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Marital Sorting, Household Inequality and Selection
\title{\textbf{Marital Sorting, Household Inequality and Selection}\thanks{
We are grateful to David Card and Stella Hong for useful comments and to
Richard Blundell, Bram De Rock and Costas Meghir for several important
references.}}
\author{Ivan Fern\'andez-Val\thanks{
\ Boston University.} \and Aico van Vuuren\thanks{
\ University of Groningen and Gothenburg University} \and Francis Vella
\thanks{
\ Georgetown University}}
\maketitle
\begin{abstract}
Using CPS data for 1976 to 2022 we explore how wage inequality has evolved
for married couples with both spouses working full time full year, and its
impact on household income inequality. We also investigate how marriage
sorting patterns have changed over this period. To determine the factors
driving income inequality we estimate a model explaining the joint
distribution of wages which accounts for the spouses' employment decisions.
We find that income inequality has increased for these households and
increased assortative matching of wages has exacerbated the inequality
resulting from individual wage growth. We find that positive sorting
partially reflects the correlation across unobservables influencing both
members' of the marriage wages. We decompose the changes in sorting patterns
over the 47 years comprising our sample into structural, composition and
selection effects and find that the increase in positive sorting primarily
reflects the increased skill premia for both observed and unobserved
characteristics. \thispagestyle{empty}
\end{abstract}
\section{Introduction}
\addtocounter{page}{-1} While most empirical studies evaluate inequality via
an examination of individuals' wages or earnings (see, for example, Katz and
Murphy 1992, Murphy and Welch 1992, Juhn, Murphy, and Pierce 1993, Welch
2000, Autor, Katz, and Kearney 2008, Blau and Kahn 2009, Acemoglu and Autor
2011, Autor, Manning, and Smith 2016, Murphy and Topel 2016, and Fern\'{a}
ndez-Val et al. 2023a, b), for many individuals their household's income is
more determinative of their economic welfare. As households are composed in
a variety of ways, it is useful for policy purposes to analyze income
inequality for different household types. One type of particular interest is
married couples comprising individuals both working full time full year
(FTFY). Although the proportion of husbands and wives in the total
population aged between 24 and 65 years decreased from 77.7\% to 58.2\% from
1976 to 2022, this group represents an increasingly larger share of married
couples. In 1976 26.2\% of married couples in which both the husband and the
wife were aged between 24 and 65 years had both members working FTFY and by
2022 this number increased to 51.3\%. As the percentage of married males in
FTFY employment increased from 82.8 to 84.1 while that for married females
went from 31.1 to 56.3, the growth of this group reflects the increasing
employment rates of married females.
Examining the patterns and determinants of income inequality across FTFY
married couples is interesting from a number of perspectives. First, studies
on wage and income inequality generally do not distinguish between married
and unmarried individuals. It is useful to examine if inequality among the
married shows similar patterns to those of their unmarried counterparts and
married individuals with non-working spouses. Second, as household income
combines the earnings of both spouses, positive sorting on wages will
exacerbate inequality while negative sorting will mitigate it. Finally, the
large increase in the participation rates of females, with its implications
for selection bias, has been shown to affect females' wages and income
inequality (see, for example, Mulligan and Rubinstein 2008, Blau, Kahn,
Boboshko, and Comey 2021 and Fern\'{a}ndez-Val et al. 2023b). While
selection bias has been largely ignored for males, the existing evidence
indicates it is not economically important. However, as increasing married
female participation rates increase the sample of males in FTFY married
couples, female selection into FTFY employment may have implications for the
wage and earnings distributions of married males with working spouses.
There is a vast literature, employing a variety of methodological
approaches, on the relationship between marriage, sorting and inequality and
it provides mixed evidence on sorting behavior. Kremer (1999) documents that
marital sorting on education in the United States, measured by the
correlation between spouse's education levels, had declined over the period
1940-1990 and via a calibration exercise finds that marital sorting had a
larger effect on intergenerational mobility than inequality. Fern\'{a}ndez
and Rogerson (2001) develop and calibrate a dynamic model of \
intergenerational education acquisition, fertility and marital sorting based
on the PSID and conclude that an increase in sorting is likely to increase
the degree of income inequality. Greenwood, Guner, Kocharkov, and Santos
(2015) examine US census data and document an increase in positive
assortative matching on education and via the comparison of counterfactual
income distributions conclude that the level of income inequality measured
by the Gini coefficient has increased. Chiappori, Salani\'{e} and Weiss
(2017) provide a model in which increased returns to education results in
higher paid couples spending more time with their children and increasing
assortative matching on education. Examining US marriages for individuals
who are married and born between 1943 and 1972, they find that this occurred
for white, but not black, individuals. Eika, Mogstad, and Zafar (2019)
examine data for Denmark, Germany, Norway, the United Kingdom, and the
United States and show that there is a considerable amount of educational
assortative matching although it has changed little since the 1980s.
Moreover, while there was an increase in sorting at the bottom of the
educational distribution, there is a remarkable decrease of assortative
matching at the top of the educational distribution. They conclude that the
increases in the Gini coefficient cannot be explained by changes in
assortative matching.\ However, they also note that assortative matching on
education contributes to the cross-sectional inequality in household income
in each country. This supports the earlier work of Breen and Salazar (2011)
for the United States that finds a very small impact of educational sorting
on changes in income inequality between the late 1970s and early 2000s.
Gihleb and Lang (2018) employ a range of statistical measures of
assortativeness to the CPS data for 1970 to 2010 and the 2010 American
Community Survey to test for increased educational homogany among married
couples in the US and matching conclude that there is no evidence of
increased assortative matching. However, they do not explore sorting on
wages due to the non trivial non participation of wives in market
employment. Chiappori, Costa-Dias and Meghir (2020,2023) provide alternative
measures of assortativeness to illustrate the difficulty in quantifying how
it varies across economies with different marginal distributions of the
variable on which it is being evaluated. They conclude that the degree to
which educational homogany has changed among married couples in the US
depends on the measure employed. The conclusion also varies on where in the
educational distribution it is measured. Chiappori, Costa-Dias, Crossman and
Meghir (2020) employed related measures in evaluating educational
assortativeness in the U.K and conclude that there are no clear patterns. \
However, they also conclude that the changes appear to have only slightly
increased income inequality. Our empirical work adds to this literature as
we provide a more rigorous and detailed analysis of sorting on wages while
accounting for selection although we do so by examining a more homogenous,
albeit large, group of workers. However, as our focus is primarily on the
role of sorting on inequality we focus on this issue rather than adopting
the measures proposed by Gihleb and Lang (2018) and Chiappori and co-authors.
While earnings appear a more appropriate measure than wages for evaluating
the welfare implications of inequality, examining different measures may
lead to substantially different conclusions. While substantial evidence
suggests female wage inequality has risen, Fern\'{a}ndez-Val, Van Vuuren,
Vella and Perrachi, hereafter FVVP (2023a), show that annual earnings
inequality has decreased for females in the United States due to the shifts
in their annual hours of work distribution. This finding is similar in
spirit to Cancian and Reed (1998a,1998b) who find household inequality
decreased due to the large shifts in the female income distribution
resulting from changes in their hours of work. Analyzing changes in annual
income is more challenging for married couples as it requires jointly
modeling annual hours and wages of both spouses. However, one could begin
by restricting attention to those with a relatively homogeneous level of
hours while accounting for the accompanying selection. As FTFY individuals
generally work a similar numbers of hours, we can then examine income
inequality via comparisons of the sum of couples' wages.
We document the changes in the earnings distributions of dual FTFY
households and examine if household inequality has been exacerbated by
sorting behavior. We do so via an examination of how the wage distributions
of husbands and wives and sorting patterns have changed over time. We report
how the probability of a male in the $j^{\mathrm{th}}$ decile of the married
male wage distribution being married to a female in the $k^{\mathrm{th}}$
decile of the married female wage distribution has evolved over our sample
period. We investigate the source of these changes by estimating the
conditional joint distribution of husbands' and wives' wages via the
bivariate distribution regression methodology of Fern\'{a}ndez-Val, Meier, Van Vuuren and Vella
(2023). While this provides insight into how the observed sorting patterns
can be explained by observable characteristics and unobservable factors, it
does not account for the selection arising from the couple's respective
employment decisions. We incorporate these employment decisions by extending
the selection model of Chernozhukov, Fern\'{a}ndez-Val, and Luo (2019)
(hereafter CFL) based on the Heckman (1974, 1979) selection model to
bivariate selection rules. We show point identification of this model under
the same exclusion restrictions as in CFL. This analysis relies on a useful
result for the multivariate standard normal distribution that might be of
independent interest. Lemma \ref{lemma1} in the appendix characterizes the
derivative of this function with respect to each element of the correlation
matrix and shows it is positive. While this result is well known in the
bivariate case (e.g., Sibuya, 1959) we have not found it for the general
multivariate case. Using the model's estimates, we evaluate the role of the
different forces generating the observed marital patterns and their
implications for the couples' earnings distribution.
Our empirical investigation uncovers a number of notable findings. First, we
confirm that wage inequality among couples for which both spouses are
working FTFY has increased for the period 1976-2022. Second, we find
increasing levels of assortative sorting on wages and that these have
contributed to increasing household inequality. Third, there is mixed
evidence regarding the role of selection from work decisions on observed
sorting patterns. Fourth, the primary factors behind the observed positive
sorting patterns are the observed characteristics of the individuals and the
correlation between the unobservables driving the wages of each spouse.
Finally, we find that positive sorting on wages has increased over the 47
years of our sample and this reflects the increasing market value of
observed and unobserved individual characteristics. Consistent with Gihleb
and Lang (2018), Eika, Mogstad, and Zafar (2019) and Chiappori, Costa-Dias
and Meghir (2020,2023) our results establish that the nature of sorting on
observed characteristics, such as education, has not substantially changed.
However, we find that the prices of these characteristics have changed to
push individuals further up, or down, their wage distributions.
We highlight three important features of our modeling approach treated as
exogenous. The first is the individual's decision to marry and the second is
their choice of spouse (see, for example, Chiappori, Costa-Dias and Meghir
2019). While each of these is interesting, it is beyond the scope of this
paper to model these decisions. We also do not address how household income
is allocated across the spouses nor the implications of this allocation for
the work or marriage decisions (see, for example, Lise and Seitz 2011, Lise
and Yamada 2018 and De Rock, Kovaleva and Potoms 2023). While the failure to
address each of these issues represents a shortcoming of our approach, the
focus on explaining the sorting pattern of spouses, and its implication for
inequality, within the context of endogenous employment decisions remains
important.
The following section briefly describes the Current Population Survey data
examined and how the sample is selected. It also presents the time series
trends in earnings and the wage distributions of FTFY households. Section
\ref{sec:marital_sorting} describes the observed patterns of marital sorting
and Section \ref{sec:econometric_model} provides our econometric model. The
measures of sorting are discussed in Section \ref{sec:sorting}, and
estimation is discussed in Section \ref{sec:estimation}. Our empirical
results are presented in Section \ref{sec:empirical}. Section \ref
{sec:conclusion} concludes.
\section{Data}
\subsection{Data}
\label{sec:data}
We employ the Annual Social and Economic Supplement (ASEC) of the Current
Population Survey (CPS), or March CPS, for the 47~survey years from 1976 to
2022 which report annual earnings and hours worked for the previous calendar
year.\footnote{
The data are taken from the IPUMS-CPS website maintained by the Minnesota
Population Center at the University of Minnesota (Flood, King, Ruggles, and
Warren 2015).} The 1976 survey is the first for which information on weeks
worked and usual hours of work per week last year are available.\footnote{
We refer to the year of the survey and not the calendar year to which it
refers.} To avoid issues related to retirement and ongoing educational
investment we restrict attention to those aged 24--65 years in the survey
year. This produces an overall sample of 2,054,502 males and 2,228,726
females. The annual sample sizes range from a minimum of 30,767 males and
33,924 females in 1976 to a maximum of 55,039 males and 59,622 females in
2001.
Annual hours worked are defined as the product of weeks worked and usual
weekly hours of work last year. Those reporting zero hours generally respond
not being in the labor force (i.e., they report themselves as doing
housework, unable to work, at school, or retired) in the week of the March
survey. We define hourly wages as the ratio of reported annual labor
earnings in the year before the survey, converted to constant 2021 prices
using the consumer price index for all urban consumers, and annual hours
worked. Hourly wages are unavailable for those not in the labor force. As
annual earnings and hours tend to be poorly measured for the Armed Forces,
self-employed, and the unpaid family workers we exclude these groups and
focus on civilian dependent employees with positive hourly wages and people
out of the labor force last year. This restricted sample comprises 1,783,599
males and 2,097,035 females (respectively 86.8\% and 94.1\% of the original
sample of those aged 24--65). The subsample of civilian dependent employees
with positive hourly wages contains 1,540,948 males and 1,465,165 females.
Married individuals aged between 24 and 65 years make up 65\% of the total
sample. While this has decreased from 77\% in 1976 to 57\% in 2022, they
still represent a substantial fraction of the total sample. The percentage
of married couples with both spouses working full time increased drastically
from 26.2 in 1976 to 51.3 in 2022.
\begin{comment}
The March CPS differs from the Outgoing Rotation Groups of the CPS, or ORG
CPS, which contains information on hourly wages in the survey week for those
paid by the hour and on weekly earnings from the primary job during the
survey week for those not paid by the hour. Lemieux (2006) and Autor et al.\
(2008) argue that the ORG CPS data are preferable because they provide a
point-in-time wage measure and workers paid by the hour (more than half of
the U.S.\ workforce) may recall their hourly wages better. However, there is
no clear evidence regarding differences in the relative reporting accuracy
of hourly wages, weekly earnings and annual earnings. In addition, many
workers paid by the hour also work overtime, so their effective hourly wage
depends on the importance of overtime work and the wage differential between
straight time and overtime. Furthermore, the failure of the March CPS to
provide a point-in-time wage measure may be an advantage as it smooths out
intra-annual variations in hourly wages.
\end{comment}
\subsection{Descriptive statistics}
Figure \ref{fig:income} presents the time series of various quantiles of
annual household labor earnings (in 2021 dollars) for dual FTFY households.
Median (Q2) household annual income increased by 29.8\% from 97.9 to 127
thousand dollars from 1976 to 2022. However, growth has been more modest at
lower quantiles. For example, at the first quartile (Q1), household income
increased by 19.0\% from 76.5 to 91.0 thousand. Moreover, household income
at this quantile was virtually constant from 1976 to 2000. There were
increases of 12.5\% and 8.9\% at the first decile (D1) and the 5th
percentile respectively. Increases at higher quantiles have been notably
larger. Income at Q3 grew by 49.1\% from 122.1 to 182 thousand. Increases at
D9 and the 95th percentile were 76.7\% and 97.8\% respectively. These
changes have drastically increased inequality. The Q3/Q1 and D9/D1 ratios
are shown in Figure \ref{fig:income1}. The former increased from 1.60 to
2.00 and the latter from 2.43 to 3.82.
\begin{figure}[!ht]
\caption{Household income of households working FTFY at various quantiles.
\emph{Note:} Px represents a percentile, \emph{i.e.} P5 means the 5th
percentile.}
\label{fig:income}\centering
\begin{tikzpicture}
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table {
1976 51.426060295105
1977 52.370735168457
1978 53.0132331848145
1979 54.0081253051758
1980 51.4872947692871
1981 49.5731927871704
1982 47.6878067016602
1983 47.127375793457
1984 47.9081928253174
1985 47.8488977432251
1986 47.824390411377
1987 49.4308929443359
1988 48.4229278564453
1989 49.4462104797363
1990 48.0780487060547
1991 47.6829261779785
1992 46.5585174560547
1993 48.2520294189453
1994 46.8699226379395
1995 47.1019968032837
1996 46.2081298828125
1997 44.9514396667481
1998 46.0580854415894
1999 46.5300827026367
2000 47.9674758911133
2001 50.3154449462891
2002 49.8287029266357
2003 50.8923606872559
2004 49.6009201049805
2005 50.1951179504395
2006 49.9317054748535
2007 48.8409111022949
2008 52.1619506835938
2009 50.3414611816406
2010 49.9048805236816
2011 49.6910591125488
2012 49.8842445373535
2013 48.4000015258789
2014 48.8292655944824
2015 49.1082893371582
2016 50.9417797088623
2017 50.7804870605469
2018 51.9674797058105
2019 51.7463455200195
2020 53.0081253051758
2021 56.5463371276855
2022 56
};
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table {
1976 60.4292678833008
1977 61.8926849365234
1978 61.4613838195801
1979 62.3170700073242
1980 60.4529090881348
1981 58.8517036437988
1982 56.6292724609375
1983 56.1300773620605
1984 57.1268272399902
1985 57.3430862426758
1986 57.8926811218262
1987 59.3170700073242
1988 58.3157550811768
1989 58.578296661377
1990 58.5678066253662
1991 57.219510269165
1992 56.2579975128174
1993 56.7443885803223
1994 56.2439079284668
1995 56.656909942627
1996 55.0943031311035
1997 55.2585334777832
1998 56.3034858703613
1999 56.5008125305176
2000 58.5365867614746
2001 60.0263328552246
2002 59.6731719970703
2003 60.2276420593262
2004 59.5975608825684
2005 60.2341499328613
2006 59.6406478881836
2007 59.0959358215332
2008 61.7053672790527
2009 60.4097595214844
2010 60.6449157714844
2011 59.9522609710693
2012 60.2439079284668
2013 59.0243911743164
2014 59.2926788330078
2015 59.525203704834
2016 61.5393627166748
2017 62.0650405883789
2018 62.1840705871582
2019 62.5527069091797
2020 65.7300796508789
2021 69.112190246582
2022 68
};
\addlegendentry{P10}
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table {
1976 76.4568729400635
1977 77.8598079681396
1978 78.2414627075195
1979 78.9349517822266
1980 77.6195087432861
1981 75.6195068359375
1982 74.5121917724609
1983 72.6779308319092
1984 74.7817344665527
1985 75.5886077880859
1986 75.6242084503174
1987 78.1008148193359
1988 78.717077255249
1989 77.8406448364258
1990 76.519495010376
1991 75.6707305908203
1992 75.5674819946289
1993 76.2382049560547
1994 74.9918746948242
1995 74.9333343505859
1996 74.4661712646484
1997 74.2536544799805
1998 75.9512176513672
1999 77.4393539428711
2000 78.8617858886719
2001 80.1902465820312
2002 81.0943069458008
2003 81.3073120117188
2004 80.9349594116211
2005 80.5846748352051
2006 80.4455337524414
2007 80.5853652954102
2008 82.360969543457
2009 81.7696380615234
2010 82.121955871582
2011 81.6517200469971
2012 81.9317092895508
2013 81.4536590576172
2014 81.3821105957031
2015 81.2747955322266
2016 84.5886077880859
2017 84.634147644043
2018 85.6910629272461
2019 86.2439041137695
2020 90.1138153076172
2021 92.6731643676758
2022 91
};
\addlegendentry{P25}
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table {
1976 97.8626899719238
1977 99.6115188598633
1978 100.296035766602
1979 100.953659057617
1980 99.2634124755859
1981 96.1321258544922
1982 95.3756103515625
1983 95.4211349487305
1984 97.9317092895508
1985 99.0471572875977
1986 101.157398223877
1987 103.804885864258
1988 104.956100463867
1989 103.024391174316
1990 102.867362976074
1991 101.585361480713
1992 101.41951751709
1993 102.294311523438
1994 101.936447143555
1995 102.347969055176
1996 99.8806457519531
1997 101.187019348145
1998 102.956092834473
1999 106.053684234619
2000 107.317077636719
2001 108.492691040039
2002 109.353382110596
2003 109.91544342041
2004 110.365859985352
2005 110.429267883301
2006 110.959342956543
2007 110.133346557617
2008 111.775611877441
2009 111.380485534668
2010 113.707313537598
2011 113.047157287598
2012 114.463417053223
2013 112.807403564453
2014 113.934959411621
2015 114.471549987793
2016 116.595123291016
2017 117.632431030273
2018 119.414642333984
2019 118.585372924805
2020 125.279945373535
2021 128.800003051758
2022 127
};
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table {
1976 122.069637298584
1977 123.785377502441
1978 125.157730102539
1979 126.711372375488
1980 124.318099975586
1981 121.648773193359
1982 120.709762573242
1983 120.679679870605
1984 125.134971618652
1985 127.718696594238
1986 130.887817382812
1987 135.267639160156
1988 135.965866088867
1989 134.602508544922
1990 135.492691040039
1991 134.756088256836
1992 133.237411499023
1993 135.105682373047
1994 136.860168457031
1995 137.073165893555
1996 135.069900512695
1997 134.692687988281
1998 138.399993896484
1999 142.082916259766
2000 146.341461181641
2001 146.337768554688
2002 147.859401702881
2003 149.063415527344
2004 148.626037597656
2005 150.585357666016
2006 149.795120239258
2007 150.426010131836
2008 152.956100463867
2009 152.282928466797
2010 156.663421630859
2011 155.284561157227
2012 156.634155273438
2013 157.004867553711
2014 160.439010620117
2015 160.260162353516
2016 164.604873657227
2017 167.011367797852
2018 170.276428222656
2019 170.116111755371
2020 178.107315063477
2021 184.299194335938
2022 182
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1976 147.119075012207
1977 152.351211547852
1978 156.000152587891
1979 156.623565673828
1980 153
1981 147.951217651367
1982 149.024383544922
1983 151.551223754883
1984 157.537295532227
1985 158.99674987793
1986 166.126831054688
1987 173.008117675781
1988 173.122465515137
1989 171.707305908203
1990 174.82926940918
1991 172.073165893555
1992 171.021133422852
1993 173.707305908203
1994 178.105682373047
1995 182.764221191406
1996 177.723556518555
1997 178.002944946289
1998 185.658538818359
1999 192.767471313477
2000 195.121948242188
2001 194.972381591797
2002 197.380477905273
2003 202.741310119629
2004 204.544723510742
2005 203.648788452148
2006 205.274795532227
2007 203.440447998047
2008 207.863403320312
2009 207.658538818359
2010 214.780487060547
2011 214.913833618164
2012 216.878051757812
2013 214.848785400391
2014 219.731704711914
2015 225.508941650391
2016 228.617874145508
2017 236.975601196289
2018 241.704055786133
2019 237.170745849609
2020 254.439010620117
2021 261.788604736328
2022 260
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table {
1976 166.918466949463
1977 173.299521636963
1978 178.385513305664
1979 178.642288208008
1980 175.390243530273
1981 170.246479797363
1982 169.887802124023
1983 176.750804901123
1984 183.382551574707
1985 187.668273925781
1986 191.297561645508
1987 200.195129394531
1988 202.756103515625
1989 201.469909667969
1990 203.239043426514
1991 203.170715332031
1992 199.98486328125
1993 202.658538818359
1994 211.31247177124
1995 217.489440917969
1996 213.268280029297
1997 214.126846313477
1998 222.790252685547
1999 230.988617706299
2000 235.832534790039
2001 240.570732116699
2002 238.692672729492
2003 248.439025878906
2004 250.162628173828
2005 250.975601196289
2006 256.593475341797
2007 252.50080871582
2008 254.92682800293
2009 257.999969482422
2010 262.790252685547
2011 263.362579345703
2012 271.097564697266
2013 270.331695556641
2014 275.536590576172
2015 283.431549072264
2016 288.058532714844
2017 299.040618896484
2018 309.593505859375
2019 309.076583862303
2020 328.650360107422
2021 330.167822265625
2022 330
};
\addlegendentry{P95}
\end{axis}
\end{tikzpicture}
\end{figure}
\begin{figure}[!ht]
\caption{Inequality measures of households working FTFY at various
quantiles. }
\label{fig:income1}\centering
\begin{tikzpicture}
\begin{axis}[
legend cell align={left},
legend style={
fill opacity=0.8,
draw opacity=1,
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tick pos=left,
x grid style={white!69.0196078431373!black},
xlabel={Year},
xmin=1976, xmax=2022,
xtick style={color=black},
y grid style={white!69.0196078431373!black},
ylabel={Inequality},
ymin=1.4749963482262, ymax=4.00176589470427,
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table {
1976 1.59658161005718
1977 1.58984950942975
1978 1.59963433416884
1979 1.60526318841712
1980 1.60163471771958
1981 1.60869566972033
1982 1.62000015973031
1983 1.66047214731134
1984 1.67333604270199
1985 1.68965536383869
1986 1.73076611398586
1987 1.73196194525063
1988 1.72727279555848
1989 1.72920598008631
1990 1.77069504995644
1991 1.78082182112807
1992 1.76315801429633
1993 1.77215193420313
1994 1.82499996184889
1995 1.82926820328372
1996 1.81384242292604
1997 1.81395365563584
1998 1.82222218650622
1999 1.834763708961
2000 1.85567013899773
2001 1.82488238647564
2002 1.82330187249399
2003 1.8333334584454
2004 1.83636389859381
2005 1.86865998992937
2006 1.8620688216232
2007 1.86666660355021
2008 1.85714302917672
2009 1.86234074256534
2010 1.90769228482382
2011 1.9017916716004
2012 1.91176476887458
2013 1.92753609070713
2014 1.97142848035927
2015 1.9718310123582
2016 1.94594612633417
2017 1.97333313381111
2018 1.98709669837128
2019 1.97250012628093
2020 1.97647069381626
2021 1.98870078078634
2022 2
};
\addlegendentry{Q3/Q1}
\addplot [very thick, black, dashed]
table {
1976 2.43456656294958
1977 2.46153825293088
1978 2.53818158481152
1979 2.51333327538377
1980 2.53089557322941
1981 2.51396660573915
1982 2.63157863537304
1983 2.70000026505075
1984 2.75767626426042
1985 2.77272746020429
1986 2.86956533771686
1987 2.91666661307477
1988 2.96870828945191
1989 2.93124443171829
1990 2.98507455687199
1991 3.00724639347854
1992 3.03994349219215
1993 3.06122438278313
1994 3.1666662032014
1995 3.22580637342348
1996 3.22580641587643
1997 3.22127522652868
1998 3.29746082233412
1999 3.41176458673683
2000 3.33333320299785
2001 3.24811415786541
2002 3.30769207165599
2003 3.36625016665806
2004 3.43209890609079
2005 3.38095231158971
2006 3.44186059006406
2007 3.44254550113948
2008 3.36864380662841
2009 3.43749984213241
2010 3.54160747571749
2011 3.58474943458552
2012 3.59999972138813
2013 3.64000002585167
2014 3.70588256487394
2015 3.78846148546783
2016 3.7149860520665
2017 3.81818168408095
2018 3.88691273350072
2019 3.79153449256733
2020 3.87096762960814
2021 3.78787886481828
2022 3.82352941176471
};
\addlegendentry{D9/D1}
\end{axis}
\end{tikzpicture}
\end{figure}
FVVP (2023a) show that there is little variation in hours for either males
or females working FTFY and the income variation reflects changes in wages.
