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Quantifying The Progress of Economic and Social Justice:
Charting Changes in Equality of Opportunity in the United
States 1960-2000
Gordon Anderson
*
University of Toronto
Teng Wah Leo
,
St. Francis Xavier University
October 16, 2015
*
Department of Economics, University of Toronto. Email Address: anderson@chass.utoronto.ca
Department of Economics, St. Francis Xavier University. Email Address: tleo@stfx.ca
The authors would like to thank the editor, and anonymous referees for their very helpful comments,
and to the participants at the 2015 Human Development and Capability Association (HDCA) conference
in Washington, D.C.
Abstract
The notion of Equality of Opportunity (EO) has pervaded much of economic
and social justice policy, and research over the last half century. The sense that dif-
ferences in agent outcomes that are the consequence of their individual choice and
effort are acceptable, whereas variation in agent outcomes that are the consequence
of circumstances beyond their control are not, has underpinned much gender, race,
education, and family law and policy over that period, making it a many dimen-
sioned issue. In this context the empirical analysis of EO has been hampered in the
sense that the usual techniques are one-dimensional in nature. Here a new approach
to evaluating levels of, and changes in EO which readily accommodates these many
dimensions is introduced, and progress in the extent of equality of opportunity for 18
year old’s in the United States is examined over the period 1960-2000. The evidence
is that gains were made in all categories throughout the period, more so for males
than females (though females were better off in an EO sense to start with), more
so for children in single parent circumstances, and more so for the poorly endowed.
Key Words: Overlap Measure; Stochastic Dominance; Equality of Opportunity
1 Introduction
Education is a primary source for the development of peoples’ capabilities, their abilities
to be and do (Nussbaum 2000, 2011). That it’s achievement should not be hindered by the
circumstances that people confront, their race, gender, inherited family, and social back-
ground, is a central tenet of notions of social justice founded upon a sense of equality of
opportunity (Roemer 1998, 2006). However assessment of the degree to which educational
attainment has been achieved in an optimally “socially just” fashion is difficult. Atkinson
(2012) in discussing public policy reform in the realm of political economy, argued that
the aim is “to remedy injustice rather than characterize perfect justice”. In this he was
following Sen (2009), who in the introduction to The Idea of Justice avows it to be “. . .
an attempt to investigate realization-based comparisons that focus on the advancement
or retreat of justice”. The objective for both is to seek progressive reform rather than
transcendental optimality. Accordingly, techniques for evaluating such progress should be
capable of measuring the degree and significance of such advances or retreats in economic
and social outcomes. Here a new method is proposed and implemented, and the effects of
the many Economic and Social Justice policy reforms in the United States over the last
half century are evaluated in terms of the extent to which various notions of economic
and social justice have advanced the imperative.
In recent years there has been considerable interest amongst economists and philoso-
phers in quantifying various notions of Economic and Social Justice in its many dimensions
(Arrow et al. 2000; Brighouse and Robeyns 2010; Dworkin 2011; Rawls 2001; Sen 2009).
One aspect that has perhaps resonated most with law and policy makers alike is the
notion of Equality of Opportunity (EO), the sense that inequalities that are the result
of differences in individual choice and effort are acceptable, whereas inequalities that de-
rive from differences in individual circumstances are not.1Roemer (2006) in referring
to equalizing opportunities as a “field levelling” exercise suggests that education is an
institution central to such an exercise. Thus EO has underpinned much gender, race, and
education law and policy in the past 50 years, all of which can be interpreted as attempts
to release one generation from the constraints of inherited circumstances.2This notion
1See Arneson (1989), Cohen (1989), Dworkin (1981a), Dworkin (1981b), Dworkin (2000), Roemer
(1998), Roemer et al. (2003) and Roemer (2006) for the philosophical foundations.
2As such it can also be construed as part of an agenda associated with advocates of a functionings
and capabilities approach to societal wellbeing, and Human Development (Sen 2009; Nussbaum 2011).
1
of justice is essentially a statement about the nature of the desired joint density function
of a collection of outcome and circumstance variables, and a measure of the degree to
which justice exists would be the proximity of the existing joint density to the desired
joint density that characterizes a particular notion of EO.
