We apply results in Hu and Schennach (2008) to achieve nonparametric identification of causal effects using noisy proxies for unobserved confounders. We call this the `triple proxy' approach because it requires three proxies that are jointly independent conditional on unobservables. We consider three different choices for the third proxy: it may be an outcome, a vector of treatments, or a collection of auxiliary variables. We compare to an alternative identification strategy introduced by Miao et. al. (2018) in which causal effects are identified using two conditionally independent proxies. We refer to this as the `double proxy' approach. We show that the conditional independence assumptions in the double and triple proxy approaches are non-nested, which suggests that either of the two identification strategies may be appropriate depending on the particular setting.

翻译：我们运用Hu和Schennach(2008年)的结果,对未观察到的困惑者使用吵闹的代理人对因果关系进行非参数识别。我们称之为“三重代理”办法,因为它要求三个共同独立的代理人以不可观察者为条件。我们考虑第三个代理人的三个不同的选择:它可能是结果、治疗矢量或一系列辅助变量。我们比较了Mioo等人(2018年)提出的替代识别战略,其中使用两个有条件的独立代理人来识别因果关系。我们称之为“双重代理”办法。我们表明,双重和三重代理办法中的有条件独立假设是不可放弃的,这表明两种识别战略中的任何一个都可能适合特定环境。