Figure \ref{fig:quantiles} presents wage growth for married males and
females at D1, Q1, Q2, Q3 and D9. For 1976-1996 the real wage for married
males at D1 decreased by 17\%. It recovers somewhat but in 2022 it remains
11\% below its 1976 value. There are large decreases during the financial
crisis and a large decrease in the 2010's. The real wage at Q1 for married
males shows a similar pattern to that at D1. There is a large decrease in
the 1976-1996 period but a recovery during the second half of the sample
period despite the two large dips. The 2022 value is 9\% lower than the 1976
value. The married male median shows a large decrease at 1996 but virtually
no change over the sample period. The overall picture at Q3 is more positive
although there are several sharp decreases. However these are offset by
large gains during periods of increasing real wages. The 2022 value is 21\%
higher than the 1976 value. At D9 there is a similar pattern to that at Q3
but with smaller dips and faster increases. This results in a dramatic gain
of 42\% over the whole sample period.
The time series pattern of the married female wage distribution at the
lowest decile in the earlier part of the sample period is similar to that of
married males although the decreases are less dramatic. Over the period
1979-1996 it decreases by 8\%. For the remaining 26 years it increases by
27\% and for the whole period it increases by 19\%. The pattern of real
wages for married females at Q1 is dissimilar to that at D1. The trend of
wage growth appears to be affected by cyclical factors but generally is
steadily increasing for the time period examined. This results in an
increase of 29\%. Growth at Q2 is even more drastic with an increase of
41\%. Moreover, with the exception of a dip in the early 1980's the median
real wage for this group follows a strong upward trend. \ At Q3 there are
periods of substantial gains and these offset the small decreases which are
incurred. An overall gain of 60\% is achieved. A similar story is observed
at D9 but the increases amount to a very substantial 84\%.
Given the existence of a marriage premium in mean wages it is interesting to
contrast this evidence on married individuals' wage distributions to that of
all individuals. Fern\'{a}ndez-Val et al. (2018) provide the corresponding
rates of wage changes for all males and all females for the period
1976-2016. Median male wages decline by 13.6\% for all working males and the
decrease at Q1 is 18.2\%. At Q3 there is very modest growth. This clearly
suggests that there is a substantial difference between the experience of
married and unmarried males. For all females, wage growth at Q1 and Q2 are
17\% and 25\% respectively and there are strong increases at Q3. Married
females have experienced favorable wage growth compared to their unmarried
counterparts.
The primary focus of our empirical work below is to model the joint
distribution of married couple's real wage rates. To motivate what follows
we report how the distribution of the sum of the husband's and wife's wages
has evolved over our sample period. This is reported in Figure \ref
{fig:wages_both}. We define this as the household hourly wage as it captures
the combined market value of an hour of work of each spouse. The household
hourly wage at D1 decreases by 12\% over the period 1976-1996 but increases
over the remainder of the sample period. By 2022 it is 9\% higher than in
1976. The household wage at Q1 shows some of the dramatic dips featured in
the married males profile but over the sample period there is an increase of
15\%. At Q2 it initially resembles the pattern of the male wage with
multiple large dips during our sample. However, towards the end of the
sample the large increases in wives' wages result in an increase of about
26\% percent over the sample period. At Q3 the large shifts in the wive's
wage distribution become more important. The household wage shows a dip in
the 1980's and a prolonged decrease in the 1990's but over the whole sample
period it increases by 43\%. The trend at D9 is consistent with the male
pattern at this decile combined with those of females at any of the higher
quantiles. This produces a steadily rising profile and an increase of almost
70\%.
\begin{figure}[tbp]
\begin{tikzpicture}
\begin{groupplot}[group style={group size=2 by 1}]
\nextgroupplot[
tick align=outside,
tick pos=left,
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x grid style={white!69.0196078431373!black},
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table {
1976 16.1402950286865
1977 16.041187286377
1978 16.1174945831299
1979 15.7124242782593
1980 15.3387680053711
1981 14.9445676803589
1982 14.3292684555054
1983 13.9659557342529
1984 13.9503860473633
1985 13.8470525741577
1986 13.9305478096008
1987 14.1631870269775
1988 13.7617263793945
1989 13.7093362808228
1990 13.6585369110107
1991 12.9573163986206
1992 12.9068946838379
1993 12.867208480835
1994 12.618824005127
1995 12.301438331604
1996 12.1757207870483
1997 12.2538461685181
1998 12.171669960022
1999 12.4634141921997
2000 12.5078172683716
2001 13.6069421768188
2002 13.3882112503052
2003 13.5565486907959
2004 13.4419956207275
2005 13.5843091964722
2006 13.3364601135254
2007 12.9143218994141
2008 13.1988735198975
2009 13.2845525741577
2010 13.3630390167236
2011 13.0308713912964
2012 12.8726291656494
2013 12.5597047805786
2014 12.5203247070312
2015 12.1075668334961
2016 13.1455278396606
2017 13.3545007705688
2018 13.2895565032959
2019 13.4756107330322
2020 14.2714185714722
2021 15.1031885147095
2022 14.4230766296387
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table {
1976 22.273609161377
1977 22.6295309066772
1978 22.3495941162109
1979 21.9707622528076
1980 21.5290813446045
1981 20.6541595458984
1982 20.2042675018311
1983 20.2025766372681
1984 19.8792991638184
1985 20.0500297546387
1986 20.1688556671143
1987 20.2001247406006
1988 20.3877429962158
1989 19.8123817443848
1990 19.9419889450073
1991 18.9912490844727
1992 19.1213264465332
1993 18.5584735870361
1994 18.026891708374
1995 18.0601348876953
1996 17.5506649017334
1997 17.4343338012695
1998 18.1763610839844
1999 18.3755474090576
2000 18.7617263793945
2001 19.0065231323242
2002 19.1260166168213
2003 19.303731918335
2004 19.2828960418701
2005 19.305814743042
2006 19.115592956543
2007 19.1017169952393
2008 19.0507555007935
2009 18.9976100921631
2010 19.3778533935547
2011 18.6775817871094
2012 18.5377445220947
2013 18.1621823310852
2014 18.4451217651367
2015 17.8861789703369
2016 18.685115814209
2017 18.9884300231934
2018 18.6053791046143
2019 19.176830291748
2020 20.3877410888672
2021 20.8951187133789
2022 20.1923084259033
};
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table {
1976 30.156078338623
1977 30.5190753936768
1978 30.83580493927
1979 30.7990131378174
1980 30.4995307922363
1981 29.2424945831299
1982 28.6585369110107
1983 28.3348960876465
1984 28.686351776123
1985 28.9266242980957
1986 29.0431518554688
1987 29.7060661315918
1988 29.2085628509521
1989 28.6178855895996
1990 28.5778617858887
1991 27.9080657958984
1992 27.6196937561035
1993 27.8377094268799
1994 27.0403385162354
1995 26.3602256774902
1996 25.974983215332
1997 25.9024391174316
1998 26.710054397583
1999 27.3581790924072
2000 27.3608512878418
2001 28.0778179168701
2002 28.2929229736328
2003 28.8971004486084
2004 28.2989368438721
2005 27.5797367095947
2006 27.3397426605225
2007 27.7657909393311
2008 28.2833003997803
2009 27.9261074066162
2010 28.3458404541016
2011 28.2335548400879
2012 27.8048782348633
2013 27.7465076446533
2014 27.8968496322632
2015 27.5171985626221
2016 27.4781112670898
2017 28.2113819122314
2018 28.1738586425781
2019 28.5060977935791
2020 30.5816116333008
2021 31.1217231750488
2022 29.914529800415
};
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1976 38.9788131713867
1977 40.0562858581543
1978 40.7611455917358
1979 40.0736713409424
1980 39.7072505950928
1981 38.7265472412109
1982 38.2113838195801
1983 38.5280513763428
1984 38.9510116577148
1985 38.8469352722168
1986 39.6923065185547
1987 40.1626014709473
1988 40.138370513916
1989 39.135570526123
1990 39.8749313354492
1991 38.8719482421875
1992 38.2426528930664
1993 37.5765018463135
1994 37.8564720153809
1995 38.2445106506348
1996 37.5953712463379
1997 37.3592872619629
1998 38.1379013061523
1999 39.1479034423828
2000 39.0869293212891
2001 40.8813018798828
2002 40.8675537109375
2003 41.5947189331055
2004 41.8824272155762
2005 41.3696060180664
2006 40.709774017334
2007 41.8424034118652
2008 41.139347076416
2009 42.3545951843262
2010 42.5187606811523
2011 41.8073806762695
2012 42.2865867614746
2013 41.7711067199707
2014 41.9207305908203
2015 41.58154296875
2016 43.9649772644043
2017 43.4021263122559
2018 44.0453872680664
2019 44.0548782348633
2020 46.5163059234619
2021 48.3302040100098
2022 47.1153831481934
};
\addplot [ultra thick, black]
table {
1976 50.7491264343262
1977 51.3438568115234
1978 51.575984954834
1979 51.9308929443359
1980 51.2709922790527
1981 49.3170738220215
1982 49.1289215087891
1983 49.9733619689941
1984 51.0060997009277
1985 50.1250762939453
1986 51.0145130157471
1987 52.8107833862305
1988 52.1678581237793
1989 52.1775588989258
1990 52.5328330993652
1991 51.1358985900879
1992 50.6715126037598
1993 51.0358009338379
1994 51.9174499511719
1995 52.7204513549805
1996 51.266414642334
1997 50.6425895690918
1998 52.7439041137695
1999 54.3277053833008
2000 55.5902976989746
2001 58.1540733337403
2002 58.8492813110352
2003 58.6957824707031
2004 60.1352424621582
2005 59.9829296112061
2006 60.0140724182129
2007 60.6973114013672
2008 60.3377075195312
2009 60.5065650939941
2010 60.7410888671875
2011 59.7248268127441
2012 60.8231735229492
2013 60.2265567779541
2014 60.9756088256836
2015 61.6385231018066
2016 64.8483428955078
2017 65.1031875610352
2018 65.7228927612305
2019 65.983757019043
2020 70.5729522705078
2021 73.2275848388672
2022 72.1153869628906
};
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table {
1976 10.0860706329346
1977 10.3499479293823
1978 10.0815544128418
1979 9.98671054840088
1980 10.0982704162598
1981 9.8634147644043
1982 9.69672527313232
1983 9.44496536254883
1984 9.73039436340332
1985 9.50750904083252
1986 9.68105030059814
1987 9.56428537368774
1988 9.66336584091187
1989 9.51931266784668
1990 9.45591068267822
1991 9.71798706054688
1992 9.5606632232666
1993 9.65040588378906
1994 9.37398338317871
1995 9.37252426147461
1996 9.25643634796143
1997 9.48807334899902
1998 9.73733615875244
1999 9.58724212646484
2000 9.67557125091553
2001 9.98447227478027
2002 10.2986240386963
2003 10.3516263961792
2004 10.3291120529175
2005 10.2439022064209
2006 10.002345085144
2007 10.1404333114624
2008 10.4752969741821
2009 10.2861166000366
2010 10.4127578735352
2011 10.7504692077637
2012 10.4268293380737
2013 10.2294940948486
2014 10.4801826477051
2015 10.2730875015259
2016 10.8221483230591
2017 10.850531578064
2018 10.6316452026367
2019 10.8841466903687
2020 11.3265228271484
2021 12.226390838623
2022 12.0192308425903
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\addlegendentry{D1}
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table {
1976 13.0149483680725
1977 13.2249326705933
1978 12.9939498901367
1979 13.3156137466431
1980 13.0609760284424
1981 12.6454029083252
1982 12.7070217132568
1983 12.7550137042999
1984 13.0784864425659
1985 12.8342127799988
1986 13.2477531433105
1987 13.3023767471313
1988 13.6466910839081
1989 13.2082548141479
1990 13.5492687225342
1991 13.2521157264709
1992 13.1117668151855
1993 13.2560529708862
1994 13.3220520019531
1995 13.1801128387451
1996 12.8668651580811
1997 13.2833023071289
1998 13.0603952407837
1999 13.5819263458252
2000 13.8975744247437
2001 14.3628835678101
2002 14.7123203277588
2003 14.4777984619141
2004 14.5645198822021
2005 14.4024448394775
2006 14.3366947174072
2007 14.3327445983887
2008 15.0844268798828
2009 14.6947174072266
2010 15.1852722167969
2011 14.931206703186
2012 14.4830837249756
2013 14.7560968399048
2014 14.5336380004883
2015 14.4661836624146
2016 15.2006568908691
2017 15.1918296813965
2018 15.4158849716187
2019 15.5487813949585
2020 16.3101921081543
2021 17.1587333679199
2022 16.8269233703613
};
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table {
1976 17.4315204620361
1977 17.8536586761475
1978 17.6881065368652
1979 17.9760780334473
1980 17.5820827484131
1981 17.1471672058105
1982 17.1951217651367
1983 17.0435466766357
1984 17.7867412567139
1985 17.7405166625977
1986 18.151969909668
1987 19.0118827819824
1988 18.6926765441895
1989 18.7116947174072
1990 18.9118213653564
1991 18.9376163482666
1992 19.1213264465332
1993 18.9944972991943
1994 18.9282360076904
1995 19.33083152771
1996 18.7976856231689
1997 19.094747543335
1998 19.4746723175049
1999 19.8846492767334
2000 19.6998119354248
2001 20.4104137420654
2002 20.5972480773926
2003 20.9928073883057
2004 21.2242031097412
2005 20.6848030090332
2006 20.6715126037598
2007 20.6643905639648
2008 21.6838645935059
2009 21.1772975921631
2010 21.478048324585
2011 21.6808090209961
2012 21.4329280853271
2013 21.5666027069092
2014 22.357723236084
2015 22.0137577056885
2016 21.9824886322021
2017 22.2435894012451
2018 22.3264541625977
2019 23.2195129394531
2020 24.2104434967041
2021 25.1719818115234
2022 24.5201930999756
};
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table {
1976 23.4221973419189
1977 23.6330633163452
1978 23.6828498840332
1979 23.9681034088135
1980 23.3231716156006
1981 22.9197940826416
1982 22.926830291748
1983 23.2353649139404
1984 24.148051738739
1985 25.0625381469727
1986 25.4127578735352
1987 26.1413383483887
1988 26.2128124237061
1989 25.9762344360352
1990 26.5429058074951
1991 26.4227733612061
1992 26.7698554992676
1993 26.9097862243652
1994 27.0403385162354
1995 27.226523399353
1996 27.3420886993408
1997 26.5666046142578
1998 27.5559997558594
1999 27.9627895355225
2000 28.7415800094604
2001 29.4817085266113
2002 29.4246406555176
2003 29.6794872283936
2004 30.3705825805664
2005 30.3377094268799
2006 30.0070362091064
2007 30.9943714141846
2008 31.4258899688721
2009 30.8583488464355
2010 31.5866713523865
2011 31.8532409667969
2012 31.8597564697266
2013 31.7823619842529
2014 32.4186973571777
2015 33.020637512207
2016 32.9737319946289
2017 33.757209777832
2018 34.5528450012207
2019 33.6890258789062
2020 35.6785469055176
2021 37.2545318603516
2022 37.5
};
\addlegendentry{Q3}
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table {
1976 29.2566318511963
1977 30.1175098419189
1978 30.9455909729004
1979 30.7813568115234
1980 30.4995307922363
1981 29.5059413909912
1982 29.8048782348633
1983 29.9161767959595
1984 31.3883686065674
1985 31.9706766128541
1986 32.9213806152344
1987 34.3402137756348
1988 34.4043159484863
1989 34.1213264465332
1990 35.7223281860352
1991 35.4388160705566
1992 35.374454498291
1993 36.189022064209
1994 36.5700736999512
1995 37.7829895019531
1996 36.7409324645996
1997 36.9363906860352
1998 37.5067749023438
1999 39.6272659301758
2000 40.2036972045898
2001 40.5184516906738
2002 41.1944961547852
2003 42.3475608825684
2004 42.4484062194824
2005 43.0381774902344
2006 42.6766738891602
2007 44.2776756286621
2008 43.9962463378906
2009 44.169792175293
2010 45.5558166503906
2011 46.4526443481445
2012 46.3414649963379
2013 45.4033737182617
2014 46.6318206787109
2015 46.7792358398438
2016 48.3614768981934
2017 49.1963092803955
2018 51.0318946838379
2019 51.570125579834
2020 53.19970703125
2021 54.3714790344238
2022 53.8461532592773
};
\addlegendentry{D9}
\end{groupplot}
\end{tikzpicture}
\caption{Time series of wages of married males and females at different
quantiles.}
\label{fig:quantiles}
\end{figure}
\begin{figure}[tbp]
\centering
\begin{tikzpicture}
\definecolor{darkgray176}{RGB}{176,176,176}
\begin{axis}[
tick align=outside,
tick pos=left,
x grid style={darkgray176},
xlabel={Year},
xmin=1976, xmax=2022,
xtick style={color=black},
y grid style={darkgray176},
ylabel={Real dollars},
ymin=20.287365, ymax=118.625935,
ytick style={color=black}
]
\addplot [semithick, black]
table {
1976 28.1161
1977 28.4162
1978 28.0581
1979 28.241
1980 27.7391
1981 27.4562
1982 26.4314
1983 25.788
1984 26.228
1985 26.3273
1986 26.5137
1987 27.1096
1988 26.7911
1989 26.8036
1990 26.5775
1991 26.186
1992 25.829
1993 25.9934
1994 25.7963
1995 25.4044
1996 24.7573
1997 24.8012
1998 25.5014
1999 25.5245
2000 26.2743
2001 27.2317
2002 27.3386
2003 27.6517
2004 27.1747
2005 27.5087
2006 27.3822
2007 26.5875
2008 28.0558
2009 27.5427
2010 27.9209
2011 27.6924
2012 27.6597
2013 27.0993
2014 27.2224
2015 27.1158
2016 27.8152
2017 28.1332
2018 28.4281
2019 28.3773
2020 29.4826
2021 31.3846
2022 30.608
};
\addplot [semithick, black, dotted]
table {
1976 35.2058
1977 35.5248
1978 35.5457
1979 35.803
1980 35.1052
1981 34.4165
1982 33.8842
1983 33.1293
1984 33.8606
1985 34.0743
1986 34.6691
1987 35.0801
1988 35.2224
1989 34.7977
1990 34.5242
1991 34.1633
1992 33.9895
1993 34.1163
1994 33.7016
1995 33.4115
1996 32.7936
1997 32.9634
1998 33.6136
1999 34.3637
2000 34.9707
2001 35.8063
2002 36.0378
2003 36.362
2004 36.2902
2005 36.2145
2006 35.8728
2007 35.765
2008 36.4868
2009 36.4026
2010 36.737
2011 36.8285
2012 36.5101
2013 36.313
2014 36.5635
2015 36.3094
2016 37.2619
2017 37.8132
2018 37.7649
2019 38.0564
2020 39.8754
2021 41.5385
2022 40.5958
};
\addplot [semithick, black, dashed]
table {
1976 44.4495
1977 44.9152
1978 45.162
1979 45.6714
1980 45.052
1981 43.7754
1982 43.4312
1983 43.4416
1984 44.4306
1985 44.5507
1986 45.8403
1987 46.5241
1988 46.7086
1989 46.2494
1990 46.0535
1991 45.3082
1992 45.1325
1993 45.4884
1994 44.9888
1995 44.8722
1996 44.2828
1997 44.398
1998 45.4611
1999 46.388
2000 47.0318
2001 47.6454
2002 48.3314
2003 48.6876
2004 48.6631
2005 48.5368
2006 48.3964
2007 48.7145
2008 49.2188
2009 49.1146
2010 50.0784
2011 50.1911
2012 49.7876
2013 49.8626
2014 50.234
2015 50.1946
2016 51.4136
2017 51.8096
2018 52.3551
2019 52.4201
2020 55.4885
2021 57.6505
2022 56.0126
};
\addplot [semithick, black, dash pattern=on 1pt off 3pt on 3pt off 3pt]
table {
1976 55.3706
1977 56.1101
1978 56.6983
1979 57.222
1980 56.1405
1981 54.8594
1982 54.736
1983 54.9769
1984 56.4528
1985 57.1423
1986 58.7356
1987 59.9183
1988 60.2343
1989 58.8629
1990 60.2007
1991 59.7059
1992 58.9526
1993 59.3713
1994 60.2529
1995 59.7539
1996 59.2687
1997 58.6688
1998 60.2704
1999 61.715
2000 63.0076
2001 64.0841
2002 64.6291
2003 65.179
2004 65.5347
2005 65.8445
2006 65.2977
2007 65.7748
2008 66.4965
2009 67.027
2010 69.2054
2011 68.4424
2012 68.8406
2013 68.5224
2014 69.5421
2015 70.4252
2016 71.7955
2017 72.5334
2018 74.1161
2019 74.243
2020 77.8796
2021 81.25
2022 79.5807
};
\addplot [semithick, black, mark=*, mark size=1, mark options={solid}]
table {
1976 67.0486
1977 68.1258
1978 68.9548
1979 70.4678
1980 68.6057
1981 66.5688
1982 66.5434
1983 68.2916
1984 69.9415
1985 71.2274
1986 73.1312
1987 75.7095
1988 75.6662
1989 74.8925
1990 76.2542
1991 76.2183
1992 75.1714
1993 75.3218
1994 77.4681
1995 77.2414
1996 76.4495
1997 76.8427
1998 79.0609
1999 81.0009
2000 83.6121
2001 83.9748
2002 85.7038
2003 87.0027
2004 87.453
2005 88.2316
2006 88.7268
2007 88.8782
2008 89.432
2009 90.1397
2010 92.8094
2011 93.031
2012 93.488
2013 93.2215
2014 95.0116
2015 96.1777
2016 99.3649
2017 101.455
2018 103.559
2019 104.201
2020 110.382
2021 114.156
2022 113.862
};
\end{axis}
\end{tikzpicture}
\caption{Time series of the households' hourly wage at different quantiles.}
\label{fig:wages_both}
\end{figure}
Consider the implications of these trends for inequality. For males, females
and households the Q3/Q1 ratios increased from 1.75, 1.80 and 1.58 in 1976
to 2.33, 2.23 and 1.96 in 2022. The D9/D1 ratios increase from 3.14. 2.90
and 2.39 in 1976 to 5.00, 4.48 and 3.72 in 2022. While we cannot directly
infer anything definitive about sorting via the relative magnitudes of these
measures we report them to reflect the extent of inequality. However, the
rate of growth over the time period is similar for each which appears to
suggest positive assortative matching. Fern\'{a}ndez-Val et al. (2018) find
that for the period 1976-2016 the D9/D1 ratio for all males increases from
3.6 to 5.4 and that of females increases from 3.7 to 5. This suggests that
there is less wage inequality for married individuals. This appears to
primarily reflect the relatively higher wages of married individuals at low
quantiles.