Hitherto most techniques for assessing such concepts of social and economic justice
have been based on generational regressions or transition matrices, each of which in its
own way maps parental outcomes into child outcomes, and seeks to evaluate the de-
gree of dependence/independence in the mapping. The regression approaches have been
extended to multiple regressions which have taken multiple circumstance factors into ac-
count (Bourguignon et al. 2007; Ferreira and Gignoux 2011, 2013; Marrero and Rodr´ıguez
2012), but evaluation of progress of equality of opportunity using these techniques remains
difficult since they are essentially based upon multiple correlation measures at a point in
time. Lefranc et al. (2008) suggest comparing the distributions of the outcomes of agents
from different circumstance classes, for the absence of stochastic dominance relationships
between the different circumstance group distributions. Each technique has its problems
for this purpose, especially when policy is pursued asymmetrically over a variety of child
and parent outcomes.
An illuminating discussion in the survey by Ramos and Van de Gaer (2015) points out
that in the theoretical literature, a policy maker confronts agents identified by outcomes
(which is the object of the policy), which depend upon jointly distributed vectors of efforts
and circumstances. Many EO measures have been founded on principles of compensation
and reward, two of which, ex-post compensation (EPC) −equalizing outcomes of indi-
viduals with the same effort, and ex-ante compensation (EAC) equalizing unambiguous
inequalities in circumstance, were shown by Fleurbaey and Peragine (2013) to be mutually
incompatible. This suggests that if one wants to evaluate outcomes from the perspective
of EO, a choice has to be made between ex-ante and ex-post compensation.
In the following discussion, effort is not observed, and only outcomes and some form
of independence of outcomes and circumstances will be the object of enquiry, so that we
need to rationalize the connection between the theoretical arguments and the empirical
results here. First note that the conditional distribution of outcomes (conditional on
circumstances) is equal to its unconditional distribution if and only if there is ex-ante EO.
This then means that when the conditional and unconditional distributions of outcomes
are not equal, ex-ante EO cannot prevail. Secondly, if efforts and circumstances are
2
independently distributed, there is ex-post EO if and only if there is ex-ante EO. This
then means that if ex-ante EO does not prevail, neither will it ex-post. Hence, when
outcomes are dependent upon circumstances, it constitutes a rejection of ex-ante EO. If
in addition, it is assumed that effort and circumstances are independently distributed,
then a rejection of ex-post EO is also implied.
A pure EO policy mandate values releasing any and every child from its circumstances
whether they be favourable or unfavourable. However the pursuit of such a pure EO goal
has not been unequivocal; Cavanagh (2002) and Roemer (2010) expressed some philo-
sophical reservations, Jencks and Tach (2006) have questioned whether an EO imperative
should require the elimination of all sources of economic resemblance between parents
and children3, and in a similar vein Dardanoni et al. (2006) question how demanding
the pursuit of EO should be in terms of the feasibility of such a pursuit. Furthermore,
most observed law and policy practice has been more qualified in its approach to reliev-
ing poorly endowed children of dependence upon their circumstance, whilst leaving the
dependence of richly endowed children upon their circumstance more or less intact4, with
an emphasis on facilitating the upward mobility of the poorly endowed. Methods for
evaluating the degree of progress should be able to reflect this asymmetric, conditional
and progressive nature of the policy objective. Here as examples, two constructions of the
EO imperative are considered, one (EO) reflecting an unadulterated requirement of inde-
pendence between outcomes and circumstances, and another more qualified view, hence
Qualified Equality of Opportunity (QEO), which reflects an emphasis on upward mobility.
In essence the QEO measure, instead, postulates that the bottom of the circumstances
domain is more important than the rest of the circumstance distribution in evaluating
changes in inequality of opportunities.
In the following, new techniques are developed and employed to examine the progress
toward EO over the last four decades of the 20th century in the United States in the context
of the educational attainment of 18 year olds and an array of circumstances they faced
3Indeed in terms of the nature versus nurture debate, it is doubtful that resemblances due to nature
can be totally eliminated or compensated for.
4Equal opportunity programs observed amongst Western societies do seem to be of this flavour. For
example, when questioned on the widening gap between the rich and poor, the British Prime Minister
responded that “. . . the issue is not in fact whether the very richest person ends up being richer. The
issue is the poorest person is given the chance they don’t otherwise have. The most important thing is to
level up, not level down.” Interview with the Prime Minister on BBC News Newsnight on June 4, 2001.
Transcript available from http://news.bbc.co.uk/2/hi/events/newsnight/1372220.stm.
3
(Note that in a general sense, EO in the work place, and the impact of “the glass ceiling”
for example, are not being considered here). The novelty is a new multivariate measure of
EO which calibrates the proximity to statistical independence between an agent’s outcome
and circumstance sets. It is a “norm” based approach (Ramos and Van de Gaer (2015))
based upon the deviation of the actual distribution from the norm of independence (or
qualified independence), which can be related to inequality averse rewards. The norm of
statistical independence can be readily interpreted as the equality of outcome distributions
of all circumstance types which implies the absence of any dominance relations between
such distributions, which are the object of examination in the studies by Lefranc et al.
