Alternative Methods to Test for Positive
Assortative Mating
Hyunjae Kang
Kyoto University

Steven Stern
Stony Brook University
August 2023
Abstract

We propose easy econometric methods to test for positive assortative mating. Compared to existing methods such as by Siow (2015), ourprocedure avoids the cost associated with imposing positive assortative
mating restrictions, and it allows for more general tests. The test does
not require numerical optimization, and the test statistics can be calculated easily. We compare the power of the tests using simulation. The
empirical application suggests that the null hypothesis of no assortative
mating is rejected for selected years of CPS data that range from 1962 to
2019.

1

Introduction

This paper proposes easy methods to test for positive assortative mating. A
number of empirical studies on sorting in the marriage market have investigated
the intensity of assortative mating and how the assortativeness of mating haschanged over time. Only a handful of econometric tests have been developed
for testing positive assortative mating (Chiappori, Salanié, and Weiss, 2011;
Anderson and Teng, 2013; Siow, 2015; Gualdani and Sinha, 2023).
A strand of literature on marriage matching market is initiated by Chooand Siow (2006b). They propose a marriage matching function that is based on
the transferable utility matching framework with random preferences.1 ;2 The
critical assumptions of Choo and Siow (2006b) are (a) additive separability ofjoint surplus, (b) large marriage market, and (c) extreme value distribution of
the shock (Galichon and Salanié, 2021). Choo and Siow (2006a) applied the
1 See Chiappori and Salanié (2016) for the survey of the empirical marriage matching models under di¤erent assumptions: transferable utility, non-transferable utility, and imperfectlytransferable utility models.
2 See Fox (2010), Fox (2018), and Fox, Yang, and Hsu (2018) for an alternative method employing a maximum score approach based on the rank-order property associated with sorting.
Suen and Liu (1999) develop an empirical model of spouse selection to examine complementarity in the marital production function.

1

marriage matching function to quantify the e¤ect of the legalization of abortion
on gains to marriage.
Based on the transferable utility framework, papers in the literature extend
Choo and Siow (2006b) in di¤erent directions. Extending the structural modelof Chiappori, Iyigun, and Weiss (2009), Chiappori, Salanié, and Weiss (2017)
build a marriage matching model with parental investment and embed it into
the Choo and Siow (2006b) marriage matching function. They measure thechanges in assortative mating by estimating the returns to education in the US
marriage market. Mouri…é and Siow (2021) incorporate peer e¤ects into themarriage matching function of Choo and Siow (2006b). Dupuy and Galichon(2014) suggest how to amend the Choo and Siow (2006b) model for continuous attributes based on the continuous choice framework developed by Dagsvik(1994) and Dagsvik (2000). Galichon and Salanié (2021) show that the extreme
value assumption of Choo and Siow (2006b) can be relaxed. Fernández, Guner,
and Knowles (2005) construct a search model with positive assortative mating
as an outcome. Instead of testing for positive assortative mating, they wantto estimate the relationship between the degree of positive assortative mating
and other variables of interest (e.g. wage gaps between skilled and unskilledworkers).
The literature on worker-…rm matching deals with a di¤erent setting but
shares a similar interest in that it investigates the strength and the directionof assortative mating between two agents (workers and …rms). Using employeremployee matched datasets, Abowd, Kramarz, and Margolis (1999) estimate thecorrelation between worker …xed e¤ect and the …rm …xed e¤ect from wage equations. Pointing out that wage does not represent the underlying types such as
job amenities, papers in this literature explore the correlation between types ina di¤erent way. One set of papers builds and estimates equilibrium search models and investigate the sign of sorting (Lise, Meghir, and Robin, 2016; Bagger
and Lentz, 2019). This approach provides a consistent behavioral framework
but comes with the cost of specifying a potentially complicated model. On
the other hand, another set of papers presents a simpler method of testing for
sorting at the expense of more restrictive assumptions. For example, Mendes,
van den Berg, and Lindeboom (2010) assume that the observed skill level canidentify the worker’s type. Then they estimate the correlation coe…cients of the…xed e¤ects based on their production function equations. Taking into accountfor the noise in the worker wage rankings, Bartolucci, Devicienti, and Monzón(2018) propose a test for the sign of sorting between workers and …rms.
Few papers develop econometric tests for positive assortative mating. Siow(2015) proposes a test for positive assortative mating based on maximum likelihood estimation. The test statistics are based on the theoretical framework of
Choo and Siow (2006b), and the test procedure requires numerical optimization
with di¤erent restrictions. Also, their test statistics have non-standard asymptotic distributions. Our method does not require numerical optimization andcan easily be calculated.
Suppose there are I types of men, i = 1; 2; ::; I, and J types of women,
j = 1; 2; ::; J, where types are vertically ordered. The likelihood ratio test of2Siow (2015) mainly uses two kinds of restrictions to de…ne the null hypothesis:
one imposes the positive assortativeness restriction for all (i; j)-type pairs, andthe other restriction solely focuses on the preference for own types (TP23 andDP24 ). It is not clear which of the restrictions in Siow (2015) are more suitable
to assess positive assortative mating.5 For example, the log-likelihood with
the TP2 restriction is rejected against the unrestricted likelihood, but the loglikelihood with the DP2 restriction is not rejected. Siow (2015) states thatthe rejection of the TP2 likelihood is driven by the couples whose completed
education is distant. As the extreme matches account only for less than 0:2%
of marriages, TP2 restriction is arguably sensitive to outliers.6 Our method
requires only the unrestricted estimates of the log odds ratios. Note that thisalso means that the researcher does not have to choose among the appropriate
restrictions to impose. Also, it exploits all types but is not as restrictive as
imposing TP2.
We describe our method in Section 2. In Section 3, we provide empiricalexamples using Current Population Survey samples between 1962 and 2019.
We conclude in Section 4.

