Higher-Order Debiased Estimators for General Treatment Models

2026-06-01

AI Summary

This work addresses the suboptimal convergence rates of classical influence functions when estimating complex, implicitly defined causal parameters such as quantile treatment effects. While existing higher-order methods are limited to explicitly defined parameters, this paper extends the higher-order influence function framework to implicit M- and Z-estimation problems for the first time. By integrating U-process theory with nonparametric estimation, the authors construct a debiased estimator that substantially relaxes the stringent Hölder smoothness assumptions typically imposed on nuisance parameters. The proposed approach achieves improved convergence rates in settings like quantile treatment effect estimation and reduces requirements on model complexity, thereby broadening the applicability of higher-order influence function methodology to a wider class of semiparametric problems.

Abstract

It is now well known that estimators based on influence functions can be sub-optimal in terms of convergence rates in various settings. To address this issue, higher-order influence functions (HOIF) are developed, generalizing the classical semiparametric theory. However, most existing results in this regard focus on treatment effect parameters defined in explicit forms, such as average treatment effects (ATE). In applications, economists are often confronted with tasks of inferring more complex parameters, such as quantile treatment effects (QTE) or effects of complicated treatment regimes/policy. These more complex parameters can often only be implicitly defined as the solution to nonlinear estimating equations, which correspond to M/Z-estimation problems. Our current understanding of these problems is mainly limited to the classical semiparametric theory. Given the foundational role of HOIF for estimating explicit parameters such as ATE, a modest step toward enriching the statistical foundation of econometrics and causal inference is to develop the corresponding higher-order estimators for those more complex parameters. To this end, we consider parameters of a class of non-separable structural models in the econometrics literature and develop a class of higher-order estimators for the target parameters. Statistical properties of these higher-order estimators are derived using recent advances in U-processes theory. Our proposed higher-order estimators relax complexity-reducing assumptions, quantified by Holder smoothness, imposed on the nuisance parameters compared to existing alternative estimators for many important parameters in this class, including QTE and quantile dose-response functions, among others.

Problem

• higher-order influence functions
• implicit parameters
• M-estimation
• quantile treatment effects
• non-separable structural models

Innovation

• higher-order influence functions
• implicit parameters
• M-estimation
• nuisance parameter relaxation
• U-processes