On the Properties of the Synthetic Control Estimator with Many Periods and Many Controls

We consider the asymptotic properties of the Synthetic Control (SC) estimator when both the number of pre-treatment periods and control units are large. If potential outcomes follow a linear factor model, we provide conditions under which the SC unit asymptotically recovers the factor structure of the treated unit, even when the pre-treatment fit is imperfect. This happens when there are weights diluted among an increasing number of control units such that a weighted average of the factor structure of the control units asymptotically reconstructs the factor structure of the treated unit. In this case, the SC estimator is asymptotically unbiased even when treatment assignment is correlated with time-varying unobservables.

本文研究当预处理时期数目与对照单元数目均较大时，合成控制法（Synthetic Control, SC）估计量的渐近性质。若潜在结果服从线性因子模型，我们给出了合成控制单元可渐近还原处理单元因子结构的条件，即便此时预处理拟合存在缺陷。当权重分散于不断增加的对照单元之间，使得对照单元因子结构的加权平均能够渐近重构处理单元的因子结构时，上述情形即可成立。在此情形下，即便处理分配与时变不可观测变量存在相关性，合成控制法估计量仍具有渐近无偏性。