\section{Marital Sorting}
\label{sec:marital_sorting}
We capture marital sorting patterns by reporting the propensity of the male
in the $j^{th}$ decile of the male wage distribution to be married to a
female in the $k^{th}$ decile of the female wage distribution. We consider $
j $,$k=1\ldots 10$ producing 100 cells. To overcome small sample sizes we
aggregate the data into five year intervals. To contrast the changes in cell
sizes over our entire sample period, we compare the beginning and end
periods corresponding to 1976-80 and 2018-22. Tables \ref{tab:frequencies_1}
and \ref{tab:frequencies_2} report the contingency tables for these two
periods. As random sorting is consistent with each cell containing 0.01 of
the data, we divide the sample frequencies by this number. Deviations from 1
are suggestive of sorting behavior. Perfect positive assortative matching
implies all couples should appear on the diagonal going from the top left to
the bottom right with each element having a value of 10. Perfect negative
assortative matching implies the elements on the diagonal going from the
bottom left to the top right have value 10. Note that dj denotes that an
observation is located between the D(j-1) to D(j) decile.
We acknowledge that a number of factors may generate departures from random
sorting. For example, individuals are likely to sort on observed factors
such as age, education, race and the individual's region of residence. As
these factors are determinants of wages this may spuriously support positive
sorting on wages. There may also be wage variations reflecting regional
specific cost of living influences. While cost of living components of wages
may also partially capture unobserved factors, we do not take a position on
whether this necessarily reflects sorting.
For the 1976-80 period there is evidence of non-random sorting. The most
striking features of Table \ref{tab:frequencies_1} are the cells associated
with the extreme values of the joint wage distribution. The two largest
observed frequencies correspond to both spouses in the bottom decile (2.19)
and both in the top decile (2.39). The next two highest values correspond to
the cells immediately adjacent these extremes (1.72 and 1.59). This is
consistent with positive sorting. If we define the bottom as (d1+d2) and the
top as (d9+d10) we obtain 6.56\% and 6.60\%. This contrasts with the 4\%
implied by random sorting. Another interesting feature of the table is
revealed by the off-diagonal values. There is an almost monotonically
decreasing relationship between distance from the diagonal and the
probability of marriage. Finally, the sum of the elements on the diagonal is
13.78, which appears to support positive assortative matching. However, as
this ignores behavior away from the diagonal and it is not invariant to the
definition of the diagonal we computed the Kendall rank correlation
coefficient. Its value of 0.18 supports positive assortative matching.
Table \ref{tab:frequencies_2} presents the corresponding values for the
2018-2022 period. As this period is associated with a substantial increase
in wage inequality, the level of household inequality is likely to have
increased even if the cell frequencies remained at the 1976-80 values.
However, it appears that increased marital sorting has exacerbated the
individual level inequality. For example, the two extreme cells have
increased to 2.94 and 3.25. This represents a particularly large increase in
the d1/d1 cell. The frequencies in the bottom 2 and top 2 have increased to
8.13 and 8.48. This reflects a large growth in the frequency in which lower
(higher) paid males are married to lower (higher) paid females. The growth
in the positive sorting at the bottom is concerning given the manner in
which wages, relative to 1976, have fallen for males and only slightly
increased for females at these locations of their distributions. The other
features of Table \ref{tab:frequencies_1} regarding sorting are also
generally supported by Table \ref{tab:frequencies_2}. There is greater
evidence that the lowly paid are married to the lowly paid while the highly
paid are married to the highly paid. The sum of the diagonal is now 17.24
and the Kendall rank correlation coefficient is 0.27. This suggests
increased positive assortative matching relative to the 1976-80 period.
\begin{table}[ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{2.191} & 1.287 & 1.158 & 0.937 & 0.833 & 0.899 & 0.77 & 0.684
& 0.66 & 0.582 \\
& d2 & 1.723 & \textbf{1.352} & 1.203 & 1.069 & 0.958 & 0.839 & 0.803 & 0.767
& 0.755 & 0.531 \\
& d3 & 1.272 & 1.394 & \textbf{1.099} & 1.108 & 1.12 & 0.872 & 0.866 & 0.86
& 0.758 & 0.654 \\
& d4 & 1.17 & 1.4 & 1.194 & \textbf{1.117} & 0.925 & 0.916 & 0.905 & 0.899 &
0.779 & 0.693 \\
& d5 & 0.899 & 1.12 & 1.224 & 1.12 & \textbf{1.105} & 0.973 & 0.955 & 0.94 &
0.911 & 0.755 \\
& d6 & 0.74 & 1.036 & 1.063 & 1.14 & 1.063 & \textbf{1.111} & 1.003 & 1.024
& 0.955 & 0.866 \\
& d7 & 0.612 & 0.782 & 1.102 & 1.036 & 1.087 & 1.099 & \textbf{1.042} & 1.122
& 1.146 & 0.97 \\
& d8 & 0.579 & 0.609 & 0.827 & 1.134 & 1.078 & 1.108 & 1.152 & \textbf{1.125}
& 1.218 & 1.17 \\
& d9 & 0.436 & 0.618 & 0.702 & 0.758 & 1.146 & 1.203 & 1.275 & 1.254 &
\textbf{1.224} & 1.385 \\
& d10 & 0.379 & 0.403 & 0.43 & 0.579 & 0.687 & 0.982 & 1.227 & 1.325 & 1.594
& \textbf{2.391} \\ \hline\hline
\end{tabular}}
\caption{Frequencies of the combination of deciles of married couples for
the period 1976-1980.}
\label{tab:frequencies_1}
\end{table}
\begin{table}[ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{2.946} & 1.56 & 1.06 & 0.881 & 0.874 & 0.774 & 0.527 & 0.579
& 0.445 & 0.353 \\
& d2 & 1.862 & \textbf{1.771} & 1.335 & 1.085 & 0.981 & 0.817 & 0.643 & 0.658
& 0.5 & 0.347 \\
& d3 & 1.268 & 1.71 & \textbf{1.408} & 1.21 & 0.987 & 0.881 & 0.85 & 0.64 &
0.57 & 0.476 \\
& d4 & 0.887 & 1.21 & 1.603 & \textbf{1.234} & 1.161 & 0.987 & 0.957 & 0.759
& 0.667 & 0.536 \\
& d5 & 0.75 & 0.975 & 1.207 & 1.353 & \textbf{1.122} & 1.082 & 1.045 & 1.03
& 0.808 & 0.628 \\
& d6 & 0.64 & 0.683 & 0.89 & 1.295 & 1.445 & \textbf{1.225} & 1.143 & 1.042
& 0.917 & 0.719 \\
& d7 & 0.488 & 0.686 & 0.932 & 1.009 & 1.222 & 1.335 & \textbf{1.2} & 1.149
& 1.109 & 0.871 \\
& d8 & 0.46 & 0.527 & 0.661 & 0.741 & 0.942 & 1.253 & 1.536 & \textbf{1.311}
& 1.411 & 1.158 \\
& d9 & 0.375 & 0.503 & 0.549 & 0.677 & 0.75 & 0.969 & 1.234 & 1.509 &
\textbf{1.774} & 1.661 \\
& d10 & 0.326 & 0.375 & 0.354 & 0.518 & 0.515 & 0.677 & 0.866 & 1.323 & 1.798
& \textbf{3.249} \\ \hline\hline
\end{tabular}}
\caption{Frequencies of the combination of deciles of married couples for
the period 2018-2022.}
\label{tab:frequencies_2}
\end{table}
The differences across the two tables are important given their potential
implications for inequality. However, the differences might reflect a
variety of factors. As employment rates have changed substantially over the
sample period, the composition, in terms of observed and unobserved
characteristics, of both husbands and wives may have changed and this may
have affected the observed patterns of sorting. It is also possible that the
prices of these observed and unobserved characteristics have changed.
Finally, the nature of marital sorting, as measured by the joint
distribution of the couples observed and unobserved characteristics, may
have changed.
Many couples share characteristics which are determinants of wages, such as
race, geographical location and age, although this does not necessarily
reflect sorting on wages. Another important characteristic is education
although this might be more reasonably interpreted as indicative of
productivity. To examine how these characteristics are allocated across
cells, Tables \ref{tab:age_wives}-\ref{tab:non_white_husbands1} report the
average value of some measure of each of these characteristics for the same
time periods.
Tables \ref{tab:age_wives}-\ref{tab:age_husbands1} report the ages of wives
and husbands. As age is a determinant of wages and spouses generally are
close in age, it is possible that the patterns in Tables \ref
{tab:frequencies_1} and \ref{tab:frequencies_2} simply reflect age
differences. For the first period the average age for both husbands and
wives increases by about 2 years as one goes from the d1/d1 to d10/d10. This
seems a remarkably small difference given the large wage discrepancies
across these cells. The highest husband age is associated with the highest
male decile and the highest ages of wives are for the lowest paid women
marrying these men. While this is an interesting result it is beyond the
scope of the paper to investigate it further. There are differences across
the cells, but it does not appear that age differences are the factors
driving the observed sorting. For the later period the average ages of the
spouses increase by approximately 2.5 years as one goes from d1/d1 to
d10/d10. In percentage terms this is a notable increase compared to the
earlier period although it does not appear to be the driving force of
increased inequality across the two periods. The oldest males continue to be
the highest paid married to the lowest paid women and the oldest wives are
those married to these men. The age differences across cells is larger than
in the earlier period but are unlikely to explain the observed wage
differences.
Tables \ref{tab:educ_wives} to \ref{tab:educ_husbands1} report the fraction
of wives and husbands with university education in the two time periods.
Unsurprisingly, educational levels in the higher wage cells are higher than
those in the lower. The large increase in individuals obtaining college
education is reflected in the substantially higher averages for the later
period. We also report the percentage of households in each cell for which
both spouses have university education. For the earlier period the product
of the two d10 cells is .353(.552*.64) although the corresponding cell in
Table \ref{tab:educ_both} is .467. This difference is supportive of couples
sorting on education at high wages. For the later period the corresponding
numbers are .868(.938*.926) and .897. Thus while there are more married
couples with both spouses university educated, there appears to be more
positive sorting on education at high wages in the earlier period. The
product of the d1 cells for the first period is .004(.064*.069) while for
Table \ref{tab:educ_both} it is .034. For the later period the corresponding
numbers are .012(.107*.115) and .059. This also suggests there is greater
positive sorting on education in the earlier period. Tables \ref
{tab:non_white_wives} to \ref{tab:non_white_husbands1} unsurprisingly
indicate that race has some association with an individual's location in
each of the wage distribution and that the strength of this relationship has
changed drastically over time. Moreover, Tables \ref{tab:non_white_wives}-
\ref{tab:non_white_husbands1} collectively confirm positive sorting on race.
This highlights the necessity to account for these factors in estimating the
sorting models below. The final characteristic we consider is the location
of residence. This is also likely to affect wage differences as it may
capture cost of living differences. Tables \ref{tab:metro_husbands} and \ref
{tab:metro_husbands1} suggest that the higher paid are living in
metropolitan areas and that the fraction of individuals living in these
areas has increased substantially over the sample period. As married
individuals generally cohabitate this may also spuriously imply sorting on
wages.
\section{Econometric model}
\label{sec:econometric_model}
\subsection{Determinants of Marital Sorting}
We now focus on estimating the determinants of the marital sorting
frequencies in Tables \ref{tab:frequencies_1} and \ref{tab:frequencies_2}.
We do so via the bivariate distribution regression (BDR) approach of Fern
\'{a}ndez-Val et al. (2023) which employs a Local Gaussian Representation
(LGR) of the joint distribution of the wives' and husbands' wages
(Chernozhukov, Fern\'{a}ndez-Val, and Luo 2019). We represent this joint
distribution as:
\begin{equation}
F_{Y_{w},Y_{h}\mid X}(y_{w},y_{h}\mid x)=\Phi _{2}(\mu _{w}(y_{w},x),\mu
_{h}(y_{h},x);\rho (y_{w},y_{h},x)),\quad (y_{w},y_{h})\in \mathbb{R}^{2},
\label{eq:bivariate}
\end{equation}
where $Y_{w}$ and $Y_{h}$ are the observed wages of wives and husbands, $X$
is a set of observed characteristics, $\Phi _{2}(\cdot ,\cdot ;\rho )$ is
the standard bivariate normal CDF with correlation $\rho ,$ and $\mu
_{j}(y,x)$ is formally defined below. In the LGR, the marginal conditional
CDFs of $Y_{w}$ and $Y_{h}$ are represented by:
\begin{equation*}
F_{Y_{j}\mid X}(y\mid x)=\Phi (\mu _{j}(y,x)),\quad y\in \mathbb{R},\quad
j\in \{w,h\},
\end{equation*}
where $\Phi $ is the standard univariate normal CDF. The parameter $\rho
(y_{w},y_{h},x)$ is the local correlation between the unobservables
influencing the spouses' wages at $(y_{w},y_{h})$ and captures sorting on
unobservables. The unconditional joint distribution of the wives' and
husbands' wages can be obtained from the LGR as:
\begin{equation*}
F_{Y_{w},Y_{h}}(y_{w},y_{h})=\int \Phi _{2}(\mu _{w}(y_{w},x),\mu
_{h}(y_{h},x);\rho (y_{w},y_{h},x))dF_{X}(x),\quad (y_{w},y_{h})\in \mathbb{R
}^{2},
\end{equation*}
where $F_{X}$ is the CDF of $X$, and the corresponding marginals are:
\begin{equation*}
F_{Y_{j}}(y)=\int \Phi (\mu _{j}(y,x))dF_{X}(x),\quad y\in \mathbb{R},\quad
j\in \{w,h\}.
\end{equation*}
Chernozhukov, Fern\'{a}ndez-Val, and Luo (2019) show that the LGR is
non-parametric as it does not impose any restrictions on the conditional
joint distribution.
The BDR model augments the LGR with two assumptions:
\begin{assumption}[BDR]
\label{ass:bdr} (1) $\mu_j(y,x) = P_{j}(x)^{\prime }\beta _{j}(y),$ $j \in
\{w,h\},$ where $P_{w}$ and $P_{h}$ are transformations of $x$ and $\beta
_{w}(y)$ and $\beta _{h}(y)$ are vectors of coefficients; and (2) $\rho
(y_{w},y_{h},x) = \rho(y_{w},y_{h})$.
\end{assumption}
We allow different specifications for $P_{w}$ and $P_{h}.$ For example, $
P_{w}$ includes the wife's education and age while $P_{h}$ does not. While
we assume that the marital sorting parameter does not depend on observed
characteristics, we allow it to vary by location in the joint distribution
of wages. This model is semiparametric as the parameters $y\mapsto \beta
_{j}(y),$ $j\in \{w,h\},$ and $(y_{w},y_{h})\mapsto \rho (y_{w},y_{h})$ are
function-valued.
The BDR model describes the joint distribution of wages conditional on the
two indices capturing the observed determinants of husbands and wives'
wages. Given a random sample of $(Y_{w},Y_{h},X)$, estimation is performed
via a series of bivariate probits of the indicators $\mathbf{1}(Y_{w}\leq
y_{w})$ and $\mathbf{1}(Y_{h}\leq y_{h})$ on $X$ for multiple values of $
y_{h}$ and $y_{w}$. This is done in two steps. First, estimate $\beta
_{j}(y_{j})$ via univariate probit of $\mathbf{1}(Y_{j}\leq y_{j})$ on $X$,
for $j\in \{w,h\}$. Second, estimate $\rho (y_{w},y_{h})$ via bivariate
probit plugging-in the estimates of $\beta _{w}(y_{w})$ and $\beta
_{h}(y_{h})$.
