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A Martingale Representation for Matching Estimators


  • Alberto Abadie
  • Guido Imbens


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.

Suggested Citation

  • Alberto Abadie & Guido Imbens, 2009. "A Martingale Representation for Matching Estimators," NBER Working Papers 14756, National Bureau of Economic Research, Inc.
  • Handle: RePEc:nbr:nberwo:14756
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    References listed on IDEAS

    1. Angrist, Joshua D. & Krueger, Alan B., 1999. "Empirical strategies in labor economics," Handbook of Labor Economics, in: O. Ashenfelter & D. Card (ed.), Handbook of Labor Economics, edition 1, volume 3, chapter 23, pages 1277-1366, Elsevier.
    2. Rajeev H. Dehejia & Sadek Wahba, 1998. "Causal Effects in Non-Experimental Studies: Re-Evaluating the Evaluation of Training Programs," NBER Working Papers 6586, National Bureau of Economic Research, Inc.
    3. Guido W. Imbens & Jeffrey M. Wooldridge, 2009. "Recent Developments in the Econometrics of Program Evaluation," Journal of Economic Literature, American Economic Association, vol. 47(1), pages 5-86, March.
    4. Alberto Abadie & Guido W. Imbens, 2008. "On the Failure of the Bootstrap for Matching Estimators," Econometrica, Econometric Society, vol. 76(6), pages 1537-1557, November.
    5. Guido W. Imbens, 2004. "Nonparametric Estimation of Average Treatment Effects Under Exogeneity: A Review," The Review of Economics and Statistics, MIT Press, vol. 86(1), pages 4-29, February.
    6. Ben B. Hansen, 2004. "Full Matching in an Observational Study of Coaching for the SAT," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 609-618, January.
    7. James J. Heckman & Hidehiko Ichimura & Petra Todd, 1998. "Matching As An Econometric Evaluation Estimator," Review of Economic Studies, Oxford University Press, vol. 65(2), pages 261-294.
    8. Alberto Abadie & Guido W. Imbens, 2006. "Large Sample Properties of Matching Estimators for Average Treatment Effects," Econometrica, Econometric Society, vol. 74(1), pages 235-267, January.
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    Cited by:

    1. Liu, Angela Xia & Xie, Ying & Zhang, Jurui, 2019. "It's Not Just What You Say, But How You Say It: The Effect of Language Style Matching on Perceived Quality of Consumer Reviews," Journal of Interactive Marketing, Elsevier, vol. 46(C), pages 70-86.
    2. Huber, Martin & Camponovo, Lorenzo & Bodory, Hugo & Lechner, Michael, 2016. "A wild bootstrap algorithm for propensity score matching estimators," FSES Working Papers 470, Faculty of Economics and Social Sciences, University of Freiburg/Fribourg Switzerland.
    3. Hirukawa, Masayuki & Prokhorov, Artem, 2018. "Consistent estimation of linear regression models using matched data," Journal of Econometrics, Elsevier, vol. 203(2), pages 344-358.
    4. Fredrik Savje, 2019. "Consistency and matching without replacement," Papers 1907.07288,
    5. Shu Yang & Guido W. Imbens & Zhanglin Cui & Douglas E. Faries & Zbigniew Kadziola, 2016. "Propensity score matching and subclassification in observational studies with multi-level treatments," Biometrics, The International Biometric Society, vol. 72(4), pages 1055-1065, December.

    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

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