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.

翻译：合成控制法中的权重估计通常会产生稀疏权重，即仅有少数控制单元具有非零权重，该过程通过优化程序选择并组合控制单元以紧密匹配处理单元。然而，控制单元预处理期结果的线性组合（采用非负权重且满足权重之和为1的约束条件）往往无法充分逼近处理单元的预处理结果。针对这一问题，本文提出了一种简单有效的方法，称为合成回归控制法。SRC方法首先通过一元线性回归将控制单元的预处理期与处理单元进行适当对齐，随后通过合成回归后的控制变量得到SRC估计量。为确定合成过程中的权重，我们提出了一种基于无偏风险估计准则的优化方法。理论上，我们证明了该合成方式具有渐近最优性，能够达到不可行最优合成估计量的最小损失。大量数值实验凸显了SRC方法的优势。