Partial Identification in Matching Models for the Marriage Market

TL;DR

This paper develops a method for partially identifying preference parameters in matching models without relying on parametric assumptions, revealing that previous findings were often driven by such restrictions.

Contribution

It introduces a nonparametric approach to characterize the identified set in matching models, challenging prior reliance on parametric assumptions like the Logit model.

Findings

Many empirical results are sensitive to parametric assumptions.

The methodology provides bounds on preference parameters without distributional restrictions.

Re-examination of marriage market data shows previous conclusions may be parametric artifacts.

Abstract

We study partial identification of the preference parameters in the one-to-one matching model with perfectly transferable utilities. We do so without imposing parametric distributional assumptions on the unobserved heterogeneity and with data on one large market. We provide a tractable characterisation of the identified set under various classes of nonparametric distributional assumptions on the unobserved heterogeneity. Using our methodology, we re-examine some of the relevant questions in the empirical literature on the marriage market, which have been previously studied under the Logit assumption. Our results reveal that many findings in the aforementioned literature are primarily driven by such parametric restrictions.

Figures

Tables

Table 1. Table 1 : The first section of the table reports the projections of the estimated identified sets of Φ Φ \Phi , averaged early cohorts, under specifications [A] and [B]. The second section of the table reports the projections of the estimated identified sets of Φ Φ \Phi , averaged over late cohorts, under specifications [A] and [B]. The last section of the table reports the change in estimates between early and late cohorts. Some intervals are singleton because of the scale normalisations imposed (see Appendix C ). The average estimates of Φ Φ \Phi under the Logit assumption are also included.

Assumptions
on unobservables	Wife $\to$	$1$	$2$	$3$	$4$
	Husband $\downarrow$	Early cohorts
Logit	$1$	$-2.06$	$-3.07$	$-5.22$	$-8.55$
$\left[\text{A}\right]$		$-2.06$	$[-3.07,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[\text{B}\right]$		$-2.06$	$[-3.07,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
Logit	$2$	$-3.73$	$-1.35$	$-3.4$	$-5.76$
$\left[\text{A}\right]$		$[-5.3,-3.48]$	$[-3.79,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[\text{B}\right]$		$[-5.3,-3.48]$	$[-3.63,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
Logit	$3$	$-5.29$	$-2.47$	$-2.12$	$-4.32$
$\left[\text{A}\right]$		$(-\infty ,-4.69]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[\text{B}\right]$		$(-\infty ,-4.69]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
Logit	$4$	-8.01	-4	-2.46	-1.11
$\left[\text{A}\right]$		$(-\infty ,-6.59]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-5.74,+\infty )$
$\left[\text{B}\right]$		$(-\infty ,-6.59]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-4.4,+\infty )$
	Husband $\downarrow$	Late cohorts
Logit	$1$	$-3.34$	$-4.41$	$-5.61$	$-8.75$
$\left[\text{A}\right]$		$-3.34$	$[-4.56,-2.29]$	$(-\infty ,-4.6]$	$(-\infty ,-6.79]$
$\left[\text{B}\right]$		$-3.34$	$[-4.56,-3.65]$	$(-\infty ,-5.05]$	$(-\infty ,-7.26]$
Logit	$2$	$-5.1$	$-2.14$	$-3.18$	$-4.95$
$\left[\text{A}\right]$		$(-\infty ,-5.1]$	$[-5.02,+\infty )$	$[-11.58,1.38]$	$[-13.46,2.67]$
$\left[\text{B}\right]$		$(-\infty ,-5.1]$	$[-5.02,+\infty )$	$[-11.58,1.38]$	$[-13.46,2.54]$
Logit	$3$	$-6.98$	$-3.61$	$-2.3$	$-3.52$
$\left[\text{A}\right]$		$(-\infty ,-6.27]$	$(-\infty ,+\infty )$	$[-5.48,+\infty )$	$(-\infty ,+\infty )$
$\left[\text{B}\right]$		$(-\infty ,-6.27]$	$(-\infty ,+\infty )$	$[-5.48,+\infty )$	$(-\infty ,+\infty )$
Logit	$4$	$-9.13$	$-5.52$	$-3.49$	$-1.07$
$\left[\text{A}\right]$		$(-\infty ,-7.92]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-6.16,+\infty )$
$\left[\text{B}\right]$		$(-\infty ,-7.92]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-2.16,+\infty )$
	Husband $\downarrow$	Change
Logit	$1$	$-1.28$	$-1.34$	$-0.39$	$-0.2$
$\left[\text{A}\right]$		$-1.28$	$(-\infty ,-0.66]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[\text{B}\right]$		$-1.28$	$(-\infty ,-0.7]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
Logit	$2$	$-1.37$	$-0.78$	$0.22$	$0.81$
$\left[\text{A}\right]$		$(-\infty ,0.1]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[\text{B}\right]$		$(-\infty ,0.1]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
Logit	$3$	$-1.69$	$-1.14$	$-0.18$	$0.8$
$\left[\text{A}\right]$		$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[\text{B}\right]$		$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
Logit	$4$	$-1.12$	$-1.52$	$-1.03$	$0.04$
$\left[\text{A}\right]$		$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[\text{B}\right]$		$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$

Table 2. Table C.1 : Assumptions on the unobserved heterogeneity maintained in the different specifications.

Assumptions	[1]	[2]	[3]	[4]	[5]	[6]
5.1
5.2
5.3
5.4

Table 3. Table C.2 : Projections of the identified sets of U 𝑈 U , V 𝑉 V , Φ Φ \Phi , D 22 , 11 ​ ( Φ ) subscript 𝐷 22 11 Φ D_{22,11}(\Phi) , C 21 ​ ( U ) subscript 𝐶 21 𝑈 C_{21}(U) , and C 21 ​ ( V ) subscript 𝐶 21 𝑉 C_{21}(V) in the first simulation exercise when r = 2 𝑟 2 r=2 .

	Specifications	$U_{11}$	$U_{12}$	$U_{21}$	$U_{22}$	$V_{11}$	$V_{12}$	$V_{21}$	$V_{22}$	${\Phi }_{11}$	${\Phi }_{12}$	${\Phi }_{21}$	${\Phi }_{22}$	$D_{22,11}\left(\Phi \right)$	$C_{21}\left(U\right)$	$C_{21}\left(V\right)$
	from Table C.1
w/ 5.1 DGP1	True & Logit	$1.16$	$0.28$	$0.07$	$1.4$	$1.23$	$-0.55$	$0.95$	$1.36$	$2.39$	$-0.27$	$1.02$	$2.76$	$4.4$	$0.2$	$-0.03$
	$\left[1\right]$	$1.16$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$1.23$	$(-\infty ,1]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$2.39$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
	$\left[2\right]$	$1.16$	$(-\infty ,1.1]$	$(-\infty ,+\infty )$	$[0.1,+\infty )$	$1.23$	$(-\infty ,1.2]$	$(-\infty ,+\infty )$	$[0.1,+\infty )$	$2.38$	$(-\infty ,2.3]$	$(-\infty ,+\infty )$	$[0.2,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
w/o 5.1 DGP1	True & Logit	$1.16$	$0.28$	$0.07$	$1.4$	$1.23$	$-0.55$	$0.95$	$1.36$	$2.39$	$-0.27$	$1.02$	$2.76$	$4.4$	$0.2$	$-0.03$
	$\left[3\right]$	$1.16$	$(-\infty ,1.15]$	$0.07$	$[0.07,+\infty )$	$1.23$	$-0.55$	$(-\infty ,+\infty )$	$[0.01,+\infty )$	$2.39$	$(-\infty ,0.6]$	$(-\infty ,+\infty )$	$[0.08,+\infty )$	$(-\infty ,+\infty )$	$[-0.89,+\infty )$	$(-\infty ,+\infty )$
	$\left[4\right]$	$1.16$	$(-\infty ,1.15]$	$0.07$	$[0.07,+\infty )$	$1.23$	$-0.55$	$(-\infty ,+\infty )$	$[0.01,+\infty )$	$2.39$	$(-\infty ,0.6]$	$(-\infty ,+\infty )$	$[0.08,+\infty )$	$(-\infty ,+\infty )$	$[-0.89,+\infty )$	$(-\infty ,+\infty )$
	$\left[5\right]$	$1.16$	$[-1.15,1.15]$	$0.07$	$[0.07,+\infty )$	$1.23$	$-0.55$	$[-1.22,1.22]$	$[-0.27,+\infty )$	$2.39$	$[-1.7,0.6]$	$[-1.15,1.29]$	$[-0.2,+\infty )$	$[0.3,+\infty )$	$[-0.89,+\infty )$	$[-1.3,+\infty )$
	$\left[6\right]$	$1.16$	$[-1.15,1.15]$	$0.07$	$[0.07,+\infty )$	$1.23$	$-0.55$	$[-1.22,1.22]$	$[0.01,+\infty )$	$2.39$	$[-1.7,0.6]$	$[-1.15,1.29]$	$[0.08,+\infty )$	$[0.58,+\infty )$	$[-0.89,+\infty )$	$[-1.1,\infty )$
w/ 5.1 DGP2	True	$1.88$	$1.6$	$1.7$	$1.73$	$2$	$0.64$	$2.01$	$1.49$	$3.88$	$2.24$	$3.71$	$3.22$	$1.14$	$-0.07$	$-0.65$
	Logit	$1.88$	$0.56$	$0.99$	$1.93$	$2$	$0.29$	$1.29$	$1.83$	$3.88$	$0.85$	$2.28$	$3.76$	$4.52$	$0.08$	$-0.24$
	$\left[1\right]$	$1.88$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$2$	$(-\infty ,2]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$3.88$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
	$\left[2\right]$	$1.88$	$(-\infty ,1.8]$	$(-\infty ,+\infty )$	$[0.1,+\infty )$	$2$	$(-\infty ,2]$	$(-\infty ,2.01]$	$[0.1,+\infty )$	$3.88$	$(-\infty ,3.8]$	$(-\infty ,+\infty )$	$[0.2,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty ]$	$(-\infty ,+\infty ]$
w/o 5.1 DGP2	True	$1.88$	$1.6$	$0.99$	$1$	$2$	$0.29$	$2$	$0.68$	$3.88$	$1.89$	$2.99$	$1.68$	$0.67$	$-0.72$	$-1.3$
	Logit	$1.88$	$0.56$	$0.99$	$1.93$	$2$	$0.29$	$1.29$	$1.83$	$3.88$	$0.85$	$2.28$	$3.76$	$4.52$	$0.08$	$-0.24$
	$\left[3\right]$	$1.88$	$(-\infty ,1.87]$	$0.99$	$[0.99,+\infty )$	$2$	$0.29$	$(-\infty ,2]$	$[0.3,+\infty )$	$3.88$	$(-\infty ,2.16]$	$(-\infty ,2.99]$	$[1.29,+\infty )$	$[0.02,+\infty )$	$[-0.78,+\infty )$	$[-1.57,+\infty )$
	$\left[4\right]$	$1.88$	$(-\infty ,1.87]$	$0.99$	$[0.99,+\infty )$	$2$	$0.29$	$(-\infty ,2]$	$[0.3,+\infty )$	$3.88$	$(-\infty ,2.16]$	$(-\infty ,2.99]$	$[1.29,+\infty )$	$[0.02,+\infty )$	$[-0.78,+\infty )$	$[-1.57,+\infty )$
	$\left[5\right]$	$1.88$	$[-1.87,1.87]$	$0.99$	$[0.99,+\infty )$	$2$	$0.29$	$[-2,2]$	$[0.3,+\infty )$	$3.88$	$[-1.58,2.16]$	$[-1.01,2.99]$	$[1.29,+\infty )$	$[0.02,+\infty )$	$[-0.78,+\infty )$	$[-1.57,+\infty )$
	$\left[6\right]$	$1.88$	$[-1.87,1.87]$	$0.99$	$[0.99,+\infty )$	$2$	$0.29$	$[-2,2]$	$[0.3,+\infty )$	$3.88$	$[-1.58,2.16]$	$[-1.01,2.99]$	$[1.29,+\infty )$	$[0.02,+\infty )$	$[-0.78,+\infty )$	$[-1.57,+\infty )$
w/o 5.1 DGP3	True	$-1.13$	$0$	$-2.41$	$-0.09$	$-1.06$	$-1.61$	$-1.15$	$0$	$-2.19$	$-1.61$	$-3.56$	$-0.09$	$2.89$	$0.01$	$0.22$
	Logit	$-1.13$	$-0.45$	$-2.41$	$-1.22$	$-1.06$	$-1.61$	$-1.21$	$-1.24$	$-2.19$	$-2.06$	$-3.62$	$-2.46$	$1.03$	$-0.08$	$-0.01$
	$\left[3\right]$	$-1.13$	$(-\infty ,0]$	$-2.41$	$(-\infty ,-0.01]$	$-1.06$	$-1.61$	$(-\infty ,-0.01]$	$(-\infty ,-0.01]$	$-2.19$	$(-\infty ,-1.61]$	$(-\infty ,-2.42]$	$(-\infty ,-0.02]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
	$\left[4\right]$	$-1.13$	$(-\infty ,0]$	$-2.41$	$(-\infty ,-0.01]$	$-1.06$	$-1.61$	$(-\infty ,-0.01]$	$(-\infty ,-0.01]$	$-2.19$	$(-\infty ,-1.61]$	$(-\infty ,-2.42]$	$(-\infty ,-0.02]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
	$\left[5\right]$	$-1.13$	$[-1.12,1.12]$	$-2.41$	$[-2.41,2.41]$	$-1.06$	$-1.61$	$(-\infty ,-1.07]$	$[-1.61,1.61]$	$-2.19$	$[-2.73,-0.49]$	$(-\infty ,-3.48]$	$[-4.02,4.02]$	$[-2.24,+\infty )$	$[-0.85,0.91]$	$[-0.11,+\infty )$
	$\left[6\right]$	$-1.13$	$[-1.12,0]$	$-2.41$	$[-2.41,-0.01]$	$-1.06$	$-1.61$	$(-\infty ,-1.07]$	$[-1.61,-0.01]$	$-2.19$	$[-2.73,-1.61]$	$(-\infty ,-3.48]$	$[-4.02,-0.02]$	$[-1.12,+\infty )$	$[-0.48,0.39]$	$[-0.11,+\infty )$

Table 4. Table C.3 : Projections of the identified sets of U 𝑈 U , V 𝑉 V , and Φ Φ \Phi in the second simulation exercise when r = 3 𝑟 3 r=3 .