(2008, 2009) (see also Andreoli et al. (2014)), i.e. it is a state wherein no further welfare
improvements can be made. In the Qualified approach, proximity to the norm is more
important for those poorly endowed in circumstance than for those richly endowed in
circumstance. Existing approaches to measuring equality of opportunity are discussed,
and the new measure is introduced in section 2. Section 3 provides some background to
the EO policies that were enacted in the preceding decades in the U.S., and section 4
reports the empirical results of the examination of these policies. Some conclusions are
drawn in section 5.
2 Measuring Equality of Opportunity
2.1 Classical Techniques
Generally EO has been studied and evaluated in the context of an agent’s outcomes
being measurably independent of its circumstances (usually measured as the agent’s cor-
responding parental outcomes, which has led to a generational mobility interpretation),
and the two dominant approaches are Generational Regressions and Transition Matrices
approach. Generational Regressions, where child outcomes are regressed on indicators
of their circumstances (usually parental characteristics), evaluate EO by the proximity
of the circumstance coefficients to zero5(note this approach weights equally trends away
5Much time is spent in introductory statistics courses stressing that, while independence implies zero
covariance, zero covariance (the basis of inference on β) does not imply independence! Think about an
exact non-independent relationship y= 0.5 + 2(x−x2) for 0 <x<1 (a fairly plausible technology),
with parent quality uniformly distributed in [0,1]. A random sample of agents from this would yield
zero covariance between yand x, and hence a zero estimate of βimplying independence for what is a
completely dependent relationship.
4
from dependence of the richly and poorly endowed upon their circumstances). Transition
Matrices between “parent outcomes” and corresponding “child when adult” outcomes,
evaluate EO by the proximity of the matrix structure to that which would be engen-
dered by independence between outcomes and circumstances (again this approach weights
equally trends away from dependence of the richly and poorly endowed upon their cir-
cumstances). Here and elsewhere (Ramos and Van de Gaer 2015; Roemer and Trannoy
2015), it is contended that both approaches present problems for evaluating the progress
of equal opportunity beyond not reflecting the asymmetric nature of policy imperatives,
and an alternative technique is proposed which does not suffer these deficiencies.
2.2 Alternative Techniques of Quantifying EO
The statistics literature abounds with types of dependence. Lehmann (1966) outlines
three types of dependence, all of which deal with monotone relations between Xand Y
(see also Bartolucci et al. (2001)). The most recent approach to measurement of social
and economic justice (Lefranc et al. 2008, 2009) focuses on dependencies by examining
the stochastic dominance relationships between the outcome variables associated with
different dependency classes, wherein the absence of a dominance relationship provides
evidence of a lack of dependency. While this approach will undoubtedly capture the ef-
fects of QEO policies in the simple two variable case, it becomes much more difficult to
employ in multivariate situations, and does not yield a statistic that readily facilitates
measurement of progress.6A more omnibus notion of dependence is required, since re-
lationships between parent and child characteristics need not be monotone. Thus here
a more general concept of “distance from independence” is employed which admits both
monotone and non-monotone relationships (as were conjectured in footnote 5), and which
will always provide consistent tests.
Letting xbe an k-dimensional vector, and fa(x) and fb(x) be two continuous mul-
tivariate distributions. The extent to which fa(x) and fb(x) overlap can be measured
as:
OV =
∞
Z
−∞
. . .
∞
Z
−∞
min {fa(x), fb(x)}dx(1)
6While parent-child education relationships are clearly monotonic it is not clear that other relation-
ships, e.g. parental income-child education are not necessarily monotonic given notions of diminishing
marginal returns to scale in investment in education.
5
If fa(x) is the unrestricted joint p.d.f. of x∈ X ⊂
R
kand fb(x) is the joint distribution
when the x’s are independent, then 0 ≤OV ≤1 is an index of independence, and 1 −OV
is a general index of dependence, be it monotone or not. The measure defined in (1) is a
very convenient index, bounded between 0 and 1, and most importantly has well defined
statistical properties that facilitate inference.
One slight wrinkle, is that xis often a mixture of discrete and continuous variables.
Denoting them by xd∈ Xdand xc∈ Xcrespectively, so that x0= [x0
d,x0
c], the appropriate
overlap measure is:
OVmix =Z
xc∈XcX
xd∈Xd
min{fa(x), fb(x)}dxc(2)
Here, summation is over the discrete components, and integration is over the contin-
uous components. The discrete version of OV has been developed in Anderson et al.
