Matching Estimators with Few Treated and Many Control Observations

Ferman, Bruno (2017): Matching Estimators with Few Treated and Many Control Observations.

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.

• Item Type: MPRA Paper
• Language: English
• Keywords: matching estimator, treatment effect, hypothesis testing, randomization inference
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C12 - Hypothesis Testing: General; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C21 - Cross-Sectional Models ; Spatial Models ; Treatment Effect Models ; Quantile Regressions

References

Abadie, Alberto, Alexis Diamond, and Jens Hainmueller, “Synthetic Control Methods for Com- parative Case Studies: Estimating the Effect of California’s Tobacco Control Program,” Journal of the American Statiscal Association, 2010, 105 (490), 493–505.

Abadie, Alberto, Alexis Diamond, and Jens Hainmueller,, “Comparative Politics and the Synthetic Control Method,” American Journal of Political Science, 2015, 59 (2), 495–510.

Abadie, Alberto and Guido W. Imbens, “Large Sample Properties of Matching Estimators for Average Treatment Effects,” Econometrica, 2006, 74 (1), 235–267.

Abadie, Alberto and Guido W. Imbens, “On the Failure of the Bootstrap for Matching Estimators,” Econometrica, 2008, 76 (6), 1537– 1557.

Abadie, Alberto and Guido W. Imbens, “Bias-Corrected Matching Estimators for Average Treatment Effects,” Journal of Business & Economic Statistics, 2011, 29 (1), 1–11.

Abadie, Alberto and Javier Gardeazabal, “The Economic Costs of Conflict: A Case Study of the Basque Country,” American Economic Review, 2003, 93 (1), 113–132.

Abadie, Alberto, Susan Athey, Guido W. Imbens, and Jeffrey M. Wooldridge, “Finite Population Causal Standard Errors,” Working Paper 20325, National Bureau of Economic Research July 2014.

Busso, Matias, John DiNardo, and Justin McCrary, “New Evidence on the Finite Sample Properties of Propensity Score Reweighting and Matching Estimators,” The Review of Economics and Statistics, December 2014, 96 (5), 885–897.

Canay, Ivan A., Joseph P. Romano, and Azeem M. Shaikh, “Randomization Tests under an Ap- proximate Symmetry Assumption?,” 2014.

Daz, Juan, Toms Rau, and Jorge Rivera, “A Matching Estimator Based on a Bilevel Optimization Problem,” The Review of Economics and Statistics, October 2015, 97 (4), 803–812.

Ferman, Bruno and Cristine Pinto, “Inference in Differences-in-Differences with Few Treated Groups and Heteroskedasticity,” MPRA Paper 67665, University Library of Munich, Germany November 2015.

Ferman, Bruno and Cristine Pinto, “Revisiting the Synthetic Control Estimator,” MPRA Paper 73982, University Library of Munich, Germany September 2016.

Ferman, Bruno and Cristine Pinto, “Placebo Tests for Synthetic Controls,” MPRA Paper 78079, University Library of Munich, Germany April 2017.

Firpo, Sergio and Vitor Possebom, “Synthetic Control Estimator: A Generalized Inference Procedure and Confidence Sets,” April 2016.

Frolich, Markus, “Finite-Sample Properties of Propensity-Score Matching and Weighting Estimators,” The Review of Economics and Statistics, 2004, 86 (1), 77–90.

Gobillon, Laurent and Thierry Magnac, “Regional Policy Evaluation: Interative Fixed Effects and Synthetic Controls,” Review of Economics and Statistics, 2016. Forthcoming.

Hahn, Jinyong and Ruoyao Shi, “Synthetic Control and Inference,” 2016.

Imbens, Guido, “Nonparametric Estimation of Average Treatment Effects under Exogeneity: A Review,” Review of Economics and Statistics, 2004.

Imbens, Guido, “Matching Methods in Practice: Three Examples,” NBER Working Papers 19959, National Bureau of Economic Research, Inc March 2014.

Imbens, Guido W. and Jeffrey M. Wooldridge, “Recent Developments in the Econometrics of Program Evaluation,” Technical Report 1 2009.

Rosenbaum, Paul R., “Conditional Permutation Tests and the Propensity Score in Observational Studies,” Journal of the American Statistical Association, 1984, 79 (387), 565–574.

Rosenbaum, Paul R., “Covariance Adjustment in Randomized Experiments and Observational Studies,” Statist. Sci., 08 2002, 17 (3), 286–327.

Rubin, Donald B., “Matching to Remove Bias in Observational Studies,” Biometrics, 1973, 29 (1), 159– 183.