Debiased Semiparametric U-Statistics - Machine Learning Inference on Inequality of Opportunity

Aug 3, 2026

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Constructs locally robust/orthogonal moments within a semiparametric U-statistics framework, yielding quadratic moments whose first derivative vanishes. This design targets reduction of model selection and regularization bias that arises when machine learning is used in the first-step nuisance estimation. Using orthogonal moments, the work proposes new debiased estimators and delivers valid inference across applications, centered on Inequality of Opportunity (IOp) and its Gini fitted-values measure. Includes a novel U-moment representation of the First Step Influence Function (U-FSIF), asymptotic regularity conditions, simulation evidence of improved finite-sample performance, and an empirical study where Spain’s income inequality is decomposed into circumstances beyond individual control.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages57

Reading Time2 h 24 min

AccessFree to download

Introduction

Locally robust/orthogonal moments in semiparametric U-statistics

Debiased inference for Inequality of Opportunity (IOp)

U-FSIF and orthogonal quadratic moment construction

arXiv :2206 .05235v1 [ econ .EM] 10 Jun 2022 Debiased Semiparametric U-Statistics: Machine Learning Inference on Inequality of Opportunity ∗ Juan Carlos Escanciano Universidad Carlos III de Madrid Jo􀁿el R. Terschuur Universidad Carlos III de Madrid June 10th, 2022 Abstract We construct locally robust/orthogonal moments in a semiparametric U-statistics setting. These are quadratic moments in the distribution of the data with a zero derivative with respect to 􀀌rst steps at their limit, which reduces model selection bias with machine learning 􀀌rst steps. We use orthogonal moments to propose new debiased estimators and valid inferences in a variety of applications ranging from Inequality of Opportunity (IOp) to distributional treatment e􀀋ects. U-statistics with machine learning 􀀌rst steps arise naturally in these and many other applications. A leading example in IOp is the Gini coe􀀎cient of machine learning 􀀌tted values. We introduce a novel U-moment representation of the First Step In􀀍uence Function (U-FSIF) to take into account the e􀀋ect of the 􀀌rst step estimation on an identifying quadratic moment. Adding the U-FISF to the identifying quadratic moment gives rise to an orthogonal quadratic moment. Our leading and motivational application is to measuring IOp, for which we propose a simple debiased estimator, and the 􀀌rst available inferential methods. We give general and simple regularity conditions for asymptotic theory, and demonstrate an improved 􀀌nite sample performance in simulations for our debiased measures of IOp. In an empirical application, we 􀀌nd that standard measures of IOp are about six times more sensitive to 􀀌rst step machine learners than our debiased measures, and that between 42% and 46% of income inequality in Spain is explained by circumstances out of the control of the individual. JEL Classi􀀌cation: C13; C14; C21; D31; D63 ∗ Research founded by Ministerio de Ciencia e Innovaci􀀓on, grant ECO2017-86675-P, MCI/AEI/FEDER/UE, grant PGC 2018-096732-B-100, and Comunidad de Madrid, grants EPUC3M11 (VPRICIT) and H2019/HUM- 589. 1 Introduction Many parameters of interest in economics can be estimated with U-statistics which depend on a 􀀌rst step estimation of possibly high dimensional nuisance parameters, here simply referred to as semiparametric U-statistics. Examples include measures of inequality of opportunity, such as the Gini coe􀀎cient of 􀀌tted values from machine learners, measures of economic polarization, optimal risk in the bipartite ranking problem, distance-based estimators in semiparametric conditional moment restrictions, and quadratic functionals of the marginal distributions of potential outcomes in treatment e􀀋ects. We give a general construction of orthogonal quadratic moment functions which can be used to obtain debiased estimators and valid inferences in these settings. Quadratic moments are the population analog to semiparametric U-statistics. The orthogonality of the proposed moment functions is such that the estimation of 􀀌rst steps has no local e􀀋ect on the parameters of interest. This orthogonality is important because it reduces model selection and regularization biases typically present when machine learning 􀀌rst steps are used, as shown in Chernozhukov et al. (2022) for GMM and here for semiparametric U-statistics. Our leading application and main motivation for this work is inference on Inequality of Opportunity (IOp) . Equality of opportunity theory, exposited in the seminal contribution by Roemer (1998), distinguishes between fair inequality and IOp. IOp is inequality due to circumstances which are out of the control of the individual, such as parental wealth/income/education, biological sex, color of the skin or social origin. Fair inequality consists in inequalities not caused by circumstances, such as those derived from exerted e􀀋ort. Roemer's theoretical framework has fueled a growing empirical literature quantifying IOp, see Roemer and Trannoy (2016), Ramos an

FAQs

What problem does the paper address in machine-learning-based inequality of opportunity measurement?

Standard IOp measures become biased when the first-step nuisance components are estimated with machine learning, because the measure lacks local robustness to first-step estimation.

How does the paper reduce model selection and regularization bias?

It constructs locally robust/orthogonal quadratic moments with zero derivative with respect to the first-step estimation, making the first-step effect locally negligible for the parameters of interest.

What is the main application focus and the proposed estimator?

The main motivation is inference on Inequality of Opportunity using a debiased version related to the Gini coefficient of fitted values, incorporating a U-moment representation of the First Step Influence Function.

Aug 3, 2026

Upload by Jiven