Synthetic Control: An Exact Geometric Formulation

The synthetic control method matches a treated unit to a convex combination of donor units, with the relative importance of each predictor determined by a weight matrix $V$. In standard practice, $V$ is selected through a bi-level optimization problem that is nonconvex and lacks convergence guarantees. This article derives an exact geometric solution for synthetic control estimators. For any positive definite $V$, a synthetic control is given by the projection of the treated unit's predictors onto the convex hull of the donor predictors under a $V$-weighted norm. As $V$ varies over its candidate class, these projections trace out the set of attainable synthetic controls. We characterize this set. When the predictor vector of the treated unit lies outside the convex hull of the donor predictors, the attainable set under general positive definite $V$ is the union of the facets of the donor hull that are visible from the treated unit. For diagonal $V$, a sign-compatibility test identifies the attainable set. When the treated unit lies inside the hull, exact matches exist. We select the one that minimizes the average of the pairwise matching discrepancies between the treated unit and each of the donors contributing to the synthetic control. These characterizations replace bi-level optimization with a fast enumeration method that is guaranteed to deliver the global optimum over the attainable set. Moreover, the Karush-Kuhn-Tucker conditions for the selection of donor weights characterize the set of positive definite predictor weight matrices that rationalize any given set of donor weights. Applications to the California tobacco control program and to the economic effects of German reunification illustrate the advantages of the proposed procedure over existing practice.