2

Alternative Tests

Let nij be the number of observations where a type-i male, iP= 1; 2; ::; I ismatched with a type-j female, j = 1; 2; ::; J, and de…ne N = ij nij as thetotal number of sampled marriages. Let pij be the probability that a randomly
sampled marriage is of type ij. Siow (2015) states that the unrestricted log
likelihood function for this problem isXL =nij log pij
i;j

st 1

=

X

pij :

ij

3 De…ne the (i; j) local log odds ratio as

log(i; j) + log(i + 1; j + 1)

log

(i; j + 1)

log

(i + 1; j) ;where ( ) is the proportion of the (i; j) pairs in the population. TP2 is when all local log
odds ratios are positive of order 2 usually denoted as “ 2 .” For two random variables A; B,
we say that A 2 B i¤
Z x
[Pr (B > t) Pr (A > t)] dt 0 8x
1

and 9x : the inequality in the equation is positive. See, for example, Shaked and Shanthikumar (2007).
4 DP2 is when all log odds ratios of the same types (i = j) are positive.
5 Chiappori, Salanié and Weiss (2017) also propose a test based on their structural modelthat incorporates returns to parental investment. Their method requires using minimum
distance of the moment conditions that come from their structural model. The main purposeof this test is to examine the changes in the intensities of educational homogamy.
6 Anderson and Teng (2013) develop a test statistic very similar to the TP2 test in Chooand Siow (2006b).

3

The unrestricted MLE of p is pbu = pbuij with pbuij = nij =N . Siow (2015) andothers cited by Siow (2015) argue that the way to test for positive assortative
mating is to also estimate pb under the restrictions imposed by positive assortative mating and then perform a likelihood ratio test. The null hypothesis in
Siow (2015) is a region with positive measure, and the alternative hypothesis isa region with positive measure. To see this, consider a simple case where there
are just two types of men and two types of women. The parameter space is theunit square, and the condition for positive assortative mating is
p11 + p22

p12

p21

p11 + p22

1:0which simpli…es to(1)

