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In a seminal paper Abadie, Diamond, and Hainmueller [2010] (ADH), see alsoAbadie and Gardeazabal [2003], Abadie et al. [2014], develop the syntheticcontrol procedure for estimating the effect of a treatment, in the presence ofa single treated unit and a number of control units, with pre-treatmentoutcomes observed for all units. The method constructs a set of weights suchthat selected covariates and pre-treatment outcomes of the treated unit areapproximately matched by a weighted average of control units (the syntheticcontrol). The weights are restricted to be nonnegative and sum to one, which isimportant because it allows the procedure to obtain unique weights even whenthe number of lagged outcomes is modest relative to the number of controlunits, a common setting in applications. In the current paper we propose ageneralization that allows the weights to be negative, and their sum to differfrom one, and that allows for a permanent additive difference between thetreated unit and the controls, similar to difference-in-difference procedures.The weights directly minimize the distance between the lagged outcomes for thetreated and the control units, using regularization methods to deal with apotentially large number of possible control units.

Balancing, Regression, Difference-In-Differences and Synthetic Control Methods: A Synthesis