Conditional Rank-Rank Regression via Deep Conditional Transformation Models

Xiaoyi Wang; Long Feng; Zhaojun Wang

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arXiv · 2603.07230

Conditional Rank-Rank Regression via Deep Conditional Transformation Models

Xiaoyi Wang·Long Feng·Zhaojun Wang

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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.

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Xiaoyi Wang, Long Feng, Zhaojun Wang. 2026-03-07. Conditional Rank-Rank Regression via Deep Conditional Transformation Models. https://arxiv.org/abs/2603.07230

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