Conditional Rank-Rank Regression via Deep Conditional Transformation Models

Xiaoyi Wang; Long Feng; Zhaojun Wang

stat.MEMar 7, 2026

Conditional Rank-Rank Regression via Deep Conditional Transformation Models

Authors: Xiaoyi Wang, Long Feng, Zhaojun Wang

Organizations: Nankai University

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 & tables

Figure 1: Schematic architecture of the DCTM network.

$i$	1	2	3	4	5	6	7	8	9	10
$Y_{i}$	3	4	7	7	10	11	15	15	15	15
Smallest rank ( $\omega=0$ )	0.1	0.2	0.3	0.3	0.5	0.6	0.7	0.7	0.7	0.7
Mid-rank ( $\omega=0.5$ )	0.1	0.2	0.35	0.35	0.5	0.6	0.85	0.85	0.85	0.85
Largest rank ( $\omega=1$ )	0.1	0.2	0.4	0.4	0.5	0.6	1	1	1	1

Table 1: An example of three tie-breaking rank definitions (adapted from Chetverikov and Wilhelm (2023) ).

Figure 2: $\rho_{C}(\omega)$ , $\rho_{S}(\omega)$ , and $\mathrm{sd}(U_{\omega})/\mathrm{sd}(V_{\omega})$ as functions of $\omega$ .

Panel of Figure 3 — Figure 3: Simple continuous setting with $\delta=0$ : conditional CDFs estimated by DR and DCTM versus the truth.

Panel of Figure 4 — Figure 4: Simple continuous setting with $\delta=12$ : conditional CDFs estimated by DR and DCTM versus the truth.

Panel of Figure 5 — Figure 5: PIT diagnostics for DR-based conditional CDF estimation.

Panel of Figure 6 — Figure 6: PIT diagnostics for DCTM-based conditional CDF estimation.

	Method	KS $p$ -value	$\chi^{2}$ $p$ -value	PIT Mean	PIT Var
$Y\mid X=0$	DR	0.9024	0.4708	0.4986	0.0845
$Y\mid X=0$	DCTM	0.8850	0.4024	0.4988	0.0845
$Y\mid X=1$	DR	0.3220	0.9870	0.4948	0.0839
$Y\mid X=1$	DCTM	0.2174	0.8696	0.4941	0.0838
$W\mid X=0$	DR	0.9180	0.9710	0.4999	0.0841
$W\mid X=0$	DCTM	0.9327	0.9117	0.4997	0.0840

Table 2: Simple continuous setting with

\delta=0

: KS

p

-values,

\chi^{2}

p

-values, and PIT mean/variance.

	Method	KS $p$ -value	$\chi^{2}$ $p$ -value	PIT Mean	PIT Var
$Y\mid X=0$	DR	0.9024	0.4708	0.4986	0.0845
$Y\mid X=0$	DCTM	0.9217	0.3770	0.4984	0.0845
$Y\mid X=1$	DR	0.3220	0.9870	0.4948	0.0839
$Y\mid X=1$	DCTM	0.3127	0.9800	0.4946	0.0838
$W\mid X=0$	DR	0.9199	0.9540	0.5000	0.0842
$W\mid X=0$	DCTM	0.9080	0.9522	0.5000	0.0840

Table 3: Simple continuous setting with

\delta=12

: KS

p

-values,

\chi^{2}

p

-values, and PIT mean/variance.

	Method	Mean	SD	MAD	RMSE
$\delta=0$	DR	0.58205	0.00080	0.00066	0.00079
$\delta=0$	DCTM	0.58216	0.00091	0.00073	0.00092
$\delta=12$	DR	0.58190	0.00085	0.00068	0.00084
$\delta=12$	DCTM	0.58206	0.00112	0.00084	0.00111

Table 4: Simple continuous setting: estimation results for the CRRR slope

\rho_{C}

Panel of Figure 7 — Figure 7: Complex continuous setting: DR, DCTM, and true conditional CDFs for two representative covariate groups.

Method	Mean	SD	MAD	RMSE
DR	0.14761	0.00158	0.43432	0.43432
DCTM	0.57646	0.00119	0.00547	0.00559

Table 5: Complex continuous setting: estimation results for the CRRR slope

\rho_{C}

Panel of Figure 8 — Figure 8: Simple discrete ordinal setting with $\delta=0$ and $X\in\{0,1\}$ : estimated vs. true step CDFs for $F(Y_{d}\mid X)$ and $F(W_{d}\mid X)$ .

Panel of Figure 9 — Figure 9: Simple discrete ordinal setting with $\delta=12$ and $X\in\{0,1\}$ : estimated vs. true step CDFs for $F(Y_{d}\mid X)$ and $F(W_{d}\mid X)$ .

