Synthetic Regressing Control

Estimating weights in the synthetic control method, typically resulting in sparse weights where only a few control units have non-zero weights, involves an optimization procedure that selects and combines control units to closely match the treated unit. However, it is not uncommon for the linear combination of pre-treatment period outcomes for the control units, using nonnegative weights with the constraint that their sum equals one, to inadequately approximate the pre-treatment outcomes for the treated unit. To address the issue, this paper proposes a simple and effective method called Synthetic Regressing Control (SRC). The SRC method begins by performing the univariate linear regression to appropriately align the pre-treatment periods of the control units with the treated unit. Subsequently, a SRC estimator is obtained by synthesizing the regressed controls. To determine the weights in the synthesis procedure, we propose an approach that utilizes a criterion of an unbiased risk estimator. Theoretically, we show that the synthesis way is asymptotically optimal in the sense of achieving the minimum loss of the infeasible best possible synthetic estimator. Extensive numerical experiments highlight the advantages of the SRC method.