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Asymptotic Properties of the Synthetic Control Method

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  • Xiaomeng Zhang
  • Wendun Wang
  • Xinyu Zhang

Abstract

This paper provides new insights into the asymptotic properties of the synthetic control method (SCM). We show that the synthetic control (SC) weight converges to a limiting weight that minimizes the mean squared prediction risk of the treatment-effect estimator when the number of pretreatment periods goes to infinity, and we also quantify the rate of convergence. Observing the link between the SCM and model averaging, we further establish the asymptotic optimality of the SC estimator under imperfect pretreatment fit, in the sense that it achieves the lowest possible squared prediction error among all possible treatment effect estimators that are based on an average of control units, such as matching, inverse probability weighting and difference-in-differences. The asymptotic optimality holds regardless of whether the number of control units is fixed or divergent. Thus, our results provide justifications for the SCM in a wide range of applications. The theoretical results are verified via simulations.

Suggested Citation

  • Xiaomeng Zhang & Wendun Wang & Xinyu Zhang, 2022. "Asymptotic Properties of the Synthetic Control Method," Papers 2211.12095, arXiv.org.
    • Handle: RePEc:arx:papers:2211.12095
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    References listed on IDEAS

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    Cited by:

    1. Lu Zhang & Xiaomeng Zhang & Xinyu Zhang, 2024. "Asymptotic Properties of the Distributional Synthetic Controls," Papers 2405.00953, arXiv.org, revised Aug 2024.
    2. Jared Amani Greathouse & Mani Bayani & Jason Coupet, 2023. "Splash! Robustifying Donor Pools for Policy Studies," Papers 2308.13688, arXiv.org.

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