When there are multiple outcome series of interest, Synthetic Control analyses typically proceed by estimating separate weights for each outcome. In this paper, we instead propose estimating a common set of weights across outcomes, by balancing either a vector of all outcomes or an index or average of them. Under a low-rank factor model, we show that these approaches lead to lower bias bounds than separate weights, and that averaging leads to further gains when the number of outcomes grows. We illustrate this via a re-analysis of the impact of the Flint water crisis on educational outcomes.

翻译：当存在多个关注的结果序列时，合成控制分析通常通过为每个结果分别估计权重来进行。本文提出了一种替代方法：通过平衡所有结果构成的向量或其索引（或平均值），跨结果估计一组共同的权重。在低秩因子模型下，我们证明这些方法相比单独权重能获得更低的偏差界，且当结果数量增加时，平均化能带来进一步的改进。我们通过对弗林特水危机教育结果影响的再分析来阐释这一方法。