Conditional Rank-Rank Regression via Deep Conditional Transformation Models

TL;DR

This paper introduces a deep conditional transformation model-based approach for conditional rank-rank regression, improving the measurement of within-group mobility with better handling of nonlinearity, interactions, and discrete outcomes.

Contribution

It extends conditional rank-rank regression by employing deep conditional transformation models and cross-fitting, enabling flexible, end-to-end distribution learning for complex socio-economic data.

Findings

Enhanced accuracy in simulations with complex data structures

Revealed significant within-group income persistence in the US

Identified gender disparities in educational mobility in India

Abstract

Intergenerational mobility quantifies the transmission of socio-economic outcomes from parents to children. While rank-rank regression (RRR) is standard, adding covariates directly (RRRX) often yields parameters with unclear interpretation. Conditional rank-rank regression (CRRR) resolves this by using covariate-adjusted (conditional) ranks to measure within-group mobility. We improve and extend CRRR by estimating conditional ranks with a deep conditional transformation model (DCTM) and cross-fitting, enabling end-to-end conditional distribution learning with structural constraints and strong performance under nonlinearity, high-order interactions, and discrete ordered outcomes where the distributional regression used in traditional CRRR may be cumbersome or prone to misconfiguration. We further extend CRRR to discrete outcomes via an $\omega$-indexed conditional-rank definition and study sensitivity to $\omega$. For continuous outcomes, we establish an asymptotic theory for the proposed estimators and verify the validity of exchangeable bootstrap inference. Simulations across simple/complex continuous and discrete ordered designs show clear accuracy gains in challenging settings. Finally, we apply our method to two empirical studies, revealing substantial within-group persistence in U.S. income and pronounced gender differences in educational mobility in India.