We present new results for nonparametric identification of causal effects using noisy proxies for unobserved confounders. Our approach builds on the results of \citet{Hu2008} who tackle the problem of general measurement error. 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 \citet{Miao2018a} in which causal effects are identified using two conditionally independent proxies. We refer to this as the `double proxy' approach. The triple proxy approach identifies objects that are not identified by the double proxy approach, including some that capture the variation in average treatment effects between strata of the unobservables. Moreover, the conditional independence assumptions in the double and triple proxy approaches are non-nested.

翻译：我们提出了利用未观测混杂因素的噪声代理变量进行因果效应非参数识别的新结果。该方法基于\citet{Hu2008}处理一般测量误差问题的研究成果，将其称为"三重代理"方法，因其需要三个在未观测变量条件下联合独立的代理变量。我们对第三个代理变量考虑了三种不同选择：结果变量、处理变量向量或辅助变量集合。我们将此与\citet{Miao2018a}提出的替代识别策略进行比较，后者通过两个条件独立的代理变量识别因果效应，我们称之为"双重代理"方法。三重代理方法能够识别双重代理方法无法识别的对象，包括捕捉未观测变量层间平均处理效应变异的一些指标。此外，双重与三重代理方法中的条件独立性假设是非嵌套的。