We begin by examining our capacity to explain the variation in Tables \ref
{tab:frequencies_1} and \ref{tab:frequencies_2} via 100 bivariate probits.
The entries correspond to estimates of:
\begin{multline}
s(\underline{y}_{w},\overline{y}_{w},\underline{y}_{w},\overline{y}_{w}):=
\frac{\Pr (\underline{y}_{w}<Y_{w}\leq \overline{y}_{w},\underline{y}
_{h}<Y_{h}\leq \overline{y}_{h})}{\Pr (\underline{y}_{w}<Y_{w}\leq \overline{
y}_{w})\Pr (\underline{y}_{h}<Y_{h}\leq \overline{y}_{h})}
\label{eq:transition} \\
=\frac{F_{Y_{w},Y_{h}}(\overline{y}_{w},\overline{y}_{h})-F_{Y_{w},Y_{h}}(
\overline{y}_{w},\underline{y}_{h})-F_{Y_{w},Y_{h}}(\underline{y}_{w},
\overline{y}_{h})+F_{Y_{w},Y_{h}}(\underline{y}_{w},\underline{y}_{h})}{
[F_{Y_{w}}(\overline{y}_{w})-F_{Y_{w}}(\underline{y}_{w})][F_{Y_{h}}(
\overline{y}_{h})-F_{Y_{h}}(\underline{y}_{h})]},
\end{multline}
where $\underline{y}_{w}$ and $\overline{y}_{w}$ are evaluated at the sample
deciles of $Y_{w}$, and $\underline{y}_{h}$ and $\overline{y}_{h}$ at the
sample deciles of $Y_{h}$. The predicted probabilities are reported in
Tables \ref{tab:bivariate_1} for 1976-1980 and Table \ref{tab:bivariate_2}
for 2018-2022. A comparison of the predicted and empirical probabilities for
each of the sample periods indicates that the estimated model reproduces the
empirical probabilities across the 100 cells despite the restrictions
imposed by Assumption \ref{ass:bdr}.\footnote{
The predicted probabilities are computed by first using univariate
distribution regression models to obtain the quantiles of the marginal
distribution of wives and husbands. That is, we estimate the quantiles for
wives, $Q_{\tau }^{w};\tau \in \lbrack 0,1]$ solving the empirical analog of
$\int \Phi (x\beta (Q_{\tau }^{w}))dF_{X_{w}}(x)=\tau $, where $\beta (\cdot
)$ is estimated by univariate distribution regression.The quantiles for
husbands are estimated similarly. Based on these quantiles, we can estimate
the bivariate distribution regression model as in (\ref{eq:bivariate}) and
estimate the empirical analog of $\int \Phi _{2}(x_{w}\beta (Q_{\tau
}^{w}),x_{h}\beta (Q_{\tau }^{h}),\rho )dF_{X_{w},X_{h}}(x_{w},x_{h})$.}
Figure \ref{fig:kernel} presents the estimated densities of the 100
estimates of $\rho (y_{w},y_{h})$ for each sample period. The average values
for the two periods are $0.18$ and $0.24$ and the figure reveals a clear
shift in the distribution to the right over time. Thus, despite the rich
nature of $X$, there remains a strong and positive correlation between the
unobservables driving wages for spouses and it has increased over time.
\footnote{\label{footnote:x}$X$ includes 3 dummy variables for education
(high school, some college, college degree or higher), age, age$^{2}$, age
interacted with 3 education dummy variables, $age^{2}$ interacted with 3
education dummy variables, a dummy variable for non-white, a dummy variable
for Hispanic, 2 dummy variables for metropolitan area (central city, outside
central city), and 7 regional dummy variables (middle Atlantic, east north
central, west north central, south atlantic, east south central, west south
central, mountain, and pacific. base: New England).} This supports the
presence of positive and increasing assortative matching on wages. While
this may partially reflect common influences on wages, such as local
adjustments related to cost of living, it may also capture unobserved
ability or other unobserved determinants of productivity.
\begin{figure}[tbp]
\caption{Estimated densities of $\protect\rho (y_{w},y_{h})$}
\label{fig:kernel}\centering
\begin{tikzpicture}
\definecolor{darkgray176}{RGB}{176,176,176}
\definecolor{steelblue31119180}{RGB}{31,119,180}
\begin{groupplot}[group style={group size=2 by 1}]
\nextgroupplot[
tick align=outside,
tick pos=left,
title={1976-1980},
x grid style={darkgray176},
xlabel={\(\displaystyle \rho (y_{w},y_{h})\)},
xmin=0, xmax=0.5,
xtick style={color=black},
y grid style={darkgray176},
ylabel={density},
ymin=0, ymax=8,
ytick style={color=black}
]
\addplot [thick, steelblue31119180]
table {
0.0200588 0.013329643832925
0.0300588 0.0442055423533545
0.0400588 0.118073769492605
0.0500588 0.259471947519481
0.0600588 0.485332750392802
0.0700588 0.80807818950166
0.0800588 1.24750277326404
0.0900588 1.81679894667106
0.1000588 2.48466901033453
0.1100588 3.18030325021522
0.1200588 3.85901875886555
0.1300588 4.53852923545341
0.1400588 5.24794549651488
0.1500588 5.95560303236278
0.1600588 6.5608258713713
0.1700588 6.96067377756324
0.1800588 7.14577543254212
0.1900588 7.2429298853014
0.2000588 7.41529052852382
0.2100588 7.64318708585818
0.2200588 7.60939188099878
0.2300588 6.91119137675457
0.2400588 5.46077937803065
0.2500588 3.63712547415058
0.2600588 2.00211012163508
0.2700588 0.899195163145697
0.2800588 0.326536888161733
0.2900588 0.0952337548354639
0.3000588 0.0221899726569547
0.3100588 0.00411371744460971
0.3200588 0.000604771489147844
0.3300588 7.03192163422005e-05
0.3400588 6.45274988578379e-06
};
\nextgroupplot[
tick align=outside,
tick pos=left,
title={2018-2022},
x grid style={darkgray176},
xlabel={\(\displaystyle \rho (y_{w},y_{h})\)},
xmin=0, xmax=0.5,
xtick style={color=black},
y grid style={darkgray176},
ymin=0, ymax=8,
ytick style={color=black}
]
\addplot [thick, steelblue31119180]
table {
0.071201 0.0164881995479213
0.081201 0.0555945912949744
0.091201 0.150304709413774
0.101201 0.330669621129432
0.111201 0.6062883260344
0.121201 0.961099905348157
0.131201 1.38176647391184
0.141201 1.88214729381961
0.151201 2.47084199232373
0.161201 3.09097246627192
0.171201 3.63025398289376
0.181201 4.02214228658895
0.191201 4.31986041037763
0.201201 4.63584408680516
0.211201 5.01473382362662
0.221201 5.40363844167821
0.231201 5.74585778778284
0.241201 6.05019050689403
0.251201 6.33713807552092
0.261201 6.55518099490341
0.271201 6.59498344616832
0.281201 6.3684546179569
0.291201 5.84777189673211
0.301201 5.06476838528119
0.311201 4.11522775346944
0.321201 3.14153422500065
0.331201 2.27154726679771
0.341201 1.56825276391304
0.351201 1.03426043901221
0.361201 0.64440333264624
0.371201 0.370156933169089
0.381201 0.189130764245107
0.391201 0.0826685957801919
0.401201 0.0298981356784806
0.411201 0.0087359447832138
0.421201 0.00203102133454942
0.431201 0.000372298790214417
0.441201 5.3523529419514e-05
0.451201 6.01677481068804e-06
};
\end{groupplot}
\end{tikzpicture}
\end{figure}
While the estimate of $\rho (y_{w},y_{h})$ is consistent with positive
sorting, it is not immediately clear from the coefficient how economically
important this parameter is in determining the observed patterns. We employ
the estimates from the BDR model to construct counterfactual contingency
tables from setting $\rho (y_{w},y_{h})=0$ for all $y_{w},y_{h}$.\footnote{
As the quantiles of the marginal distributions are not affected by setting $
\rho (y_{w},y_{h})$ to zero, we can simply estimate the corresponding
distributions in \eqref{eq:bivariate} by setting $\rho (y_{w},y_{h})$ equal
to zero.} This corresponds to no correlation or sorting on unobservables
while retaining the sorting on the observed characteristics. We implement
this counterfactual in the two periods and report the estimates in Tables
\ref{tab:biv2_1976} and \ref{tab:biv2_2018}. The predicted probabilities
indicate that sorting on unobservables accounts for a large fraction of the
observed positive marital sorting. This is remarkable given the positive
sorting that occurs on race, age, educational attainments and location of
residence highlighted above. Even though it is hard to make any conjectures
about the source of the unobserved heterogeneity, Eika, Mogstad, and Zafar
(2019) show for Norwegian data that the choice of college major is an
important source of educational sorting. This would be captured in $\rho
(y_{w},y_{h}).$
\begin{table}[ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{1.326} & 1.233 & 1.08 & 1.065 & 1.008 & 0.944 & 0.914 & 0.871
& 0.828 & 0.697 \\
& d2 & 1.271 & \textbf{1.223} & 1.088 & 1.093 & 1.04 & 0.98 & 0.956 & 0.912
& 0.87 & 0.734 \\
& d3 & 1.132 & 1.124 & \textbf{1.018} & 1.039 & 1.005 & 0.949 & 0.936 & 0.906
& 0.871 & 0.746 \\
& d4 & 1.097 & 1.135 & 1.046 & \textbf{1.083} & 1.058 & 1.001 & 0.992 & 0.972
& 0.94 & 0.813 \\
& d5 & 1.036 & 1.091 & 1.027 & 1.086 & \textbf{1.077} & 1.032 & 1.023 & 1.024
& 0.993 & 0.9 \\
& d6 & 0.914 & 0.977 & 0.925 & 0.994 & 0.995 & \textbf{0.96} & 0.966 & 0.978
& 0.957 & 0.916 \\
& d7 & 0.914 & 0.98 & 0.949 & 1.027 & 1.046 & 1.026 & \textbf{1.042} & 1.058
& 1.058 & 1.059 \\
& d8 & 0.814 & 0.892 & 0.877 & 0.977 & 1.005 & 1.01 & 1.026 & \textbf{1.065}
& 1.073 & 1.122 \\
& d9 & 0.784 & 0.861 & 0.854 & 0.965 & 1.015 & 1.047 & 1.089 & 1.14 &
\textbf{1.175} & 1.28 \\
& d10 & 0.656 & 0.708 & 0.723 & 0.844 & 0.905 & 0.977 & 1.069 & 1.156 & 1.266
& \textbf{1.601} \\ \hline\hline
\end{tabular}}
\caption{Results of estimated sorting measures in 1976-1980 at different
quantiles when $\protect\rho_{Y^*_w, Y^*_h} = 0$.}
\label{tab:biv2_1976}
\end{table}
\begin{table}[ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{1.525} & 1.375 & 1.236 & 1.131 & 0.991 & 0.933 & 0.834 & 0.739
& 0.683 & 0.553 \\
& d2 & 1.351 & \textbf{1.301} & 1.229 & 1.15 & 1.021 & 0.978 & 0.884 & 0.781
& 0.724 & 0.581 \\
& d3 & 1.212 & 1.218 & \textbf{1.189} & 1.134 & 1.028 & 1.01 & 0.926 & 0.833
& 0.792 & 0.659 \\
& d4 & 1.097 & 1.132 & 1.141 & \textbf{1.114} & 1.025 & 1.031 & 0.967 & 0.897
& 0.873 & 0.724 \\
& d5 & 0.965 & 1.024 & 1.059 & 1.069 & \textbf{1.02} & 1.028 & 1.009 & 0.962
& 0.989 & 0.875 \\
& d6 & 0.889 & 0.966 & 1.019 & 1.039 & 1.01 & \textbf{1.047} & 1.041 & 1.005
& 1.046 & 0.938 \\
& d7 & 0.809 & 0.885 & 0.958 & 0.991 & 0.986 & 1.048 & \textbf{1.061} & 1.069
& 1.154 & 1.039 \\
& d8 & 0.73 & 0.803 & 0.879 & 0.927 & 0.962 & 1.046 & 1.081 & \textbf{1.115}
& 1.253 & 1.205 \\
& d9 & 0.66 & 0.717 & 0.786 & 0.845 & 0.922 & 1.009 & 1.092 & 1.174 &
\textbf{1.386} & 1.408 \\
& d10 & 0.569 & 0.627 & 0.691 & 0.77 & 0.876 & 0.962 & 1.093 & 1.218 & 1.488
& \textbf{1.706} \\ \hline\hline
\end{tabular}}
\caption{Results of estimated sorting measures in 2018-2022 at different
quantiles when $\protect\rho_{Y^*_w, Y^*_h} = 0$.}
\label{tab:biv2_2018}
\end{table}
\subsection{Marital Sorting with Employment Selection}
\label{sec:model}
The evidence above is based on the subpopulation of working married couples
as the BDR model above does not account for selection into employment.
As our sample period witnessed drastic increases in the market participation
of married females it is possible that both the compositions of the
subpopulations of working married females and the working married males with
working spouses have changed. We now incorporate the participation decision
of each spouse, while allowing for a relationship between these decisions,
by extending the BDR model to incorporate endogenous employment decisions.
This represents an application of the CFL approach with some provisos we
outline below. We begin with some necessary preliminaries.
\subsection{The sample selection model}
Consider the vector of random variables $(D_{w}^{\ast },D_{h}^{\ast
},Y_{w}^{\ast },Y_{h}^{\ast })$, where $D_{w}^{\ast }$ and $D_{h}^{\ast }$
are latent variables that determine the employment decision of the wife and
husband, and $Y_{w}^{\ast }$ and $Y_{h}^{\ast }$ are the offered wages to
the wife and husband. $Z$ is a vector of observed characteristics of which $
X $ is a subset . We make the following assumptions regarding the joint CDF
of $(D_{w}^{\ast },D_{h}^{\ast },Y_{w}^{\ast },Y_{h}^{\ast })$ conditional
on $Z $:
\begin{assumption}[LGR, Relevance and Exclusion]
\label{ass:lgr} (1) For $(d_{w},d_{h},y_{w},y_{h}) \in \mathbb{R}^4,$
\begin{equation*}
F_{D_{w}^{\ast },D_{h}^{\ast },Y_{w}^{\ast },Y_{h}^{\ast } \mid
Z}(d_{w},d_{h},y_{w},y_{h} \mid z) = \Phi _{4}(\boldsymbol{\mu}
(d_{w},d_{h},y_{w},y_{h},z); \boldsymbol{\Sigma}
(d_{w},d_{h},y_{w},y_{h},z)),
\end{equation*}
where $\Phi _{4}(\cdot; \boldsymbol{\Sigma})$ is the standard tetravariate
normal CDF with correlation matrix $\boldsymbol{\Sigma}$,
\begin{equation*}
\boldsymbol{\mu} (d_{w},d_{h},y_{w},y_{h},z)=\left(
\begin{array}{c}
\mu _{D_{w}^{\ast}}(d_{w},z) \\
\mu _{D_{h}^{\ast}}(d_{h},z) \\
\mu _{Y_{w}^{\ast}}(y_{w},z) \\
\mu _{Y_{h}^{\ast}}(y_{h},z)
\end{array}
\right)
\end{equation*}
and
\begin{multline*}
\boldsymbol{\Sigma} (d_{w},d_{h},y_{w},y_{h},z)= \\
\left(
\begin{array}{cccc}
1 & \rho _{D_{w}^{\ast },D_{h}^{\ast }}(d_{w},d_{h},z) & - \rho
_{D_{w}^{\ast },Y_{w}^{\ast }}(d_{w},y_{w},z) & - \rho _{D_{w}^{\ast
},Y_{h}^{\ast }}(d_{w},y_{h},z) \\
\rho _{D_{w}^{\ast },D_{h}^{\ast }}(d_{w},d_{h},z) & 1 & - \rho
_{D_{h}^{\ast },Y_{w}^{\ast }}(d_{h},y_{w},z) & - \rho _{D_{h}^{\ast
},Y_{h}^{\ast }}(d_{h},y_{h},z) \\
- \rho _{D_{w}^{\ast },Y_{w}^{\ast }}(d_{w},y_{w},z) & - \rho _{D_{h}^{\ast
},Y_{w}^{\ast }}(d_{h},y_{w},z) & 1 & \rho _{Y_{w}^{\ast },Y_{h}^{\ast
}}(y_{w},y_{h},z) \\
- \rho _{D_{w}^{\ast },Y_{h}^{\ast }}(d_{w},y_{h},z) & - \rho _{D_{h}^{\ast
},Y_{h}^{\ast }}(d_{h},y_{h},z) & \rho _{Y_{w}^{\ast },Y_{h}^{\ast
}}(y_{w},y_{h},z) & 1
\end{array}
\right)
\end{multline*}
is non-singular almost everywhere in $z$; (2) $\mu _{D_{w}^{\ast}}(d_{w},z)
\neq \mu _{D_{w}^{\ast}}(d_{w},z^{\prime }) $ and $\mu
_{D_{h}^{\ast}}(d_{h},z^{\prime \prime }) \neq \mu
_{D_{h}^{\ast}}(d_{h},z^{\prime \prime \prime })$ for some $z = (z_1,x)$ , $
z^{\prime } = (z_1^{\prime },x)$, $z^{\prime \prime }= (z_1^{\prime \prime
},x)$, and $z^{\prime \prime \prime} = (z_1^{\prime \prime \prime},x)$; and
(3) $\mu _{Y_{w}^{\ast }}(y_{w},z) = \mu _{Y_{w}^{\ast }}(y_{w},x)$, $\mu
_{Y_{h}^{\ast }}(y_{h},z) = \mu _{Y_{h}^{\ast }}(y_{h},x)$, $
\rho_{D_{w}^{\ast },Y_{w}^{\ast }}(d_{w},y_{w},z) = \rho _{D_{w}^{\ast
},Y_{w}^{\ast }}(d_{w},y_{w},x)$,
and $\rho _{D_{h}^{\ast},Y_{h}^{\ast }}(d_{h},y_{h},z) = \rho _{D_{h}^{\ast
},Y_{h}^{\ast }}(d_{h},y_{h},x)$,
for $z = (z_1,x)$.
\end{assumption}
Assumption \ref{ass:lgr}(1) is similar to the LGR of a joint CDF. In
contrast to the bivariate case, this representation restricts some features
of the joint tetravariate distribution. While the univariate and bivariate
marginals remain unrestricted, some restrictions are imposed on the
trivariate marginals and joint tetravariate distributions. To highlight
this, note that the local dependence between any pair of random variables,
as measured by the corresponding component of the matrix $\boldsymbol{\Sigma
}(d_{w},d_{h},y_{w},y_{h},z)$, does not depend on the value of the other
components. For example, local pairwise independence of all the components,
that is $\boldsymbol{\Sigma }(d_{w},d_{h},y_{w},y_{h},z)$ equal to the
identity matrix, implies joint local independence of all the components.
Assumption \ref{ass:lgr}(3) embodies exclusion restrictions on the marginal
distributions of $Y_w^*$ and $Y_h^*$, and the local dependence matrix $
\boldsymbol{\Sigma} (d_{w},d_{h},y_{w},y_{h},z)$. Thus, $Y_w^*$ and $Y_h^*$
are independent of the components of $Z$ not included in $X$. Moreover,
these components do not affect the local dependence between all the
components in $(D_{w}^{\ast },D_{h}^{\ast },Y_{w}^{\ast },Y_{h}^{\ast })$.
Assumption \ref{ass:lgr}(2) is a relevance condition of these excluded
components of $Z$ on the expectations of $D_w^*$ and $D_h^*$.
The observed variables $(D_{w},D_{h},Y_{w},Y_{h})$ are related to the latent
variables as:
\begin{equation*}
\begin{split}
D_{w}& =\mathbf{1}(D_{w}^{\ast }\leq 0), \\
D_{h}& =\mathbf{1}(D_{h}^{\ast }\leq 0),
\end{split}
\end{equation*}
where $D_{w}$ ($D_{h})$ equals 1 when the wife (husband) is working FTFY and
0 otherwise. Moreover,
\begin{equation*}
\begin{split}
Y_{w}& =Y_{w}^{\ast }~\mbox{ if}~D_{w}=1~\mbox{and}~D_{h}=1 \\
Y_{h}& =Y_{h}^{\ast }~\mbox{ if}~D_{w}=1~\mbox{and}~D_{h}=1
\end{split}
\end{equation*}
where $Y_{w}$ and $Y_{h}$ are the FTFY hourly wages of the wife and husband.
These are only observed for FTFY working couples.
We show in Appendix \ref{app:id} that all the parameters of the LGR of the
latent variables are identified from the distribution of the observed
variables.
\begin{theorem}[Identification Under Employment Selection]
\label{thm:id} Under Assumption \ref{ass:lgr}, $\boldsymbol{\mu}
(0,0,y_{w},y_{h},z)$ and $\boldsymbol{\Sigma} (0,0,y_{w},y_{h},z)$ are
identified from the joint distribution of $(D_{w},D_{h},Y_{w},Y_{h},Z)$.
