Imperfect Synthetic Controls

The synthetic control method assumes the existence of a perfect synthetic control, which cannot exist if the outcomes are functions of transitory shocks with nonzero asymptotic variance and may not exist even in expectation for the treated unit. This paper first shows the benefits of estimating synthetic controls for all units. If the treated unit composes part of the synthetic control for any untreated unit, the treatment effect is independently identified by the synthetic outcome minus the outcome of the untreated unit in the post-treatment period (divided by the synthetic control weight on the treated unit outcome). This paper introduces an estimator which generates synthetic controls for all units and develops moment conditions which are valid given transitory shocks. I also introduce a weighting metric which asymptotically excludes units without appropriate synthetic controls. The paper exploits the estimator's construction of multiple estimates of the treatment effect to produce valid inference even when the number of control units is small. The estimator is used to evaluate the repeal of Wisconsin's handgun purchase waiting period on suicide rates.