(2010) so the properties of OVmix can be derived as a mixture of the two cases as in
Anderson and Hachem (2012). Moreover, OVmix lends itself quite naturally to a measure
of the degree of independence7, as well as other notions of social justice by letting fa(x)
be the empirical distribution, and fb(x) the desired distribution under a given particular
definition of social justice. To see how, with some abuse of notation, let ybe a vector
of agent outcomes with joint distribution g(y), and xbe a vector of their circumstances
with joint distribution f(x), and the joint distribution of outcomes and circumstances is
denoted by h(y,x). Under independence, h(y,x) = f(x)g(y), and the following measure
of their independence will be:
OV =ZXmin{h(x,y),(f(x)g(y))}dz∈(0,1) (3)
or alternatively in its conditional form,
OV =ZXmin h(x,y)
f(x), g(y)dz∈(0,1) (4)
where integration is over z, the continuous components of yand x, and summation is over
the discrete components of yand x. It should be noted that the marginal distribution
7The Overlap Measure proposed in this paper can be adapted to the three conceptions of intergener-
ational mobility suggested by Van de Gaer et al. (2001), since each transition matrix implies a particular
structure for the joint density matrix, which the empirical joint density can be measured against. Fur-
ther, the third mobility measure for Markov chains proposed by Van de Gaer et al. (2001) is related
to the Overlap Measure in the sense that it measures the complement to the overlapping region of the
conditional probabilities.
6
of attainment outcomes is used in constructing the dependence free reference distribution
as the basis of the norm, since if this state were the optimal EO outcome, this would
have to be the achievement distribution for all circumstance classes. A greater degree of
dependence between yand ximplies less overlap between h(x,y) and f(x)g(y), leading
to lower values of OV. Furthermore, the statistic can be calculated conditionally on
particular aspects of circumstances to check for example whether equality of opportunity
improvements are symmetric with respect to poorly or richly endowed children, or on
the marital status of the parents to check whether equality of opportunity policies have
affected those groups differentially.8
The simple “pure” version of EO, as a justice imperative has met with its critics9, who
refer to it as “Luck Egalitarianism”. Basically the critics’ concern are that because good
outcomes are strongly correlated with good circumstances, if there is insufficient capacity
in the system (Anderson et al. 2014) to upgrade the poorly endowed to the status of
the richly endowed, high type inheritors have to be disinherited to achieve the just equal
opportunity outcome. So for example, high achieving children, who are so because they
have genetically inherited benefits from their high achieving parents, have to be penalized
or disinherited, in essence destroying inherited social capital. A resulting compromise
policy is to follow a “Qualified” Equality of Opportunity imperative which seeks equality
of opportunity for the poorly endowed whilst preserving the outcomes of those who have
been more fortunate in their inheritance. This may be interpreted as following a second,
Pareto-like imperative, wherein no child should be made worse off by an EO policy, so
that the focus is on elevationg the outcomes of the poorly endowed. A simple way of
characterizing the just outcome in this case can be achieved by modifying the previous
measures of equation (3) and (4) by considering the target joint density of the “just”
society as:
h∗∗(x,y) = (1 −F(x∗))f(x)g(y) + F(x∗)h(x,y) (5)
Here F(.) is the cumulative density of y∗which is a monotonic non-decreasing function of
a subset or the entire set of parental qualities that can be ordered (perhaps a mixture of
8Appendix A.1 provides a brief description of how the measure is estimated. These indices are
confined to the unit interval with proximity to one representing the ultimate in social justice however
defined.
9Anderson (1999), Cavanagh (2002), Hurley (1993), Piketty (2000), and Swift (2005) are some oppo-
nents.
7
income and educational status), so that high type circumstance outcomes tend to preserve
the status quo, whilst low type outcomes engender independence between outcome and
circumstance. The same measures of distance (that is the overlap) between h(x,y) and
h∗∗(x,y) provide indicators of the extent of this sense of justice in this society.
3 Background of Policies with Effect on EO in the
U.S.
Legislation promoting EO in the United States has fallen into two broad categories, poli-
cies promoting outcomes of the poorly endowed, and more general anti-discrimination
policies. The former group is comprised of family law policies (usually a state level is-
sue) which promote the outcomes of children in disadvantaged home circumstances and
schooling legislation, which has largely been a federal matter manifesting in policies such
“No Child Left Behind”. The latter group have generally appeared under a civil rights
banner.
