Synthetic control methods are widely used to estimate the treatment effect on a single treated unit in time series settings. A common approach for estimating synthetic controls is to regress the pre-treatment outcomes of the treated unit on those of untreated control units via ordinary least squares. However, this approach can perform poorly if the pre-treatment fit is not near perfect, whether the weights are normalized or not. In this paper, we introduce a single proxy synthetic control approach, which essentially views the outcomes of untreated control units as proxies of the treatment-free potential outcome of the treated unit, a perspective we formally leverage to construct a valid synthetic control. Under this framework, we establish alternative identification and estimation methodology for synthetic controls and, in turn, for the treatment effect on the treated unit. Notably, unlike a recently proposed proximal synthetic control approach which requires two types of proxies for identification, ours relies on a single type of proxy, thus facilitating its practical relevance. Additionally, we adapt a conformal inference approach to perform inference on the treatment effect, obviating the need for a large number of post-treatment data. Lastly, our framework can accommodate time-varying covariates and nonlinear models, allowing binary and count outcomes. We demonstrate the proposed approach in a simulation study and a real-world application.

翻译：合成控制方法广泛用于时间序列场景中估计单个处理单元的处理效应。一种常见的合成控制估计方法是通过普通最小二乘法将处理单元的预处理结果回归到未处理控制单元的结果上。然而，若预处理的拟合效果不够接近完美，无论权重是否归一化，该方法的效果可能较差。本文提出了一种单一代理合成控制方法，其核心思想是将未处理控制单元的结果视为处理单元无处理潜在结果的代理变量，我们正式利用这一视角构建有效的合成控制。在此框架下，我们建立了合成控制的替代识别与估计方法，进而推导出处理单元的处理效应。值得注意的是，与近期提出的需要两种代理变量进行识别的近端合成控制方法不同，我们的方法仅依赖单一类型的代理变量，从而增强了其在实践中的适用性。此外，我们采用共形推断方法对处理效应进行推断，避免了大量处理后期数据的需求。最后，我们的框架可容纳时变协变量与非线性模型，支持二元及计数结果。我们通过模拟研究与实际应用验证了所提方法的有效性。