Conditional Rank-Rank Regression via Deep Conditional Transformation Models

TL;DR

This work improves 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.

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 $ω$-indexed conditional-rank definition and study sensitivity to $ω$. 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.

Figures (18)

• Figure 1: Schematic architecture of the DCTM network.
• Figure 2: $\rho_C(\omega)$, $\rho_S(\omega)$, and $\mathrm{sd}(U_\omega)/\mathrm{sd}(V_\omega)$ as functions of $\omega$.
• Figure 3: Simple continuous setting with $\delta=0$: conditional CDFs estimated by DR and DCTM versus the truth.
• Figure 4: Simple continuous setting with $\delta=12$: conditional CDFs estimated by DR and DCTM versus the truth.
• Figure 5: PIT diagnostics for DR-based conditional CDF estimation.

Theorems & Definitions (12)

• Remark 1
• Lemma 1: Consistency of fold-specific conditional CDF estimators
• Theorem 1: Consistency of $\rho_C$ estimators
• Lemma 2: Joint asymptotic linearity of $(A_n,B_{n,U},B_{n,V})$
• Theorem 2: Asymptotic normality of $\widehat{\rho}_C$, $\widetilde{\rho}_C$, and $\breve\rho_C$
• Lemma 3: Exchangeable bootstrap CLT; van1996weak
• Theorem 3: Bootstrap consistency
• proof : Proof of Lemma \ref{['lem:dctm_consistency_fun']}
• proof : Proof of Theorem \ref{['thm:consistency_rho_fun']}
• proof : Proof of Lemma \ref{['lem:ABlin_fun']}