\end{theorem}
\begin{comment}
\subsection{Identification}
We define $\nu _{D_{w}^{\ast }}:=\nu _{D_{w}^{\ast }}(1)$, $\nu
_{D_{h}^{\ast }}:=\nu _{D_{h}^{\ast }}(1)$ and $\rho _{D_{w}^{\ast
},D_{h}^{\ast }}:=\rho _{D_{w}^{\ast },D_{h}^{\ast }}(d_{w},d_{h})$. The
parameters $\nu _{D_{w}^{\ast }}(z)$, $\nu _{D_{h}^{\ast }}(z)$ and $\rho
_{D_{w}^{\ast },D_{h}^{\ast }}(z)$ are identified through the following
system of equations:
\begin{equation*}
\begin{split}
\mathbb{P}(D_{w}& =1;D_{h}=1)=1-\Phi _{2}(\nu _{D_{w}^{\ast }}(z),\nu
_{D_{h}^{\ast }}(z),\rho _{D_{w}^{\ast },D_{h}^{\ast }}(z)) \\
\mathbb{P}(D_{w}& =1;D_{h}=0)=\Phi (\nu _{D_{h}^{\ast }}(z))-\Phi _{2}(\nu
_{D_{w}^{\ast }}(z),\nu _{D_{h}^{\ast }}(z),\rho _{D_{w}^{\ast },D_{h}^{\ast
}}(z)) \\
\mathbb{P}(D_{w}& =0;D_{h}=1)=\Phi (\nu _{D_{w}^{\ast }}(z))-\Phi _{2}(\nu
_{D_{w}^{\ast }}(z),\nu _{D_{h}^{\ast }}(z),\rho _{D_{w}^{\ast },D_{h}^{\ast
}}(z))
\end{split}
.
\end{equation*}
We have the following three restrictions for the distribution of $Y_{w}$ and
$Y_{h}$ for the selected sample:
\begin{equation*}
\begin{split}
\mathbb{P}(D_{w}& =1;D_{h}=1,Y_{w}\leq y_{w},Y_{h}\leq y_{h}) \\
& =\int_{\nu _{D_{w}^{\ast }}(z)}^{\infty }\int_{\nu _{D_{h}^{\ast
}}(z)}^{\infty }\int_{-\infty }^{\mu _{Y_{w}^{\ast }}(y_{w},z)}\int_{-\infty
}^{\mu _{Y_{h}^{\ast }}(y_{h},z)}d\Phi _{4}(u_{1},u_{2},u_{3},u_{4},\Sigma
(y_{1},y_{2},z)) \\
\mathbb{P}(D_{1}& =1;D_{2}=1,Y_{w}\leq y_{1},Y_{h}>y_{2})= \\
& \int_{\nu _{D_{w}^{\ast }}(z)}^{\infty }\int_{\nu _{D_{h}^{\ast
}}(z)}^{\infty }\int_{-\infty }^{\mu _{Y_{w}^{\ast }}(y_{w}(z))}\int_{\mu
_{Y_{h}^{\ast }}(y_{h}(z))}^{\infty }d\Phi
_{4}(u_{1},u_{2},u_{3},u_{4},\Sigma (y_{w},y_{h},z)) \\
\mathbb{P}(D_{1}& =1;D_{2}=1,Y_{w}>y_{w},Y_{h}\leq y_{h})= \\
& \int_{\nu _{D_{w}^{\ast }}(z)}^{\infty }\int_{\nu _{D_{h}^{\ast
}}(z)}^{\infty }\int_{\mu _{Y_{w}^{\ast }}(y_{w},z)}^{\infty }\int_{-\infty
}^{\mu _{Y_{h}^{\ast }}(y_{h},z)}d\Phi _{4}(u_{1},u_{2},u_{3},u_{4},\Sigma
(y_{w},y_{h},z))
\end{split}
\end{equation*}
where $\Sigma (y_{w},y_{h},z):=\Sigma (0,0,y_{w},y_{h},z)$. The 7 remaining
unknowns are: $\mu _{Y_{w}^{\ast }}(y_{w},z)$, $\mu _{Y_{h}^{\ast }}(y_{2},z)
$, $\rho _{D_{w}^{\ast },Y_{w}^{\ast }}(y_{w},z)$, $\rho _{D_{w}^{\ast
},Y_{h}^{\ast }}(y_{h},z)$, $\rho _{D_{h}^{\ast },Y_{w}^{\ast }}(y_{w},z)$, $
\rho _{D_{h}^{\ast },Y_{h}^{\ast }}(y_{h},z)$ and $\rho _{Y_{w}^{\ast
},Y_{h}^{\ast }}(y_{w},y_{h},z)$ and these parameters are not point
identified without further exclusion restrictions. Define $X\subset Z$ as
the subset of the characteristics $Z$. Define $Z:=(X,\widetilde{Z})$. We
make the following assumptions about and excluded random variable $
\widetilde{Z}$:
\begin{enumerate}
\item $0 < \mathbb{P}(D_w=1;D_h=1|X=x) < 1$ and $0 < \mathbb{P}(\widetilde{Z}
=z|D_w=1,D_h=1,X=x) < 1$ for $z \in {z_0,z_1,z_2}; z_0 < z_1 < z_2$.
\item $\mathbb{P}(D_w=1;D_h=1|\widetilde{Z}=z_0, X=x) < \mathbb{P}
(D_w=1;D_h=1|\widetilde Z=z_1,X=x) < \mathbb{P}(D_w=1;D_h=1|\widetilde
Z=z_2) < 1$.
\item $\mu_{Y^*_w}(y_h, z) = \mu_{Y^*_w}(y_1,x)$ and $\mu_{Y^*_h}(y_h, z) =
\mu_{Y^*_h}(y_2,x)$ for all $y_h \in \mathbb{R}$, $y_w \in \mathbb{R}$ and $
z \in \{z_0,z_1,z_2\}$.
\item $\rho_{D^*_w, Y^*_w}(y_h,z) = \rho_{D^*_h, Y^*_w}(y_1,x)$, $
\rho_{D^*_w, Y^*_h}(y_2,z) = \rho_{D^*_w Y^*_h}(y_h,x) $, $\rho_{D^*_h,
Y^*_w}(y_w,z) = \rho_{D^*_h Y^*_w}(y_1,x)$, $\rho_{D^*_h, Y^*_w}(y_h,z) =
\rho_{D^*_h, Y^*_w}(y_w,x)$ and $\rho_{Y^*_w, Y^*_h}(y_w,y_h,z) =
\rho_{Y^*_w, Y^*_h}(y_w,y_h,x)$ for all $y_w \in \mathbb{R}$, $y_h \in
\mathbb{R}$ and $z \in \{z_0,z_1,z_2\}$.
\end{enumerate}
The first restriction implies that $Z$ has at least three mass points on the
selected sample of both partners working. The second restriction is a
combined monotonicity assumption of the exclusion restriction and a
restriction that implies relevance. The third and fourth assumptions form
the basis of the exclusion restriction implying that the distribution of the
wages are not directly affected by the outcome of $Z$.
Using these restrictions we write the model as:
\begin{equation*}
F_{D_{w}^{\ast },D_{h}^{\ast },Y_{w}^{\ast },Y_{h}^{\ast
}|Z}(0,0,y_{w},y_{h}|z)=\Phi _{4}(\mu (y_{w},y_{h},z),\Sigma (y_{w},y_{h},z))
\end{equation*}
with
\begin{equation*}
\mu (y_{w},y_{h},z)=\left(
\begin{array}{c}
\nu _{D_{w}^{\ast }}(z) \\
\nu _{D_{h}^{\ast }}(z) \\
\mu _{Y_{w}^{\ast }}(y_{w},x) \\
\mu _{Y_{h}^{\ast }}(y_{h},x)
\end{array}
\right)
\end{equation*}
and
\begin{equation*}
\Sigma (y_{w},y_{h}|z)=\left(
\begin{array}{cccc}
1 & \rho _{D_{w}^{\ast },D_{h}^{\ast }}(z) & \rho _{D_{w}^{\ast
},Y_{w}^{\ast }}(y_{w},x) & \rho _{D_{w}^{\ast },Y_{h}^{\ast }}(y_{h},x) \\
\rho _{D_{w}^{\ast },D_{h}^{\ast }}(z) & 1 & \rho _{D_{h}^{\ast
},Y_{w}^{\ast }}(y_{w},x) & \rho _{D_{h}^{\ast },Y_{h}^{\ast }}(y_{h},x) \\
\rho _{D_{w}^{\ast },Y_{w}^{\ast }}(y_{w},x) & \rho _{D_{h}^{\ast
},Y_{w}^{\ast }}(y_{w},x) & 1 & \rho _{Y_{w}^{\ast },Y_{h}^{\ast
}}(y_{w},y_{h},x) \\
\rho _{D_{w}^{\ast },Y_{h}^{\ast }}(y_{h},x) & \rho _{D_{h}^{\ast
},Y_{h}^{\ast }}(y_{h},x) & \rho _{Y_{w}^{\ast },Y_{h}^{\ast
}}(y_{w},y_{h},x) & 1
\end{array}
\right) .
\end{equation*}
The parameters $\nu _{D_{w}^{\ast }}(z_{j})$, $\nu _{D_{h}^{\ast }}(z_{j})$
and $\rho _{D_{w}^{\ast },D_{h}^{\ast }}(z_{j})$ for $j=0,1,2$ are
identified by the following set of equations:
\begin{equation*}
\begin{split}
\mathbb{P}(D_{w}& =1;D_{h}=1|Z=z_{j})=1-\Phi _{2}(\nu _{D_{w}^{\ast
}}(z_{j}),\nu _{D_{h}^{\ast }}(z_{j}),\rho _{D_{w}^{\ast },D_{h}^{\ast }}(z))
\\
\mathbb{P}(D_{w}& =1;D_{h}=0|Z=z_{j})=\Phi (\nu _{D_{h}^{\ast
}}(z_{j}))-\Phi _{2}(\nu _{D^{\ast }w}(z),\nu _{D_{h}^{\ast }}(z),\rho
_{D_{w}^{\ast },D_{h}^{\ast }}(z)) \\
\mathbb{P}(D_{w}& =0;D_{h}=1|Z=z_{j})=\Phi (\nu _{D_{w}^{\ast
}}(z_{j}))-\Phi _{2}(\nu _{D_{w}^{\ast }}(z_{j}),\nu _{h}(z_{j}),\rho
_{D_{w}^{\ast },D_{h}^{\ast }}(z_{j}))
\end{split}
\end{equation*}
for $j=0,1,2$. Even so we have the following three restrictions for the
distribution of $Y_{w}$ and $Y_{h}$ for the selected sample:
\begin{equation*}
\begin{split}
& \mathbb{P}(D_{w}=1;D_{h}=1,Y_{w}\leq y_{w},Y_{h}\leq y_{h}|Z=z_{j})= \\
& \int_{\nu _{D_{w}^{\ast }}(z_{j})}^{\infty }\int_{\nu _{D_{h}^{\ast
}}(z_{j})}^{\infty }\int_{-\infty }^{\mu _{Y_{w}^{\ast
}}(y_{w},x)}\int_{-\infty }^{\mu _{Y_{h}^{\ast }}(y_{h},x)}d\Phi
_{4}(u_{1},u_{2},u_{3},u_{4},\Sigma (y_{w},y_{h},z_{j})) \\
& \mathbb{P}(D_{w}=1;D_{h}=1,Y_{w}\leq y_{w},Y_{h}>y_{h}|Z=z_{j})= \\
& \int_{\nu _{D_{w}^{\ast }}(z_{j})}^{\infty }\int_{\nu _{D_{h}^{\ast
}}(z_{j})}^{\infty }\int_{-\infty }^{\mu _{Y_{w}^{\ast }}(y_{w},x)}\int_{\mu
_{Y_{h}^{\ast }}(y_{h},x)}^{\infty }d\Phi
_{4}(u_{1},u_{2},u_{3},u_{4},\Sigma (y_{w},y_{h},z_{j})) \\
& \mathbb{P}(D_{w}=1;D_{h}=1,Y_{w}>y_{w},Y_{h}\leq y_{h}|Z=z_{j})= \\
& \int_{\nu _{D_{w}^{\ast }}(z_{j})}^{\infty }\int_{\nu _{D_{h}^{\ast
}}(z_{j})}^{\infty }\int_{\mu _{Y_{w}^{\ast }}(y_{w},x)}^{\infty
}\int_{-\infty }^{\mu _{Y_{h}^{\ast }}(y_{h},x)}d\Phi
_{4}(u_{1},u_{2},u_{3},u_{4},\Sigma (y_{w},y_{h},z_{j}))
\end{split}
\end{equation*}
for $j=0,1,2$. This gives a system of 9 equations and 7 unknowns.
\subsection{The model}
We assume that
\begin{equation*}
\begin{split}
F_{W_w^*, W_h^*, H_w^*, H_h^*|X, Z}(w_w, w_h, 0,0|x,z) = \\
\Phi_4(-\beta^T_w(w_w, w_h) x, -\beta^T_h(w_w, w_h) x, - \pi^T_w z, -\pi^T_h
z, \Sigma_4(w_w, w_h))
\end{split}
\end{equation*}
where
\begin{equation*}
\begin{split}
&\Sigma_4(w_w, w_h,x,z) = \\
&\left(
\begin{array}{cccc}
1 & \rho_{10}(w_w, w_h, x, z) & \rho_{20}(w_w, w_h, x, z) & \rho_{30}(w_w,
w_h, x, z) \\
\rho_{10}(w_w, w_h,x,z) & 1 & \rho_{21}(w_w, w_h,x,z) & \rho_{31}(w_w,
w_h,x,z) \\
\rho_{20}(w_w, w_h,x,z) & \rho_{21}(w_w, w_h,x,z) & 1 & \rho_{32}(x,z) \\
\rho_{30}(w_w, w_h,x,z) & \rho_{31}(w_w, w_h,x,z) & \rho_{32}(x,z) & 1
\end{array}
\right)
\end{split}
\end{equation*}
and where $W_w^*$ and $W_h^*$ are the latent wages of the wife and the
husband while $H^*_w$ and $H^*_h$ are their corresponding latent working
hours. The vector $X$ represents the observed characteristics that determine
both the latent wage as well as the latent number of working hours for the
wife as well as the husband. The vector $Z$ represents the observed
characteristics that only affect the number of working hours. Define
\begin{equation*}
D_{w} = \mathbf{1}(H^*_w > 0)
\end{equation*}
as a dummy variable in the case that the wife works and $D_h$ is defined
likewise for the husband. Moreover, we define $W_w$ as the observed wage for
the case that the wife works, \emph{i.e.} $W_w = W^*_w$ in case that $D_w =1$
and otherwise it is undefined. These assumption imply that the joint wage
distribution of the wife and the husband among families where both partners
work equals
\begin{equation}
\begin{split}
\mathbb{P}(W_w \leq w_w; W_h\leq y_w|D_w=1;D_h=1,X=x , Z = z) = \\
\frac{\mathbb{P}(W_w \leq w_w; W_h\leq w_h,D_w=1;D_h=1|Z = z)}{\mathbb{P}
(D_w=1;D_h=1_h|X=x, Z = z)} = \\
\frac{\int^{-\beta_w(w_w) x}_{-\infty} \int^{-\beta_h(w_h)x}_{-\infty}
\int_{-\gamma_w (x,z)}^{\infty} \int_{-\gamma_h (x,z)}^{\infty}
\varphi_4(u_1, u_2, u_3, u_4, \Sigma_4(w_w, w_h, x,z)) du_4 du_3 du_2 du_1} {
\int_{-\gamma^T_w (x,z)}^{\infty} \int_{-\gamma^T_h(x,z)}^{\infty}
\varphi_2(u_1, u_2, \rho_{32}(x,z)) du_2 du_1}
\end{split}
\label{eq:prob}
\end{equation}
\end{comment}
\section{Measures of sorting in the presence of selection}
\label{sec:sorting}
We consider how the expressions for the contingency tables entries change
when we account for endogenous sample selection. They are now written:
\begin{equation}
\begin{split}
& s(\underline{y}_{w},\overline{y}_{w},\underline{y}_{h},\overline{y}
_{h}\mid D_{w}=1,D_{h}=1)= \\
& \frac{\Pr (\underline{y}_{w}<Y_{w}\leq \overline{y}_{w},\underline{y}
_{h}<Y_{h}\leq \overline{y}_{h}\mid D_{w}=1,D_{h}=1)}{\Pr (\underline{y}
_{w}<Y_{w}\leq \overline{y}_{w}\mid D_{w}=1,D_{h}=1)\Pr (\underline{y}
_{h}<Y_{h}\leq \overline{y}_{h}\mid D_{w}=1,D_{h}=1)},
\end{split}
\label{eq:sorting}
\end{equation}
which corresponds to the ratio of the joint distribution of $(Y_{w},Y_{h})$
to the product of the marginals in the selected population. It is equal to 1
when the wages of the wives and husbands are independent from each other in
the selected population.
The values of $s$ are identified from the wage data among couples in which
both partners work and does not depend on the identification of the sample
selection model. Nevertheless, it is interesting to understand how changes
in the sorting measure can be attributed to changes in the model's
parameters. This can be conducted via counterfactuals. Note that the
numerator of this sorting measure can be written as:
\begin{equation}
\int_{\mathcal{Z}}\Pr (\underline{y}_{w}<Y_{w}\leq \overline{y}_{w},
\underline{y}_{h}<Y_{h}\leq \overline{y}_{h}\mid
D_{w}=1,D_{h}=1,Z=z)dF_{Z\mid D_{w},D_{h}}(z\mid 1,1) \label{eq:integral}
\end{equation}
where the integrand equals:
\begin{equation*}
\begin{split}
\Pr (\underline{y}_{w}& <Y_{w}\leq \overline{y}_{w},\underline{y}
_{h}<Y_{h}\leq \overline{y}_{h}\mid D_{w}=1,D_{h}=1,Z=z)= \\
& \frac{\Pr (\underline{y}_{w}<Y_{w}\leq \overline{y}_{w},\underline{y}
_{h}<Y_{h}\leq \overline{y}_{h},D_{w}=1,D_{h}=1\mid Z=z)}{\Pr
(D_{w}=1,D_{h}=1\mid Z=z)}= \\
& \frac{\Pr (\underline{y}_{w}<Y_{w}^{\ast }\leq \overline{y}_{w},\underline{
y}_{h}<Y_{h}^{\ast }\leq \overline{y}_{h},D_{w}^{\ast }\leq 0,D_{h}^{\ast
}\leq 0\mid Z=z)}{\Phi _{2}(\mu _{D_{w}^{\ast }}(0,z),\mu _{D_{h}^{\ast
}}(0,z),\rho _{D_{w}^{\ast },D_{h}^{\ast }}(0,0,x))},
\end{split}
\end{equation*}
with
\begin{multline*}
\Pr (\underline{y}_{w}<Y_{w}^{\ast }\leq \overline{y}_{w},\underline{y}
_{h}<Y_{h}^{\ast }\leq \overline{y}_{h},D_{w}^{\ast }\leq 0,D_{h}^{\ast
}\leq 0\mid Z=z)= \\
\Phi _{4}(\boldsymbol{\mu }(0,0,\overline{y}_{w},\overline{y}_{h},z);
\boldsymbol{\Sigma }(0,0,\overline{y}_{w},\overline{y}_{h},x))-\Phi _{4}(
\boldsymbol{\mu }(0,0,\underline{y}_{w},\overline{y}_{h},z);\boldsymbol{
\Sigma }(0,0,\underline{y}_{w},\overline{y}_{h},x)) \\
-\Phi _{4}(\boldsymbol{\mu }(0,0,\overline{y}_{w},\underline{y}_{h},z);
\boldsymbol{\Sigma }(0,0,\overline{y}_{w},\underline{y}_{h},x))+\Phi _{4}(
\boldsymbol{\mu }(0,0,\underline{y}_{w},\underline{y}_{h},z);\boldsymbol{
\Sigma }(0,0,\underline{y}_{w},\underline{y}_{h},x)).
\end{multline*}
\subsection{Counterfactuals based on a specific time period}
For each time period counterfactuals can be obtained by replacing $
\boldsymbol{\Sigma }(d_{w},d_{h},y_{w},y_{h},x)$ with an alternative
positive definite $\widetilde{\boldsymbol{\Sigma }}
(d_{w},d_{h},y_{w},y_{h},x)$ reflecting different model parameters. For
example, setting $\rho _{D_{w}^{\ast },D_{h}^{\ast }}(0,0,x)$ to zero
examines the impact of making the employment decisions for husbands and
wives conditionally independent. Setting $\rho _{D_{w}^{\ast },Y_{w}^{\ast
}}(0,y_{w},x)$, $\rho _{D_{h}^{\ast },Y_{h}^{\ast }}(0,y_{h},x)$, $\rho
_{D_{w}^{\ast },Y_{h}^{\ast }}(0,y_{h},x)$, and $\rho _{D_{h}^{\ast
},Y_{w}^{\ast }}(0,y_{w},x)$ to zero eliminates the role of selection while
setting $\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}(y_{w},y_{h},x)$ to zero
eliminates sorting on the unobservables correlated with wages. Sorting may
still arise here on the basis of observed characteristics. Another potential
counterfactual distribution integrates (\ref{eq:integral}) over the
distribution of $Z$ for the whole population of married couples in our
sample rather than the population of FTFY working couples.
