[Paper Review] Distribution Regression with Sample Selection, with an Application to Wage Decompositions in the UK

Victor Chernozhukov, Iván Fernández‐Val+1 more|arXiv (Cornell University)|Nov 28, 2018

Spatial and Panel Data Analysis4 citations

TL;DR

This paper develops a semi-parametric distribution regression model with endogenous sample selection, generalizing the classical Heckman model to allow for non-Gaussian errors and heterogeneous treatment effects. It enables uniform inference and wage decomposition in the UK, revealing a persistent 21–40% unexplained gender wage gap after controlling for selection, with significant positive sorting among single men and negative sorting among married women contributing substantially to top-of-distribution inequality.

ABSTRACT

We develop a distribution regression model under endogenous sample selection. This model is a semi-parametric generalization of the Heckman selection model. It accommodates much richer effects of the covariates on outcome distribution and patterns of heterogeneity in the selection process, and allows for drastic departures from the Gaussian error structure, while maintaining the same level tractability as the classical model. The model applies to continuous, discrete and mixed outcomes. We provide identification, estimation, and inference methods, and apply them to obtain wage decomposition for the UK. Here we decompose the difference between the male and female wage distributions into composition, wage structure, selection structure, and selection sorting effects. After controlling for endogenous employment selection, we still find substantial gender wage gap -- ranging from 21% to 40% throughout the (latent) offered wage distribution that is not explained by composition. We also uncover positive sorting for single men and negative sorting for married women that accounts for a substantive fraction of the gender wage gap at the top of the distribution.

Motivation & Objective

• To address sample selection bias in wage analysis when outcomes and selection are driven by unobserved heterogeneity.
• To generalize the classical Heckman selection model by relaxing parametric and homogeneity assumptions on error distributions and covariate effects.
• To enable uniform inference and decomposition of wage distributions, accounting for selection and sorting effects.
• To apply the model to UK labor market data to decompose the male-female wage gap into composition, structure, selection, and sorting components.
• To quantify the role of endogenous employment selection and assortative matching in shaping observed wage inequality.

Proposed method

• Proposes a semi-parametric distribution regression model with sample selection, where outcome and selection equations are modeled nonparametrically.
• Uses instrumental variables to identify selection effects and allows for non-Gaussian, heteroskedastic, and dependent error structures.
• Employs a latent variable framework with observed wages only for employed individuals, modeling offered wages as latent outcomes.
• Applies uniform inference procedures to construct confidence bands for nonparametric coefficients and decomposition components.
• Uses a four-component decomposition: composition, wage structure, selection structure, and sorting effects.
• Calibrates Monte Carlo simulations to empirical data to validate inference procedures and robustness.

Experimental results

Research questions

• RQ1 To what extent does endogenous employment selection contribute to the observed gender wage gap in the UK?
• RQ2 How do heterogeneous effects of covariates on the wage distribution differ from classical parametric models?
• RQ3 What is the contribution of positive or negative sorting (assortative matching) to wage inequality at different quantiles?
• RQ4 How do the components of the wage gap—composition, structure, selection, and sorting—vary across different specifications and time periods?
• RQ5 To what extent do non-Gaussian error structures and non-linear effects invalidate classical Heckman model assumptions?

Key findings

• After controlling for endogenous sample selection, a substantial unexplained gender wage gap remains, ranging from 21% to 40% across the latent offered wage distribution.
• Positive sorting—where higher-ability men are more likely to work—accounts for a significant fraction of the wage gap at the top of the distribution.
• Negative sorting among married women, where higher-ability women are less likely to work, contributes substantially to the gender wage gap at higher quantiles.
• The decomposition reveals that selection structure and sorting effects are quantitatively important, especially in the upper tail of the wage distribution.
• Confidence bands from uniform inference confirm the robustness of the decomposition results, particularly for the sorting and selection components.
• The model’s flexibility captures non-Gaussian patterns such as wage bunching at minimum wage levels, which the classical Heckman model fails to address.