Debiased inference for proximal dose-response function

Abstract

We study nonparametric inference for a continuous-treatment causal dose-response curve and its first derivative under unmeasured confounding, using treatment- and outcome-inducing proxies. We establish a proximal doubly robust pseudo-outcome whose conditional mean given treatment equals the dose-response curve whenever either bridge function is correctly specified. Their (smoothed) efficient influence function representations lead to cross-fitted local-polynomial estimators with explicit smoothing-bias correction. We establish pointwise and finite-dimensional asymptotic normality and oracle uniform Gaussian approximations over compact interior treatment intervals, allowing mean-squared-error-optimal bandwidths without undersmoothing. The derivative inference reuses the estimated bridges and the same form of bandwidth-dependent nuisance-rate conditions, evaluated at the bandwidth rate appropriate for derivative estimation, without requiring derivative-specific bridge estimation. Employing cross-fitting accommodates flexible bridge estimators under product convergence rate conditions without entropy restrictions. We further develop tests against fixed-dimensional parametric dose-response null hypotheses, including constancy, linearity, and quadratic specifications, and derive their limiting distributions under the null and local alternatives, together with practical bootstrap procedures. We also develop practical bandwidth selectors, pointwise confidence intervals, and simultaneous confidence bands for each causal estimand. Extensive simulations and a data analysis, presented in the appendix, demonstrate the finite-sample performance of the proposed methods under latent confounding and multiple proxies.