A Method of Moments Approach to Asymptotically Unbiased Synthetic Controls

Author:

Joseph Fry

Keyword:

Economics, Econometrics, Econometrics (econ.EM)

journal:

date:

2023-12-02 00:00:00

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 (2021) 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 if the Synthetic Control satisfies the moment conditions it must have the same loadings on latent factors as the treated unit. 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 set of controls 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 and asymptotically valid inference can be conducted using a subsampling method. I conduct simulations and an empirical application to compare the performance of this method with existing approaches in the literature.

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