\subsection{Counterfactuals based on different periods of time}
To investigate intertemporal changes in sorting we examine counterfactuals
over time. We start by rewriting the elements of the contingency matrix in
\eqref{eq:sorting} as:
\begin{multline*}
s(\underline{y}_{w},\overline{y}_{w},\underline{y}_{h},\overline{y}_{h}\mid
D_{w}=1,D_{h}=1) \\
=\scriptstyle{\ \frac{F_{Y_{w},Y_{h}\mid D_{w},D_{h}}(\overline{y}_{w},
\overline{y}_{h}\mid 1,1)-F_{Y_{w},Y_{h}\mid D_{w},D_{h}}(\overline{y}_{w},
\underline{y}_{h}\mid 1,1)-F_{Y_{w},Y_{h}\mid D_{w},D_{h}}(\underline{y}_{w},
\overline{y}_{h}\mid 1,1)+F_{Y_{w},Y_{h}\mid D_{w},D_{h}}(\underline{y}_{w},
\underline{y}_{h}\mid 1,1)}{[F_{Y_{w}\mid D_{w},D_{h}}(\overline{y}_{w}\mid
1,1)-F_{Y_{w}\mid D_{w},D_{h}}(\underline{y}_{w}\mid 1,1)][F_{Y_{h}\mid
D_{w},D_{h}}(\overline{y}_{h}\mid 1,1)-F_{Y_{h}\mid D_{w},D_{h}}(\underline{y
}_{h}\mid 1,1)]}},
\end{multline*}
where
\begin{equation*}
F_{Y_{w},Y_{h}\mid D_{w},D_{h}}(y_{w},y_{h}\mid 1,1)=\Pr (Y_{w}\leq
y_{w},Y_{h}\leq y_{h}\mid D_{w}=1,D_{h}=1),
\end{equation*}
and
\begin{equation*}
F_{Y_{j}\mid D_{w},D_{h}}(y_{j}\mid 1,1)=\Pr (Y_{j}\leq y_{j}\mid
D_{w}=1,D_{h}=1),\quad j=w,h.
\end{equation*}
We calculate counterfactual joint distributions assuming employment
selection is as in year $q$, the wage structure is as in year $r,$ and the
composition of the work force is as in year $s$ by:
\begin{multline}
F_{Y_{w},Y_{h}\mid D_{w},D_{h}}^{q,r,s}(y_{w},y_{h}\mid 1,1)
\label{eq:cdist} \\
:=\int {\Pr }^{q,r}(Y_{w}\leq y_{w},Y_{h}\leq y_{h}\mid
D_{w}=1,D_{h}=1,Z=z)dF_{Z}^{s}(z)
\end{multline}
where $F_{Z}^{s}$ is the distribution of $Z$ in year $s$, and
\begin{multline*}
{\Pr }^{qr}(Y_{w}\leq y_{w},Y_{h}\leq y_{h}\mid D_{w}=1,D_{h}=1,Z=z) \\
=\frac{\Phi _{4}(\boldsymbol{\mu }^{q,r}(0,0,y_{w},y_{h},z);\boldsymbol{
\Sigma }^{q,r}(0,0,y_{w},y_{h},x))}{\Phi _{2}(\mu _{D_{w}^{\ast
}}^{q}(0,z),\mu _{D_{h}^{\ast }}^{q}(0,z),\rho _{D_{w}^{\ast },D_{h}^{\ast
}}^{q}(0,0,x))},
\end{multline*}
with
\begin{equation*}
\boldsymbol{\mu }^{q,r}(0,0,y_{w},y_{h},z)=\left(
\begin{array}{c}
\mu _{D_{w}^{\ast }}^{q}(0,z) \\
\mu _{D_{h}^{\ast }}^{q}(0,z) \\
\mu _{Y_{w}^{\ast }}^{r}(y_{w},x) \\
\mu _{Y_{h}^{\ast }}^{r}(y_{h},x)
\end{array}
\right)
\end{equation*}
and\footnote{
A practical problem is that there is no guarantee that $\boldsymbol{\Sigma }
^{q,r}(0,0,y_{w},y_{h},x)$ is positive definite. Nevertheless, we did not
encounter this problem in our empirical analysis.}
\begin{multline*}
\boldsymbol{\Sigma }^{q,r}(0,0,y_{w},y_{h},x)= \\
\left(
\begin{array}{cccc}
1 & \rho _{D_{w}^{\ast },D_{h}^{\ast }}^{q}(0,0,x) & -\rho _{D_{w}^{\ast
},Y_{h}^{\ast }}^{q}(0,y_{h},x) & -\rho _{D_{w}^{\ast },Y_{h}^{\ast
}}^{q}(0,y_{h},x) \\
\rho _{D_{w}^{\ast },D_{h}^{\ast }}^{q}(0,0,x) & 1 & -\rho _{D_{h}^{\ast
},Y_{w}^{\ast }}^{q}(0,y_{w},x) & -\rho _{D_{h}^{\ast },Y_{h}^{\ast
}}^{q}(0,y_{h},x) \\
-\rho _{D_{w}^{\ast },Y_{w}^{\ast }}^{q}(0,y_{1},x) & -\rho _{D_{h}^{\ast
},Y_{w}^{\ast }}^{q}(0,y_{w},x) & 1 & \rho _{Y_{w}^{\ast },Y_{h}^{\ast
}}^{r}(y_{w},y_{h},x) \\
-\rho _{D_{w}^{\ast },Y_{h}^{\ast }}^{q}(0,y_{h},x) & -\rho _{D_{h}^{\ast
},Y_{h}^{\ast }}^{q}(0,y_{h},x) & \rho _{Y_{w}^{\ast },Y_{h}^{\ast
}}^{r}(y_{w},y_{h},x) & 1
\end{array}
\right) .
\end{multline*}
Comparable counterfactual marginal distributions can be computed for the
denominator.
Comparisons should be based on wage levels of husbands and wives which
incorporate the changes in the wage distributions. Since the real wages of
males decreased at the bottom of the distribution while the real wages
increased over the whole distribution for females, using fixed wage levels
is not appropriate. For example, we are more likely to find a husband
earning more than 20 dollars an hour with a wife earning more than 15
dollars an hour in 2020 than in 1976. This, however, does not imply a change
in sorting behavior. Rather, it reflects the changes in the likelihoods from
the respective marginal distributions. As we use quantiles rather than fixed
wages to make these comparisons, we need the counterfactual quantiles of the
corresponding marginal wage distributions. For example, the $\tau $-th
quantile of the marginal wage distribution of the wives is:
\begin{multline}
Q_{Y_{w}}^{q,r,s}(\tau ) \\
=\inf \left\{ y\in \mathbb{R}:\int {\Pr }^{q,r}(Y_{w}\leq y,Y_{h}\leq \infty
\mid D_{w}=1,D_{h}=1,Z=z)dF_{Z}^{s}(z)\geq \tau \right\} .
\label{eq:quantile_w}
\end{multline}
\section{Estimation}
\label{sec:estimation}
We consider a semiparametric BDR model with selection that imposes:
\begin{assumption}[BDR with Selection]
\label{ass:bdr-selection} (1) $\mu_{Y^*_j}(y,x) = P_{j}(x)^{\prime }\beta
_{j}(y),$ $\mu_{D^*_j}(0,z) = Q_{j}(z)^{\prime }\gamma_j,$ where $P_{j}$ and
$Q_{j}$ are transformations of $x$ and $z$, and $\beta _{j}(y)$ and $\gamma
_{j}$ are vectors of coefficients, $j \in \{w,h\}$; and (2) $\boldsymbol{
\Sigma} (d_{w},d_{h},y_{w},y_{h},x) = \boldsymbol{\Sigma}
(d_{w},d_{h},y_{w},y_{h})$.
\end{assumption}
\subsection{Estimation of the local model parameters}
Using Assumption \ref{ass:bdr-selection} with $P_{j}(x)=x$ and $Q_{j}(z)=z$
for $j\in \{w,h\}$ we estimate the parameters in 2 steps:
\begin{enumerate}
\item Bivariate probit to obtain $\gamma _{w}$, $\gamma _{h}$ and $\rho
_{D_{w}^{\ast },D_{h}^{\ast }}:=\rho _{D_{w}^{\ast },D_{h}^{\ast }}(0,0)$
using:
\begin{equation*}
\mathbb{P}(D_{w}=1,D_{h}=1\mid Z=z)=\Phi _{2}(z^{\prime }\gamma
_{w},z^{\prime }\gamma _{h};\rho _{D_{w}^{\ast },D_{h}^{\ast }}).
\end{equation*}
We denote the estimators as $\widehat{\gamma }_{w},$ $\widehat{\gamma }_{h}$
and $\widehat{\rho }_{D_{w}^{\ast },D_{h}^{\ast }}$.
\item Multivariate probit with sample selection correction to estimate the
remaining parameters using:
\begin{multline*}
\mathbb{P}(Y_{w}\leq y_{w};Y_{h}\leq y_{w}|D_{w}=1;D_{h}=1,X=x,Z=z)\propto \\
\Phi _{4}(z^{\prime }\gamma _{w},z^{\prime }\gamma _{h},x^{\prime }\beta
_{w}(y_{w}),x^{\prime }\beta _{h}(y_{h});\boldsymbol{\Sigma }
(0,0,y_{w},y_{h})),
\end{multline*}
which can be estimated by a small adaption of the multivariate probit model
after plugging in the first-stage estimators $\widehat{\gamma }_{w},$ $
\widehat{\gamma }_{h}$ and $\widehat{\rho }_{D_{w}^{\ast },D_{h}^{\ast }}$.
\end{enumerate}
The first step is standard. The second step is straightforward, but the
calculation of higher-order integrals in the multivariate probit is both
computationally intensive and imprecise. The imprecision is especially
unfortunate when combined with a numerical optimization method that assumes
smoothness of the first and second order derivative of the criterion
function. Therefore, we employ the GHK importance sampling simulator of
Geweke (1991), Hajivassiliou and McFadden (1990) and Keane (1990) to
simulate these probabilities. This importance sampling simulator uses the
result that multivariate normal distributions, conditional on realizations
of one or more elements of the outcome vector, are also normally distributed
but with a lower dimension.
\subsection{Estimation of the measures of sorting}
Estimation of the sorting measures requires estimates of the quantiles of
the marginal distributions of wives' and husbands' wage distributions. These
are obtained via application of the generalized inverse or rearrangement
operator to the plug-in estimator of the distribution. For example:
\begin{equation}
\widehat{Q}_{Y_{w}}^{q,r,s}(\tau )=\int_{0}^{\infty }\boldsymbol{1}\left\{
\frac{1}{n_{s}}\sum_{i=1}^{n_{s}}\widehat{\Pr }^{q,r}(Y_{w}\leq y\mid
D_{w}=1,D_{h}=1,Z^{s}=Z_{i})\leq \tau \right\} dy, \label{eq:quant1}
\end{equation}
with:
\begin{multline*}
\widehat{\Pr }^{q,r}(Y_{w}\leq y_{w}\mid D_{w}=1,D_{h}=1,Z^{s}=z)= \\
\frac{\Phi _{4}(z^{\prime }\widehat{\gamma }_{w}^{q},z^{\prime }\widehat{
\gamma }_{h}^{q},x^{\prime }\widehat{\beta }_{w}^{r}(y_{w}),\infty ;\widehat{
\boldsymbol{\Sigma }}^{q,r}(0,0,y_{w},y_{h}))}{\Phi _{2}(z^{\prime }\widehat{
\gamma }_{w}^{q},z^{\prime }\widehat{\gamma }_{h}^{q};\widehat{\rho }
_{D_{w}^{\ast },D_{h}^{\ast }}^{q})},
\end{multline*}
for any $y_{h}$, where $\widehat{\boldsymbol{\Sigma }}
^{q,r}(0,0,y_{w},y_{h}) $ is the plug-in estimator of $\boldsymbol{\Sigma }
^{q,r}(0,0,y_{w},y_{h})$, and the integrals are calculated using the GHK
importance sampling method.
Equation \eqref{eq:quant1} is solved using bisection implying that we need
to calculate the term on the right-hand side of that equation for different
trial values of the quantile. The counterfactual quantiles for husbands can
be estimated in a similar way.
The estimator of the measures of sorting is: \vspace{-0.5cm}
\begin{multline*}
\widehat{s}^{q,r,s}(\underline{y}_{w},\overline{y}_{w},\underline{y}_{h},
\overline{y}_{h}\mid D_{w}=1,D_{h}=1) \\
=\scriptstyle{\ \frac{\widehat{F}_{Y_{w},Y_{h}\mid D_{w},D_{h}}^{q,r,s}(
\overline{y}_{w},\overline{y}_{h}\mid 1,1)-\widehat{F}_{Y_{w},Y_{h}\mid
D_{w},D_{h}}^{q,r,s}(\overline{y}_{w},\underline{y}_{h}\mid 1,1)-\widehat{F}
_{Y_{w},Y_{h}\mid D_{w},D_{h}}^{q,r,s}(\underline{y}_{w},\overline{y}
_{h}\mid 1,1)+\widehat{F}_{Y_{w},Y_{h}\mid D_{w},D_{h}}^{q,r,s}(\underline{y}
_{w},\underline{y}_{h}\mid 1,1)}{[\widehat{F}_{Y_{w}\mid
D_{w},D_{h}}^{q,r,s}(\overline{y}_{w}\mid 1,1)-\widehat{F}_{Y_{w}\mid
D_{w},D_{h}}^{q,r,s}(\underline{y}_{w}\mid 1,1)][\widehat{F}_{Y_{h}\mid
D_{w},D_{h}}^{q,r,s}(\overline{y}_{h}\mid 1,1)-\widehat{F}_{Y_{h}\mid
D_{w},D_{h}}^{q,r,s}(\underline{y}_{h}\mid 1,1)]}},
\end{multline*}
where $\widehat{F}_{Y_{w},Y_{h}\mid D_{w},D_{h}}^{q,r,s}$, $\widehat{F}
_{Y_{w}\mid D_{w},D_{h}}^{q,r,s}$ and $\widehat{F}_{Y_{h}\mid
D_{w},D_{h}}^{q,r,s}$ are plug-in estimators of $F_{Y_{w},Y_{h}\mid
D_{w},D_{h}}^{q,r,s}$, $F_{Y_{w}\mid D_{w},D_{h}}^{q,r,s}$ and $F_{Y_{h}\mid
D_{w},D_{h}}^{q,r,s}$ in \eqref{eq:cdist}, respectively; and $\underline{y}
_{j}$ and $\overline{y}_{j}$ evaluated at $\widehat{Q}
_{Y_{j}}^{q,r,s}((i-1)/10)$ and $\widehat{Q}_{Y_{k}}^{q,r,s}(i/10)$, $
i=1,\ldots ,10$ and $j\in \{w,h\}$.
\section{Empirical Results}
\label{sec:empirical}
\subsection{Parameter Estimates}
We estimate the model at each decile for both of the males and females wage
distribution for 5 year periods starting from 1976 and ending at 2022. As
the total number of years is not divisible by 5 we use two periods of 6
years. These are 1991-1996 and 1997-2002. We estimate one hundred different
sets of coefficients for each time period corresponding to the different
combinations of deciles for each spouse. As our primary focus is on the role
of selection and sorting, we do not report the 100 sets of parameter
estimates but Figures \ref{fig:covariance_q1}-\ref{fig:covariance_q3}
present the estimates of the different $\rho ^{\prime }$s at the same
quartile of both distributions along with their bootstrapped standard
errors. Results for the other quantiles are available from the authors. Note
that we employ the 100 sets of parameter estimates in conducting the
counterfactuals reported below and the discussion of the $\rho ^{\prime }$s
which follows is only to provide some insight into the behavior of these
selection and sorting parameters.
\begin{figure}[tbp]
\caption{Kernel density estimates of $\protect\rho (y_{w},y_{h})$.}
\label{fig:kernelModelComplete}\centering
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We discuss our model specification before proceeding to the results.
Footnote \ref{footnote:x} listed the variables in $X$ employed in the BDR
specification. We continue to use these variable but identification now
requires some variables in $Z$ which are excluded from $X.$ We follow
Mulligan and Rubinstein (2008) and employ a dummy variable for children at
home, family size, and a dummy variable for children at home under 5 years
of age in this role. These variables are seen as contentious and we discuss
the implications of these choices below.
Consider the correlation between the unobservables in the spouses'
employment equations, $\rho _{D_{w}^{\ast },D_{h}^{\ast }},$ recalling it is
invariant to the location in the wage distributions. The positive value
indicates that the unobservables affecting the spouses' work decisions are
positively correlated. The coefficient is small in magnitude but is
statistically significantly different from zero. It generally increases over
time, except for a dip in the middle of the sample period, and reveals that
the unobserved factors which make spouses both work FTFY are increasingly
correlated over time. We leave a discussion of the marginal impact of
changing this, and other parameters, to the following section.
The estimated selection coefficients for wives ($\rho _{D_{w}^{\ast
},Y_{w}^{\ast }}$) and husbands ($\rho _{D_{h}^{\ast },Y_{h}^{\ast }}$)
capture the correlation between the unobservables affecting the individual's
respective work decision and their own wages. While in conventional
selection models they capture the relationship between the selection
decision and the mean wage, we evaluate this correlation at different
quantiles of the wage distribution. The estimate for wives is particularly
interesting and consistent with earlier evidence using similar
identification approaches (see, for example, Mulligan and Rubinstein, 2008,
and FVVP, 2023b) for FTFY workers. Similar to these earlier papers we find
evidence of negative selection changing to positive selection. At each
quantile there is negative selection in the earlier period with an estimate
around -0.3 which is statistically different from zero. For the latter
sample period the estimate has increased. At Q1/Q1 the estimate has
increased to zero, while at Q2/Q2 and Q3/Q3 the estimates are approximately
0.2 and 0.3. As discussed in FVVP (2023b), the sign of $\rho _{D_{w}^{\ast
},Y_{w}^{\ast }}$ is contentious given its interpretation and its
implication for selection. They note that the sign of the selection terms
appears to reflect the impact of the variables used to explain participation
which are excluded from the wage equation. The change of sign appears to
capture the changing impact of these variables on the participation decision.
\footnote{
FVVP (2023b) show that this negative coefficient for the earlier period is
likely to be due to the exclusion restrictions. The reader is referred to
that paper for a detailed discussion of how the exclusion restrictions may
be generating the selection related results. One could reproduce the
counterfactual that follows using the FVVP (2023b) approach which employs an
alternative identification strategy but that is not feasible without making
a number of additional assumptions.} While the sign of $\rho _{D_{w}^{\ast
},Y_{w}^{\ast }}$ is controversial we explore the impact of changing its
value in counterfactual exercises below.
It is not typical to account for selection when estimating male wage
equations. However, it is important to do so here as we do not know apriori
the impact of selection in this model. Moreover, as we account for the role
of male selection on the female wage it is necessary to model the male work
decision. The parameter $\rho _{D_{h}^{\ast },Y_{h}^{\ast }}$ is very poorly
estimated at each of the quantiles reported. At Q1/Q1 and Q2/Q2 it is
negative and large but very imprecisely estimated. At Q3/Q3 it is both
negative and positive but generally imprecisely estimated. We attribute this
result to our inability to identify this parameter from these data.
The parameters $\rho _{D_{w}^{\ast },Y_{h}^{\ast }}$ and $\rho _{D_{h}^{\ast
},Y_{w}^{\ast }}$ also vary by location in the respective wage distributions
and capture the correlation between the unobservables affecting the work
decision of the individual with the unobservables affecting the wage of
their spouse at a certain quantile in the spouses wage distribution. First
consider $\rho _{D_{w}^{\ast },Y_{h}^{\ast }}$. Figures \ref
{fig:covariance_q1}-\ref{fig:covariance_q3} suggest that at each quartile
the estimate is both negative and positive at each quantile depending on the
time period. However, the confidence bands suggest that we cannot reject
that the estimate is zero for many of the periods with the exception of some
middle years at the first quartile. $\rho _{D_{h}^{\ast },Y_{w}^{\ast }}$
captures how the unobservables driving the husband's participation decision
affects the wife's wage. Given the evidence above on male selection it is
unsurprising that this parameter is imprecisely estimated although it
appears to be generally negative and statistically different from zero.
\begin{figure}[tbp]
\caption{Results for covariance matrix at Q1.}
\label{fig:covariance_q1}
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\end{figure}
\begin{figure}[tbp]
\caption{Results for covariance matrix at Q2.}
\label{fig:covariance_q2}
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\end{figure}
\begin{figure}[tbp]
\caption{Results for covariance matrix at Q3.}
\label{fig:covariance_q3}
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\end{figure}
$\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}$ is an economically important parameter
as it captures sorting via the correlation between the unobservables
generating the husband's and wife's wages. If individuals are sorting with
respect to these unobservables we expect this parameter to be positive.
Moreover, as this parameter also captures the prices of these unobservables
it will capture similar structural effects associated with the implicit
prices of observed characteristics. As noted above, this parameter will
capture factors related to influences, such as cost of living or wage
premia, which are not captured by the conditioning variables but which are
shared by spouses. For example, while the data measure various aspects of
the location in which the spouses live we are unable to distinguish between
those living in costly urban areas. It will also capture other factors
specific to unobserved issues shared by the spouses. For example, it
captures that both may work for the same firm and or went to the same
college and share the costs and/or benefits associated with their wages. The
estimate of this parameter is positive and generally ranges between 0.2 and
0.3. Figure \ref{fig:kernelModelComplete} plots the estimated densities for
this parameter for the starting and ending time periods for all 100 models
while Figures \ref{fig:covariance_q1}-\ref{fig:covariance_q3} present the
estimates at the various quantiles we consider. There is some evidence that
it is increasing at these different quantiles over time but the confidence
intervals do not appear to reject that the impact is constant.
With respect to sorting the only clear and consistent evidence is associated
with $\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}.$ The evidence from the
counterfactuals based on the BDR estimates revealed that setting $\rho
_{Y_{w}^{\ast },Y_{h}^{\ast }}$ to zero drastically reduced the presence of
positive sorting. However, this may now be offset by allowing for selection
even though the estimates of $\rho _{D_{w}^{\ast },D_{h}^{\ast }},$ $\rho
_{D_{w}^{\ast },Y_{w}^{\ast }}$, $\rho _{D_{h}^{\ast },Y_{h}^{\ast }}$, $
\rho _{D_{w}^{\ast },Y_{h}^{\ast }}$ and $\rho _{D_{h}^{\ast },Y_{w}^{\ast
}} $ do not individually or collectively present a clear story regarding
sorting.