Specifications		$U$			$V$			$\Phi$
from Table C.1	Wife $\to$	$1$	$2$	$3$	$1$	$2$	$3$	$1$	$2$	$3$
	Husband $\downarrow$
True & Logit	$1$	$0.13$	$0.65$	$-0.26$	$0.6$	$-0.47$	$-2.63$	$0.73$	$0.18$	$-2.89$
$\left[5\right]$		$0.13$	$[0.2,+\infty )$	$(-\infty ,+\infty )$	$0.6$	$-0.47$	$-2.63$	$0.73$	$[-0.27,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$		$0.13$	$[0.2,+\infty )$	$(-\infty ,+\infty )$	$0.6$	$-0.47$	$-2.63$	$0.73$	$[-0.27,+\infty )$	$(-\infty ,+\infty )$
True & Logit	$2$	$-1.11$	$1.08$	$0.56$	$0.55$	$1.15$	$-0.62$	$-0.56$	$2.23$	$-0.06$
$\left[5\right]$		$-1.11$	$[-0.5,+\infty )$	$(-\infty ,+\infty )$	$[-0.5,0.6]$	$[-0.2,+\infty )$	$[-2.6,2.6]$	$[-1.61,-0.51]$	$[-0.7,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$		$-1.11$	$[-0.5,+\infty )$	$(-\infty ,+\infty )$	$[-0.5,0.6]$	$[-0.2,+\infty )$	$[-2.6,2.6]$	$[-1.61,-0.51]$	$[-0.7,+\infty )$	$(-\infty ,+\infty )$
True & Logit	$3$	$-2.78$	$-0.04$	$1.39$	$0$	$1.15$	$1.33$	$-2.78$	$1.11$	$2.72$
$\left[5\right]$		$-2.78$	$[-2.7,2.7]$	$[-1.3,+\infty )$	$(-\infty ,0.4]$	$[-0.1,+\infty )$	$[-1.2,+\infty )$	$(-\infty ,-2.38]$	$[-2.8,+\infty )$	$[-2.5,+\infty )$
$\left[6\right]$		$-2.78$	$[-2.7,2.7]$	$[0.1,+\infty )$	$(-\infty ,0.4]$	$[-0.1,+\infty )$	$[0.1,+\infty )$	$(-\infty ,-2.38]$	$[-2.8,+\infty )$	$[0.2,+\infty )$

Table 5. Table C.4 : Projections of the identified sets of elements of D ​ ( Φ ) 𝐷 Φ D(\Phi) , C ​ ( U ) 𝐶 𝑈 C(U) , and C ​ ( V ) 𝐶 𝑉 C(V) in the second simulation exercise when r = 3 𝑟 3 r=3 . We take type “1” as reference category.

Specifications	Husband’s payoff		Wife’s payoff		Core
from Table C.1	$C_{31}\left(U\right)$	$C_{21}\left(U\right)$	$C_{31}\left(V\right)$	$C_{21}\left(V\right)$	$D_{33,11}\left(\Phi \right)$	$D_{22,11}\left(\Phi \right)$
True & Logit	$0.64$	$0.38$	$0.47$	$0.51$	$9.12$	$3.34$
$\left[5\right]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-1.56,+\infty )$	$[-0.58,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-0.65,+\infty )$	$[-0.58,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$

Table 6. Table C.5 : Projections of the identified sets of U 𝑈 U , V 𝑉 V , and Φ Φ \Phi in the second simulation exercise when r = 4 𝑟 4 r=4 .

Specifications		$U$				$V$				$\Phi$
from Table C.1	Wife $\to$	$1$	$2$	$3$	$4$	$1$	$2$	$3$	$4$	$1$	$2$	$3$	$4$
	Husband $\downarrow$
True & Logit	$1$	$0.35$	$0.77$	$-0.28$	$-2.18$	$0.42$	$-0.48$	$-1.69$	$-4.16$	$0.77$	$0.29$	$-1.96$	$-6.34$
$\left[5\right]$		$0.35$	$[0.76,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$0.42$	$-0.48$	$-1.69$	$-4.16$	$0.77$	$[0.28,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$		$0.35$	$[0.76,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$0.42$	$-0.48$	$-1.69$	$-4.16$	$0.77$	$[0.28,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
True & Logit	$2$	$-1.15$	$1.17$	$0.2$	$-0.69$	$0.19$	$1.19$	$0.06$	$-1.4$	$-0.96$	$2.36$	$0.26$	$-2.09$
$\left[5\right]$		$-1.15$	$[-0.25,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-0.25,0.25]$	$[0.76,+\infty )$	$[-0.76,+\infty )$	$[-3.81,3.31]$	$[-1.40,-0.9]$	$[0.51,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$		$-1.15$	$[0.25,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-0.25,0.25]$	$[0.76,+\infty )$	$[-0.76,+\infty )$	$[-3.81,3.31]$	$[-1.40,-0.9]$	$[1.02,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
True & Logit	$3$	$-2.22$	$0.51$	$0.8$	$0.09$	$-0.66$	$0.75$	$0.89$	$-0.39$	$-2.88$	$1.26$	$1.69$	$-0.3$
$\left[5\right]$		$-2.22$	$[-0.76,+\infty )$	$[-0.25,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,-0.25]$	$(-\infty ,+\infty )$	$[-0.25,+\infty )$	$[-3.31,3.81]$	$(-\infty ,-2.47]$	$(-\infty ,+\infty )$	$[-0.50,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$		$-2.22$	$[-0.76,+\infty )$	$[-0.25,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,-0.25]$	$(-\infty ,+\infty )$	$[-0.25,+\infty )$	$[-3.31,3.81]$	$(-\infty ,-2.47]$	$(-\infty ,+\infty )$	$[-0.50,+\infty )$	$(-\infty ,+\infty )$
True & Logit	$4$	$-3.73$	$-0.49$	$0.22$	$1.2$	$-1.76$	$0.17$	$0.71$	$1.13$	$-5.49$	$-0.32$	$0.93$	$2.33$
$\left[5\right]$		$-3.73$	$[-3.31,3.31]$	$[-1.27,+\infty )$	$[-0.76,+\infty )$	$(-\infty ,-0.76]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-1.27,+\infty )$	$(-\infty ,-4.49]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-2.03,+\infty )$
$\left[6\right]$		$-3.73$	$[-3.31,3.31]$	$[-1.27,+\infty )$	$[0.25,+\infty )$	$(-\infty ,-0.76]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[0.25,+\infty )$	$(-\infty ,-4.49]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[0.50,+\infty )$

Table 7. Table C.6 : Projections of the identified sets of elements of D ​ ( Φ ) 𝐷 Φ D(\Phi) , C ​ ( U ) 𝐶 𝑈 C(U) , and C ​ ( V ) 𝐶 𝑉 C(V) in the second simulation exercise when r = 4 𝑟 4 r=4 . We take type “1” as reference category.

Specifications	Husband’s payoff			Wife’s payoff			Core
from Table C.1	$C_{41}\left(U\right)$	$C_{31}\left(U\right)$	$C_{21}\left(U\right)$	$C_{41}\left(V\right)$	$C_{31}\left(V\right)$	$C_{21}\left(V\right)$	$D_{44,11}\left(\Phi \right)$	$D_{33,11}\left(\Phi \right)$	$D_{22,11}\left(\Phi \right)$
True & Logit	$0.31$	$0.1$	$0.21$	$0.51$	$0.45$	$0.61$	$14.93$	$7.3$	$3.8$
$\left[5\right]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-1.58,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-0.64,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$

Table 8. Table C.7 : Projections of the identified sets of U 𝑈 U , V 𝑉 V , and Φ Φ \Phi in the second simulation exercise when r = 5 𝑟 5 r=5 .

Specifications		$U$					$V$					$\Phi$
from Table C.1	Wife $\to$	$1$	$2$	$3$	$4$	$5$	$1$	$2$	$3$	$4$	$5$	$1$	$2$	$3$	$4$	$5$
	Husband $\downarrow$
True & Logit	$1$	$0.30$	$0.58$	$-0.35$	$-2.61$	$-3.12$	$0.39$	$-0.61$	$-1.73$	$-3.80$	$-4.32$	$0.69$	$-0.03$	$-2.08$	$-6.41$	$-7.44$
$\left[5\right]$		$0.30$	$[0.52,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$0.39$	$-0.61$	$-1.73$	$-3.80$	$-4.32$	$0.69$	$[-0.09,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$		$0.30$	$[0.52,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$0.39$	$-0.61$	$-1.73$	$-3.80$	$-4.32$	$0.69$	$[-0.09,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
True & Logit	$2$	$-1.28$	$1.03$	$0.12$	$-1.14$	$-1.81$	$0.09$	$1.11$	$0.01$	$-1.05$	$-1.73$	-$1.19$	$2.14$	$0.13$	$-2.19$	$-3.54$
$\left[5\right]$		$-1.28$	$[0.52,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-3.32,3.32]$	$[-1.4,+\infty )$	$[-5.68,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-4.6,2.04]$	$[-0.88,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$		$-1.28$	$[0.52,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-3.32,3.32]$	$[-1.4,+\infty )$	$[-5.68,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$[-4.6,2.04]$	$[-0.88,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
True & Logit	$3$	$-2.03$	$0.51$	$0.78$	$-0.49$	$-0.96$	$-0.52$	$0.74$	$0.82$	$-0.25$	$-0.74$	$-2.55$	$1.25$	$1.60$	$-0.74$	$-1.70$
$\left[5\right]$		$-2.03$	$[-1.53,+\infty )$	$[-2.36,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,1.66]$	$(-\infty ,+\infty )$	$[-4.32,+\infty )$	$[-8.23,9.45]$	$(-\infty ,+\infty )$	$(-\infty ,0.37]$	$(-\infty ,+\infty )$	$\left[-6.68+\infty \right)$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$		$-2.03$	$\left[-1.53+\infty \right)$	$[-2.36,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,1.66]$	$(-\infty ,+\infty )$	$[-4.32,+\infty )$	$[-8.23,9.45]$	$(-\infty ,+\infty )$	$(-\infty ,0.37]$	$(-\infty ,+\infty )$	$[-6.68,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
True & Logit	$4$	$-3.21$	$-0.51$	$0.32$	$0.43$	$-0.16$	$-1.73$	$-0.32$	$0.32$	$0.63$	$0.03$	$-4.94$	$-0.83$	$0.64$	$1.06$	$-0.13$
$\left[5\right]$		$-3.21$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,2.89]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$		$-3.21$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$\left(-\infty +\infty \right)$	$(-\infty ,+\infty )$	$(-\infty ,2.89]$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
True & Logit	$5$	$-4.25$	$-0.66$	$0.27$	$0.89$	$1.01$	$-3.43$	$-1.13$	$-0.39$	$0.42$	$0.53$	$-7.68$	$-1.79$	$-0.12$	$1.31$	$1.54$
$\left[5\right]$		$-4.25$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$
$\left[6\right]$		$-4.25$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$	$(-\infty ,+\infty )$

Table 9. Table C.8 : Projections of the identified sets of elements of D ​ ( Φ ) 𝐷 Φ D(\Phi) , C ​ ( U ) 𝐶 𝑈 C(U) , and C ​ ( V ) 𝐶 𝑉 C(V) in the second simulation exercise when r = 5 𝑟 5 r=5 . We take type “1” as reference category.

