Beware of Hinges in Proximal Variables Regression: Adjusting for Colliding-Mediators with Nuisance PV

Hubert Marek Drazkowski

Keywords: Proximal causal inference, Proximal variables regression, Unobserved confounding, Causal identification, Graphical models

TL;DR: Proximal variables regression estimates effects in the presence of unobserved confounding by using conditionally separated proxies. We study a structural violation caused by variables that act simultaneously as mediators and colliders between them.

Abstract:

Conditional proximal variables regression (PV) enables estimation of causal effects in the presence of unobserved confounding by using treatment-side and outcome-side proxies that are conditionally separated given observed covariates. We study a structural violation of this separation causedby hinges—observed variables that act simultaneously as mediators and colliders between the two proxy blocks—so that no single conditioning choice blocks all bias-inducing paths. Ignoring a hinge can introduce mediator leakage into proximal moment equations, while conditioning on it can activate collider paths and induce cross-side noise endogeneity. We formalize hinges in a structural
causal model for continuous, possibly multivariate treatments and proxies, and develop complementary identification results via a outcome bridge and a novel regression based treatment bridge. We express identification and uniqueness through transparent moment equations and rank requirements on conditional covariance operators, enabling direct comparison with instrumental variables regression. Finally, we propose Nuisance Proximal Variables (NPV): a hinge-robust correction that augments the moment system with a nuisance block and identifies the target effect after projecting out the resulting nuisance span. Synthetic experiments illustrate the two hinge failure modes and
show that NPV recovers the causal effect when nuisance-span conditions hold.

Submission Number: 79