Efficiently and flexibly estimating treatment effect heterogeneity is an important task in a wide To identify causal effects, the aforementioned approaches variety of settings ranging from medicine to marketing, operate under the exchangeability assumption, i.e., the assertion and there are a considerable number of that conditional on observed covariates, the treatment promising conditional average treatment effect assignment is as good as random. We propose a CATE estimators currently available. These, however, estimator, which using the framework of Tchetgen Tchetgen typically rely on the assumption that the measured et al. (2020), allows one to estimate causal effects in covariates are enough to justify conditional settings where conditional exchangeability fails, but one has exchangeability. We propose the P-learner, motivated measured a set of sufficient proxy variables. Our practical by the Rand DR-learner, a tailored twostage approach is motivated by the generic Neyman-orthogonal loss function for learning heterogeneous (Chernozhukov et al., 2018a) loss function from Nie & Wager treatment effects in settings where exchangeability (2021) and Kennedy (2020) that decouples nuisance given observed covariates is an implausible assumption, estimation and CATE estimation into two stages that can be and we wish to rely on proxy variables estimated (and tuned with cross-validation) by flexible lossminimizing for causal inference.

We address the problem of causal effect estimation in the presence of unobserved confounding, but where proxies for the latent confounder(s) are observed. We propose two kernel-based methods for nonlinear causal effect estimation in this setting: (a) a two-stage regression approach, and (b) a maximum moment restriction approach. We focus on the proximal causal learning setting, but our methods can be used to solve a wider class of inverse problems characterised by a Fredholm integral equation. In particular, we provide a unifying view of two-stage and moment restriction approaches for solving this problem in a nonlinear setting. We provide consistency guarantees for each algorithm, and we demonstrate these approaches achieve competitive results on synthetic data and data simulating a real-world task. In particular, our approach outperforms earlier methods that are not suited to leveraging proxy variables.