sample properties have not been established in many cases. The absence of formal results in this area may be partly due to the fact that standard asymptotic expansions do not apply to matching estimators with a fixed number of matches because such estimators are highly nonsmooth functionals of the data. In this article we develop new methods for analyzing the large sample properties of matching estimators and establish a number of new results. We focus on matching with replacement with a fixed number of matches. First, we show that matching estimators are not N1/2-consistent in general and describe conditions under which matching estimators do attain N1/2-consistency. Second, we show that even in settings where matching estimators are N1/2-consistent, simple matching estimators with a fixed number of matches do not attain the semiparametric efficiency bound. Third, we provide a consistent estimator for the large sample variance that does not require consistent nonparametric estimation of unknown functions. Software for implementing these methods is available in Matlab, Stata, and R.