Synthetic Design: An Optimization Approach to Experimental Design with Synthetic Controls

TL;DR

This paper presents an optimization-based method for experimental design using synthetic controls, improving treatment unit selection and weighting to enhance estimation accuracy and statistical power.

Contribution

It introduces a mixed-integer programming approach for jointly selecting treated units and weights, addressing NP-hardness and outperforming traditional methods.

Findings

Improved mean squared error in treatment effect estimation.

Enhanced statistical power over standard randomized trials.

Qualitatively different units selected for treatment with the new method.

Abstract

We investigate the optimal design of experimental studies that have pre-treatment outcome data available. The average treatment effect is estimated as the difference between the weighted average outcomes of the treated and control units. A number of commonly used approaches fit this formulation, including the difference-in-means estimator and a variety of synthetic-control techniques. We propose several methods for choosing the set of treated units in conjunction with the weights. Observing the NP-hardness of the problem, we introduce a mixed-integer programming formulation which selects both the treatment and control sets and unit weightings. We prove that these proposed approaches lead to qualitatively different experimental units being selected for treatment. We use simulations based on publicly available data from the US Bureau of Labor Statistics that show improvements in terms of mean squared error and statistical power when compared to simple and commonly used alternatives such as randomized trials.