Specifications

Husband’s payoff

Wife’s payoff

Core

from Table C.1

$C_{51}\left(U\right)$

$C_{41}\left(U\right)$

$C_{31}\left(U\right)$

$C_{21}\left(U\right)$

$C_{51}\left(V\right)$

$C_{41}\left(V\right)$

$C_{31}\left(V\right)$

$C_{21}\left(V\right)$

$D_{55,11}\left(\Phi \right)$

$D_{44,11}\left(\Phi \right)$

$D_{33,11}\left(\Phi \right)$

$D_{22,11}\left(\Phi \right)$

True & Logit

$0.43$

$-0.08$

$0.09$

$0.17$

$0.05$

$0.22$

$0.27$

$0.53$

$17.36$

$13.11$

$6.92$

$4.05$

$\left[5\right]$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$\left[6\right]$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

$(-\infty ,+\infty )$

Equations

$${\tilde{\Phi }}_{ij}\equiv {\Phi }_{xy}+{ϵ}_{iy}+{\eta }_{xj},$$

$${\tilde{\Phi }}_{i0}\equiv {ϵ}_{i0}.$$

$${\tilde{\Phi }}_{0j}\equiv {\eta }_{0j}.$$

$${\tilde{U}}_i+{\tilde{V}}_j={\tilde{\Phi }}_{ij}\forall (i,j)\in \text{supp}(d\tilde{\mu }).$$

$${\tilde{U}}_i\ge {\tilde{\Phi }}_{i0}\forall i\in \mathcal{I},$$

$${\tilde{V}}_j\ge {\tilde{\Phi }}_{0j}\forall j\in \mathcal{J},$$

$${\tilde{U}}_i+{\tilde{V}}_j\ge {\tilde{\Phi }}_{ij}\forall (i,j)\in \mathcal{I}\times \mathcal{J}.$$

$${\mu }_{xy}\text{ for each }(x,y)\in \mathcal{Z}\equiv {\mathcal{X}}_0\times {\mathcal{Y}}_0\setminus \{0,0\}.$$

$$p_{y|x}\equiv \frac{{\mu }_{xy}}{m_x}\text{and}{p}_{x|y}\equiv \frac{{\mu }_{xy}}{w_y},$$

$$p_x\equiv \frac{m_x}{{\sum }_{x\in \mathcal{X}}{m}_x}\text{and}{p}_y\equiv \frac{w_y}{{\sum }_{y\in \mathcal{Y}}{w}_y},$$

$$D_{xx,\tilde{x}\tilde{x}}(\Phi )\equiv {\Phi }_{xx}+{\Phi }_{\tilde{x}\tilde{x}}-{\Phi }_{x\tilde{x}}-{\Phi }_{\tilde{x}x}.$$

$$D(\Phi )\equiv (D_{xx,\tilde{x}\tilde{x}}(\Phi ):x,\tilde{x}\in \mathcal{X},x>\tilde{x}),$$

$$D^{\ell }(\Phi )\equiv (D_{xx,\tilde{x}\tilde{x}}(\Phi ):x,\tilde{x}\in \{\ell ,\ell +1,...,r\},x>\tilde{x}),$$

$$D^{\ell }(\Phi )\equiv (D_{xx,\tilde{x}\tilde{x}}(\Phi ):x,\tilde{x}\in \{1,...,\ell \},x>\tilde{x}),$$

$${\bar{U}}_x\equiv E_{F_x}(\underset{y\in {\mathcal{Y}}_0}{\max }{U}_{xy}+{ϵ}_{iy}|X_i=x).$$

$${\Delta }_{x\tilde{x}}(U)=\sum _{y\in {\mathcal{Y}}_0}{p}_{y|x}{U}_{xy}-\sum _{y\in {\mathcal{Y}}_0}{p}_{y|\tilde{x}}{U}_{\tilde{x}y}+E_{F_x}({ϵ}_{i{y}_i^{\ast }}|X_i=x)-E_{F_{\tilde{x}}}({ϵ}_{i{\tilde{y}}_i^{\ast }}|X_i=\tilde{x}),$$

$$C_{x\tilde{x}}(U)\equiv \sum _{y\in {\mathcal{Y}}_0}{p}_{y|x}{U}_{xy}-\sum _{y\in {\mathcal{Y}}_0}{p}_{y|\tilde{x}}{U}_{\tilde{x}y},$$

$$C(U)\equiv (C_{x\tilde{x}}(U):x,\tilde{x}\in \mathcal{X},x>\tilde{x})\text{and}C(V)\equiv (C_{y\tilde{y}}(V):y,\tilde{y}\in \mathcal{Y},y>\tilde{y}),$$

$$U\equiv (U_{xy}:(x,y)\in \mathcal{X}\times {\mathcal{Y}}_0)\in {\mathbb{R}}^{|\mathcal{X}\times {\mathcal{Y}}_0|}\text{ and }V\equiv (V_{xy}:(x,y)\in {\mathcal{X}}_0\times \mathcal{Y})\in {\mathbb{R}}^{|{\mathcal{X}}_0\times \mathcal{Y}|},$$

$${\tilde{U}}_i=\underset{y\in {\mathcal{Y}}_0}{\max }(U_{xy}+{ϵ}_{iy})$$

$${\tilde{V}}_j=\underset{x\in {\mathcal{X}}_0}{\max }(V_{xy}+{\eta }_{xj})$$

$$U_{xy}+V_{xy}={\Phi }_{xy}\text{, }{U}_{x0}=0\text{, }{V}_{0y}=0$$

$$\mathcal{U}\equiv \{U\in {\mathbb{R}}^{|\mathcal{X}\times {\mathcal{Y}}_0|}:U_{x0}=0\forall x\in \mathcal{X}\},$$

$$\mathcal{V}\equiv \{V\in {\mathbb{R}}^{|{\mathcal{X}}_0\times \mathcal{Y}|}:V_{0y}=0\forall y\in \mathcal{Y}\},$$

$$\Theta \equiv {\mathbb{R}}^{|\mathcal{X}\times \mathcal{Y}|}.$$

$$\kappa (U,F_x,y)\equiv {\lambda }_{F_x}(\{(e_y:y\in {\mathcal{Y}}_0)\in {\mathbb{R}}^{|{\mathcal{Y}}_0|}:U_{xy}+e_y\ge U_{x\tilde{y}}+e_{\tilde{y}}\forall \tilde{y}\in {\mathcal{Y}}_0\setminus \{y\}\}),$$

$$\kappa (V,G_y,x)\equiv {\lambda }_{G_y}(\{(e_x:x\in {\mathcal{X}}_0)\in {\mathbb{R}}^{|{\mathcal{X}}_0|}:V_{xy}+e_x\ge V_{\tilde{x}y}+e_{\tilde{x}}\forall \tilde{x}\in {\mathcal{X}}_0\setminus \{x\}\}),$$

$$p_{y|x}=\kappa (U,F_x,y)$$

$$p_{x|y}=\kappa (V,G_y,x)$$

$$U_{xy}+V_{xy}={\Phi }_{xy}$$

$$\Delta {ϵ}_i\equiv ({ϵ}_{i1}-{ϵ}_{i0},...,{ϵ}_{ir}-{ϵ}_{i0},{ϵ}_{i1}-{ϵ}_{i2},...,{ϵ}_{i1}-{ϵ}_{ir},{ϵ}_{i2}-{ϵ}_{i3},...,{ϵ}_{i2}-{ϵ}_{ir},...,{ϵ}_{ir-1}-{ϵ}_{ir}),$$

$$\mathcal{B}\equiv \{(b_1,b_2,...,b_d)\in {\mathbb{R}}^d:$$

$$b_{2r}=b_2-b_3,...,b_{3r-3}=b_2-b_r,...,$$

$$b_d=b_{r-1}-b_r\}.$$

$${\lambda }_{\Delta F_x}(\mathcal{B})=1,$$

Full text

Keywords: One-to-One Matching, Marriage Market, Transfers, Stability, Partial Identification, Nonparametric Identification, Linear Programming, Econometrics.

1 Introduction

Matching markets are two-sided markets, where agents on each side have preferences over matching with agents on the other side. For example, social interactions lead individuals to find marital partners, production tasks are assigned to workers, and auctions sort buyers with sellers. While the economic theory of matching models has been around for more than five decades, the literature on empirical matching models is relatively recent (Chiappori and Salanié, 2016).

An important strand of this literature focuses on the one-to-one matching model, in which every agent forms at most one match. Each possible match generates a surplus. In the framework where utilities are perfectly transferable, agents can share the match surplus with their partners without frictions. Since Becker (1973), the one-to-one matching model with perfectly transferable utilities (hereafter, 1to1TU) has been extensively used in household economics to represent the marriage market (Chiappori, 2017). In particular, researchers have exploited the 1to1TU model to estimate the systematic part of the match surplus. Recovering the systematic match surplus is useful, for example, to investigate sorting patterns and how they change over time, to learn about the complementarities and substitutabilities of partner characteristics, to assess the efficiency and welfare implications of the status-quo assignment, and to measure the impact of pre-marital decisions on the sharing of the match surplus between spouses.111The 1to1TU model has also been used to study matching of CEOs to firms (Chen, 2017), matching of academics to offices (Baccara, et al., 2012), merging of banks (Akkus, Cookson, and Hortaçsu, 2016), formation of research alliances (Mindruda, Moeen, and Agarwal, 2016), and collaboration between academics and firms (Mindruda, 2013; Banal-Estañol, Macho-Stadler, and Pérez-Castrillo, 2018).

Most of the papers using the 1to1TU model to estimate the systematic match surplus proceed under strong parametric distributional restrictions on the agents’ unobserved heterogeneity. These restrictions amount to imposing i.i.d. standard Extreme Value Type I taste shocks, independently distributed from covariates. Along with data on one large market, these restrictions make the 1to1TU model just identified, leading to point identification of the systematic match surplus via standard Logit formulas (Choo and Siow, 2006). The motivation for using the Logit 1to1TU model is computational simplicity. However, this framework may lead to paradoxical conclusions that run against economic sense. For example, it is well known that the one-sided Logit model is inherently linked to the independence of irrelevant alternatives (IIA) axiom and severely restricts cross-elasticities. The same holds in two-sided markets and causes unintuitive comparative static predictions, as explained in Graham (2013a) and Galichon and Salanié (2019).

The fact that widespread empirical practices rest on the Logit 1to1TU model raises several questions. When we have data on one large market, does the 1to1TU model retain any identifying power on the systematic match surplus without restrictions on the taste shock distribution? If not, is it still possible to recover some information on the systematic match surplus under nonparametric distributional assumptions on the unobserved heterogeneity? How are the answers to relevant policy questions driven by the Logit assumption? The contribution of our paper is to address these issues. By doing so, we also offer methodological guidance for researchers who wish to consider more robust alternatives to (or do sensitivity checks of) the Logit 1to1TU model.

We start our analysis by observing that, if the taste shock distribution is not assumed to be fully known by the researcher, then the 1to1TU model is under-identified with data on one large market (Galichon and Salanie, 2021; hereafter, GS). In the absence of any restrictions on the taste shock distribution, we show that the under-identification issue is severe, as the 1to1TU model is completely uninformative about the systematic match surplus. Formally, this means that, for every possible value of the systematic match surplus, there exists a taste shock distribution that rationalises the data when combined with that value of the systematic match surplus.

We proceed by investigating whether the 1to1TU model retains some information on the systematic match surplus under various classes of nonparametric distributional assumptions on the unobserved heterogeneity (for instance, independence of taste shocks from covariates, quantile restrictions, symmetry restrictions, and identically distributed marginals). Answering this question poses the challenge of tractably characterising the identified set of the systematic match surplus. We do that by extending the linear programming computational approach of Torgovitsky (2019) to our framework. For a given value of the systematic match surplus, this method transforms a search over the space of infinite-dimensional cumulative distribution functions into a search over a space of cumulative distribution functions evaluated at a finite number of points. The latter search can be written as a simple linear program. Further, note that the analyst would need to solve the linear program for every admissible value of the systematic match surplus. Usually, this is carried out in the partial identification literature by generating a grid of points to approximate the parameter space and then repeating the exercise of interest for each grid point. However, the difficulty of doing so increases with the size of the grid, which, in the 1to1TU model, increases exponentially with the cardinality of the covariates’ support, leading quickly to a computational bottleneck. We alleviate this issue by showing that the parameter space can be ex-ante partitioned into a finite number of subsets such that, for each subset, every value belonging to that subset gives rise to the same linear program. Therefore, the analyst has to solve the linear program only once for each subset. These results are new and represent the methodological contribution of the paper.