The region consistent with the null hypothesis is all values of (p11 ; p22 ) thatsatisfy equation (1), and the region consistent is the part of the unit squarethat does not satisfy equation (1). This is a quite uncommon problem in theliterature. Meanwhile, our testing problem is standard: the null hypothesis isa point, and the alternative is a region surrounding the point.
A similar example of interest in the literature is testing a unit root hypothesis for a time series. Almost all of the literature on testing for a unit root
assumes that the null hypothesis is nonstationarity. A strong reason for making this assumption instead of stationarity being the null hypothesis is that,when nonstationarity is the null hypothesis, it can be represented by a single
AR(1) parameter equalling 1. Thus, the situation is turned into a standard
case where the null hypothesis corresponds to a point.
However, from an
economic point of view, the null hypothesis should be stationarity in that themodel behaves better when there is stationarity (we would like to not reject thenull hypothesis and then rely on the nice properties of stationary processes).
Alternatively, Kwiatkowski et al. (1992), Leybourne and McCabe (1994), andXiao (2001) all use a null hypothesis of stationarity by constructing a time series model with a random walk where the random walk is the only source ofnonstationarity. The null hypothesis is then that the variance of the random
walk error is equal to zero.
A second example, much closer to our problem, is to test whether the Slutsky
substitution matrix is negative semide…nite. Gill and Lewbel (1992) choose anull hypothesis of semide…niteness (which has positive measure) and show how
to construct a consistent test statistic. Gill and Lewbel (1992) state that theasymptotics of this problem are very similar to that in Kodde and Palm (1986)
where the null and alternative hypotheses have positive measure.7 An easierapproach is to construct a Wald test, requiring only the unrestricted estimates.8
This avoids the cost associated with imposing the positive assortative mating
restrictions, and it allows for more general tests.
7 Unfortunately, Cragg and Donald (1996, 1997) show that the asymptotics for a related
rank condition are incorrect, and neither paper addresses the correctness of the argument ofGill and Lewbel (1992) about testing for semide…niteness.
8 It’s not clear which should be considered the restricted model and which the alternative.

4Researchers have developed di¤erent methods to test hypotheses related toassortative mating, although not all directly focus on assessing the positivity ofassortative mating. For instance, Chiappori, Salanie, and Weiss (2017) focuseson the change of the degree of educational assortative mating over time. They
reject the null hypothesis that the degree of educational assortative mating has
remained constant over time. Mouri…é and Siow (2021) propose a Cobb-Douglas
marriage matching function and emphasize the signi…cance of peer and scale effects in marital matching. Gualdani and Sinha (2023) employ a set identi…cation
approach with nonparametric distributional assumptions on unobserved heterogeneity. Their …ndings indicate the presence of positive educational sorting
among less educated couples.

2.1

Wald Test

In the next two subsections, we develop a Wald test, a likelihood ratio test,
and a Lagrange multiplier test. Starting with the Wald test, the condition for
positive assortative mating is9
Rij = (log pij + log pi+1;j+1 )(log pi+1;j + log pi;j+1 ) > 0 8ij;

and the proposed test statistic for testing for positive assortative mating isXT =log pbuij + log pbui+1;j+1
log pbui+1;j + log pbui;j+1

(2)

i;j

where a superscript u indicates “unrestricted.”
One might worry that the distribution of T under either the null or alternative would be di¢ cult to evaluate. However, the same is true of Siow (2015)
because the restrictions are inequalities instead of equality restrictions (Kudo,
1962; Gourieroux, Holly, and Monfort, 1982; Kodde and Palm, 1986; Wolak,1989). Siow (2015) uses parametric bootstrapping to simulate critical values.
The same can be done for T . Under positive assortative mating, T > 0, and,with total positive assortative mating (TP2 in Siow, 2015), T ! 1 because
pi+1;j pi;j+1 = 0. Under an alternative speci…cation, random sorting (Agresti,
2002), where log pij = log xi + log yj , using the de…nition of Rij in equation (2),
Rij = (log xi + log yj + log xi+1 + log yj+1 )(log xi+1 + log yj + log xi + log yj+1 )0:

One can simulate a critical value bc such that
9 Note that R

ij > 0 8ij )Pr [T > bc j Rij = 0 8ij] = :log pij + log pi+k;j+k > log pi+k;j + log pi;k+258k:Note that one could test the hypothesis that Rij = 0 8ij with power against
the alternative of (positive or negative) assortative mating as a Wald test with
b2
XR
ij
s2ij

i;j

where

s2ij

2
bij
R
= log pbuij + log pbui+1;j+1= V arlog pbuij + log pbui+1;j+1

2
K

log pbui+1;j + log pbui;j+1 ;log pbui+1;j + log pbui;j+1(1 pij ) (1 pi+1j+1 ) (1 pi+1;j ) (1 pi;j+1 )
+
+
+
;
N pij
N pi+1j+1
N pi+1;jN pi;j+1
using a …rst-order Taylor series approximation, andK = (I