		Method	$\rho_{C}^{\mathrm{True}}$	Mean $(\widehat{\rho}_{C})$	SD $(\widehat{\rho}_{C})$	MAD	RMSE
$\omega=0.0$	$\delta=0$	DR	0.56920	0.56767	0.00255	0.00234	0.00293
	$\delta=0$	dDCTM	0.56920	0.56762	0.00252	0.00234	0.00293
	$\delta=12$	DR	0.29507	0.29415	0.00179	0.00166	0.00199
	$\delta=12$	dDCTM	0.29507	0.29413	0.00210	0.00195	0.00227
$\omega=0.5$	$\delta=0$	DR	0.56920	0.56760	0.00250	0.00241	0.00293
	$\delta=0$	dDCTM	0.56920	0.56759	0.00251	0.00242	0.00294

Table 6: Simple discrete ordinal setting: CRRR slope

\rho_{C}

estimation for

\omega\in\{0.0,0.5,1.0\}

Panel of Figure 10 — Figure 10: Complex discrete ordinal setting: estimated vs. true step CDFs for $F(Y_{d}\mid X=x)$ and $F(W_{d}\mid X=x)$ at a randomly selected covariate value $X=x$ .

Figure 11: Complex discrete ordinal setting: estimated $\rho_{C}(\omega)$ by traditional CRRR (DR) and our method (dDCTM) versus the truth.

	Method	$\rho_{C}^{\mathrm{True}}$	Mean	SD	MAD	RMSE
$\omega=0.0$	DR	0.15556	0.14742	0.00312	0.00814	0.00870
$\omega=0.0$	dDCTM	0.15556	0.15517	0.00324	0.00039	0.00321
$\omega=0.5$	DR	0.35002	0.11976	0.00336	0.23026	0.23028
$\omega=0.5$	dDCTM	0.35002	0.35066	0.00360	0.00169	0.00398
$\omega=1.0$	DR	0.18949	0.08571	0.00351	0.10378	0.10384
$\omega=1.0$	dDCTM	0.18949	0.19670	0.00294	0.00722	0.00777

Table 7: Complex discrete ordinal setting: CRRR slope

\rho_{C}

estimation for

\omega\in\{0.0,0.5,1.0\}

	Father-Child		Father-Son		Father-Daughter
	Mean	SD	Mean	SD	Mean	SD
log (Child Income)	8.90	3.24	9.58	2.59	8.30	3.63
log (Father Income)	5.36	4.09	5.42	4.09	5.30	4.10
Higher Education	0.53	0.50	0.55	0.50	0.51	0.50
At least 2 children	0.75	0.43	0.75	0.44	0.76	0.43
log (Expand)	9.94	0.07	9.94	0.06	9.94	0.08

Table 8: Descriptive statistics

Panel of Figure 12 — Figure 12: Transition matrices: Father-Child

Panel of Figure 13 — Figure 13: Transition matrices: Father-Son

Panel of Figure 14 — Figure 14: Transition matrices: Father-Daughter

	Father-Child
	$\widehat{\rho}_{C}$	SE	95% CI
RRR	0.180	0.022	[0.142, 0.218]
CRRR	0.121	0.019	[0.081, 0.160]
CRRR, by Father:
Higher Education	0.073	0.030	[0.018, 0.129]
Age $>26$ at birth	0.108	0.034	[0.039, 0.178]

Table 9: Father-Child intergenerational income mobility

	Father-Son			Father-Daughter
	$\widehat{\rho}_{C}$	SE	95% CI	$\widehat{\rho}_{C}$	SE	95% CI
RRR	0.099	0.031	[0.033, 0.165]	0.248	0.026	[0.188, 0.308]
CRRR	0.062	0.029	[0.007, 0.117]	0.180	0.028	[0.124, 0.237]
CRRR, by Father:
Higher Education	-0.027	0.037	[-0.092, 0.039]	0.179	0.045	[0.101, 0.258]
Age $>26$ at birth	0.036	0.054	[-0.078, 0.151]	0.171	0.047	[0.059, 0.283]

Table 10: Father-Son and Father-Daughter intergenerational income mobility

	Father-Child		Father-Son		Father-Daughter
	Mean	SD	Mean	SD	Mean	SD
Child Edu Level	3.69	1.88	4.30	1.66	3.32	1.92
Father Edu Level	2.72	1.58	3.19	1.93	2.43	1.25
Group	2.28	0.89	2.30	0.88	2.27	0.89
Nperson	5.88	2.50	6.29	2.63	5.64	2.38
log (Mpce)	9.89	0.65	9.90	0.65	9.89	0.65

Table 11: Descriptive statistics

Panel of Figure 15 — Figure 15: Transition matrices: Father-Child

Panel of Figure 16 — Figure 16: Transition matrices: Father-Son

Panel of Figure 17 — Figure 17: Transition matrices: Father-Daughter

Figure 18: Estimated CRRR slope $\widehat{\rho}_{C}(\omega)$

		Father-Child	Father-Son	Father–Daughter
$\omega=0.0$	RRR	0.353	0.365	0.312
	CRRR	0.246	0.223	0.251
	CRRR by:
	Muslim	0.270	0.286	0.279
	Nperson $\geq 4$	0.241	0.221	0.252
	Urban	0.213	0.246	0.178

Table 12: Educational mobility analysis for

\omega\in\{0.0,0.5,1.0\}

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