Divorce law changes, associated with facilitating Unilateral Divorce (either party in the
marriage having the ability to leave the marriage without consent from the other party),
and No-Fault Divorce (the party leaving not needing to prove he/she is leaving because
the other party has transgressed during the marriage) occurred first in California in 1969,
and by 1980s, almost all states had them.10 Although these legal changes might have
empowered parties to a marriage, it has been found that they have increased the incidence
of children living in the context of disadvantageous circumstances (See Gruber 2004, and
Johnson and Mazingo 2000). Child custody law changes which also began in California
in 1980, (but at least 3 states had acknowledge the possibility of joint custody in the
1970s) can be construed to have an opposing intent in securing more and better resources
for investment in child development (Del Boca and Ribero 1998). Nonetheless, the latter
view is contested by opponents to the law (Maccoby and Mnookin 1994; Mason 1999). By
2000, only 7 states had not implemented such laws, namely Nebraska, New York, North
Dakota, Rhode Island, Vermont, West Virginia, and Wyoming. The impact of the law’s
10Studies into the effect of divorce law changes include the following, (a.) Divorce rates (Peters 1986;
Allen 1992; Peters 1992; Friedberg 1998; Wolfers 2006), (b.) Marriage rates (Rasul 2006), (c.) Child
outcomes (Gruber 2004; Johnson and Mazingo 2000), (d.) Marriage specific investments (Stevenson
2007), and (e.) Domestic violence rates (Stevenson and Wolfers 2006).
8
adoption is exemplified in California, where joint custody decisions rose from 2.2% of all
final decrees in 1979, to 13% in 1981 (Maccoby and Mnookin 1994). Further, its breadth
of influence was evident among states which permitted divorced parents to reevaluate
custodial arrangements made prior to the regime shift, obtaining fresh judgments based
on current application of the new law (Mason 1999). Basically, the change in statute
allows for both parents to share in the custody whereas previously, the law acknowledged
maternal preference in their rulings.11
With respect to schooling policies, Title 1 was the centerpiece of the Elementary and
Secondary Education Act signed into Law by Lyndon Johnson in 1965 to provide financial
assistance to local education agencies in districts with high incidences of poverty. Its
implementation and impact have been critically discussed in Kosters and Mast (2003),
and Cohen and Moffitt (2009). Over one billion dollars were assigned over and above the
regular school budget (i.e. this was not a reallocation of the school budget) in the first
year. In the ensuing 35 years almost
$
200 billion was allocated. The asymmetric nature of
these family and schooling laws may well reflect the presence of a second policy imperative
which modifies or qualifies a pure equal opportunity policy so that poor in circumstance
children are relieved of their circumstance connection, whereas the connection between
richly endowed children and their circumstance is maintained12.
Title IV of the Civil Rights Act of 1964 desegregated public education in the U.S.,
and was largely a response to the Supreme Court’s ruling on Brown versus the Board of
Education of Topeka 1954 (Brown). Brown was the best known of a sequence of cases
initiated by the National Association for the Advancement of Colored People (NAACP)
Legal Defense and Educational Fund to break down racial segregation in the field of
education and beyond. In the ruling, Justice Warren declared that “in the field of public
education, the doctrine of ‘separate but equal’ has no place. Separate educational facilities
are inherently unequal.”. Previously, Court decisions held that educational segregation
was acceptable as long as conditions and curriculum in the separate schools were equal13.
11Studies of the effect of custodial law changes include, (a.) Implications for a non-custodial parent’s
willingness to make child custody payment (Weiss and Willis 1985; Del Boca and Ribero 1998), and (b.)
Implications for divorce and marriage rates, and consequent impact on child investments (Rasul 2006;
Brinig and Buckley 1998; Halla 2008; Halla and Holzl 2007; Leo 2008; Nunley and Seals 2009).
12Anderson et al. (2014) showed that in a constrained world with no growth in average child outcomes,
movement toward an equal opportunity outcome for one group of children must necessarily make another
group of children worse off.
13Interestingly some school boards met the equality mandate by penalizing white schools. King George
9
The 1964 Act enforced the assignment of students to schools without regard to their race,
colour, religion or national origin, and was explicit in averring that desegregation did
not mean assigning students to schools to overcome racial imbalance. It offered technical
assistance, training assistance, and grants to school boards to facilitate desegregation.
