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 simultaneously selects and aligns control units to closely match the treated unit. However, this simultaneous selection and alignment of control units may lead to a loss of efficiency. Another concern arising from the aforementioned procedure is its susceptibility to under-fitting due to imperfect pre-treatment fit. It is not uncommon for the linear combination, using nonnegative weights, of pre-treatment period outcomes for the control units to inadequately approximate the pre-treatment outcomes for the treated unit. To address both of these issues, 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 (taking a weighted average) the fitted controls. To determine the weights in the synthesis procedure, we propose an approach that utilizes a criterion of unbiased risk estimator. Theoretically, we show that the synthesis way is asymptotically optimal in the sense of achieving the lowest possible squared error. Extensive numerical experiments highlight the advantages of the SRC method.

翻译：合成控制方法中的权重估计通常产生稀疏权重，即仅少数控制单元具有非零权重，该过程涉及同时选择并调整控制单元以密切匹配处理单元。然而，这种同时选择与调整控制单元的做法可能导致效率损失。上述过程的另一个问题是，由于预处理期拟合不完美，其容易欠拟合。常见情况是，使用非负权重的线性组合对控制单元预处理期结果进行拟合时，无法充分近似处理单元的预处理期结果。为解决这两个问题，本文提出一种简单有效的方法——合成回归控制（SRC）。SRC方法首先执行单变量线性回归，将控制单元的预处理期结果适当对齐至处理单元；随后通过综合（加权平均）拟合后的控制单元得到SRC估计量。在综合过程的权重确定上，我们提出基于无偏风险估计准则的方法。理论上，我们证明该方法在实现最低平方误差的意义上渐近最优。大量数值实验凸显了SRC方法的优势。