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A Martingale Representation for Matching Estimators

27 Pages Posted: 26 Feb 2009  

Alberto Abadie

Harvard University - Harvard Kennedy School (HKS); National Bureau of Economic Research (NBER)

Guido W. Imbens

Stanford Graduate School of Business

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Date Written: February 2009

Abstract

Matching estimators (Rubin, 1973a, 1977; Rosenbaum, 2002) are widely used in statistical data analysis. However, the large sample distribution of matching estimators has been derived only for particular cases (Abadie and Imbens, 2006). This article establishes a martingale representation for matching estimators. This representation allows the use of martingale limit theorems to derive the large sample distribution of matching estimators. As an illustration of the applicability of the theory, we derive the asymptotic distribution of a matching estimator when matching is carried out without replacement, a result previously unavailable in the literature. In addition, we apply the techniques proposed in this article to derive a correction to the standard error of a sample mean when missing data are imputed using the “hot deck�, a matching imputation method widely used in the Current Population Survey (CPS) and other large surveys in the social sciences. We demonstrate the empirical relevance of our methods using two Monte Carlo designs based on actual data sets. In these realistic Monte Carlo exercises the large sample distribution of matching estimators derived in this article provides an accurate approximation to the small sample behavior of these estimators. In addition, our simulations show that standard errors that do not take into account hot deck imputation of missing data may be severely downward biased, while standard errors that incorporate the correction proposed in this article for hot deck imputation perform extremely well. This result demonstrates the practical relevance of the standard error correction for the hot deck proposed in this article.

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Suggested Citation

Abadie, Alberto and Imbens, Guido W., A Martingale Representation for Matching Estimators (February 2009). NBER Working Paper Series, Vol. w14756, pp. -, 2009. Available at SSRN: https://ssrn.com/abstract=1349589

Alberto Abadie (Contact Author)

Harvard University - Harvard Kennedy School (HKS) ( email )

79 John F. Kennedy Street
Cambridge, MA 02138
United States
617-496-4547 (Phone)
617-495-2575 (Fax)

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Guido W. Imbens

Stanford Graduate School of Business ( email )

655 Knight Way
Stanford, CA 94305-5015
United States

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