The Synthetic Control method (SC) has become a valuable tool for estimating causal effects. Originally designed for single-treated unit scenarios, it has recently found applications in high-dimensional disaggregated settings with multiple treated units. However, challenges in practical implementation and computational efficiency arise in such scenarios. To tackle these challenges, we propose a novel approach that integrates the Multivariate Square-root Lasso method into the synthetic control framework. We rigorously establish the estimation error bounds for fitting the Synthetic Control weights using Multivariate Square-root Lasso, accommodating high-dimensionality and time series dependencies. Additionally, we quantify the estimation error for the Average Treatment Effect on the Treated (ATT). Through simulation studies, we demonstrate that our method offers superior computational efficiency without compromising estimation accuracy. We apply our method to assess the causal impact of COVID-19 Stay-at-Home Orders on the monthly unemployment rate in the United States at the county level.

翻译：合成控制方法已成为估计因果效应的有力工具。该方法最初针对单一处理单元场景设计，近年来已扩展至具有多个处理单元的高维细分数据场景。然而，在此类场景中，实际实施与计算效率方面存在挑战。为应对这些挑战，我们提出一种将多元平方根Lasso方法整合至合成控制框架的新颖方法。我们严格建立了使用多元平方根Lasso拟合合成控制权重的估计误差界，该方法能够适应高维性与时间序列依赖性。此外，我们量化了处理组平均处理效应的估计误差。通过模拟研究，我们证明该方法在保持估计精度的同时具有卓越的计算效率。我们将该方法应用于评估美国县级层面COVID-19居家令对月度失业率的因果影响。