Debiased inference for proximal dose-response function

Technical Summary: Debiased Inference for Proximal Dose-Response Function

Problem Statement
This paper addresses the challenge of nonparametric inference for the causal dose-response curve, $\theta(a) = E\{Y(a)\}$, under continuous treatments in the presence of unmeasured confounding. While extensive literature exists for continuous treatments under the assumption of no unmeasured confounding (e.g., Hirano and Imbens, 2004; Calonico et al., 2018), such assumptions are often untenable in observational studies due to incomplete recording of socioeconomic, behavioral, or biological factors. Latent confounding invalidates standard estimation and uncertainty quantification.

The paper leverages the proximal causal inference framework, which utilizes treatment-inducing ($Z$) and outcome-inducing ($W$) confounding proxies to identify causal effects when observed covariates ($X$) are insufficient. While Wu et al. (2024) recently proposed a kernel-localized augmented estimator for this setting, their work established pointwise bias-variance rates but did not provide a feasible asymptotic distribution, standard errors, confidence intervals, or simultaneous confidence bands. Furthermore, at the mean-squared-error (MSE) optimal bandwidth, the leading bias is typically of the same order as the standard error, complicating inference.

Methodology
The authors propose a novel framework for debiased inference that separates proximal adjustment from curve learning. The methodology proceeds in three main stages:

1. Proximal Doubly Robust Pseudo-Outcome:
   The authors introduce a population pseudo-outcome, $\xi_{\text{prox}}(O; \tilde{q}, \tilde{h})$, constructed by adding a treatment-bridge-weighted outcome-bridge residual to the marginal outcome-bridge mean.
   $$\xi_{\text{prox}}(O; \tilde{q}, \tilde{h}) = \tilde{q}(Z, A, X)\{Y - \tilde{h}(W, A, X)\} + S_{\tilde{h}}(A)$$
   where $S_{\tilde{h}}(a) = E\{\tilde{h}(W, a, X)\}$. Theorem 1 establishes that the conditional mean of this pseudo-outcome given treatment $A=a$ equals the true dose-response curve $\theta(a)$ if either the outcome bridge $\tilde{h}$ or the treatment bridge $\tilde{q}$ is correctly specified. This "proximal doubly robust" property allows for flexible estimation of the bridges.

2. Debiased Estimation via Local-Polynomial Fitting:
   To estimate $\theta(a)$, the authors employ a local-linear fit with bandwidth $h$ for the curve and a local-quadratic fit with bandwidth $b$ to estimate the second derivative required for bias correction.

   • Influence Function: They derive the efficient influence function for the smoothed target parameter, accounting for the stochastic contribution of the bias correction.
   • Cross-Fitting: To accommodate flexible, high-complexity estimators for the bridge functions without imposing entropy or Donsker conditions, the authors utilize a cross-fitted estimator. The data is partitioned into folds; bridge functions are estimated on training folds and applied to validation folds. This avoids the need for uniform convergence rates or complexity restrictions on the nuisance classes.
   • Bias Correction: Unlike previous approaches requiring undersmoothing (where $h \to 0$ faster than the MSE-optimal rate), this method uses a local-quadratic correction that accounts for the estimation error of the second derivative in the limiting variance. This permits $h \asymp b \asymp n^{-1/5}$, the MSE-optimal order, without undersmoothing.

3. Inference and Bandwidth Selection:
   The paper develops practical tools for inference, including:

   • Pointwise and finite-dimensional asymptotic normality.
   • Uniform Gaussian approximation over compact treatment intervals.
   • Data-adaptive bandwidth selectors based on plug-in rules (minimizing estimated integrated MSE or using a direct scale rule) and leave-one-out cross-validation.

Key Contributions
The paper makes four primary contributions:

1. Theoretical Identity: It establishes the proximal doubly robust regression identity for continuous treatments, formulating a pseudo-outcome whose conditional mean equals the dose-response curve under single-bridge correctness.
2. Debiased Estimator Construction: It constructs a cross-fitted, local-linear estimator with a local-quadratic bias correction. Crucially, it derives an influence function representation that incorporates the stochastic contribution of the bias correction, allowing for valid inference at the MSE-optimal bandwidth without undersmoothing.
3. Relaxed Regularity Conditions: The theoretical framework requires smoothness only for the target curve $\theta$, the marginal treatment density, and the conditional pseudo-outcome variance. It avoids imposing sup-norm convergence or Lipschitz conditions on the fitted bridge estimators, which were required in prior work (e.g., Colangelo and Lee, 2026; Takatsu and Westling, 2025) for uniform inference. The use of cross-fitting further removes entropy restrictions on the bridge estimation classes.
4. Practical Implementation: It develops and compares practical bandwidth selectors and provides methods for constructing pointwise confidence intervals and simultaneous confidence bands.

Results

• Asymptotic Theory: The authors establish pointwise and finite-dimensional asymptotic normality for the estimator. They prove a uniform Gaussian approximation over compact treatment intervals, showing that the studentized estimator process converges to a tight Gaussian process.
• Simulation Studies: Extensive simulations under various nuisance specification regimes (both bridges correct, one bridge correct, neither correct) demonstrate that the proposed estimators attain nominal pointwise and simultaneous coverage. The estimators exhibit robustness, maintaining coverage even when only one bridge function is correctly specified. The bias-corrected estimators (PDR) consistently yield narrower confidence intervals and lower integrated mean squared error compared to an undersmoothed local-linear competitor (ProxLL-US).
• Real Data Application: The method is applied to a study of legalized abortion and crime rates (Donohue and Levitt, 2001). Using state-year panel data with multiple proxies, the proposed estimators successfully recover the ground-truth dose-response curve (derived from a generative model) with high accuracy (MSE between 0.0025 and 0.0052), outperforming benchmarks reported in Wu et al. (2024). The results support a negative dose-response relationship between abortion rates and subsequent murder rates.

Significance and Claims
The paper claims to fill a critical gap in proximal causal inference for continuous treatments by providing the first feasible inferential framework (standard errors, confidence intervals, and bands) that does not rely on undersmoothing. By leveraging cross-fitting and a novel bias-correction strategy, the method achieves MSE-optimal convergence rates while maintaining valid uncertainty quantification under unmeasured confounding. The authors emphasize that their approach allows for the use of flexible, high-dimensional nuisance estimators without restrictive entropy conditions, making the method applicable to modern, complex data structures. The simulation and real-data results suggest the method is robust to model misspecification of the bridge functions and performs competitively in practical settings.