The estimates of the parameters related to the husband's selection decision
warrant further discussion. As male participation is typically treated as
exogenous there is little empirical work on the impact of male selection on
wages. Moreover, as methods to analyze the role of selection at different
quantiles have only recently been developed there is little existing
evidence on how it varies across the wage distribution. However, the
parameters are not precisely estimated in this setting and in some instances
may be unreasonable in magnitude. This is likely to reflect the form of
identification employed. Alternatively, husbands' work decisions may be
exogenous. Given the restricted version of the model is a special case of
the richer model we decided to proceed with the results with this full model
but we also estimated the model treating the husbands' work decisions as
exogenous and conducted the corresponding counterfactuals. We comment on
those results below although to anticipate the main finding, the treatment
of husbands' work decisions as exogenous did not alter the substantive
results. We report the estimates of $\rho _{D_{w}^{\ast },Y_{w}^{\ast }}$, $
\rho _{D_{w}^{\ast },Y_{h}^{\ast }}$ and $\rho _{D_{h}^{\ast },Y_{w}^{\ast
}} $ from this restricted model in Figures \ref{fig:results_bootstraps_25_1}-
\ref{fig:results_bootstraps_75_1}.
\pgfplotsset{/pgfplots/group/.cd,
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\begin{figure}[tbp]
\caption{Results of covariance matrix at Q1 (exogenous selection of
husbands).}
\label{fig:results_bootstraps_25_1}
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\begin{figure}[tbp]
\caption{Results of covariance matrix at Q2 (exogenous selection of
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\label{fig:results_bootstraps_50_1}
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\begin{figure}[tbp]
\caption{Results of covariance matrix at Q3 (exogenous selection of
husbands).}
\label{fig:results_bootstraps_75_1}
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1991 -0.180127286732687
1997 -0.090816600197043
2003 -0.166235690599964
2008 -0.169643812061593
2013 -0.177036126326048
2018 -0.0603171951978475
};
\nextgroupplot[
scaled x ticks=manual:{}{\pgfmathparse{#1}},
tick align=outside,
tick pos=left,
title={\(\displaystyle \rho_{Y^*_w, Y^*_h}\)},
x grid style={white!69.0196078431373!black},
xlabel={Year},
xmin=1975, xmax=2022,
xtick style={color=black},
xticklabels={},
ylabel={Correlation coefficient},
y grid style={white!69.0196078431373!black},
ymin=0.156892633501126, ymax=0.315950976104346,
ytick style={color=black}
]
\path [draw=color0, semithick]
(axis cs:1976,0.164122558164908)
--(axis cs:1976,0.204832753857966);
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(axis cs:1981,0.19598484102107)
--(axis cs:1981,0.242336150126219);
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(axis cs:1986,0.21633062542162)
--(axis cs:1986,0.256205462178187);
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(axis cs:1991,0.210508790293773)
--(axis cs:1991,0.246778198031691);
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(axis cs:1997,0.21268039206127)
--(axis cs:1997,0.246805228733967);
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(axis cs:2003,0.184642172656086)
--(axis cs:2003,0.228911247377712);
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(axis cs:2008,0.209316703396153)
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--(axis cs:2013,0.24194154034401);
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(axis cs:2018,0.242110434823934)
--(axis cs:2018,0.308721051440563);
\addplot [semithick, color0, mark=-, mark size=3, mark options={solid}, only marks]
table {
1976 0.164122558164908
1981 0.19598484102107
1986 0.21633062542162
1991 0.210508790293773
1997 0.21268039206127
2003 0.184642172656086
2008 0.209316703396153
2013 0.186710407212868
2018 0.242110434823934
};
\addplot [semithick, color0, mark=-, mark size=3, mark options={solid}, only marks]
table {
1976 0.204832753857966
1981 0.242336150126219
1986 0.256205462178187
1991 0.246778198031691
1997 0.246805228733967
2003 0.228911247377712
2008 0.264433597315541
2013 0.24194154034401
2018 0.308721051440563
};
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table {
1976 0.184477656011437
1981 0.219160495573644
1986 0.236268043799904
1991 0.228643494162732
1997 0.229742810397619
2003 0.206776710016899
2008 0.236875150355847
2013 0.214325973778439
2018 0.275415743132248
};
\end{groupplot}
\end{tikzpicture}
}
\end{figure}
\subsection{Counterfactual Sorting Patterns}
\label{ss:counterfactual_results}
The estimated model's capacity to reproduce the frequencies in Tables \ref
{tab:frequencies_1} and \ref{tab:frequencies_2} is reflected in Tables \ref
{tab:results_original_1976} and \ref{tab:results_original_2018} and,
although the large number of cells makes it difficult to draw a conclusion
by a visual inspection, the estimated cells appear close to the true values.
We now examine the role of various model parameters in generating the
observed data by changing selected model parameters and examining the
predicted allocation across cells. We compare these counterfactuals to
Tables \ref{tab:results_original_1976} and \ref{tab:results_original_2018}
as they represent the predictions of our estimated model.
\begin{table}[!ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{2.135} & 1.37 & 1.14 & 1.006 & 0.848 & 0.939 & 0.757 & 0.691
& 0.728 & 0.386 \\
& d2 & 1.623 & \textbf{1.43} & 1.2 & 1.034 & 0.974 & 0.845 & 0.758 & 0.735 &
0.724 & 0.678 \\
& d3 & 1.368 & 1.184 & \textbf{1.122} & 1.067 & 1.092 & 0.898 & 0.842 & 0.8
& 0.737 & 0.89 \\
& d4 & 1.154 & 1.344 & 1.193 & \textbf{1.111} & 0.925 & 0.976 & 0.87 & 0.897
& 0.756 & 0.774 \\
& d5 & 0.955 & 1.185 & 1.225 & 1.178 & \textbf{1.105} & 1.012 & 0.982 & 0.969
& 0.936 & 0.452 \\
& d6 & 0.756 & 1.042 & 1.086 & 1.147 & 1.108 & \textbf{1.151} & 1.007 & 1.005
& 0.946 & 0.751 \\
& d7 & 0.597 & 0.776 & 1.074 & 1.033 & 1.078 & 1.131 & \textbf{1.031} & 1.096
& 1.096 & 1.088 \\
& d8 & 0.601 & 0.603 & 0.832 & 1.131 & 1.093 & 1.132 & 1.135 & \textbf{1.13}
& 1.197 & 1.144 \\
& d9 & 0.427 & 0.597 & 0.676 & 0.756 & 1.136 & 1.209 & 1.245 & 1.239 &
\textbf{1.167} & 1.546 \\
& d10 & 0.381 & 0.544 & 0.374 & 0.535 & 0.655 & 0.696 & 1.373 & 1.434 & 1.714
& \textbf{2.294} \\ \hline\hline
\end{tabular}}
\caption{Results of estimated sorting measures in 1976-1980 at different
quantiles when $\protect\rho_{D^*_w, D^*_h} = 0$.}
\label{tab:results_1_1976}
\end{table}
\begin{table}[!ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{2.189} & 1.235 & 0.987 & 0.848 & 0.986 & 0.705 & 0.935 & 0.675
& 0.646 & 0.795 \\
& d2 & 1.752 & \textbf{1.525} & 1.067 & 1.087 & 0.813 & 0.865 & 0.844 & 0.755
& 0.684 & 0.61 \\
& d3 & 1.272 & 1.354 & \textbf{1.218} & 1.1 & 0.932 & 1.081 & 0.809 & 0.912
& 0.699 & 0.623 \\
& d4 & 1.068 & 1.316 & 1.286 & \textbf{1.103} & 1.024 & 0.916 & 1.041 & 0.885
& 0.837 & 0.524 \\
& d5 & 0.859 & 1.226 & 1.272 & 1.137 & \textbf{1.045} & 0.995 & 1.022 & 0.915
& 0.909 & 0.619 \\
& d6 & 0.648 & 1.041 & 1.241 & 1.117 & 1.156 & \textbf{1.075} & 1.029 & 1.032
& 0.912 & 0.749 \\
& d7 & 0.658 & 0.625 & 1.207 & 1.16 & 1.13 & 1.17 & \textbf{1.015} & 1.187 &
1.121 & 0.726 \\
& d8 & 0.484 & 0.707 & 0.764 & 1.2 & 1.182 & 1.169 & 1.153 & \textbf{1.098}
& 1.237 & 1.005 \\
& d9 & 0.539 & 0.573 & 0.512 & 0.722 & 1.129 & 1.248 & 1.16 & 1.215 &
\textbf{1.493} & 1.41 \\
& d10 & 0.523 & 0.403 & 0.442 & 0.53 & 0.6 & 0.766 & 1.01 & 1.249 & 1.538 &
\textbf{2.94} \\ \hline\hline
\end{tabular}}
\caption{Results of estimated sorting measures in 1976-1980 at different
quantiles when $\protect\rho_{D^*_w, D^*_h} = 0, \protect\rho_{D^*_w Y^*_w}
= 0, \protect\rho_{D^*_w, Y^*_h} = 0, \protect\rho_{D^*_h, Y^*_w} = 0,
\protect\rho_{D^*_h, Y^*_h} = 0$.}
\label{tab:results_3_1976}
\end{table}
\begin{table}[!ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{2.209} & 1.217 & 0.94 & 0.826 & 0.981 & 0.731 & 0.956 & 0.601
& 0.733 & 0.806 \\
& d2 & 1.722 & \textbf{1.521} & 1.155 & 1.019 & 0.924 & 0.844 & 0.837 & 0.723
& 0.631 & 0.623 \\
& d3 & 1.388 & 1.327 & \textbf{1.139} & 1.185 & 0.842 & 1.065 & 0.844 & 0.762
& 0.915 & 0.533 \\
& d4 & 1.058 & 1.407 & 1.31 & \textbf{1.11} & 0.982 & 0.903 & 1.017 & 0.934
& 0.7 & 0.58 \\
& d5 & 0.822 & 1.237 & 1.282 & 1.112 & \textbf{1.088} & 1.056 & 0.98 & 1.007
& 0.792 & 0.624 \\
& d6 & 0.673 & 1.024 & 1.266 & 1.118 & 1.161 & \textbf{1.013} & 1.117 & 1.02
& 0.893 & 0.716 \\
& d7 & 0.641 & 0.594 & 1.182 & 1.181 & 1.095 & 1.155 & \textbf{0.973} & 1.363
& 1.02 & 0.796 \\
& d8 & 0.452 & 0.722 & 0.769 & 1.188 & 1.214 & 1.202 & 1.095 & \textbf{1.201}
& 1.191 & 0.967 \\
& d9 & 0.571 & 0.511 & 0.496 & 0.75 & 1.118 & 1.236 & 1.173 & 1.368 &
\textbf{1.299} & 1.477 \\
& d10 & 0.476 & 0.435 & 0.458 & 0.519 & 0.579 & 0.785 & 1.014 & 1.022 & 1.818
& \textbf{2.894} \\ \hline\hline
\end{tabular}}
\caption{Results of estimated sorting measures in 1976-1980 at different
quantiles when $\protect\rho_{D^*_w, D^*_h} = 0, \protect\rho_{D^*_w, Y^*_w}
= 0, \protect\rho_{D^*_w, Y^*_h} = 0, \protect\rho_{D^*_h, Y^*_w} = 0,
\protect\rho_{D^*_h, Y^*_h} = 0$ and distribution of $Z$ as in total
population.}
\label{tab:results_4_1976}
\end{table}
\begin{table}[!ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{1.353} & 1.2 & 1.095 & 0.973 & 1.045 & 0.919 & 0.946 & 0.856
& 0.82 & 0.795 \\
& d2 & 1.328 & \textbf{1.255} & 1.125 & 1.001 & 1.051 & 0.941 & 0.942 & 0.827
& 0.779 & 0.752 \\
& d3 & 1.209 & 1.048 & \textbf{1.02} & 1.124 & 0.832 & 1.024 & 0.904 & 0.912
& 0.998 & 0.93 \\
& d4 & 1.051 & 1.055 & 1.046 & \textbf{1.04} & 0.994 & 0.988 & 0.993 & 0.99
& 0.944 & 0.901 \\
& d5 & 0.985 & 1.02 & 1.031 & 1.046 & \textbf{1.011} & 1.007 & 0.998 & 1.01
& 0.962 & 0.93 \\
& d6 & 0.944 & 0.996 & 1.011 & 1.013 & 1.014 & \textbf{1.012} & 1.024 & 1.012
& 1.006 & 0.968 \\
& d7 & 0.86 & 0.935 & 0.974 & 1.002 & 1.026 & 1.018 & \textbf{1.03} & 1.076
& 1.035 & 1.043 \\
& d8 & 0.786 & 0.884 & 0.946 & 0.98 & 1.029 & 1.045 & 1.06 & \textbf{1.102}
& 1.075 & 1.093 \\
& d9 & 0.734 & 0.797 & 0.878 & 0.938 & 0.975 & 1.017 & 1.067 & 1.171 &
\textbf{1.182} & 1.242 \\
& d10 & 0.759 & 0.809 & 0.872 & 0.892 & 1.008 & 1.022 & 1.041 & 1.048 & 1.191
& \textbf{1.357} \\ \hline\hline
\end{tabular}}
\caption{Results of estimated sorting measures in 1976-1980 at different
quantiles when all correlations are zero and distribution of $Z$ as in total
population.}
\label{tab:results_5_1976}
\end{table}
We begin by setting $\rho _{D_{w}^{\ast },D_{h}^{\ast }}$ to zero.
Table \ref{tab:results_1_1976} presents the results \ for 1976-1980 and
Table \ref{tab:results_1_2018} those for 2018-2022. They appear similar to
Table \ref{tab:results_original_1976} and Table \ref
{tab:results_original_2018} respectively and we conclude that this parameter
is unimportant for determining the joint wage distribution.
\begin{table}[!ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{2.896} & 1.573 & 1.044 & 0.891 & 0.886 & 0.82 & 0.576 & 0.646
& 0.523 & 0.146 \\
& d2 & 1.636 & \textbf{1.702} & 1.254 & 1.02 & 0.884 & 0.818 & 0.632 & 0.655
& 0.502 & 0.897 \\
& d3 & 1.161 & 1.721 & \textbf{1.373} & 1.123 & 1.004 & 0.872 & 0.842 & 0.608
& 0.588 & 0.708 \\
& d4 & 0.85 & 1.295 & 1.604 & \textbf{1.26} & 1.078 & 1.023 & 0.954 & 0.808
& 0.7 & 0.428 \\
& d5 & 0.779 & 0.998 & 1.269 & 1.336 & \textbf{1.122} & 1.13 & 1.054 & 1.042
& 0.797 & 0.472 \\
& d6 & 0.663 & 0.766 & 0.924 & 1.302 & 1.413 & \textbf{1.27} & 1.144 & 1.071
& 0.897 & 0.549 \\
& d7 & 0.549 & 0.761 & 0.964 & 1.062 & 1.234 & 1.386 & \textbf{1.294} & 1.254
& 1.211 & 0.285 \\
& d8 & 0.454 & 0.465 & 0.609 & 0.635 & 0.812 & 1.125 & 1.445 & \textbf{1.111}
& 1.25 & 2.093 \\
& d9 & 0.472 & 0.575 & 0.604 & 0.761 & 0.748 & 1.031 & 1.274 & 1.575 &
\textbf{1.803} & 1.156 \\
& d10 & 0.537 & 0.144 & 0.354 & 0.609 & 0.816 & 0.526 & 0.789 & 1.23 & 1.73
& \textbf{3.265} \\ \hline\hline
\end{tabular}}
\caption{Results of estimated sorting measures in 2018-2022 at different
quantiles when $\protect\rho_{D^*_w, D^*_h} = 0$.}
\label{tab:results_1_2018}
\end{table}
\begin{table}[!ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{2.757} & 1.275 & 0.984 & 0.945 & 0.919 & 0.679 & 0.777 & 0.619
& 0.446 & 0.599 \\
& d2 & 1.633 & \textbf{1.959} & 1.287 & 1.074 & 0.911 & 0.862 & 0.793 & 0.597
& 0.524 & 0.36 \\
& d3 & 0.776 & 1.885 & \textbf{1.385} & 1.212 & 1.067 & 1.143 & 0.904 & 0.701
& 0.545 & 0.381 \\
& d4 & 0.867 & 1.28 & 1.517 & \textbf{1.061} & 1.191 & 1.076 & 0.901 & 0.9 &
0.668 & 0.54 \\
& d5 & 0.853 & 0.762 & 1.267 & 1.637 & \textbf{1.198} & 1.103 & 1.146 & 0.717
& 0.697 & 0.619 \\
& d6 & 0.584 & 0.81 & 1.075 & 1.148 & 1.336 & \textbf{1.181} & 1.057 & 1.068
& 1.019 & 0.721 \\
& d7 & 0.735 & 0.625 & 0.804 & 1.018 & 1.19 & 1.188 & \textbf{1.238} & 1.254
& 1.082 & 0.865 \\
& d8 & 0.709 & 0.512 & 0.609 & 0.774 & 0.946 & 1.304 & 1.196 & \textbf{1.41}
& 1.242 & 1.299 \\
& d9 & 0.524 & 0.53 & 0.604 & 0.699 & 0.712 & 0.82 & 1.131 & 1.652 & \textbf{ \ 1.813} & 1.515 \\
& d10 & 0.558 & 0.36 & 0.467 & 0.436 & 0.529 & 0.638 & 0.862 & 1.086 & 1.961
& \textbf{3.104} \\ \hline\hline
\end{tabular}}
\caption{Results of estimated sorting measures in 2018-2022 at different
quantiles when $\protect\rho_{D^*_w, D^*_h} = 0, \protect\rho_{D^*_w Y^*_w}
= 0, \protect\rho_{D^*_w, Y^*_h} = 0, \protect\rho_{D^*_h, Y^*_w} = 0,
\protect\rho_{D^*_h, Y^*_h} = 0$.}
\label{tab:results_3_2018}
\end{table}
\begin{table}[!ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{2.88} & 1.375 & 0.903 & 0.819 & 0.958 & 0.582 & 0.741 & 0.621
& 0.453 & 0.669 \\
& d2 & 1.771 & \textbf{1.803} & 1.329 & 1.116 & 0.961 & 0.792 & 0.812 & 0.59
& 0.486 & 0.339 \\
& d3 & 0.76 & 1.978 & \textbf{1.486} & 1.138 & 1.108 & 1.117 & 0.834 & 0.651
& 0.538 & 0.391 \\
& d4 & 0.767 & 1.195 & 1.65 & \textbf{1.104} & 1.184 & 1.095 & 1.04 & 0.743
& 0.659 & 0.564 \\
& d5 & 0.762 & 0.847 & 1.133 & 1.609 & \textbf{1.273} & 1.101 & 1.068 & 0.905
& 0.73 & 0.572 \\
& d6 & 0.521 & 0.766 & 1.075 & 1.197 & 1.328 & \textbf{1.217} & 1.048 & 1.074
& 1.017 & 0.758 \\
& d7 & 0.793 & 0.571 & 0.792 & 1.0 & 1.115 & 1.399 & \textbf{1.184} & 1.236
& 1.038 & 0.873 \\
& d8 & 0.579 & 0.536 & 0.583 & 0.805 & 0.949 & 1.201 & 1.324 & \textbf{1.44}
& 1.305 & 1.278 \\
& d9 & 0.523 & 0.546 & 0.611 & 0.637 & 0.744 & 0.849 & 1.122 & 1.668 &
\textbf{1.813} & 1.488 \\
& d10 & 0.638 & 0.38 & 0.445 & 0.494 & 0.457 & 0.651 & 0.831 & 1.076 & 1.95
& \textbf{3.077} \\ \hline\hline
\end{tabular}}
\caption{Results of estimated sorting measures in 2018-2022 at different
quantiles when $\protect\rho_{D^*_w, D^*_h} = 0, \protect\rho_{D^*_w, Y^*_w}
= 0, \protect\rho_{D^*_w, Y^*_h} = 0, \protect\rho_{D^*_h, Y^*_w} = 0,
\protect\rho_{D^*_h, Y^*_h} = 0$ and distribution of $Z$ as in total
population.}
\label{tab:results_4_2018}
\end{table}
\begin{table}[!ht]
\resizebox{15cm}{!}{
\begin{tabular}{l||l|rrrrrrrrrr}
\hline\hline
& & \multicolumn{10}{c}{Husbands' quantiles} \\ \hline\hline
& & d1 & d2 & d3 & d4 & d5 & d6 & d7 & d8 & d9 & d10 \\ \hline
\parbox[t]{2mm}{\multirow{10}{*}{\rotatebox[origin=c]{90}{Wives' quantiles}}}
& d1 & \textbf{1.542} & 1.265 & 1.058 & 0.96 & 0.945 & 0.879 & 0.841 & 0.817
& 0.8 & 0.894 \\
& d2 & 1.215 & \textbf{1.204} & 1.108 & 1.056 & 1.029 & 0.966 & 0.908 & 0.896
& 0.839 & 0.78 \\
& d3 & 1.089 & 1.153 & \textbf{1.12} & 1.072 & 1.068 & 1.0 & 0.942 & 0.893 &
0.867 & 0.794 \\
& d4 & 1.031 & 1.077 & 1.096 & \textbf{1.066} & 1.064 & 1.012 & 0.977 & 0.941
& 0.901 & 0.836 \\
& d5 & 0.908 & 0.984 & 1.042 & 1.05 & \textbf{1.065} & 1.045 & 1.023 & 1.005
& 0.972 & 0.907 \\
& d6 & 0.859 & 0.926 & 0.991 & 1.004 & 1.039 & \textbf{1.041} & 1.061 & 1.049
& 1.035 & 0.995 \\
& d7 & 0.825 & 0.881 & 0.95 & 0.992 & 1.014 & 1.051 & \textbf{1.067} & 1.081
& 1.075 & 1.064 \\
& d8 & 0.803 & 0.864 & 0.939 & 0.975 & 1.014 & 1.051 & 1.094 & \textbf{1.103}
& 1.096 & 1.062 \\
& d9 & 0.797 & 0.821 & 0.88 & 0.909 & 0.954 & 1.007 & 1.073 & 1.13 & \textbf{ \ 1.186} & 1.242 \\
& d10 & 0.926 & 0.821 & 0.822 & 0.836 & 0.887 & 0.953 & 1.016 & 1.088 & 1.218
& \textbf{1.433} \\ \hline\hline
\end{tabular}}
\caption{Results of estimated sorting measures in 2018-2022 at different
quantiles when all correlations are zero and distribution of $Z$ as in total
population.}
\label{tab:results_5_2018}
\end{table}
Tables \ref{tab:results_3_1976} and \ref{tab:results_3_2018} present the
results when $\rho _{D_{w}^{\ast },Y_{w}^{\ast }},$ $\rho _{D_{h}^{\ast
},Y_{h}^{\ast }}$, $\rho _{D_{w}^{\ast },Y_{h}^{\ast }}$ and $\rho
_{D_{h}^{\ast },Y_{w}^{\ast }}$ are also set to zero. We characterize these
parameters as collectively capturing the selection process. For the earlier
period the d10/d10 cell increases from 2.29 to 2.94 while the sum in the
(d9+d10)/(d9+d10) cells increases from 6.7 to 7.4. This suggests that
negative selection bias operating in both labor markets is reducing
household inequality. Negative selection is removing the relatively higher
paid males and females from FTFY employment. Setting these parameters to
zero inserts more higher paid workers into the market. This results in a
higher propensity of the higher paid males and females to marry. This is an
interesting result although it may reflect the identification strategy.
