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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 pretreatment periods and control units are large. If potential outcomes follow a linear factor model, we provide conditions under which the SC unit asymptotically recovers the factor structure of the treated unit, even when the pretreatment fit is imperfect. This happens when there are weights diluted among an increasing number of control units such that a weighted average of the factor structure of the control units asymptotically reconstructs the factor structure of the treated unit. In this case, the SC estimator is asymptotically unbiased even when treatment assignment is correlated with time-varying unobservables. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

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  • Bruno Ferman, 2021. "On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1764-1772, October.
    • Handle: RePEc:taf:jnlasa:v:116:y:2021:i:536:p:1764-1772
    DOI: 10.1080/01621459.2021.1965613
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    Cited by:

    1. Ignacio Martinez & Jaume Vives-i-Bastida, 2022. "Bayesian and Frequentist Inference for Synthetic Controls," Papers 2206.01779, arXiv.org, revised Feb 2023.
    2. Ferman, Bruno, 2021. "Matching estimators with few treated and many control observations," Journal of Econometrics, Elsevier, vol. 225(2), pages 295-307.
    3. Joseph Fry, 2023. "A Method of Moments Approach to Asymptotically Unbiased Synthetic Controls," Papers 2312.01209, arXiv.org, revised Mar 2024.
    4. Alberto Abadie & Jinglong Zhao, 2021. "Synthetic Controls for Experimental Design," Papers 2108.02196, arXiv.org, revised Dec 2023.
    5. Xiaomeng Zhang & Wendun Wang & Xinyu Zhang, 2022. "Asymptotic Properties of the Synthetic Control Method," Papers 2211.12095, arXiv.org.
    6. Yiping Lu & Jiajin Li & Lexing Ying & Jose Blanchet, 2022. "Synthetic Principal Component Design: Fast Covariate Balancing with Synthetic Controls," Papers 2211.15241, arXiv.org.
    7. David Gilchrist & Thomas Emery & Nuno Garoupa & Rok Spruk, 2023. "Synthetic Control Method: A tool for comparative case studies in economic history," Journal of Economic Surveys, Wiley Blackwell, vol. 37(2), pages 409-445, April.
    8. Alberto Abadie & Jaume Vives-i-Bastida, 2022. "Synthetic Controls in Action," Papers 2203.06279, arXiv.org.
    9. Wei Tian & Seojeong Lee & Valentyn Panchenko, 2023. "Synthetic Controls with Multiple Outcomes: Estimating the Effects of Non-Pharmaceutical Interventions in the COVID-19 Pandemic," Papers 2304.02272, arXiv.org.
    10. Guido Imbens & Nathan Kallus & Xiaojie Mao, 2021. "Controlling for Unmeasured Confounding in Panel Data Using Minimal Bridge Functions: From Two-Way Fixed Effects to Factor Models," Papers 2108.03849, arXiv.org.
    11. Bruno Ferman & Cristine Pinto, 2021. "Synthetic controls with imperfect pretreatment fit," Quantitative Economics, Econometric Society, vol. 12(4), pages 1197-1221, November.
    12. Luis Alvarez & Bruno Ferman, 2023. "Extensions for Inference in Difference-in-Differences with Few Treated Clusters," Papers 2302.03131, arXiv.org.
    13. Timo Schenk, 2023. "Time-Weighted Difference-in-Differences: Accounting for Common Factors in Short T Panels," Tinbergen Institute Discussion Papers 23-004/III, Tinbergen Institute.
    14. Zongwu Cai & Ying Fang & Ming Lin & Zixuan Wu, 2023. "A Quasi Synthetic Control Method for Nonlinear Models With High-Dimensional Covariates," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 202305, University of Kansas, Department of Economics, revised Aug 2023.
    15. Guido W. Imbens & Davide Viviano, 2023. "Identification and Inference for Synthetic Controls with Confounding," Papers 2312.00955, arXiv.org.

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