Controlling for Latent Confounding with Triple Proxies

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.