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A Method of Moments Approach to Asymptotically Unbiased Synthetic Controls

Joseph Fry

arXiv 2 Dec 2023 · Econometrics · publishedJournal of Econometrics (2024) · 2 citations (OpenAlex)

arXiv:2312.01209 · PDF · DOI · OpenAlex · Extracted main text

Abstract

A common approach to constructing a Synthetic Control unit is to fit on the outcome variable and covariates in pre-treatment time periods, but it has been shown by Ferman and Pinto (2019) that this approach does not provide asymptotic unbiasedness when the fit is imperfect and the number of controls is fixed. Many related panel methods have a similar limitation when the number of units is fixed. I introduce and evaluate a new method in which the Synthetic Control is constructed using a General Method of Moments approach where units not being included in the Synthetic Control are used as instruments. I show that a Synthetic Control Estimator of this form will be asymptotically unbiased as the number of pre-treatment time periods goes to infinity, even when pre-treatment fit is imperfect and the number of units is fixed. Furthermore, if both the number of pre-treatment and post-treatment time periods go to infinity, then averages of treatment effects can be consistently estimated. I conduct simulations and an empirical application to compare the performance of this method with existing approaches in the literature.

Citation extraction

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appendix boundary found by appendix_titled_section at “Appendix A. Proofs of the Main Results” · 57% of the source is main text. Read the extracted text to check this.

Most heavily cited references

The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.

ReferenceIntensityMentionsSectionsMain text
1Abadie, Diamond, and Hainmueller (2010) Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California's Tobacco Control Program1.000115100%
Powell2021unmatched citation key Powell20211.000113100%
3Ferman and Pinto (2021) Synthetic controls with imperfect pretreatment fit1.00064100%
4Abadie, Diamond, and Hainmueller (2015) Comparative Politics and the Synthetic Control Method1.00063100%
5Ferman (2021) On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls0.9619489%
6Andrews and Lu (2001) Consistent model and moment selection procedures for GMM estimation with application to dynamic panel data models0.89414371%
7Arkhangelsky, Athey, Hirshberg, Imbens, and Wager (2021) Synthetic Difference-in-Differences0.8746467%
8Chernozhukov, Wüthrich, and Zhu (2021) An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls0.87462100%
9Li (2020) Statistical Inference for Average Treatment Effects Estimated by Synthetic Control Methods0.8558462%
10Andrews (2003) End-of-Sample Instability Tests0.81142100%

Showing the top 10 of 59 scored citations. 1 of these could not be matched to a bibliography entry, so only the citation key is shown.

Cited by, within the corpus

arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.

Citing paperIntensityMentionsSections
1Robust Inference when Nuisance Parameters may be Partially Identified with Applications to Synthetic Controls0.69351
2Difference-in-differences with as few as two cross-sectional units – A new perspective to the democracy–growth debate0.51132
3Using Multiple Outcomes to Improve the Synthetic Control Method0.40511
4Identification of Average Treatment Effects in Nonparametric Panel Models0.40511