Title IX of the Education Amendment Act of 1972 addressed discrimination with
respect to gender in education. Modeled on Title IV of the 1964 act, the preamble to Title
IX declared that: “No person in the United States shall, on the basis of sex, be excluded
from participation in, be denied the benefits of, or be subject to discrimination under any
educational programs or activity receiving federal financial assistance . . . ”. With respect
to employment opportunities, Section 703 (a) of Title VII of the 1964 Civil Rights Act,
made it unlawful for an employer to “fail or refuse to hire or to discharge any individual, or
otherwise to discriminate against any individual with respect to his compensation, terms,
conditions or privileges or employment, because of such individual’s race, color, religion,
sex, or national origin.”. The final bill also allowed gender to be a consideration when it
is a bona fide occupational qualification for the job. Title VII of the act also created the
Equal Employment Opportunity Commission (EEOC) to implement the law.
Generally all of these legislative changes took place in the early part of our data period,
in the 1960s and 1970s. However the policies often took some time to implement, and
clearly very often they took some time to have an effect. Hutchinson (2011) notes that
“. . . it was only in the 1980s that . . . school district were obligated by federal courts
to implement mandatory busing plans . . . that high schools long formally desegregated
still had different bells for black and white students . . . separate basketball teams . . .
at Mississippi’s Charleston High School . . . only in 2009 . . . the first integrated prom
dance occurred.”. Here the long term effects of these policy changes will be quantified in
terms of the extent to which a young person’s academic achievement was influenced by
the circumstance they confronted.
Since so much of an individual’s circumstance can be associated with their parents,
much “Equal Opportunity” research and policy, under the banner of Generational Mobil-
ity, has had a one dimensional focus on the degree to which an individual’s outcome can be
considered independent of the corresponding parental outcome. However the “level play-
County, Virginia, for example, chose to equalize its curriculum by dropping several advanced courses
from its white high school rather than add them at the black school, an example of a symmetric equal
opportunity policy which we argue is not generally observed in the 1960-2000 period.
10
ing field” motif suggests that, given similar effort and choices, all should have the same
chance of success (or failure) regardless of color, gender and socioeconomic background,
namely a multitude of circumstances that are not purely parental. Thus if progress in EO
is to be assessed in response to all of these various policy changes, a technique is required
which will relate an outcome measure (or a collection of outcome measures) to a variety
of circumstance measures simultaneously, so that the distance of the existing joint density
from that of one reflecting independence from circumstances can be assessed.
4 Results
4.1 Descriptive Statistics
Census data from 1960-2000 drawn from the Integrated Public Use Microdata Series
(IPUMS) was employed, utilizing the educational status of the child as an outcome vari-
able, and the parental marital status, educational attainment and income variables, and
the child’s gender and race as circumstance variables. To accommodate the gradual na-
ture of policy adoption, measures of EO for 1960 (prior to enactment of the policies),
and 2000 (post adoption of policies) will be compared, and where they are of interest,
some intermediate stage results will be reported. Using the parental marital status14, the
observations were separated into three family structures; intact, divorced or separated,
and widowed parent families. The grade attainment indicator is: 1 if preschool or had no
education, 2 if grade 1-4, 3 if grade 5-8, 4 if grade 9, 5 if grade 10, 6 if grade 11, 7 if grade
12, 8 if 1-3 years of college and 9 if more than 4 years of college. For intact families the
maximum of the parental educational attainments was employed, while family income is
in constant dollars and is family size deflated according to the square root rule (Brady and
Barber 1948). The analysis below focuses on children of age 18 because for most states,
compulsory education ceased to be binding then, consequently it should be stressed that
the notions of EO reported here, apply to the achievements of 18 year old’s, and not the
14The respective coded parental marital status responses are as follows: Married, spouse present is
1; Married, spouse absent is 2; Separated is 3; Divorced is 4; Widowed is 5; Never married/single is
6. This paper does not examine children born outside of wedlock, nor marriages where one parent is
“missing”(responses 2 and 6).
