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On the Misspecification of Linear Assumptions in Synthetic Control

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  • Achille Nazaret
  • Claudia Shi
  • David M. Blei

Abstract

The synthetic control (SC) method is a popular approach for estimating treatment effects from observational panel data. It rests on a crucial assumption that we can write the treated unit as a linear combination of the untreated units. This linearity assumption, however, can be unlikely to hold in practice and, when violated, the resulting SC estimates are incorrect. In this paper we examine two questions: (1) How large can the misspecification error be? (2) How can we limit it? First, we provide theoretical bounds to quantify the misspecification error. The bounds are comforting: small misspecifications induce small errors. With these bounds in hand, we then develop new SC estimators that are specially designed to minimize misspecification error. The estimators are based on additional data about each unit, which is used to produce the SC weights. (For example, if the units are countries then the additional data might be demographic information about each.) We study our estimators on synthetic data; we find they produce more accurate causal estimates than standard synthetic controls. We then re-analyze the California tobacco-program data of the original SC paper, now including additional data from the US census about per-state demographics. Our estimators show that the observations in the pre-treatment period lie within the bounds of misspecification error, and that the observations post-treatment lie outside of those bounds. This is evidence that our SC methods have uncovered a true effect.

Suggested Citation

  • Achille Nazaret & Claudia Shi & David M. Blei, 2023. "On the Misspecification of Linear Assumptions in Synthetic Control," Papers 2302.12777, arXiv.org.
    • Handle: RePEc:arx:papers:2302.12777
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    References listed on IDEAS

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    1. Sylvia Allegretto & Arindrajit Dube & Michael Reich & Ben Zipperer, 2017. "Credible Research Designs for Minimum Wage Studies," ILR Review, Cornell University, ILR School, vol. 70(3), pages 559-592, May.
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    3. 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.
    4. Victor Chernozhukov & Kaspar Wüthrich & Yinchu Zhu, 2021. "An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 116(536), pages 1849-1864, October.
    5. Kathleen T. Li, 2020. "Statistical Inference for Average Treatment Effects Estimated by Synthetic Control Methods," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 115(532), pages 2068-2083, December.
    6. Heersink, Boris & Peterson, Brenton D. & Jenkins, Jeffery A., 2017. "Disasters and Elections: Estimating the Net Effect of Damage and Relief in Historical Perspective," Political Analysis, Cambridge University Press, vol. 25(2), pages 260-268, April.
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    Cited by:

    1. Masahiro Kato & Akari Ohda & Masaaki Imaizumi & Kenichiro McAlinn, 2023. "Synthetic Control Methods by Density Matching under Implicit Endogeneity," Papers 2307.11127, arXiv.org, revised Jul 2023.

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