Interestingly the frequencies in Table \ref{tab:results_3_2018} provide a
somewhat comparable story although the changes are less drastic. Setting the
other parameters to zero appears to have little affect on the lower cells.
However, while the d10/d10 cell decreases marginally the sum in the
(d9+d10)/(d9+d10) cells increases from 7.9 to 8.4. This is similar to the
earlier period although the estimated positive selection effects for females
at upper quantiles is offsetting the negative selection effects for males
throughout the period.
We also examine the counterfactual in which the above parameters are set to
zero and we use the $Z^{\prime }$s of the whole sample rather than only
those who are working. The results are in Tables \ref{tab:results_4_1976}
and \ref{tab:results_4_2018} and a comparison with Tables \ref
{tab:results_3_1976} and \ref{tab:results_3_2018} indicates they do not
affect the observed sorting patterns. Although previous studies (see, for
example, Chernozhukov, Fern\'{a}ndez-Val, and Luo 2019) have analyzed the
impact of selection on the distribution of wages and have found it to have
some impact, our results indicate it would not change the sorting patterns
even if it is changing the distribution of both, or either of, the male and
female wages.
Finally we set $\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}$ to zero. For the BDR
model the corresponding counterfactual produced a substantial reduction in
sorting. Table \ref{tab:results_5_1976} reports the results for the earlier
time period. Given the large number of cells it is useful to focus on the
extreme cases as these are the outcomes more closely associated with
inequality. First consider the d1/d1 and d10/d10 cells. The former decreases
from 2.21 to 1.35 while the latter decreases even more dramatically from
2.89 to 1.36. Expanding the 4 lower and upper combinations to include d2 and
d9 respectively produces reductions from 6.6 and 7.4 to 5.1 and 4.96
respectively. It is interesting that the more substantial reductions appear
to occur at the cells for the higher wage deciles. Turning to the 2018-2022
time period we see a similar pattern. The d1/d1 cell decreases from 2.88 to
1.54 and d10/d10 decreased from 3.07 to 1.43. Expanding to the four lowest
and highest we see reductions of 6.82 to 5.21 and 8.3 to 5.07 respectively.
This clearly suggests that the correlation in the unexplained components of
wages is largely driving the observed sorting patterns. That is, sorting on
unobservables is an important factor in driving inequality.
\begin{table}[tbp]
\caption{Kendall rank correlation coefficient.}
\label{tab:kendall}\centering
\begin{tabular}{lrr}
\hline\hline
& 1976-1980 & 2018-2022 \\ \hline\hline
Data & 0.1833 & 0.2758 \\
Estimated & 0.1885 & 0.2588 \\
Participation & 0.1877 & 0.2582 \\
Selection & 0.1853 & 0.2421 \\
$X$ as in whole population & 0.1869 & 0.2483 \\
$\rho_{Y^*_w, Y^*_h} = 0$ & 0.0743 & 0.0768 \\ \hline\hline
\end{tabular}
\end{table}
To capture the sorting behavior which includes the off diagonals we compute
the Kendall rank correlation coefficient for these counterfactuals. These
are reported in Table \ref{tab:kendall} and confirm the patterns described
above. For the 1976-1980 period neither the selection parameters nor the use
of the working or total population composition of $Z^{\prime }$s affect the
rank correlation coefficient. For each of these experiments the rank
correlation coefficient does not differ greatly from its value for the data
of .18. However, when we additionally set the value of $\rho _{Y_{w}^{\ast
},Y_{h}^{\ast }}$ to zero the value falls to .07. For the 2018-2022 period
the results are similar despite the rank correlation coefficient of .27
being 50\% higher than the 1976-1980 value. For this period the only
counterfactual which generates a different pattern of sorting is that
corresponding to $\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}=0.$ For this latter
period the reduced rank correlation coefficient is also .07. This is
consistent with Tables \ref{tab:results_5_1976} and \ref{tab:results_5_2018}
.
\begin{figure}[tbp]
\caption{Decompositions of changes in the sorting measures}
\label{fig:decomposition_sorting}\centering
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table {
1976 -0.0527799999999998
1981 0.24096
1986 0.39696
1991 0.37299
1997 0.40544
2003 0.36423
2008 0.39794
2013 0.40929
2018 0.60543
};
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table {
1976 -0.00271999999999961
1981 0.00143999999999966
1986 -0.09504
1991 -0.02542
1997 0.00601999999999991
2003 0.0585499999999999
2008 0.00845000000000029
2013 0.03485
2018 0.0266900000000003
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table {
1976 -0.00302000000000024
1981 0.14099
1986 0.4233
1991 0.46178
1997 0.35499
2003 0.29672
2008 0.22845
2013 0.48379
2018 0.38876
};
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table {
1976 -0.0561500000000001
1981 0.13083
1986 0.17747
1991 0.0126900000000001
1997 0.14838
2003 0.15056
2008 0.31819
2013 0.0897899999999998
2018 0.39803
};
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table {
1976 0.00911000000000017
1981 -0.0323000000000002
1986 -0.10877
1991 -0.07606
1997 -0.10395
2003 -0.1416
2008 -0.15715
2013 -0.19914
2018 -0.20805
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table {
1976 -0.01877
1981 0.0749299999999999
1986 0.16905
1991 0.20427
1997 0.23405
2003 0.20383
2008 0.22546
2013 0.20971
2018 0.27565
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table {
1976 0.00038999999999989
1981 0.00430999999999981
1986 0.00697999999999999
1991 -0.0198200000000002
1997 -0.00767999999999991
2003 0.00566999999999984
2008 0.00280999999999998
2013 0.00479999999999992
2018 0.0114399999999999
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table {
1976 -0.00688999999999984
1981 0.0585800000000001
1986 0.08325
1991 0.25408
1997 0.26913
2003 0.24644
2008 0.26588
2013 0.49002
2018 0.33171
};
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table {
1976 -0.00540000000000007
1981 0.02203
1986 0.08772
1991 -0.00235999999999992
1997 0.0128000000000001
2003 0.00441000000000003
2008 0.01902
2013 -0.2207
2018 -0.0142899999999999
};
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table {
1976 -0.00686999999999993
1981 -0.00998999999999994
1986 -0.00889999999999991
1991 -0.02763
1997 -0.0402
2003 -0.0526899999999999
2008 -0.0622499999999999
2013 -0.0644099999999999
2018 -0.05321
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1976 0.0353300000000001
1981 0.0183600000000002
1986 0.12264
1991 0.22154
1997 0.20024
2003 0.18737
2008 0.31358
2013 0.27027
2018 0.35866
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table {
1976 0.000399999999999956
1981 -0.0314699999999999
1986 -0.01511
1991 -0.0108999999999999
1997 -0.0205200000000001
2003 -0.00923000000000007
2008 0.00295000000000001
2013 -0.01308
2018 0.0130399999999999
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table {
1976 0.00971000000000011
1981 0.0850299999999999
1986 0.19362
1991 0.37764
1997 0.30745
2003 0.25825
2008 0.35081
2013 0.34718
2018 0.23342
};
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table {
1976 0.02572
1981 -0.0398399999999999
1986 -0.0497999999999998
1991 -0.14238
1997 -0.0906100000000001
2003 -0.0748500000000001
2008 -0.0536699999999999
2013 -0.06602
2018 0.10124
};
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table {
1976 -0.000499999999999945
1981 0.00463999999999998
1986 -0.00607000000000002
1991 -0.00282000000000004
1997 0.00392000000000015
2003 0.0132000000000001
2008 0.01349
2013 0.00219000000000014
2018 0.0109600000000001
};
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table {
1976 0.0381300000000002
1981 -0.0582099999999999
1986 0.141
1991 0.14807
1997 0.31916
2003 0.45728
2008 0.45837
2013 0.37116
2018 0.633
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table {
1976 0.00887000000000038
1981 -0.04006
1986 -0.12723
1991 -0.0881599999999998
1997 0.0251900000000003
2003 0.0538399999999997
2008 -0.0414000000000003
2013 -0.0322299999999998
2018 0.00768000000000013
};
\addlegendentry{Selection}
\addplot [very thick, color2]
table {
1976 -0.0476100000000002
1981 0.0862400000000001
1986 0.30024
1991 0.2019
1997 0.26355
2003 0.21145
2008 0.33575
2013 0.14262
2018 0.0986399999999996
};
\addlegendentry{Structural}
\addplot [very thick, color3]
table {
1976 0.0764100000000001
1981 -0.0986400000000001
1986 -0.00784000000000029
1991 0.0752600000000001
1997 0.09083
2003 0.26904
2008 0.28074
2013 0.33525
2018 0.60937
};
\addlegendentry{$\rho_{23}$}
\addplot [very thick, color4]
table {
1976 0.000459999999999905
1981 -0.00574999999999992
1986 -0.0241699999999998
1991 -0.0409299999999999
1997 -0.0604100000000001
2003 -0.0770499999999998
2008 -0.11672
2013 -0.0744799999999999
2018 -0.0826899999999999
};
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\end{figure}
We noted that we estimated the model treating the male employment decision
as exogenous. Although we do not report the results here, we conducted the
corresponding counterfactuals to those for the full model. The general
flavor of the simulations were similar although there was stronger evidence
of selection affecting the observed sorting for the earlier period. However,
the result that setting $\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}$ to zero
reduced the observed level of positive sorting was also clearly supported.
\subsection{Decomposing Changes in the Sorting Patterns}
We now focus on the changes in the sorting patterns in the earlier tables
and examine how the probability of both spouses being in a specific quantile
of their wage distribution changes over our sample period. We decompose the
total change into structural, selection and composition effects. Note that
unlike the previous section, we now also examine the impact of the changes
in the $\beta ^{\prime }$s over time. As we employ a multivariate normal
approximation for each point of the four dimensional partition of the data
we can use the model parameter estimates to decompose the observed changes
into the various components. While the composition effects are defined as
the changes in the conditioning variables, the allocation of the other
parameters are less straightforward.
The changes in the $\beta ^{\prime }$s can be assigned the interpretation of
conventional structural effects. We assign $\rho _{D_{w}^{\ast },Y_{w}^{\ast
}}$, $\rho _{D_{h}^{\ast },Y_{h}^{\ast }},$ $\rho _{D_{w}^{\ast
},Y_{h}^{\ast }}$ and $\rho _{D_{h}^{\ast },Y_{w}^{\ast }}$ to the selection
effects. We isolate the impact of $\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}$
parameter as it captures the correlation between the unobservables which
impact the spouses' wages. While this is a type of structural effect we
separate it from the structural effects operating through the $\beta
^{\prime }$s. Although we can evaluate the sorting behavior observed in 100
cells it appears that the interesting behavior occurs in the top and bottom
cells. Accordingly we conduct the decompositions for both members of the
couple being located in the bottom and top deciles and quartiles. These
plots are presented in Figure \ref{fig:decomposition_sorting}. The
decompositions represent how the change in the measures can be explained by
changes in the various components. If a specific component appears to be
close to zero then this indicates that its contribution has remained
unchanged and not that its contribution is zero.
We start with the probability of both spouses being located in the bottom
decile and quartile. Over the 47 years of our sample the former increases by
almost 60 percent and the latter by almost 40 percent. For d1 the large
positive change is driven by changes in the structural effect and $\rho
_{Y_{w}^{\ast },Y_{h}^{\ast }}.$ Changes in the selection effect are
unimportant and the changes in the composition effect are negative. For Q1
the results are essentially the same although the changes in the
contribution of $\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}$ are generally
unimportant except for a decrease towards the end of the sample. At the top
of the distribution there is a similar finding. The probability of both in
the top decile and quartile increases by around 60\% and 35\% respectively.
The change in the probability of both spouses being in d10 is again driven
by changes in the structural effect and $\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}$
although the change in the structural effect diminishes towards the end of
the sample. The change in the probability of both spouses being in Q3 is due
to changes in the structural effects. Changes in the composition and
selection effects appear unimportant for either d10 or Q3. The results on
the composition effects are consistent with the evidence as noted above in
Breen and Salazar (2011), Gihleb and Lang (2018), Eika, Mogstad, and Zafar
(2019) and Chiappori, Costa-Dias and Meghir (2020,2023). Education is a
component of the composition effect and there is no impact at either Q1 or
Q3. At D1 and D9 it is less clear as both probabilities decline.
The increasing presence of married females into the FTFY in the 1970's and
1980's had little impact on the sorting process in the marriage market.
Overall the selection effects are small. Second, composition effects are
generally unimportant and have reduced positive sorting. This reflects that
the increased level of acquired education has increased everyone's
probability of marrying a relatively highly educated spouse. This is
consistent with the findings of others noted above regarding the impact of
composition effects on positive sorting as measured by education. The
clearest evidence is associated with the structural effects. These are the
most important contributors to the total effect\ and their interpretation is
also quite clear. Sorting behavior has not greatly changed in terms of the
observed characteristics of the spouses. However, the higher skill premia,
and the heterogeneity of the skill premia, have resulted in the highly
(lowly) paid being even more highly (lowly) paid. This is consistent with
empirical evidence on decompositions of changes in wage inequality. The
evidence on the role of the $\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}$ provides a
similar story. While this parameter captures the impact of unobservables it
also reflects how they are priced. Our evidence suggests that the increased
observed level of positive assortative sorting reflects the increases in
prices of observables and unobservables. Individuals appear to marry the
\textquotedblleft same" spouses but the value of their characteristics has
changed due to the increased skill premia. It is also possible that $\rho
_{Y_{w}^{\ast },Y_{h}^{\ast }}$ partially captures how an individual's wage
is influenced by the value of their spouse's characteristics. For example,
individuals with highly successful spouses may benefit from exposure to
their spouse's network and this may produce some relationship between their
wage and, for example, their spouse's education level. Some preliminary
investigations of the data suggest that this relationship exists, and that
it changes over time, but we do not pursue it here.
\subsection{Household Income Inequality}
We now explore the implication of the model's estimates for household income
equality. We assume that all individuals comprising the FTFY couples are
working the same number of hours and we employ the household wage, defined
as the sum of the spouses' hourly wages, as our measure of family income. We
examine how inequality has evolved by examining the D8/D2 ratio under
different counterfactuals. We focus on this ratio as the manner in which its
components have changed is clear from the tables above. The results are
shown in Table \ref{tab:inequality}. Note that we do not isolate the role of
structural effects operating through the $\beta ^{\prime }$s nor the
composition effects reflecting the changes in the $X^{\prime }$s over time.
As listed at the start of our Introduction, there is a large existing
literature which has clearly established the impact of these factors on wage
inequality. We repeat the same exercise as in the counterfactual sorting
exercises of Section \ref{ss:counterfactual_results} in which we
incrementally set the parameters capturing different features of the model
to zero. We also explore the impact of employing the population $X^{\prime }$
s rather than those of the working sample.
\begin{table}[tbp]
\caption{Decomposition of changes in inequality.}
\label{tab:inequality}\centering
\begin{tabular}{lrr}
\hline\hline
& 1976-1980 & 2018-2022 \\ \hline\hline
Data & 1.7987 & 2.3198 \\
Estimated & 1.8002 & 2.2797 \\
Participation & 1.7997 & 2.2841 \\
Selection & 2.0661 & 2.2776 \\
$Z$ as in whole population & 2.066 & 2.2692 \\
$\rho_{Y^*_w, Y^*_h} = 0$ & 1.735 & 2.1192 \\
Random sorting & 1.6812 & 2.0505 \\ \hline\hline
\end{tabular}
\end{table}
The first row of Table \ref{tab:inequality} presents the D8/D2 ratio for our
two extreme sample periods. There is a large increase of 29.6\%, from 1.79
to 2.32, noting that it is similar to the growth in comparable measures of
inequality presented in the discussion of the data. The second row presents
the comparable estimates from the predictions of our model and indicates
that the level of inequality observed in the data are maintained. To examine
the role of sorting we first estimate the corresponding measure in a setting
of unconditional random sorting in which husbands are assigned to wives in a
random manner. This is shown in the table's bottom row. The results
represent the D8/D2 ratio based on 100,000 draws. The estimates are 1.68 and
2.05 and reflect an increase of 22\% over the sample period. This large
impact reflects the large increase in wage inequality which occurs for both
husbands and wives. More interestingly, comparing the top and bottom rows of
this table indicates that sorting increases inequality by 7.0\% in the
earlier period and 13.1\% in the latter. This suggests that sorting has
substantially increased household inequality.
To investigate the role of the various model parameters, rows 3 to 6 repeat
the counterfactuals conducted above. As with the tables above, each of the
rows reflects the incremental change in the inequality measure as the
parameters associated with some feature of the model are set to zero. For
the earlier period selection has reduced inequality although we have
highlighted our concerns regarding this result as it is reliant on the form
of identification. The results regarding the use of the population $
Z^{\prime }$s rather than the working sample $Z^{\prime }$s confirms our
earlier results that this is does not appear to affect the results regarding
sorting. Row 6 also highlights the recurring result that positive sorting is
operating through the correlation of the spouses' unobservables. For the
2018-2022 period we have similar findings although that related to selection
does not hold. However the importance of sorting on unobservables for
inequality is again highlighted. Setting $\rho _{Y_{w}^{\ast },Y_{h}^{\ast
}}=0$ decreases the D8/D2 ratio for the earlier period by 16\% from 2.06 to
1.73 and for the later period by 7\% from 2.26 to 2.12.
While there is an important role for $\rho _{Y_{w}^{\ast },Y_{h}^{\ast }}=0,$
the remaining large increase in inequality in the household wage appears due
to changes in composition and structural effects. However, the existing
evidence, here and in the existing literature, suggests the sorting on
observables has not changed for this time period and composition effects are
unimportant. This suggests that the increase in wage inequality across
households is due to structural effects capturing changes in the skill
premia.
\section{Conclusion}
\label{sec:discusion}
We examine the role of marital sorting on household income inequality in a
period of substantial and increasing wage inequality. We do so by examining
a sample of married couples from the CPS for the years 1976 to 2022 in which
each of the spouses works full time/full year. To investigate the
determinants of marital sorting and its impact on inequality we estimate a
model explaining the spouse's employment decisions and their location in the
gender specific wage distributions. This is done via a multivariate
distribution regression approach which incorporates selection. We provide a
number of important empirical findings.
First, we confirm that wage inequality has increased for both spouses in
dual FTFY couples. Changes in inequality for these groups do not precisely
correspond to those for all males and females but the trends are similar.
Second, we find clear evidence of positive sorting defined as higher
incidences of high (low) wage males marrying high (low) wage females than
expected under random sorting. However, we acknowledge that random sorting
is not a realistic alternative to positive sorting as couples are frequently
similar in terms of age, race and location of residence and each of these
are known to determine wages. However, the disproportionate fraction of
males in the extremes of the wage distribution marrying females in the
corresponding extremes of their distribution is supportive of positive
sorting. Moreover, this positive sorting has increased over our time period.
Combining this increasing sorting with increasing spouse specific inequality
has increased inequality across couples.
Third, a series of counterfactual experiments do not provide any evidence
that unobservables related to either of the spouses' work decisions have any
implications for the observed sorting behavior. However, there is compelling
evidence that the correlation between the unobservables influencing the
spouses wages is substantially contributing to the observed sorting
behavior. While some of this correlation may be due to factors, such as cost
of living premia, which are incurred by both members of the couple it is
also possible that this captures factors corresponding to unobserved ability.
Finally, we find that positive sorting associated with the increased share
of extreme cells has increased over the 47 years of our sample. A
decomposition of the changes in these cells shares over the period examined
reveals that their growth is almost entirely due to the prices of observable
and unobservable characteristics. This suggests that the increase in
observed sorting pattern in not due to different behavior nor the increase
of more females in the labor market. Rather as the skill premia has
increased this has increased the wages of both members of some couples while
couples without these skills appear to be both negatively affected.
\label{sec:conclusion}
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\end{description}