Single Proxy Synthetic Control
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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.
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