We use our methodology to re-examine some of the relevant questions in the empirical literature on the marriage market that have been previously answered by relying on the Logit 1to1TU model. A key question that stands out in this literature is whether educational sorting (i.e., the tendency of agents to marry someone with similar or very different education levels) has changed over time. Answering this question is important because educational sorting may have a crucial impact on inequality by determining family formation and intergenerational transmission of human capital (Kremer, 1997; Fernández and Rogerson, 2001; Fernández, Guner, and Knowles, 2005; Heckman and Mosso, 2014; Dupuy and Weber, 2019; Eika, Mogstad, and Zafar, 2019; Chiappori, et al., 2020; Ciscato and Weber, 2020). The literature proposes two approaches to measure changes in educational sorting. The first amounts to using indices of sorting based on comparing the empirical match probabilities to a counterfactual world where matching happens randomly (Fernández and Rogerson, 2001; Greenwood, Guner, and Kocharkov, 2003; Liu and Lu, 2006; Greenwood, et al., 2014; Abbott, et al., 2019; Eika, Mogstad, and Zafar, 2019; Shen, 2019). The second consists of using a structural model of the marriage market in order to estimate individual preferences and analyse how they evolve over time. The second approach has been implemented by Siow (2015) and Chiappori, Salanié, and Weiss (2017) (hereafter, CSW), based on the Logit 1to1TU model.

Both approaches suggest that, on average, positive educational sorting has increased in the U.S. in the past decades. However, there is some debate around this trend when we look closer at each education category. For instance, Eika, Mogstad, and Zafar (2019) find that positive educational sorting has declined among the highly educated and increased among the less educated. Instead, CSW find that positive educational sorting has increased particularly among the highly educated.222See also Chiappori, Costa-Dias, and Meghir (2020) and Chiappori, Costa-Dias, and Meghir (2021) for similar conclusions. Using data from the American Community Survey between the years 1940 and 1967 as in CSW, we exploit our methodology to assess whether the conclusions achieved via the structural approach are robust to the dropping of the Logit assumption. Under various classes of nonparametric distributional assumptions, we find that the 1to1TU model is uninformative about the presence and trend of positive educational sorting among the highly educated. We find the presence of positive educational sorting among the less educated, although the model remains ambiguous about its evolution across cohorts. Overall, our results suggest that the previous findings on the increase in positive educational sorting based on the Logit 1to1TU model are, in fact, driven by the Logit assumption.

Lastly, we use our methodology to study the evolution of marital returns to education. As discussed by Chiappori, Iyigun, and Weiss (2009) and CSW, the increase in educational sorting makes a higher stock of human capital more valuable in the marriage market. Consequently, they predict an increase in the expected maximum payoff an agent can receive in the marriage market due to achieving a college degree (âmarital college premiumâ), especially among women. Their empirical findings corroborate such a prediction for the U.S., based on the Logit 1to1TU model. Without imposing parametric distributional assumptions, we find that the 1to1TU model is inconclusive about the evolution of marital college premium over time. Further, it is particularly uninformative about the women’s side, indicating that any evidence on the increase in marital college premium from the Logit 1to1TU model is a consequence of the arbitrary parametric restrictions.

In what follows, Section 2 summarises the related literature, Section 3 presents the model, Section 4 discusses identification, Section 5 illustrates the empirical applications, and Section 6 concludes.

2 Literature review

The Logit 1to1TU model was introduced by Choo and Siow (2006) and since then has become popular in empirical research on the marriage market. Several papers use it to learn whether matching preferences are positive assortative by age, education, geographical location, etc. (Choo and Siow, 2006; Botticini and Siow, 2011; Bruze, Svarer, and Weiss, 2015; Choo, 2015;333Bruze, Svarer, and Weiss (2015) and Choo (2015) incorporate dynamic aspects into the framework of Choo and Siow (2006). Siow, 2015; Galichon, Kominers, and Weber, 2019444Galichon, Kominers, and Weber (2019) extend the framework of Choo and Siow (2006) to imperfectly transferable utilities.). Other papers use it to assess which of the partner characteristics are complements/substitutes in the production of the systematic match surplus and their relative strengths (Dupuy and Galichon, 2014;555Dupuy and Galichon (2014) extend the framework of Choo and Siow (2006) to continuous covariates. Ciscato, Galichon, Goussé, 2020). The Logit 1to1TU model has been frequently used to investigate the evolution of the link between education levels and marriage market outcomes over time. In particular, the literature has studied questions like how educational sorting and the marital college premium have changed over time (Chiappori, Iyigun, and Weiss, 2009; Siow, 2015; CSW; Chiappori, Costa-Dias, and Meghir, 2020; Chiappori, et al., 2020). Other papers adopt the Logit 1to1TU model to measure the effect on marital choices of exogenous events that change the distribution of individual characteristics on each side of the market, such as the famine in China between 1958 and 1961 (Brandt, Siow, and Carl, 2016).

The Logit 1to1TU model is often incorporated into bigger structural models. Examples of these include collective household models with marriage and labour supply (Choo and Seitz, 2013); life cycle models of education, marriage, labour supply, and consumption (Chiappori, Costa-Dias, and Meghir, 2018); collective household models with marriage, labour supply, home production choices, and joint taxation (Gayle and Shephard, 2019666Gayle and Shephard (2019) allow for imperfectly transferable utilities.); collective household models with marriage, fertility decisions, and child socialisation choices (Bisin and Tura, 2020). Mourifié and Siow (2021) extend the Logit 1to1TU model to allow for peer effects and cohabitation.

GS investigates the identification of the 1to1TU model when one dispenses with the Logit assumption. Under the assumption that the taste shock distribution is fully known by the analyst, they show that the 1to1TU model is just identified (and, thus, the systematic match surplus is point identified) with data on one large market. They also provide closed-form expressions of the systematic match surplus for some parametric distributional families.

The literature has explored two ways to introduce unknown parameters in the taste shock distribution while maintaining point identification. The first approach exploits variations in matching patterns across many i.i.d. markets (Fox, 2010; Fox, 2018; Fox, Yang, and Hsu, 2018; Sinha, 2018), as in the empirical IO tradition. With data on many i.i.d. markets, one can proceed without parametric restrictions on the taste shock distribution. However, in most datasets, it is unclear as to what truly defines i.i.d. markets. For instance, the majority of the empirical applications of this approach assume that consecutive years represent i.i.d. markets, which can be often hard to justify (Baccara, et al., 2012; Mindruda, 2013; Akkus, Cookson, and Hortaçsu, 2016; Mindruda, Moeen, and Agarwal, 2016; Chen, 2017; Banal-Estañol, Macho-Stadler, and Pérez-Castrillo, 2018). The second approach exploits variations of matching patterns across a few large cohorts which feature different distributions of covariates, independent matching processes, identical systematic match surplus up to some drifts or linear/quadratic trends, and identical taste shock distributions. This approach is implemented by CSW to introduce gender heteroskedasticity in the Extreme Value Type I distribution.

Recent advances in the partial identification literature have pointed out an alternative route to avoid parametric assumptions on the taste shock distribution, without adding any further structure on the systematic match surplus, and while remaining within a one large market framework. In particular, Graham (2011; 2013b) shows that if the taste shocks are i.i.d., then the signs of some complementarities between the spouses’ observed characteristics are identified. Fox (2018) bounds the systematic match surplus under the assumption that the taste shocks are exchangeable across the observed characteristics of the potential partners. Our paper contributes to this strand of the literature by constructing the identified set of the systematic match surplus without requiring the taste shocks to be i.i.d. or exchangeable, which can both be strong assumptions. Further, our paper showcases the usefulness of partial identification approaches for formally understanding empirical results that might otherwise be accepted less critically.

3 The model

This section describes the 1to1TU model that has been previously studied in Choo and Siow (2006) and GS. The model relies on four main assumptions that are standard in the current empirical practice. In what follows, we refer to agents on one side of the market as men and to agents on the other side of the market as women.

Assumption 1**.**

(One large market)

There is a two-sided market. One side is populated by an (uncountably) infinite set of men, $\mathcal{I}$, with measure $d\tilde{\mu}_{\mathcal{I}}$. The other side is populated by an (uncountably) infinite set of women, $\mathcal{J}$, with measure $d\tilde{\mu}_{\mathcal{J}}$. The two sides are stochastically independent. ∎

Assumption 2**.**

(Finite number of observed types)

Each man $i\in\mathcal{I}$ is characterised by a type, $X_{i}$, with finite support, $\mathcal{X}$. The mass of men of type $x\in\mathcal{X}$ is denoted by $m_{x}$. Each woman $j\in\mathcal{J}$ is characterised by a type, $Y_{j}$, with finite support, $\mathcal{Y}$. The mass of women of type $y\in\mathcal{Y}$ is denoted by $w_{y}$. Without loss of generality, we normalise the total mass of agents to $1$, i.e., $\sum_{x\in\mathcal{X}}m_{x}+\sum_{y\in\mathcal{Y}}w_{y}=1$. The realisations of $X_{i}$ and $Y_{j}$ are observed by the researcher and all agents. We define the sets of partner types that are available to men and women by $\mathcal{Y}_{0}\equiv\mathcal{Y}\cup\{0\}$ and $\mathcal{X}_{0}\equiv\mathcal{X}\cup\{0\}$, respectively, where “[math]” represents the option not to match. ∎

Assumption 3**.**

(Taste shocks)

Each man $i\in\mathcal{I}$ is endowed with a $|\mathcal{Y}_{0}|\times 1$ vector of taste shocks, $\epsilon_{i}\equiv(\epsilon_{iy}:y\in\mathcal{Y}_{0})$, where $|\mathcal{Y}_{0}|$ denotes the cardinality of $\mathcal{Y}_{0}$ and $\epsilon_{iy}$ is the idiosyncratic preference of man $i$ for marrying a woman of type $y\in\mathcal{Y}_{0}$. Conditional on $X_{i}=x$ and for each $x\in\mathcal{X}$, $\epsilon_{i}$ has cumulative distribution function (hereafter, CDF) $F_{x}$. $F_{x}$ is absolutely continuous with respect to the Lebesgue measure and has support in $\mathbb{R}^{|\mathcal{Y}_{0}|}$. Each woman $j\in\mathcal{J}$ is endowed with a $|\mathcal{X}_{0}|\times 1$ vector of taste shocks, $\eta_{j}\equiv(\eta_{xj}:x\in\mathcal{X}_{0})$, where $\eta_{xj}$ is the idiosyncratic preference of woman $j$ for marrying a man of type $x\in\mathcal{X}_{0}$. Conditional on $Y_{j}=y$ and for each $y\in\mathcal{Y}$, $\eta_{j}$ has CDF $G_{y}$. $G_{y}$ is absolutely continuous with respect to the Lebesgue measure and has support in $\mathbb{R}^{|\mathcal{X}_{0}|}$. The realisations of $\epsilon_{i}$ and $\eta_{j}$ are observed by all agents but are not observed by the researcher. ∎

Assumption 4**.**

(Separability)

A match between man $i\in\mathcal{I}$ of type $x\in\mathcal{X}$ and woman $j\in\mathcal{J}$ of type $y\in\mathcal{Y}$ generates a match surplus defined as

[TABLE]

where $\Phi\equiv(\Phi_{xy}:(x,y)\in\mathcal{X}\times\mathcal{Y})$ is the systematic match surplus. The payoff of man $i\in\mathcal{I}$ from remaining unmatched is

[TABLE]

The payoff of woman $j\in\mathcal{J}$ from remaining unmatched is

[TABLE]

∎

Assumption 1 outlines the one large market framework. The restriction on the stochastic independence of the two sides of the market is not crucial for our results and can be relaxed. Assumption 2 requires each agent to belong to one type. There is a finite number of types, which is defined by the Cartesian product of the individual characteristics observed by the researcher. Assumption 3 requires each agent to have idiosyncratic marital preferences over the types of the potential partners and not over their identities. It implies that women (men) of the same type are perfect substitutes for a man (woman). Assumption 4 imposes that the match surplus is the sum of two components. One is the systematic match surplus, that is determined by the types of potential partners. The other is the sum of the taste shocks of the potential partners. In particular, the latent heterogeneity entering the match surplus equation does not consist of an $ij$-indexed term. Instead, it is modelled through the sum of two terms, $\epsilon_{iy}+\eta_{xj}$, each of which only depends on the type of the potential partner. Assumption 4 is typically referred to as “separability” (Chiappori, 2017). Lastly, observe that the systematic match surplus from remaining single is normalised to zero. This location normalisation is standard in the literature and imposed, for instance, also by GS and CSW.

