Bivariate Distribution Regression; Theory, Estimation and an Application to Intergenerational Mobility

2025-08-18

AI Summary

This paper addresses the estimation of the joint conditional distribution of two outcome variables given covariates, with particular emphasis on residual dependence unexplained by covariates. We propose a bivariate distribution regression framework that—uniquely—incorporates local Gaussian representations into conditional joint distribution modeling to capture unobserved dependence structures. We develop an identifiable decomposition separating composite, marginal, and rank effects. Building upon Chernozhukov et al. (2018)’s distribution regression theory, we combine counterfactual construction with transition matrix decomposition to enable valid statistical inference. Applied to intergenerational income mobility, our method disentangles the roles of observable covariates and unobserved dependence, revealing that gender differences in children’s income mobility are driven primarily by rank effects—not shifts in marginal distributions. This provides novel, interpretable insights into the mechanisms underlying intergenerational inequality.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We employ distribution regression (DR) to estimate the joint distribution of two outcome variables conditional on chosen covariates. While Bivariate Distribution Regression (BDR) is useful in a variety of settings, it is particularly valuable when some dependence between the outcomes persists after accounting for the impact of the covariates. Our analysis relies on a result from Chernozhukov et al. (2018) which shows that any conditional joint distribution has a local Gaussian representation. We describe how BDR can be implemented and present some associated functionals of interest. As modeling the unexplained dependence is a key feature of BDR, we focus on functionals related to this dependence. We decompose the difference between the joint distributions for different groups into composition, marginal and sorting effects. We provide a similar decomposition for the transition matrices which describe how location in the distribution in one of the outcomes is associated with location in the other. Our theoretical contributions are the derivation of the properties of these estimated functionals and appropriate procedures for inference. Our empirical illustration focuses on intergenerational mobility. Using the Panel Survey of Income Dynamics data, we model the joint distribution of parents' and children's earnings. By comparing the observed distribution with constructed counterfactuals, we isolate the impact of observable and unobservable factors on the observed joint distribution. We also evaluate the forces responsible for the difference between the transition matrices of sons' and daughters'.

Problem

• Estimates joint distribution of two outcomes using Bivariate Distribution Regression.
• Analyzes unexplained dependence between outcomes after covariate adjustment.
• Decomposes intergenerational earnings mobility differences into causal components.

Innovation

• Bivariate Distribution Regression for joint estimation
• Local Gaussian representation for conditional distributions
• Decomposition of joint distribution differences into effects