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We present new results for nonparametric identification of causal effectsusing noisy proxies for unobserved confounders. Our approach builds on theresults of \citet{Hu2008} who tackle the problem of general measurement error.We call this the `triple proxy' approach because it requires three proxies thatare jointly independent conditional on unobservables. We consider threedifferent choices for the third proxy: it may be an outcome, a vector oftreatments, or a collection of auxiliary variables. We compare to analternative identification strategy introduced by \citet{Miao2018a} in whichcausal effects are identified using two conditionally independent proxies. Werefer to this as the `double proxy' approach. The triple proxy approachidentifies objects that are not identified by the double proxy approach,including some that capture the variation in average treatment effects betweenstrata of the unobservables. Moreover, the conditional independence assumptionsin the double and triple proxy approaches are non-nested.

Controlling for Latent Confounding with Triple Proxies