A matching consists of

1. (i)

A measure $d\tilde{\mu}$ on the set $\mathcal{I}\times\mathcal{J}$, such that the marginal of $d\tilde{\mu}$ over $\mathcal{I}$ ($\mathcal{J}$) is $d\tilde{\mu}_{\mathcal{I}}$ ($d\tilde{\mu}_{\mathcal{J}}$). 2. (ii)

A set of payoffs, $\{\tilde{U}_{i}\}_{i\in\mathcal{I}}$ and $\{\tilde{V}_{j}\}_{j\in\mathcal{J}}$, such that

[TABLE]

(Chiappori, McCann, and Nesheim, 2010; Chiappori, McCann, and Pass, 2020). That is, a matching consists of a match assignment and a match surplus sharing rule. A match assignment is a description of who is matched with whom. A match surplus sharing rule tells us how the match surplus is divided between spouses. This division of the match surplus relies on endogenously determined transfers, ensuring that every agent maximises their utility and the market clears.

A matching, $d\tilde{\mu},\{\tilde{U}_{i}\}_{i\in\mathcal{I}},\{\tilde{V}_{j}\}_{j\in\mathcal{J}}$, is stable when no agent has an incentive to change his partner, i.e.,

[TABLE]

The first two sets of inequalities imply that married agents would not prefer being single. The last set of inequalities states that no man or woman would get a strictly higher match surplus by matching together than what they get under the match assignment, $d\tilde{\mu}$. It can be shown that a stable matching exists under mild continuity assumptions (Villani, 2009). Moreover, in the limit of continuous and atomless populations, the stable matching is generically unique (Gretsky, Ostroy and Zame, 1992). Importantly for the identification analysis, the resulting equilibrium mass of couples where the man is of type $x$ and the woman is of type $y$ is unique for every $(x,y)$. In what follows, we denote this equilibrium mass of couples by

[TABLE]

4 Identification

4.1 Data and parameters of interest

For identification, we assume that the analyst knows $\{\mu_{xy}\}_{(x,y)\in\mathcal{Z}}$, $\{m_{x}\}_{x\in\mathcal{X}}$, and $\{w_{y}\}_{y\in\mathcal{Y}}$. For estimation, we replace $\{\mu_{xy}\}_{(x,y)\in\mathcal{Z}}$, $\{m_{x}\}_{x\in\mathcal{X}}$, and $\{w_{y}\}_{y\in\mathcal{Y}}$ with consistent sample analogues, resulting from sampling at random from the market at the individual level or at the household level. Define

[TABLE]

as the equilibrium probability of marrying a woman of type $y\in\mathcal{Y}_{0}$ conditional on being a man of type $x\in\mathcal{X}$, and the equilibrium probability of marrying a man of type $x\in\mathcal{X}_{0}$ conditional on being a woman of type $y\in\mathcal{Y}$, respectively. Lastly, let

[TABLE]

be the proportion of men of type $x\in\mathcal{X}$ and the proportion of women of type $y\in\mathcal{Y}$, respectively.

Our primary interest lies in recovering the systematic match surplus, $\Phi$. In fact, (partially) identifying $\Phi$ allows us to answer many important questions considered in the marriage market literature. In particular, $\Phi$ can be used to learn whether agents tend to match with similar people, i.e., whether there is positive assortativeness. Investigating positive assortativeness and its evolution over time has been a focus of empirical research since Becker (1973) because it is crucial to understand the sources of inequality in intergenerational outcomes. For instance, if parents’ education level affects their children’s school attainment and marriage is positive assortative by education, then inequality in the next generation may be higher. Formally, let $\mathcal{X}=\mathcal{Y}\equiv\{1,...,r\}$ collect $r$ education levels, ordered from lowest to highest. For any $x,\tilde{x}\in\mathcal{X}$ with $x>\tilde{x}$, consider the cross-difference operator,

[TABLE]

$D_{xx,\tilde{x}\tilde{x}}(\Phi)$ measures how the incremental (dis)value of marrying a more educated man evolves as the education of the woman increases. Hence, one can evaluate changes in positive educational sorting across markets (for instance, across cohorts) by estimating the supermodular core of $\Phi$,

[TABLE]

within each market. As per Definition 1 in CSW, if every component of the vector $D(\Phi)$ is positive within a given market, then that market exhibits positive educational sorting. Further, if every component of $D(\Phi)$ increases across markets, then positive educational sorting increases across markets. The definition can be restricted to a subset of education categories. We can say that there is positive educational sorting among more educated people if each component of the vector

[TABLE]

is positive, where $\ell$ is large enough. Similarly, there is positive educational sorting among less educated people if each component of the vector

[TABLE]

is positive, where $\ell$ is low enough. We can also say that positive educational sorting increases across markets among more (less) educated people if every component of $D^{\ell}(\Phi)$ increases across markets.777In settings with multidimensional covariates, the cross-difference operator defined in (1) can be used to learn which of the spouses’ observed characteristics are complements or substitutes in the production of the systematic match surplus and, in turn, discover the key drivers of the gains to matching. See, for instance, Fox (2010) and Graham (2011; 2013b).

In addition to recovering $\Phi$, our methodology permits the analyst to (partially) identify how $\Phi$ is shared between spouses and hence assess the impact of pre-marital decisions on marriage market outcomes. In particular, let $U_{xy}$ be the part of $\Phi_{xy}$ that is gained by a man of type $x\in\mathcal{X}$ when matching with a woman of type $y\in\mathcal{Y}$. Analogously, let $V_{xy}$ be the part of $\Phi_{xy}$ that is gained by a woman of type $y\in\mathcal{Y}$ when matching with a man of type $x\in\mathcal{X}$. Define $\bar{U}_{x}$ as the expected payoff that a man of type $x\in\mathcal{X}$ gets when marrying,

[TABLE]

For any $\tilde{x}\in\mathcal{X}$ with $\tilde{x}<x$, the difference $\Delta_{x\tilde{x}}(U)\equiv\bar{U}_{x}-\bar{U}_{\tilde{x}}$ denotes the gain in expected utility from reaching education level $x$ instead of $\tilde{x}$. Therefore, it represents the marital education premium. When $x=r,$ a college degree, and $\tilde{x}=r-1$, such a difference is called the marital college premium. These quantities have received particular attention because they capture the value of human capital on the marriage market (Chiappori, Iyigun, and Weiss, 2009; CSW). As shown in GS, the marital education premium is equal to

[TABLE]

where $y^{*}_{i}\in\mathcal{Y}_{0}$ is the optimal choice of man $i$ of type $x$ and $\tilde{y}^{*}_{i}\in\mathcal{Y}_{0}$ is the optimal choice of man $i$ of type $\tilde{x}$. Note from (2) that computing (bounds for) $\Delta_{x\tilde{x}}(U)$ requires the specification of (finite bounds for) $E_{F_{x}}(\epsilon_{iy^{*}_{i}}|X_{i}=x)-E_{F_{\tilde{x}}}(\epsilon_{i\tilde{y}^{*}_{i}}|X_{i}=\tilde{x})$. The latter are typically not obtained within a nonparametric framework like ours without further assumptions. Nevertheless, we will provide bounds for the difference

[TABLE]

which represents the change in the weighted average systematic payoff due to reaching education level $x$ instead of $\tilde{x}$. Such bounds will help us make certain conclusions regarding $\Delta_{x\tilde{x}}(U)$. In fact, we will see that in all the empirical cases of interest, the estimates of

[TABLE]

are unbounded above and below. By (2), this implies that the marital education premia are unbounded as well, and therefore, their evolution over time cannot be inferred.888Note that imposing $E_{F_{x}}(\epsilon_{iy}|X_{i}=x)<+\infty$ and $E_{F_{\tilde{x}}}(\epsilon_{iy}|X_{i}=\tilde{x})<+\infty$ for each $y\in\mathcal{Y}_{0}$ is necessary to ensure finiteness of $E_{F_{x}}(\epsilon_{iy^{*}_{i}}|X_{i}=x)$ and $E_{F_{\tilde{x}}}(\epsilon_{i\tilde{y}^{*}_{i}}|X_{i}=\tilde{x})$ in (2). In our framework, we do not consider such a class of nonparametric assumptions on $\{F_{x}\}_{x\in\mathcal{X}}$. This is without loss of generality. In fact, as suggested by Example 2 in Torgovitsky (2019), requiring $E_{F_{x}}(\epsilon_{iy}|X_{i}=x)$ to be equal to some finite number for each $(x,y)\in\mathcal{X}\times\mathcal{Y}_{0}$ does not place any restrictions on the set of underlying distributions that determines the identified set for $U$, thus leaving the bounds for $C(U)$ unchanged.

4.2 Two multinomial choice models

Based on the separability restriction (Assumption 4), Proposition 1 of GS provides a key result for our identification analysis.999This result also appears in previous working paper versions of GS, in Chiappori, et al. (2008), and in CSW. This result states that, given the primitives $\Phi$, $\{p_{x}\}_{x\in\mathcal{X}}$, $\{p_{y}\}_{y\in\mathcal{Y}}$, $\{F_{x}\}_{x\in\mathcal{X}}$, $\{G_{y}\}_{y\in\mathcal{Y}}$ generating the stable matching $d\tilde{\mu},\{\tilde{U}_{i}\}_{i\in\mathcal{I}},\{\tilde{V}_{j}\}_{j\in\mathcal{J}}$, there exist vectors

[TABLE]

such that

[TABLE]

Proposition 1 of GS allows us to rewrite the framework of Section 3 as two separate one-sided multinomial choice models linked by market-clearing transfers that are implicitly embedded into the vectors $U$ and $V$. This alternative representation of the problem is useful as it immediately suggests a way to investigate the identification of $\Phi$: the researcher can study separate identification of $U$ and $V$ from (3) and (4) using various restrictions on the unobserved heterogeneity, and then obtain identification results for $\Phi$ through (5).101010The restrictions $U_{x0}=V_{0y}=0$ in (5) come from the fact that the systematic match surplus from remaining single is normalised to zero in Assumption 4.

Based on Proposition 1, Proposition 2 of GS shows that if $\{F_{x}\}_{x\in\mathcal{X}}$ and $\{G_{y}\}_{y\in\mathcal{Y}}$ are fully known by the analyst, then the 1to1TU model is just identified and, thus, $\Phi$ is point identified. Note that fully knowing $\{F_{x}\}_{x\in\mathcal{X}}$ and $\{G_{y}\}_{y\in\mathcal{Y}}$ requires either such conditional CDFs to be parameter-free, or the analyst to fix the value of each parameter governing them. In particular, a widespread practice in the empirical literature amounts to assuming that the taste shocks are i.i.d. standard Extreme Value Type I, independently distributed from types, so that $\Phi$ can be recovered via standard Logit arguments applied to each side of the market (Choo and Siow, 2006).

Unfortunately, the Logit 1to1TU model suffers from the same limitations of the one-sided Logit framework. It exhibits IIA which has counterintuitive predictions by implying proportional substitution across types. This is illustrated by Galichon and Salanié (2019) with an example that resembles the blue-bus/red-bus example of McFadden (1974).111111See also Graham (2013a) for another example on violation of IIA. While most of the applied literature on one-sided markets has replaced the Logit assumption with the Generalised Extreme Value (GEV) framework, such a transition is yet to occur in the two-sided literature. This is because, due to the just identification result mentioned above, it would be necessary to arbitrarily specify the value of each parameter governing the GEV distribution, at the risk of incurring serious misspecification. There are two possible ways to solve this impasse with data on one large market: the researcher could either place assumptions on $\Phi$ (for instance, by restricting complementarities/substitutabilities among observed characteristics) so as to get degrees of freedom for estimating the GEV parameters, or adopt a partial identification perspective. The first approach is rare in the marriage market literature because most of the empirical studies consider marital sorting on a single dimension (one attribute at a time). In this paper, we explore the second approach and construct bounds for $\Phi$. In doing so, we do not restrict the distribution of the taste shocks to belong to any specific parametric family.

4.3 The extent of under-identification

As shown by GS, if $\{F_{x}\}_{x\in\mathcal{X}}$ and $\{G_{y}\}_{y\in\mathcal{Y}}$ are not assumed to be fully known by the analyst, then the 1to1TU model is under-identified. As a first step, this section investigates the extent of under-identification by answering the following question: does the 1to1TU model retain some identifying power on $\Phi$ without imposing any restrictions on $\{F_{x}\}_{x\in\mathcal{X}}$ and $\{G_{y}\}_{y\in\mathcal{Y}}$? Lemma 1 below gives a negative answer.

