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Matching Estimators with Few Treated and Many Control Observations

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  • Ferman, Bruno

Abstract

We analyze the properties of matching estimators when the number of treated observations is fixed while the number of treated observations is large. We show that, under standard assumptions, the nearest neighbor matching estimator for the average treatment effect on the treated is asymptotically unbiased, even though this estimator is not consistent. We also provide a test based on the theory of randomization tests under approximate symmetry developed in Canay et al. (2014) that is asymptotically valid when the number of control observations goes to infinity. This is important because large sample inferential techniques developed in Abadie and Imbens (2006) would not be valid in this setting.

Suggested Citation

  • Ferman, Bruno, 2017. "Matching Estimators with Few Treated and Many Control Observations," MPRA Paper 78940, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:78940
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    References listed on IDEAS

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    1. James G. MacKinnon & Matthew D. Webb, 2016. "Randomization Inference for Difference-in-Differences with Few Treated Clusters," Carleton Economic Papers 16-11, Carleton University, Department of Economics.
    2. LaLonde, Robert J, 1986. "Evaluating the Econometric Evaluations of Training Programs with Experimental Data," American Economic Review, American Economic Association, vol. 76(4), pages 604-620, September.
    3. Laurent Gobillon & Thierry Magnac, 2016. "Regional Policy Evaluation: Interactive Fixed Effects and Synthetic Controls," The Review of Economics and Statistics, MIT Press, vol. 98(3), pages 535-551, July.
    4. Bruno Ferman & Cristine Pinto, 2019. "Inference in Differences-in-Differences with Few Treated Groups and Heteroskedasticity," The Review of Economics and Statistics, MIT Press, vol. 101(3), pages 452-467, July.
    5. Timothy G. Conley & Christopher R. Taber, 2011. "Inference with "Difference in Differences" with a Small Number of Policy Changes," The Review of Economics and Statistics, MIT Press, vol. 93(1), pages 113-125, February.
    6. Matias Busso & John DiNardo & Justin McCrary, 2014. "New Evidence on the Finite Sample Properties of Propensity Score Reweighting and Matching Estimators," The Review of Economics and Statistics, MIT Press, vol. 96(5), pages 885-897, December.
    7. Alberto Abadie & Javier Gardeazabal, 2003. "The Economic Costs of Conflict: A Case Study of the Basque Country," American Economic Review, American Economic Association, vol. 93(1), pages 113-132, March.
    8. Rajeev H. Dehejia & Sadek Wahba, 2002. "Propensity Score-Matching Methods For Nonexperimental Causal Studies," The Review of Economics and Statistics, MIT Press, vol. 84(1), pages 151-161, February.
    9. Nikolay Doudchenko & Guido W. Imbens, 2016. "Balancing, Regression, Difference-In-Differences and Synthetic Control Methods: A Synthesis," NBER Working Papers 22791, National Bureau of Economic Research, Inc.
    10. Victor Chernozhukov & Kaspar Wüthrich & Yu Zhu, 2017. "An exact and robust conformal inference method for counterfactual and synthetic controls," CeMMAP working papers CWP62/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    11. repec:wly:emetrp:v:85:y:2017:i::p:1013-1030 is not listed on IDEAS
    12. Juan Díaz & Tomás Rau & Jorge Rivera, 2015. "A Matching Estimator Based on a Bilevel Optimization Problem," The Review of Economics and Statistics, MIT Press, vol. 97(4), pages 803-812, October.
    13. Ferman, Bruno & Pinto, Cristine, 2017. "Placebo Tests for Synthetic Controls," MPRA Paper 78079, University Library of Munich, Germany.
    14. Abadie, Alberto & Imbens, Guido W., 2011. "Bias-Corrected Matching Estimators for Average Treatment Effects," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(1), pages 1-11.
    15. Guido W. Imbens, 2015. "Matching Methods in Practice: Three Examples," Journal of Human Resources, University of Wisconsin Press, vol. 50(2), pages 373-419.
    16. 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.
    17. Ferman, Bruno & Ponczek, Vladimir, 2017. "Should we drop covariate cells with attrition problems?," MPRA Paper 80686, University Library of Munich, Germany.
    18. 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.
    19. 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.
    20. Taisuke Otsu & Yoshiyasu Rai, 2017. "Bootstrap Inference of Matching Estimators for Average Treatment Effects," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(520), pages 1720-1732, October.
    21. repec:wly:emetrp:v:85:y:2017:i::p:645-660 is not listed on IDEAS
    22. Bruno Ferman, 2019. "On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls," Papers 1906.06665, arXiv.org, revised Oct 2019.
    23. repec:bpj:causin:v:6:y:2018:i:2:p:26:n:1 is not listed on IDEAS
    24. Ferman, Bruno & Pinto, Cristine Campos de Xavier, 2016. "Revisiting the synthetic control estimator," Textos para discussão 421, FGV EESP - Escola de Economia de São Paulo, Fundação Getulio Vargas (Brazil).
    25. Huber, Martin & Lechner, Michael & Wunsch, Conny, 2013. "The performance of estimators based on the propensity score," Journal of Econometrics, Elsevier, vol. 175(1), pages 1-21.
    26. Jinyong Hahn & Ruoyao Shi, 2017. "Synthetic Control and Inference," Econometrics, MDPI, Open Access Journal, vol. 5(4), pages 1-12, November.
    27. Abadie, Alberto & Diamond, Alexis & Hainmueller, Jens, 2010. "Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California’s Tobacco Control Program," Journal of the American Statistical Association, American Statistical Association, vol. 105(490), pages 493-505.
    28. Imbens,Guido W. & Rubin,Donald B., 2015. "Causal Inference for Statistics, Social, and Biomedical Sciences," Cambridge Books, Cambridge University Press, number 9780521885881.
    29. 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.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

    matching estimator; treatment effect; hypothesis testing; randomization inference;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models

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