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