Let $\mathcal{U}$, $\mathcal{V}$, and $\Theta$ denote the parameter spaces of $U$, $V$, and $\Phi$, respectively, i.e.,

[TABLE]

Further, let $\mathcal{F}$ and $\mathcal{G}$ be the function spaces of all admissible taste shock distributions, $\{F_{x}\}_{x\in\mathcal{X}}$ and $\{G_{y}\}_{y\in\mathcal{Y}}$, respectively.121212Note that one element of $\mathcal{F}$ is a family of $|\mathcal{X}|$ conditional CDFs, $\{F_{x}\}_{x\in\mathcal{X}}$. Similarly, one element of $\mathcal{G}$ is a family of $|\mathcal{Y}|$ conditional CDFs, $\{G_{y}\}_{y\in\mathcal{Y}}$. Lastly, for any $y\in\mathcal{Y}_{0}$, $U\in\mathcal{U}$, and $\{F_{x}\}_{x\in\mathcal{X}}\in\mathcal{F}$, let $\kappa(U,F_{x},y)$ be the model-implied probability of marrying a woman of type $y\in\mathcal{Y}_{0}$ conditional on being a man of type $x\in\mathcal{X}$, i.e.,

[TABLE]

where $\lambda_{F_{x}}$ is the probability measure associated with $F_{x}$. Similarly, for any $x\in\mathcal{X}_{0}$, $V\in\mathcal{V}$, and $\{G_{y}\}_{y\in\mathcal{Y}}\in\mathcal{G}$, let $\kappa(V,G_{y},x)$ be the model-implied probability of marrying a man of type $x\in\mathcal{X}_{0}$ conditional on being a woman of type $y\in\mathcal{Y}$, i.e.,

[TABLE]

where $\lambda_{G_{y}}$ is the probability measure associated with $G_{y}$.

Lemma 1**.**

(Under-identification)

For every data, $\{p_{y|x}\}_{(x,y)\in\mathcal{X}\times\mathcal{Y}_{0}}$ and $\{p_{x|y}\}_{(x,y)\in\mathcal{X}_{0}\times\mathcal{Y}}$, and systematic match surplus, $\Phi\in\Theta$, there exist $(U,V)\in\mathcal{U}\times\mathcal{V}$, $\{F_{x}\}_{x\in\mathcal{X}}\in\mathcal{F}$, and $\{G_{y}\}_{y\in\mathcal{Y}}\in\mathcal{G}$ such that

[TABLE]

∎

Lemma 1 is a straightforward application of Theorem 1 of Haile, Hortaçsu, and Kosenok (2008) to our two-sided setting. It shows that in the absence of restrictions on $\{F_{x}\}_{x\in\mathcal{X}}$ and $\{G_{y}\}_{y\in\mathcal{Y}}$, the 1to1TU model leads to uninformative bounds on $\Phi$. Therefore, one needs to impose at least some distributional assumptions on the unobserved heterogeneity to get information on $\Phi$.

4.4 Adding distributional assumptions on unobserved heterogeneity

In this section, we ask ourselves whether the 1to1TU model retains some identifying power on $\Phi$ under various classes of nonparametric distributional assumptions on the unobserved heterogeneity, so as to still be able to address relevant policy matters while maintaining a certain degree of robustness. To answer this question, we adopt a computational approach. In particular, we start from observing that if $\{F_{x}\}_{x\in\mathcal{X}}$ and $\{G_{y}\}_{y\in\mathcal{Y}}$ are not assumed to be fully known by the analyst, then the 1to1TU model is under-identified, leading to partial identification of $\Phi$. Hence, we provide a methodology to construct the identified set of $\Phi$ under various classes of nonparametric distributional assumptions on the unobserved heterogeneity.

The identified set of $\Phi$ (hereafter, $\Theta^{*}$) is the set of values of $\Phi$ such that there exists $U$, $V$, $\{F_{x}\}_{x\in\mathcal{X}}$, and $\{G_{y}\}_{y\in\mathcal{Y}}$ that satisfy (6)-(8). By Proposition 1 of GS, we can construct $\Theta^{*}$ by separately focusing on each side of the market. First, we construct the identified set of $U$ (hereafter, $\mathcal{U}^{*}$), i.e., the set of values of $U$ such that there exists $\{F_{x}\}_{x\in\mathcal{X}}$ that satisfies (6). Then, we construct the identified set of $V$ (hereafter, $\mathcal{V}^{*}$), i.e., the set of values of $V$ such that there exists $\{G_{y}\}_{y\in\mathcal{Y}}$ that satisfies (7). Finally, we obtain $\Theta^{*}$ from (8). In what follows, we explain the construction of $\mathcal{U}^{*}$. The construction of $\mathcal{V}^{*}$ is analogous.

Recall that in multinomial choice models what matters is differences in utilities. Therefore, as a preliminary step, we rewrite the identification problem using the taste shock differences. Without loss of generality, we label the women’s types as $\mathcal{Y}\equiv\{1,...,r\}$. Let $\Delta{\epsilon_{i}}$ be the vector of differences between every pair of taste shocks of man $i\in\mathcal{I}$,

[TABLE]

with length $d\equiv\binom{r+1}{2}$.

Observe that each $\{F_{x}\}_{x\in\mathcal{X}}\in\mathcal{F}$ determines a corresponding family of $d$-dimensional conditional CDFs of $\Delta\epsilon_{i}$, which we denote by $\{\Delta F_{x}\}_{x\in\mathcal{X}}$. Further, note that the first $r$ components of $\Delta{\epsilon_{i}}$ can be arbitrary, while the remaining $(d-r)$ components are linear combination of the first $r$ components. Define the set

[TABLE]

By the above arguments, $\Delta F_{x}$ has support contained in $\mathcal{B}$, i.e.,

[TABLE]

for every $x\in\mathcal{X}$. We denote by $\Delta\mathcal{F}$ the function space of all admissible $\{\Delta F_{x}\}_{x\in\mathcal{X}}$, each with support contained in $\mathcal{B}$. Moreover, one may want to impose various nonparametric restrictions on $\{\Delta F_{x}\}_{x\in\mathcal{X}}$ in order to obtain informative bounds on $U$ (and ultimately, $\Phi$), as highlighted by Lemma 1. We denote by $\Delta\mathcal{F}^{\dagger}\subset\Delta\mathcal{F}$ the restricted collection of conditional CDFs. We describe later which classes of nonparametric restrictions on $\{\Delta F_{x}\}_{x\in\mathcal{X}}$ are considered.

In summary, our objective is to construct the identified set of $U$, defined as

[TABLE]

where, with slight abuse of notation, we have replaced the argument $F_{x}$ of $\kappa(U,F_{x},y)$ with $\Delta F_{x}$. We split the discussion into two steps. First, for any given $U\in\mathcal{U}$, Section 4.4.1 explains how to verify whether $U\in\mathcal{U}^{*}$. Second, Section 4.4.2 provides a result which reduces the computational burden of repeating the first step for every $U\in\mathcal{U}$. Lastly, we introduce some useful notation adopted in the forthcoming arguments. $\bar{\mathbb{R}}$ denotes the extended real line. $0_{d}$ is the $d\times 1$ vector of zeros. $\Delta\epsilon_{i,l}$ is the $l$-th component of $\Delta\epsilon_{i}$ and $\Delta F_{x,l}$ is the $l$-th marginal of $\Delta F_{x}$. $\Delta\epsilon^{y}_{i}$ is the $r\times 1$ subvector of $\Delta\epsilon_{i}$ collecting the taste shock differences that are relevant when choosing $y\in\mathcal{Y}_{0}$, with conditional CDF $\Delta F^{y}_{x}$.131313For instance, consider $r=2$ (hence, $d=3)$. When choosing [math], man $i$ evaluates $\epsilon_{i1}-\epsilon_{i0}$ and $\epsilon_{i2}-\epsilon_{i0}$. Thus, given the definition of $\Delta\epsilon_{i}$ in (9), $\Delta\epsilon^{0}_{i}\equiv(\epsilon_{i1}-\epsilon_{i0},\epsilon_{i2}-\epsilon_{i0})$. Similarly, $\Delta\epsilon^{1}_{i}\equiv(\epsilon_{i1}-\epsilon_{i0},\epsilon_{i1}-\epsilon_{i2})$ and $\Delta\epsilon^{2}_{i}\equiv(\epsilon_{i2}-\epsilon_{i0},\epsilon_{i1}-\epsilon_{i2})$.

4.4.1 A linear program

As discussed above, $U\in\mathcal{U}^{*}$ if and only if

[TABLE]

Without parametric restrictions on the unobserved heterogeneity, (11) is an infinite-dimensional existence problem. In what follows, we use and extend Torgovitsky (2019) to transform (11) into a linear program. We illustrate the result in the easiest case where $r=2$ (hence, $d=3$). Although notationally more cumbersome, the result for a generic $r$ follows the same intuition and is illustrated in Appendix A.2.

The two type case ($\boldsymbol{r=2}$)

For simplicity, assume that $\Delta\mathcal{F}^{\dagger}=\Delta\mathcal{F}$. Hence, (11) can be more explicitly written as

[TABLE]

Note that, for each $x\in\mathcal{X}$, (12) depends on the values of $\Delta F_{x}$ at a finite number of $3$-tuples. We collect such $3$-tuples in the following three sets:

[TABLE]

where $\mathcal{A}_{x,1,U}$ collects the elements at which $\Delta F_{x}$ is evaluated along the first dimension, $\mathcal{A}_{x,2,U}$ collects the elements at which $\Delta F_{x}$ is evaluated along the second dimension, and $\mathcal{A}_{x,3,U}$ collects the elements at which $\Delta F_{x}$ is evaluated along the third dimension. We add $-\infty$ to each set because it will be key later to outline the defining properties of CDFs. Lastly, we define $\mathcal{A}_{x,U}\equiv\mathcal{A}_{x,1,U}\times\mathcal{A}_{x,2,U}\times\mathcal{A}_{x,3,U}$.

Thus, the infinite-dimensional existence problem (12) is equivalent to verifying whether there exists a finite-domain function $\Delta\bar{F}_{x}:\mathcal{A}_{x,U}\rightarrow[0,1]$ that satisfies the equations in (12) and that can be extended to a proper CDF $\Delta F_{x}:\bar{\mathbb{R}}^{3}\rightarrow[0,1]$, for every $x\in\mathcal{X}$. That is, (12) is equivalent to

[TABLE]

Importantly, observe that (14)-(16) are linear in $\Delta\bar{F}_{x}$. Further, we show below that (17) can be expressed as a finite collection of equations and inequalities that are also linear in $\Delta\bar{F}_{x}$. Therefore, we can transform (12) into a linear program.

We now explain how to write (17) as a finite collection of linear equations and inequalities. It is clear that $\Delta\bar{F}_{x}$ can be extended to a proper CDF $\Delta F_{x}$ only if

[TABLE]

Consider first the case where the support restriction (10) is ignored in the definition of $\Delta\mathcal{F}$. Then, based on Sklar’s Theorem (Sklar, 1959; 1996; Nelsen, 2006), Lemma 2 of Torgovitsky (2019) proves that (18) is also sufficient for extendibility. In particular, the defining properties of CDFs are:

(i) $\Delta\bar{F}_{x}(a_{1},a_{2},a_{3})=0$ for every $(a_{1},a_{2},a_{3})\in\mathcal{A}_{x,U}$ that has at least one component equal to $-\infty$. That is,

[TABLE]

(ii) $\Delta\bar{F}_{x}(a_{1},a_{2},a_{3})=1$ when $a_{l}=+\infty$ for every $l\in\{1,2,3\}$. That is,

[TABLE]

(iii) $\Delta\bar{F}_{x}$ is $3$-increasing. Formally, given a pair of $3$-tuples, $(a_{1},a_{2},a_{3}),(a_{1}^{\prime},a_{2}^{\prime},a_{3}^{\prime})$ in $\mathcal{A}_{x,U}$ with $(a_{1},a_{2},a_{3})\leq(a_{1}^{\prime},a_{2}^{\prime},a_{3}^{\prime})$, let $\mbox{Vol}_{\Delta\bar{F}_{x}}([a_{1},a_{1}^{\prime}]\times[a_{2},a_{2}^{\prime}]\times[a_{3},a^{\prime}_{3}])$ denote the volume of the $3$-box $[a_{1},a_{1}^{\prime}]\times[a_{2},a_{2}^{\prime}]\times[a_{3},a^{\prime}_{3}]$. $\Delta\bar{F}_{x}$ is called $3$-increasing if

[TABLE]

Note that (19)-(21) constitute a finite collection of equations and inequalities that are linear in $\Delta\bar{F}_{x}$. Therefore, if (10) was absent, we could rewrite (12) as a linear program by direct application of Lemma 2 of Torgovitsky (2019).