11
Table 1: Summary Statistics by Gender & Family Structure
Year Boys Girls
Panel A
Intact Parents Intact Parents
Child’s
Education
Parent’s
Education
Parent’s
Income
Child’s
Education
Parent’s
Education
Parent’s
Income
1960 Mean 5.7447 5.5721 12.7190 6.1364 5.6210 13.0830
s.d. (1.3724) (2.0841) (9.3442) (1.1734) (2.0530) (9.4965)
N 5819 5819 5819 4766 4766 4766
2000 Mean 6.3644 8.0600 27.7640 6.4862 8.0280 27.4920
s.d. (0.9263) (0.9496) (20.6680) (0.9784) (1.0249) (20.6220)
N 7312 7312 7312 6052 6052 6052
Panel B
Divorced & Separated Parent Divorced & Separated Parent
Child’s
Education
Parent’s
Education
Parent’s
Income
Child’s
Education
Parent’s
Education
Parent’s
Income
1960 Mean 5.3201 4.7385 5.8884 5.7585 4.9488 6.6655
s.d. (1.5333) (1.9607) (5.9571) (1.3100) (2.0348) (5.2244)
N 334 334 334 261 261 261
2000 Mean 6.3029 7.5558 14.4500 6.4158 7.5350 14.1170
s.d. (1.0232) (1.2095) (14.0700) (0.9888) (1.2053) (13.9870)
N 1780 1780 1780 1448 1448 1448
Panel C
Widowed Parent Widowed Parent
Child’s
Education
Parent’s
Education
Parent’s
Income
Child’s
Education
Parent’s
Education
Parent’s
Income
1960 Mean 5.2145 4.3028 5.7357 5.9312 4.7319 6.0581
s.d. (1.5934) (2.1370) (6.8933) (1.3892) (2.1304) (6.1696)
N 423 423 423 375 375 375
2000 Mean 6.2419 7.4312 12.7840 6.4548 7.2876 12.5830
s.d. (1.1013) (1.5429) (12.6070) (1.1309) (1.4812) (9.9644)
N 266 266 266 197 197 197
Note: The means and standard deviations are all weighted statistics.
12
Table 1 (Continued): Summary Statistics by Race & Family Structure
Year White Child Black Child
Panel D
Intact Parents Intact Parents
Child’s
Education
Parent’s
Education
Parent’s
Income
Child’s
Education
Parent’s
Education
Parent’s
Income
1960 Mean 5.9830 5.7061 13.4960 5.1926 4.2787 5.6810
s.d. (1.2617) (2.0436) (9.4663) (1.5207) (1.9232) (4.5663)
N 9755 9755 9755 830 830 830
2000 Mean 6.4163 8.0745 28.3200 6.4535 7.7203 20.0030
s.d. (0.9472) (0.9770) (20.8910) (1.0054) (1.0089) (15.7750)
N 12402 12402 12402 962 962 962
Panel E
Single Parent Single Parent
Child’s
Education
Parent’s
Education
Parent’s
Income
Child’s
Education
Parent’s
Education
Parent’s
Income
1960 Mean 5.7082 4.9042 6.8455 4.9432 3.7566 3.2660
s.d. (1.4271) (2.0941) (6.6097) (1.6121) (1.7998) (3.1938)
N 1077 1077 1077 316 316 316
2000 Mean 6.3514 7.5968 15.5590 6.3494 7.2934 9.4821
s.d. (0.9979) (1.2246) (14.6400) (1.1021) (1.3110) (8.9945)
N 2895 2895 2895 796 796 796
Note: Single parent families include divorced & separated, and widowed parent families.
The means and standard deviations are all weighted statistics.
career achievements of adults in general15 ,16 .
As background, consider the summary statistics of the data for the years 1960 and 2000
for 18 year old children by gender presented in table 1, panels A to C, while panels D and
E presents similar information by race. For intact families the boy−girl differences in 1960
15Nonetheless, the results for children of age 16 and 17 are similar, and are available from the authors
upon request.
16However this is quite different from more traditional approaches to generational relationships where
the outcomes of offspring are related to parental outcomes at similar points in the life cycle. If rates and
periods of development differ by gender or race, some of the offspring differences observed here could be
quite different if observed at a later stage of the life cycle.
13
were significant with girls significantly outperforming boys (a “t” statistic of −15.8201,
Pr(T < t) = 0). The gap was still significant in 2000 (t=−6.4057, Pr(T < t) = 0),
albeit had been substantially reduced (the “t” for the difference-in-difference is −8.6458,
Pr(T < t) = 0). None of the boy−girl parental differences are particularly significant as
is to be expected.
With respect to boy−girl differences, the results for children of divorced or separated,
and widowed parents were qualitatively the same as those for children of intact families,
with girls significantly out-performing boys in both 1960 and 2000, though the gap had
substantially narrowed over the period (indeed for boys and girls of widowed parents,
there was no significant difference in 2000). Again generally there are no substantive
differences in the parental characteristics of parents of boys and girls. The increase in (a
multiple of over 5) the numbers of children in divorced or separated households between
1960 and 2000 is noteworthy (possibly a result of the increased ease with which divorce
was obtained over the period). Similarly the numbers of children in widowed parental
circumstances almost halved over the period, undoubtedly the result of improved health
circumstances of parents over the 40 year period.
