Degrees of Freedom and Information Criteria for the Synthetic Control Method

TL;DR

This paper analytically characterizes the degrees of freedom in the synthetic control method, introduces estimable information criteria for model selection, and demonstrates their effectiveness in empirical applications like Tianjin's car license rationing.

Contribution

It provides a novel analytical framework for understanding SCM's flexibility and proposes information criteria for tuning parameter selection, improving over cross-validation.

Findings

Information criteria outperform cross-validation in model selection.

Penalized SCM variants are effective with many candidate donors.

Model flexibility can be quantified via degrees of freedom.

Abstract

We provide an analytical characterization of the model flexibility of the synthetic control method (SCM) in the familiar form of degrees of freedom. We obtain estimable information criteria, which may be used to circumvent cross-validation when selecting either the tuning parameter in penalized variants of SCM or the weighting matrix in the SCM with covariates. We assess the impact of car license rationing in Tianjin; while a natural match is available, both it and other donors are noisy, inviting the use of SCM to average over approximately matching donors. The very large number of candidate donors calls for penalized variants of SCM and we observe that model selection using information criteria outperforms that based on cross-validation.