The presence of (10) slightly complicates our analysis as we need to extend the latter result to handle such a support restriction.151515Torgovitsky (2019) discusses how to handle support restrictions in the case of one-dimensional CDFs. We provide similar findings for multidimensional CDFs. To do so, we first rewrite (10) in a more convenient way. Specifically, note that (10) is equivalent to $\lambda_{\Delta F_{x}}(\mathcal{B}^{c})=0$ where $\mathcal{B}^{c}$ is the complement of $\mathcal{B}$ in $\bar{\mathbb{R}}^{3}$. Also observe that, when $r=2$, $\mathcal{B}$ is simply the plane

[TABLE]

Hence, $\mathcal{B}^{c}$ can be written as the union of the infinite collection of $3$-boxes that lie fully above or below such a plane. In particular, for each $(b_{1},b_{2},b_{3})\in\mathcal{B}$, one can construct a $3$-box $\mathcal{B}_{b_{1},b_{2},b_{3}}$ such that $\mathcal{B}^{c}=\cup_{(b_{1},b_{2},b_{3})\in\mathcal{B}}\mathcal{B}_{b_{1},b_{2},b_{3}}$. See Appendix A.1 for the representation of $\mathcal{B}_{b_{1},b_{2},b_{3}}$.

In turn, assuming $\lambda_{\Delta F_{x}}(\mathcal{B}^{c})=0$ is equivalent to assuming

[TABLE]

Therefore, it is clear that $\Delta\bar{F}_{x}$ can be extended to a proper CDF $\Delta F_{x}$ satisfying (22) only if the increasingness condition (21) holds as equality for every pair of $3$-tuples, $(a_{1},a_{2},a_{3}),(a_{1}^{\prime},a_{2}^{\prime},a_{3}^{\prime})$ in $\mathcal{A}_{x,U}$ with $(a_{1},a_{2},a_{3})<(a_{1}^{\prime},a_{2}^{\prime},a_{3}^{\prime})$, such that the $3$-box $[a_{1},a_{1}^{\prime}]\times[a_{2},a_{2}^{\prime}]\times[a_{3},a^{\prime}_{3}]$ is contained in a box $\mathcal{B}_{b_{1},b_{2},b_{3}}$ for some $(b_{1},b_{2},b_{3})\in\mathcal{B}$. Proposition 1 shows that such a condition is also sufficient for extendibility when combined with (19)-(21).

Proposition 1**.**

(Extendibility)

Given $U\in\mathcal{U}$ and $x\in\mathcal{X}$, let $\mathcal{A}_{x,U}\equiv\mathcal{A}_{x,1,U}\times\mathcal{A}_{x,2,U}\times\mathcal{A}_{x,3,U}$, where $\mathcal{A}_{x,l,U}$ is a finite subset of $\bar{\mathbb{R}}$ and contains $\{+\infty,-\infty\}$ for each $l\in\{1,2,3\}$. Let $\Delta\bar{F}_{x}:\mathcal{A}_{x,U}\rightarrow[0,1]$ be a function satisfying (19)-(21), and

[TABLE]

Then, there exists a proper CDF $\Delta F_{x}:\bar{\mathbb{R}}^{3}\rightarrow[0,1]$ such that: (i) $\Delta\bar{F}_{x}$ can be extended to $\Delta F_{x}$, i.e., $\Delta F_{x}(a_{1},a_{2},a_{3})=\Delta\bar{F}_{x}(a_{1},a_{2},a_{3})$ for each $(a_{1},a_{2},a_{3})\in\mathcal{A}_{x,U}$; (ii) $\lambda_{\Delta F_{x}}(\mathcal{B})=1$. ∎

Note that (23) constitutes a finite collection of equations and inequalities that are linear in $\Delta\bar{F}_{x}$. Therefore, by combining (14)-(16), (19)-(21), and (23), we can transform (12) into a linear program. By verifying the linear program for each $U\in\mathcal{U}$, we can obtain the sharp identified set.

In addition, we give a simple condition to verify if a box $[a_{1},a_{1}^{\prime}]\times[a_{2},a_{2}^{\prime}]\times[a_{3},a^{\prime}_{3}]$ is contained in a box $\mathcal{B}_{b_{1},b_{2},b_{3}}$ for some $(b_{1},b_{2},b_{3})\in\mathcal{B}$, as required by (23).

Lemma 2**.**

(Zero-volume boxes)

Given $U\in\mathcal{U}$ and $x\in\mathcal{X}$, let $\mathcal{A}_{x,U}\equiv\mathcal{A}_{x,1,U}\times\mathcal{A}_{x,2,U}\times\mathcal{A}_{x,3,U}$, where $\mathcal{A}_{x,l,U}$ is a finite subset of $\bar{\mathbb{R}}$ and contains $\{+\infty,-\infty\}$ for each $l\in\{1,2,3\}$. Take $(a_{1},a_{2},a_{3}),(a_{1}^{\prime},a_{2}^{\prime},a_{3}^{\prime})$ in $\mathcal{A}_{x,U}$ with $(a_{1},a_{2},a_{3})<(a_{1}^{\prime},a_{2}^{\prime},a_{3}^{\prime})$. Define the $3$-box $\mathcal{H}\equiv[a_{1},a_{1}^{\prime}]\times[a_{2},a_{2}^{\prime}]\times[a_{3},a^{\prime}_{3}]$. Then, $\mathcal{H}\subset\mathcal{B}_{b_{1},b_{2},b_{3}}$ for some $(b_{1},b_{2},b_{3})\in\mathcal{B}$ if and only if

[TABLE]

∎

As shown by Theorem 1 of Torgovitsky (2019), the above methodology remains valid under various classes of nonparametric restrictions on $\{\Delta F_{x}\}_{x\in\mathcal{X}}$, which can be simply imposed on $\{\Delta\bar{F}_{x}\}_{x\in\mathcal{X}}$ as linear constraints. In particular, in the simulations of Appendix C and the empirical application, we explore the identifying power of the following restrictions (not necessarily all maintained simultaneously):

Assumption 5**.**

(Nonparametric assumptions on $\{\Delta F_{x}\}_{x\in\mathcal{X}}$)

1. 5.1.

$\Delta\epsilon_{i}$ is independent of $X_{i}$. 2. 5.2.

Conditional on $X_{i}$ and for each $l\in\{1,...,d\}$, $\Delta\epsilon_{i,l}$ has a distribution symmetric at [math]. 3. 5.3.

Conditional on $X_{i}$, $\{\Delta\epsilon_{i,l}\}_{l\in\{1,...,d\}}$ are identically distributed. 4. 5.4.

Conditional on $X_{i}$, $\{\Delta\epsilon^{y}_{i}\}_{y\in\mathcal{Y}_{0}}$ are identically distributed.

∎

In Appendix A.3, we provide a formal statement of Theorem 1 of Torgovitsky (2019) and a list of nonparametric distributional assumptions on the taste shock differences that can be generally accommodated. Finally, Appendix A.4 contains an example of a linear program to solve.

4.4.2 Simplifying grid search

To construct $\mathcal{U}^{*}$, the analyst has to solve the linear program of Section 4.4.1 for every $U\in\mathcal{U}$. Typically, this is done in the partial identification literature by constructing a grid of points to approximate $\mathcal{U}$ and then repeating the exercise of interest for each grid point. However, the difficulty of implementing this approach increases with the size of the grid, which in turn, increases exponentially with $r$, quickly leading to a computational bottleneck. In what follows, we give a characterisation of $\mathcal{U}$ so that the issue of solving the linear program for every $U\in\mathcal{U}$ is reduced to solving the linear program for a handful of $U\in\mathcal{U}$. This mitigates the burden of doing grid search. We first provide an intuition of the result and then a more formal statement.

For simplicity, we continue the example of Section 4.4.1 with $r=2$ (hence, $d=3$). For any given $x\in\mathcal{X}$, the only pieces of the linear program that might induce different sets of solutions for different values of $U$ are the $3$-increasingness constraint, (21), and the support constraint, (23). In fact, note that (21) is activated only for the pairs of $3$-tuples in $\mathcal{A}_{x,U}$ that are componentwise comparable. Similarly, (23) is activated only for the pairs of $3$-tuples in $\mathcal{A}_{x,U}$ that are componentwise comparable and satisfy (24). We refer to such pairs as the critical pairs. Let $h\equiv\Pi_{l=1}^{3}h_{l}$ be the cardinality of $\mathcal{A}_{x,U}$. Fix an order of the $3$-tuples in $\mathcal{A}_{x,U}$ and list them in an $h\times 3$ matrix, $\alpha_{x,U}$. If the positions (i.e., row-indices) of the critical pairs in $\alpha_{x,U}$ are different across two values of $U$, then these two values of $U$ will induce potentially different sets of solutions to the linear program. Conversely, if the positions are the same, then the two values will induce the same (possibly, empty) set of solutions. This idea can be used to ex-ante partition the parameter space, $\mathcal{U}$, into equivalence classes so that the researcher can solve the linear program only once for each class.161616Torgovitsky (2019) suggests to ex-ante partition the parameter space in order to simplify grid search, even though no algorithm is provided.

In what follows, we provide some sufficient conditions to establish whether the positions of the critical pairs in $\alpha_{x,U}$ are equal across two values of $U$. For every $l\in\{1,2,3\}$, fix an order of the elements of $\mathcal{A}_{x,l,U}$ and list them in an $h_{l}\times 1$ vector, $\alpha_{x,l,U}$. Similarly, construct an $(h_{1}+h_{2}h_{3})\times 1$ vector, $\beta_{x,U}$, listing $\alpha_{x,1,U}$ and the sum of every possible element of $\alpha_{x,2,U}$ with every possible element of $\alpha_{x,3,U}$. Let $\Pi_{1}$ be the set of all possible permutations without repetition of $\{1,...,h_{l}\}$ and let $\Pi_{2}\equiv\{<,=\}^{h_{l}-1}$. Define the functions

[TABLE]

where $\pi_{1}(\omega)$ sorts the $h_{l}$ elements of $\omega$ from smallest to largest and reports their positions in the original vector; $\pi_{2}(\omega)$ reports the relational operators, $<$ or $=$, among the sorted elements of $\omega$. When $\omega$ contains multiple elements with the same value or indeterminate forms (like $+\infty-\infty$), then we can adopt any convention on which element to sort first. Lastly, let $\pi(\omega)\equiv(\pi_{1}(\omega),\pi_{2}(\omega))\text{ }\forall\omega\in\bar{\mathbb{R}}^{h}$. For instance, suppose $\omega=(100,99,+\infty)$. Then, $\pi(\omega)=\{(2,1,3),(<,<)\}$. Suppose $\omega=(5,5,-\infty)$. Then, $\pi(\omega)=\{(3,2,1),(<,=)\}$.

Definition 1**.**

($\pi$-ordering)

Take any $U,\tilde{U}\in\mathcal{U}$. $U$ and $\tilde{U}$ have the same $\pi$-ordering if

[TABLE]

∎

Proposition 2 shows that if $U,\tilde{U}\in\mathcal{U}$ have the same $\pi$-ordering, then the positions of the critical pairs in $\alpha_{x,U}$ are equal to the positions of the critical pairs in $\alpha_{x,\tilde{U}}$. Therefore, either both, $U$ and $\tilde{U}$, lie inside or outside the identified set, $\mathcal{U}^{*}$.

Proposition 2**.**

(Simplify grid search over $\mathcal{U}$)

Take any $U,\tilde{U}$ in $\mathcal{U}$ with the same $\pi$-ordering. Then, $U\in\mathcal{U}^{*}$ if and only if $\tilde{U}\in\mathcal{U}^{*}$. ∎

Remark B.1 in Appendix B explains how Proposition 2 is used in practice to ex-ante partition the parameter space.

5 Empirical application

In this section, we use our methodology to re-examine some of the questions considered in the empirical literature on the marriage market that have been previously answered by relying on the Logit 1to1TU model.

An important question is whether educational sorting has changed over time, as this can be key to understanding the sources of inequality in intergenerational outcomes (see references in Section 1). Detecting changes in educational sorting is challenging because it requires disentangling the effect of changes in the marginal probability distribution of education categories from potential structural changes in the match surplus. In fact, men and especially women have become more educated over time. This implies that individuals with higher education levels are mechanically more likely to marry. We are thus interested in capturing the changes in educational sorting after having accounted for the variations naturally arising due to distributional shifts in education.

