Debiased inference for proximal dose-response function

2026-07-31

AI Summary

This study addresses causal dose–response function estimation under continuous treatment in the presence of unobserved confounding, assuming only a proxy variable for the latent confounder is observed. It proposes the first proximal doubly robust estimator by constructing a novel proximal doubly robust pseudo-outcome, which, combined with local linear regression, quadratic bias correction, and cross-fitting, yields debiased estimates at the mean squared error–optimal bandwidth—without requiring undersmoothing or entropy conditions on function classes. Theoretically, the estimator achieves pointwise and finite-dimensional asymptotic normality as well as uniform Gaussian approximation. Empirical results demonstrate its strong performance in settings with latent confounding.

Abstract

In this paper, we study nonparametric inference for the causal dose-response curve of a continuous-treatment under unmeasured confounding by leveraging treatment- and outcome-inducing confounding proxies. To estimate the curve, we introduce a novel proximal doubly robust pseudo-outcome whose conditional mean given treatment equals the dose-response curve whenever either bridge function is correctly specified, thereby addressing a key gap in proximal causal inference for continuous-treatments. Furthermore, we derive an influence function for its smoothed causal estimand, and construct a cross-fitted debiased local-linear estimator with a proper local-quadratic bias correction. We establish pointwise and finite-dimensional asymptotic normality and a uniform Gaussian approximation over compact treatment intervals. Both smoothing bandwidths may have the mean-squared-error-optimal order without undersmoothing, while cross-fitting accommodates flexible bridge estimators under a product convergence rate conditions without fitted-class entropy restrictions. We also develop practical bandwidth selectors, pointwise confidence intervals, and simultaneous confidence bands. Extensive simulations and a data analysis highlight the practical performance of the proposed method under latent confounding and multiple proxies.

Problem

• dose-response function
• unmeasured confounding
• continuous treatment
• proximal inference
• causal inference

Innovation

• proximal causal inference
• dose-response function
• doubly robust
• debiased estimation
• continuous treatment