A Relaxation Approach to Synthetic Control

TL;DR

This paper introduces SCM-relaxation, a novel machine learning algorithm that enhances synthetic control methods by relaxing constraints, improving counterfactual predictions especially with group-structured donor pools.

Contribution

The paper proposes a relaxation-based algorithm for synthetic control that improves prediction accuracy and handles group structures in donor pools, achieving oracle performance asymptotically.

Findings

Outperforms traditional SCM in simulations

Achieves oracle performance asymptotically

Effectively handles group-structured donor pools

Abstract

The synthetic control method (SCM) is widely used for constructing the counterfactual of a treated unit based on data from control units in a donor pool. Allowing the donor pool contains more control units than time periods, we propose a novel machine learning algorithm, named SCM-relaxation, for counterfactual prediction. Our relaxation approach minimizes an information-theoretic measure of the weights subject to a set of relaxed linear inequality constraints in addition to the simplex constraint. When the donor pool exhibits a group structure, SCM-relaxation approximates the equal weights within each group to diversify the prediction risk. Asymptotically, the proposed estimator achieves oracle performance in terms of out-of-sample prediction accuracy. We demonstrate our method by Monte Carlo simulations and by an empirical application that assesses the economic impact of Brexit on the United Kingdom's real GDP.