Kernel Single Proxy Control for Deterministic Confounding
Liyuan Xu, Arthur Gretton
TL;DR
This work tackles causal effect estimation under unobserved confounding when only a single proxy is available, showing that deterministic confounding enables recovery of the dose-response f_struct(a). It develops two kernel-based estimators, SKPV (two-stage regression) and SPMMR (maximum moment restriction), and proves their consistency, extending proxy causal learning to continuous treatments. By identifying a bridge function h_0 whose partial average over the proxy yields f_struct, the paper connects single-proxy methods to the broader PCL framework and demonstrates practical stability and numerical advantages. Empirical results on synthetic benchmarks, including high-dimensional proxies and sensitivity analyses, indicate robust recovery of causal effects even when deterministic assumptions are relaxed to bounded noise or informative-outcome conditions, highlighting the approach’s practical relevance for causal inference with limited proxies.
Abstract
We consider the problem of causal effect estimation with an unobserved confounder, where we observe a single proxy variable that is associated with the confounder. Although it has been shown that the recovery of an average causal effect is impossible in general from a single proxy variable, we show that causal recovery is possible if the outcome is generated deterministically. This generalizes existing work on causal methods with a single proxy variable to the continuous treatment setting. We propose two kernel-based methods for this setting: the first based on the two-stage regression approach, and the second based on a maximum moment restriction approach. We prove that both approaches can consistently estimate the causal effect, and we empirically demonstrate that we can successfully recover the causal effect on challenging synthetic benchmarks.