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On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls

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  • Bruno Ferman

Abstract

We consider the asymptotic properties of the Synthetic Control (SC) estimator when both the number of pre-treatment periods and control units are large. If potential outcomes follow a linear factor model, we provide conditions under which the factor loadings of the SC unit converge in probability to the factor loadings of the treated unit. This happens when there are weights diluted among an increasing number of control units such that a weighted average of the factor loadings of the control units asymptotically reconstructs the factor loadings of the treated unit. In this case, the SC estimator is asymptotically unbiased even when treatment assignment is correlated with time-varying unobservables. This result can be valid even when the number of control units is larger than the number of pre-treatment periods.

Suggested Citation

  • Bruno Ferman, 2019. "On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls," Papers 1906.06665, arXiv.org, revised May 2020.
  • Handle: RePEc:arx:papers:1906.06665
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    File URL: http://arxiv.org/pdf/1906.06665
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Carvalho, Carlos & Masini, Ricardo & Medeiros, Marcelo C., 2018. "ArCo: An artificial counterfactual approach for high-dimensional panel time-series data," Journal of Econometrics, Elsevier, vol. 207(2), pages 352-380.
    4. Susan Athey & Guido W. Imbens, 2017. "The State of Applied Econometrics: Causality and Policy Evaluation," Journal of Economic Perspectives, American Economic Association, vol. 31(2), pages 3-32, Spring.
    5. Jushan Bai, 2009. "Panel Data Models With Interactive Fixed Effects," Econometrica, Econometric Society, vol. 77(4), pages 1229-1279, July.
    6. 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.
    7. 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.
    8. Irene Botosaru & Bruno Ferman, 2019. "On the role of covariates in the synthetic control method," Econometrics Journal, Royal Economic Society, vol. 22(2), pages 117-130.
    9. Alberto Abadie & Alexis Diamond & Jens Hainmueller, 2015. "Comparative Politics and the Synthetic Control Method," American Journal of Political Science, John Wiley & Sons, vol. 59(2), pages 495-510, February.
    10. 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.
    11. 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.
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    Cited by:

    1. Ferman, Bruno, 2017. "Matching Estimators with Few Treated and Many Control Observations," MPRA Paper 78940, University Library of Munich, Germany.
    2. Bruno Ferman & Cristine Pinto, 2019. "Synthetic Controls with Imperfect Pre-Treatment Fit," Papers 1911.08521, arXiv.org.

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