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Generalized Synthetic Control Method: Causal Inference with Interactive Fixed Effects Models

  • Yiqing Xu
Abstract

Difference-in-differences (DID) is commonly used for causal inference in time-series cross-sectional data. It requires the assumption that the average outcomes of treated and control units would have followed parallel paths in the absence of treatment. In this paper, we propose a method that not only relaxes this often-violated assumption, but also unifies the synthetic control method (Abadie, Diamond, and Hainmueller 2010) with linear fixed effects models under a simple framework, of which DID is a special case. It imputes counterfactuals for each treated unit using control group information based on a linear interactive fixed effects model that incorporates unit-specific intercepts interacted with time-varying coefficients. This method has several advantages. First, it allows the treatment to be correlated with unobserved unit and time heterogeneities under reasonable modeling assumptions. Second, it generalizes the synthetic control method to the case of multiple treated units and variable treatment periods, and improves efficiency and interpretability. Third, with a built-in cross-validation procedure, it avoids specification searches and thus is easy to implement. An empirical example of Election Day Registration and voter turnout in the United States is provided.

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Copyright
Corresponding author
* Email: yiqingxu@ucsd.edu
Footnotes
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Author’s note: The author is indebted to Matt Blackwell, Devin Caughey, Justin Grimmer, Jens Hainmueller, Danny Hidalgo, Simon Jackman, Jonathan Katz, Luke Keele, Eric Min, Molly Roberts, Jim Snyder, Brandon Stewart, Teppei Yamamoto, as well as seminar participants at the 2015 MPSA Annual Meeting and 2015 APSA Annual Meeting for helpful comments and suggestions. The author thanks the editor, Mike Alvarez, and two anonymous reviewers for their extremely helpful suggestions. He also thanks Jushan Bai for generously sharing the Matlab codes used in Bai (2009) and Melanie Springer for kindly providing the state-level voter turnout data (1920–2000). The source code and data used in the paper can be downloaded from the Political Analysis Dataverse at dx.doi.org/10.7910/DVN/8AKACJ (Xu 2016) as well as the author’s website.
Contributing Editor: R. Michael Alvarez
Footnotes
References
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