2.2

1) (J

1) :

Likelihood Ratio Test and Lagrange Multiplier Test

Also note that the MLE of p under the restriction of random mating is pbr =pbrij where
P
P
nij
j nij
r
; ybj = i:pbij = xbi ybj ; xbi =NNThus, one could test the null hypothesis of random mating against a generalalternative with a likelihood ratio (LR) test,
2
3
X
X
TLR = 2 4nij log pbuij
nij log pbrij 5
i;j

i;j

2
3
X
pbuij
= 24
nij log r 5
pbij
i;j
2
3
X
pbuij
= 24
nij log P
P u 5
u )
(
p
b
bkj
i;j
k ik
kp

(which requires only estimation of pbu ) with
K = MF

2
K

1

which is larger than the degrees of freedom in for the Wald test. This di¤erence
exists because, for the Wald test, one loses degrees of freedom at m = M andf = F which does not happen for the LR test.
Alternatively, one could construct a Lagrange Multiplier (LM) test. LetXL=
nij log pij :i;j6Then,@L@pij

jpij =bprij =

=

P

=

nij

k nik
N

P

nij
pbrij

k nkj

N

N pbuijP
P u( k pbuik )bkj
kp

implying that the LM has the same form and the same distribution as the LR
test.

3

Distribution of Test Statistics

In order to simulate the distribution of our test statistics, we start with a modelof assortative mating. De…ne a man as having an integer value of xm between
1 and 10, and de…ne a woman as having an integer value of xw between 1 and
10. Assume that the density of matches in an economy is10 ;11f (xm ; xw ) = P10

x0m =1

exp f
P10

x0w =1

jxm

exp f

xw jg

jx0m

x0w jg:Figure 1 shows the contour of f (xm ; xw ) as (xm ; xw ) moves from (1; 10) to(10; 1) for = 0:1, 0:4, and 0:7. The larger , the more assortative mating
there is.
Figure 2 shows the distributions of the Wald test statistics for samples of500, 1; 000, and 10; 000 when = 0 along with the 281 distribution.12 The 281
distribution exhibits less variance than the Wald test statistics for samples of atleast 1; 000, and it stochastically dominates the Wald test statistic distribution
for a sample of 500. The 5% critical value for the 281 distribution is o¤ by ameaningful amount relative to the simulated Wald test statistic distributions.
The 5% critical value for a test statistic with a 281 distribution is 103:0, while
the Wald test statistics with sample sizes 500, 1; 000, and 10; 000 are respectively 99:0, 113:9, and 114:0. The di¤erence between the two occurs because
the simulated Wald test statistic provides the small-sample distribution of thetest statistic, while the simulated LR test relies signi…cantly on asymptotic approximations.
Figure 3 shows the distributions of the LR test statistic for samples of 500,
1; 000, and 10; 000 when
= 0 along with the 299 distribution. The 299
distribution stochastically dominates the LR test statistic distributions for all
three sample sizes. The 5% critical value for the 299 distribution is o¤ by
P10
P10
1 0 Note that
xm =1
xw =1 f (xm ; xw ) = 1.
1 1 Also, note that this somewhat like a triangle kernel function withbandwidth (Kokonendji and Zocchi, 2010).
1 2 The degrees of freedom are (M
1)(F

1).

