Conditional Rank-Rank Regression via Deep Conditional Transformation Models
This work provides a more robust and interpretable method for economists and social scientists to analyze intergenerational mobility, particularly in the presence of complex covariate relationships and discrete outcomes.
This paper addresses the challenge of interpreting covariate-adjusted parameters in intergenerational mobility studies by proposing a Conditional Rank-Rank Regression (CRRR) framework. It estimates conditional ranks using a deep conditional transformation model (DCTM) and cross-fitting, demonstrating improved accuracy in simulations across various complex continuous and discrete ordered designs. The method reveals substantial within-group persistence in U.S. income and gender differences in Indian educational mobility.
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.