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

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 * We are grateful to Alberto Abadie, participants at the 2021 Conference on Digital Experimentation @ MIT (CODE@MIT), the 2021 Design and Analysis of Experiments (DAE) conference and New Economic School economics seminar, as well as the anonymous referees at NeurIPS 2021 for comments and suggestions. † Equal contributions.