7

=b

1 where b is theDensity from (1,10) to (10,1)
0.045
0.04
0.035
0.03
0.025
Alpha=.1

Density
0.02

Alpha=.4
Alpha=.7

0.015
0.01
0.005
0
-10

-8

-6

-4

-2

0

2

4

6

8

10

Difference in Coordinates

Figure 1: Density from (1,10) to (10,1)

Figure 2: Distribution of Pseudo-Wald Test Statistic for

8=0Distribution of LR Test Statistics for a = 0
1
0.9
0.8
0.7
0.6
n=500

Cumulative
0.5
Distribution

n=10000.4

n=10000

0.3

Chi-sq Distn(99)

0.2
0.1
0
0

50

100

150

200

LR Test Statistic

Figure 3: Distribution of LR Test Statistic for=0

a meaningful amount relative to the simulated LR test statistic distributions.
The 5% critical value for a test statistic with a 299 distribution is 123:2, while,for the LR test statistic distributions with sample sizes 500, 1; 000, and 10; 000,
they are respectively 111:3, 107:3, and 107:4.
Figure 4 shows the power functions for both test statistics for di¤erent valuesof using the 5% critical values under the null hypothesis of = 0. The Wald
test statistic has weak power for sample sizes less than 10; 000, and the LRtest statistic has very strong power. Part of the good performance of the
LR test statistic relative to the Wald test statistic is due to it having moredegrees of freedom. Also, the LR test statistic distribution is dominated by
the 299 distribution while the Wald test statistic is not dominated by the 281
distribution; i.e., the LR test statistic has more power than implied by its 299
asymptotic distribution while the Wald test statistic does not have more power
than its 281 asymptotic distribution.

4

Empirical Example

4.1

Wage Equations

Carroll, Kang, and Stern (2021) collect data from 18 Current Population Survey(CPS) samples between 1962 and 2019. For each sample, they estimate log wage
rate equations using Heckman (1979), then compute log predicted wages using
the inverse Mills’ratio ( ) = [1( )] for observed log wages and( )= ( )for unobserved log wages, and then measure the amount of assortative matingby log predicted wage rates among married couples. They also measure theamount of assortative mating by level of education. Among other analyses,
they use the Wald test statistic and the LR test statistic presented in this paper
to test for the existence of positive assortative mating. Test results are reported

9Power
1
0.9
0.8
0.7
0.6

Wald Test, n=500
Wald Test, n=1000

0.5

Wald Test, n=10000

0.4

LR Test, n=500

0.3

LR Test, n=1000

0.2
0.1
0
0

0.2

0.4

0.6

0.8

1

1.2

Alpha

Figure 4: Powerin Table 1. As can be seen in the table, the LR test statistics are an order ofmagnitude larger than the Wald test statistics, both for log predicted wages andyears of education. However, for all four test statistics, the null hypothesis of
no assortative mating is rejected at any reasonable signi…cance level for every
year of CPS data in favor of the existence of positive assortative mating.

Table 1: Assortative Mating Test Statistics
Year
1962
1967
1970
1972
1975
1978
1980
1985
1990
1995
2000
2005
2007
2010
2014
2015
2018
2019
# CellsDF

log Predicted Wage
Years of Education
Wald Test
LR test Wald Test
LR test525.5
7314.02443.65642.5773.3
6298.72981.75968.1604.15317.94453.58447.1497.97318.33952.18404.5269.1
4329.73456.17756.3318.5
2382.03857.48507.5672.93119.25022.210139.1
871.8
3101.75121.09977.7564.95818.56061.810134.3
451.1
5098.84344.78250.6544.95764.13799.67642.9631.38087.45372.711930.9654.3
7502.85108.611536.9807.47046.44525.010983.2
1004.08168.83654.210317.1
935.5
7930.83946.910395.8
1043.65283.82964.68854.51183.95208.02897.48604.3
10
5
81
99
16
24

10

4.2

Education

As an additional exercise, we use the Census 1970 and 2000 datasets, whichwere employed in Siow’s (2015) analysis. The results of the tests are presented
in Table 2. It is worth noting that the LR test statistics yield smaller values
compared to the Wald test statistics. Across all four test statistics, the null
hypothesis of no assortative mating is rejected at a signi…cant level for each yearof the Census data, providing evidence in support of the presence of positive
assortative mating.

Table 2: Test Resultsfor Siow (2015)
Year

Wald Test LR test
1970 1,664,919
90,7582000 5,906,027 186,646
#cell
5
df
16

5

Conclusion

We develop empirical tests on positive assortative mating. Our tests are straightforward to use and do not require numerical optimization, which o¤er advantages
over existing tests on positive assortative mating. The empirical applications
using CPS and Census data suggest that the null hypothesis of no assortative
mating is rejected for selected years.

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