Bayesian and Frequentist Inference for Synthetic Controls

TL;DR

This paper introduces a Bayesian approach to synthetic control methods, providing valid inference and linking Bayesian and frequentist perspectives, with theoretical guarantees and practical applications.

Contribution

It proposes a Bayesian synthetic control method that maintains key features of the standard approach and offers a new framework for inference, supported by theoretical and simulation results.

Findings

Bayesian method provides valid inference for synthetic controls.

Bayes estimator asymptotically close to MLE in total variation.

Method performs well in simulations and real-world case studies.

Abstract

The synthetic control method has become a widely popular tool to estimate causal effects with observational data. Despite this, inference for synthetic control methods remains challenging. Often, inferential results rely on linear factor model data generating processes. In this paper, we characterize the conditions on the factor model primitives (the factor loadings) for which the statistical risk minimizers are synthetic controls (in the simplex). Then, we propose a Bayesian alternative to the synthetic control method that preserves the main features of the standard method and provides a new way of doing valid inference. We explore a Bernstein-von Mises style result to link our Bayesian inference to the frequentist inference. For linear factor model frameworks we show that a maximum likelihood estimator (MLE) of the synthetic control weights can consistently estimate the predictive function of the potential outcomes for the treated unit and that our Bayes estimator is asymptotically close to the MLE in the total variation sense. Through simulations, we show that there is convergence between the Bayes and frequentist approach even in sparse settings. Finally, we apply the method to re-visit the study of the economic costs of the German re-unification and the Catalan secession movement. The Bayesian synthetic control method is available in the bsynth R-package.