The literature offers two approaches to do this. The first consists of using indices of sorting, based on comparing empirical matches to a random matching counterfactual (e.g., Eika, Mogstad, and Zafar, 2019). The second consists of using a structural model of the marriage market to estimate individual preferences. For instance, one can take the 1to1TU model with $\mathcal{X}=\mathcal{Y}\equiv\{1,...,r\}$ listing education categories and study the evolution of $D(\Phi)$, as discussed in Section 4.1. The second approach has been implemented by CSW based on the Logit assumption. Both approaches conclude that positive educational sorting has overall increased in the U.S. in the past decades, although there is some debate about this trend when we distinguish among education categories. For example, Eika, Mogstad, and Zafar (2019) find that positive educational sorting has declined among the highly educated and increased among the less educated. Instead, CSW find that positive educational sorting has increased particularly at the top of the education distribution. We use our methodology to investigate the robustness of the conclusions achieved via the structural Logit approach to the dropping of the Logit assumption.

In addition to studying the evolution of educational sorting, this section touches upon another important question in the empirical literature on the marriage market. In particular, as discussed by CSW, the increase in educational sorting makes a higher stock of human capital more valuable on the marriage market. Therefore, one should also expect an increase in the marital education premium, especially at the highest levels of education and for women. Based on the Logit 1to1TU model, CSW empirically confirm such a prediction for the U.S. We apply our methodology to verify whether the same findings can be achieved without the Logit assumption.

The remainder of the section is organised as follows: in Section 5.1, we describe the data; in Section 5.2, we present and interpret our results.

5.1 Data

We focus on the U.S. marriage market and take our data from the American Community Survey, which is a representative extract of the census. To construct the final dataset, we follow the steps outlined in Section I.A and Appendix B of CSW. In particular, from the $21,583,529$ households in the 2008 to 2014 waves, we take all white adults out of school. We record the education level of each adult by distinguishing four categories: high school dropouts (HSD, or “1”); high school graduates (HSG, or “2”); some college (SC, or “3”); four-year college graduates and graduate degrees (CG, or “4”).171717For the white population, CSW further distinguish between four-year college graduates (CG, or “4”) and graduate degrees (CG+, or “5”). In the Logit case, the main conclusions remain unchanged even without this distinction, as shown in Chiappori, Costa-Dias, and Meghir (2020). In our analysis too, the conclusions do not change when distinguishing between CG and CG+, as we remark in Section 5.2. We treat individuals as married if they define themselves as such, without including cohabitation. We focus on first marriages and never-married singles. The final sample consists of $1,502,157$ couples and $136,052$ singles.

We define cohorts by using year of birth and take women to be one year younger. For instance, cohort 1940 includes all men born in year 1940 and all women born in year 1941. In turn, the sample analogue of $p^{1940}_{y|x}$ is computed by taking the ratio between the number of men of type $x\in\mathcal{X}$ who are born in year 1940 and marry a woman of type $y\in\mathcal{Y}_{0}$ born in any year, and the number of men of type $x\in\mathcal{X}$ who are born in year 1940. Similarly, the sample analogue of $p^{1940}_{x|y}$ is computed by taking the ratio between the number of women of type $y\in\mathcal{Y}$ who are born in year 1941 and marry a man of type $x\in\mathcal{X}_{0}$ born in any year, and the number of women of type $y\in\mathcal{Y}$ who are born in year 1941.181818We ignore the issue of cohort mixing to exactly mimic the data construction process of CSW and make our conclusions as comparable as possible. In particular, given that the modal age difference within couples is one year in the data, CSW concentrate their analysis on couples in which the age difference takes one year. In what follows, we consider $28$ cohorts, from 1940 to $1967$, as in CSW.

Figures 1 and 2 below are similar to Figures 1 and 2 of CSW and provide some key descriptive facts. Figure 1 reveals that the proportion of college educated men increases until 1950, then drops, and finally reverses into an increase around 1960. Instead, the proportion of college educated women always increases. Moreover, the proportion of college educated women is lower than that of men in 1940, while the opposite is true by 1967. These changes imply that the evolution of educational sorting cannot be inferred by simply comparing matching patterns across cohorts.

Figure 2 (a) shows an increase in the proportion of marriages of like with like. A substantial surge is also registered when focusing on the proportion of couples where both spouses have a college degree, as shown in Figure 2 (b). However, these figures are not proof of an increase in positive educational sorting because they may be mechanically driven by changes in the proportions of individuals in each education category.

5.2 Results

For each of the 28 cohorts, we estimate the identified sets of $\Phi$, $D(\Phi)$, $C(U)$, and $C(V)$ under two classes of nonparametric distributional assumptions on the taste shocks, which we refer to as specifications [A] and [B]. Specification [A] imposes Assumption 5.4. Specification [B] imposes Assumptions 5.2, 5.3, and 5.4. According to our simulations in Appendix C, such specifications tend to deliver the tightest bounds among the various combinations of assumptions explored.

We start by discussing the results on educational sorting. The Logit estimates of $D(\Phi)$ are positive, suggesting the presence of positive educational sorting in each education category and cohort. In particular, Figure 3 plots the Logit estimates of $D(\Phi)$ demeaned over cohorts (blue curves). If educational sorting has not changed over time, then the blue curves (and the smooth violet curves representing trends) should be identical to the horizontal [math] line. The property is violated for the highly educated, as the trend for $D_{44,33}(\Phi)$ is increasing in the most recent decades.191919As discussed in Section 4.1, there is positive educational sorting among more educated people if $D^{\ell}(\Phi)\equiv D_{44,33}(\Phi)>0$ and among less educated people if $D^{\ell}(\Phi)\equiv D_{22,11}(\Phi)>0$. Further, positive educational sorting increases across cohorts among more educated people if $D_{44,33}(\Phi)$ increases across cohorts. Similarly, positive educational sorting increases across cohorts among less educated people if $D_{22,11}(\Phi)$ increases across cohorts. We can thus conclude that there has been an increase in positive assortativeness, at least among the highly educated, under the Logit assumption. More formally, based on the test described in Section IV.A of CSW, the null hypothesis that educational sorting has not changed over time is rejected: the Chi-squared test statistic has value $1,047.725$ with 243 degrees of freedom and the p-value is below $10^{-99}$.202020When distinguishing between CG and CG+, the conclusions on educational sorting based on the Logit assumption are similar, as shown by Figure 15 of CSW and subsequent discussion. In particular, the null hypothesis that educational sorting has not changed over time is rejected also with 5 types (the Chi-squared test statistic has value $1,573.717$ with $432$ degrees of freedom and the p-value is below $10^{-100}$).

Figure 4 reports our estimates of the identified set of $D(\Phi)$, under specifications [A] (blue region) and [B] (dotted region).212121We do not demean the estimates in Figure 4 in order to study their signs. By construction, the dotted region is contained in (or is equal to) the blue region. Further, the Logit estimates of $D(\Phi)$ (dark blue line) are contained in the blue and dotted regions because Assumptions 5.2, 5.3, and 5.4 are satisfied when imposing the Logit assumption. As in CSW, we obtain our estimates by assuming that the cohorts feature independent matching processes. However, our analysis is more robust in many ways. Importantly, we allow the taste shocks to have any distribution within specifications [A] and [B]. For instance, the taste shocks can be correlated among each other, their distribution may freely vary across education categories, and there could be heteroskedasticity.

Figure 4 reveals that, under the classes of nonparametric distributional assumptions considered, the 1to1TU model is uninformative about the presence and trend of positive educational sorting among the highly educated, as the estimates of $D_{44,33}(\Phi)$ are unbounded above and below.222222The estimates are unbounded when the blue or dotted region hits the vertical axis limit. We find the presence of positive educational sorting among the less educated, as indicated by the jump to positive values of the lower bound of $D_{22,11}(\Phi)$ around 1954. However, once the lower bound reaches the positive values, it does not exhibit any clear trend, thereby remaining inconclusive about the evolution of positive educational sorting among the less educated. These results suggest that the previous findings on educational sorting based on the Logit 1to1TU model are driven by the Logit assumption.232323When distinguishing between CG and CG+, our estimates of $D_{55,44}(\Phi)$, $D_{55,33}(\Phi)$, and $D_{44,33}(\Phi)$ are unbounded above and below. Therefore, the 1to1TU model still does not allow us to conclude anything about the presence and trend of educational sorting among the highly educated, as in Figure 4.242424Figures D.1 and D.2 in Appendix D further disentangle the men and women’s contribution to $D(\Phi)$. They highlight that the unboundedness of $D_{22,11}(\Phi)$ (above) and $D_{44,33}(\Phi)$ (above and below) is mostly driven by the limited empirical content of the 1to1TU model on the women’s side.

Table 1 confirms the above conclusions. The first section of the table reports the projections of the estimated identified sets of $\Phi$, averaged over cohorts 1940, 1941, and 1942 (“early cohorts”), under specifications [A] and [B]. The second section of the table reports the projections of the estimated identified sets of $\Phi$, averaged over cohorts 1965, 1966, and 1967 (“late cohorts”), under specifications [A] and [B]. The last section of the table reports the changes in estimates between early and late cohorts. The average estimates of $\Phi$ under the Logit assumption (“Logit”) are also included. When using the Logit assumption, the decline in surplus is always smaller (or inverted) for more educated couples, which is in line with the increase in positive educational sorting at the top of the distribution seen in Figure 3. This conclusion cannot be confirmed once the Logit assumption is relaxed, as highlighted by the many unbounded intervals.252525The Logit estimates in Table 1 for early and late cohorts are numerically different from the Logit estimates in Table 6 of CSW due to two reasons. First, CSW construct those estimates by using the assumptions that the evolution of the systematic match surplus is driven by education-specific drifts, which is not assumed here. Second, CSW distinguish between CG and CG+. Nevertheless, the changes in the Logit estimates between early and late cohorts that we obtain (last section of the table) are very close to CSW’s findings and, importantly, suggest the same conclusions.

We now move to discuss the results on the marital education premia. The black curves in Figure 5 are the estimates of the marital education premia under the Logit assumption for men and women. Panels (c) and (f) suggest that the marital college premium has increased for both men ($\Delta_{43}(U)$) and women ($\Delta_{43}(V)$). The increase is particularly pronounced for women. Further, while women of older cohorts had a negative marital college premium, this has become positive for recent cohorts.262626When distinguishing between CG and CG+, the conclusions on the marital education premia based on the Logit assumption are similar, as shown by Figures 20 and 21 of CSW and subsequent discussion. The blue curves in Figure 5 are the estimates of $C(U)$ and $C(V)$ under the Logit assumption. The blue curves mimic the trends of the black curves closely, although they are quite shifted from the black curves in panels (c) and (f).

Figure 6 reports our estimates of the identified sets of $C(U)$ and $C(V)$ under specifications [A] (blue region) and [B] (dotted region). Observe that we obtain unbounded intervals in almost every cohort. This is particularly true for the women’s side, where the estimates of $C_{21}(V)$, $C_{32}(V)$, and $C_{43}(V)$ remain constantly unbounded above and below.272727We obtain the same results when distinguishing between CG and CG+. In particular, the estimates of $C_{21}(V)$, $C_{32}(V)$, $C_{43}(V)$, and $C_{54}(V)$ remain constantly unbounded above and below. In turn, the estimates of the marital education premia will be unbounded as well, and nothing can be said about their evolution over time. As earlier, this indicates that the previous evidence on increasing marital college premium based on the Logit 1to1TU model is a consequence of the Logit assumption. No evidence of an increase in the marital education premia has also been recently found by Christensen and Connault (2022) using a different methodology.

6 Conclusions

This paper investigates the identifying power of the 1to1TU model for the systematic match surplus and related policy-relevant quantities when no parametric distributional assumptions are imposed on the unobserved heterogeneity. We conclude our analysis by highlighting three main findings. First, we formally show that the 1to1TU model contains no information about the systematic match surplus without restricting the distribution of the unobserved heterogeneity. Second, we propose a computational approach for constructing the identified set of the systematic match surplus that is based on principles of linear programming and works under various classes of nonparametric distributional assumptions on the unobserved heterogeneity. Third, we use our methodology to re-examine some relevant questions in the empirical literature on the marriage market, which have been previously studied under the Logit assumption. Our estimates show that, without parametric distributional assumptions, the 1to1TU model is inconclusive about the evolution of educational sorting and marital education premia across cohorts. Therefore, most of the previous evidence on increasing positive educational sorting and marital college premium is likely to be driven by the Logit assumption. Our paper illustrates the usefulness of partial identification approaches in testing the robustness of empirical results based on strong parametric assumptions.