On the other hand from panels D and E of table 1, for both intact and single parent
family structures, white children’s dominance in educational attainment has dwindled,
and is no longer statistically significant by 2000. This is despite significant parental
educational attainment and income differences, providing some initial evidence of the
fruitfulness of the educational and civil rights policies over the five decades that transpired.
One interesting feature that is ubiquitous across family types is that in 1960, children of
age 18 were on average more educated than their parents, whereas in 2000 they were on
average less educated than their parents.
As for differences between family types, table 2 indicates that children from intact
families clearly do better than children from single parent families, whether the head of
household is divorced/separated or bereaved, and this is the case for both genders in 1960.
However, this gap had narrowed by 2000, particularly so amongst children of bereaved
families. Using these children of widowed parents as a comparison group, it is clear that
the education outcomes of children of divorced/separated parents had improved by 2000,
with no significant differences between them17.
17When considering pure EO measures, and mobility by family structure, a first concern is whether
differing familial household structures have different transition structures. By comparing the overlap
14
Table 2: Difference in Means Tests. (Standard Normal Tests & Lower Tail Probabilities)
Boys Girls Boys Girls Boys Girls
Intact −
Divorced
Intact −
Divorced
Intact −
Widowed
Intact −
Widowed
Widowed −
Divorced
Widowed −
Divorced
1960 4.9482 4.5606 6.6639 2.7847 -0.9246 1.5954
1.0000 1.0000 1.0000 0.9973 0.1776 0.9447
2000 2.0231 2.1653 1.5475 0.3490 -0.7350 0.4174
0.9785 0.9848 0.9391 0.6364 0.2312 0.6618
Diff−Diff 3.9886 3.4546 3.6325 1.4944 -0.3159 0.9349
1.0000 0.9997 0.9999 0.9325 0.3760 0.8251
Note: Divorced refers to both Divorced & Separated Parent
4.2 Progress Towards Equality of Opportunity
Table 3 reports the overall Social Justice Indices for two definitions of Social Justice,
accommodating all of the race, gender, family type, parental type circumstances that
confront an 18 year old. The first definition, Pure Equality of Opportunity (EO), char-
acterizes social justice as independence of effort from circumstance, where all agents are
weighted the same. The second definition, Qualified Equality of Opportunity (QEO),
weighs more heavily independence of circumstance for those poorly endowed in circum-
stance relative to those richly endowed in circumstance. The weighting is based on only
those circumstances that are ordered, namely parental education and incomes. Tech-
nically, using the joint cumulative density function of parental education and income to
provide an ordinal ranking of all parental education and income pairings, and consequently
providing a measure of the advantage given by such a circumstance pair. Both EO and
QEO indices represent proximity to the ideal state in terms of the circumstances of race,
gender, household type, household income and parental education. Since the measures
reported here involve parental educational attainment and income, the densities estimated
are a mixture of both continuous and discrete variables. Note that each of the discrete
of the joint densities of intact versus single, and widowed versus divorced/separated parent families,
the possibility of common transit structures was examined. Here, an overlap of one implies a common
transit structure, and a value of less than one implies otherwise. The hypothesis of common transitional
structures were rejected in every case, the results are available from the authors on request.
15
dimension of the densities estimated utilizes the cross-validated kernel smoothing method
prescribed by Li and Racine (2003) and Ouyang et al. (2006).
As may be seen from Table 3, there has been a statistically significant progression over
the 1960-2000 period in both EO and QEO. Note that most of the improvements in Social
and Economic Justice took place over the first 20 years. In the following, the changes in
Social and Economic Justice with respect to the specific circumstances mentioned above
will be examined more closely. It is of interest to evaluate these progressions in EO
by circumstance type (Lefranc et al. (2008) do so by seeking an absence of dominance
relations of outcomes across types, but obviously this does not yield a measure of the
degree of change). Here changes in the levels of EO are compared across family income
types which speaks to the partial effects of other circumstances within the context of a
parental income circumstance types (which may be construed as a form of decomposition).
The results demonstrate that circumstances such as race and gender are more deleterious
with regards to EO for those with lower parental income circumstances.
The raw EO indices by Parental Income and Gender circumstance for the years 1960,
1980 and 2000 are reported in Table 4. As may be seen, EO is ubiquitously and sig-
nificantly lower for those in the lowest income circumstance quartile than for those in
the highest income quartile for all three observation years. However the gaps are much
narrower in 2000 than they were in the 1960’s, reflecting the much bigger gains that were
made in EO outcomes for the poor in income circumstance relative to the rich in income
circumstance18. In the two lower quartiles, girls typically enjoyed significantly greater EO
than boys, with the exception of the poorest group in 2000.
The raw EO indices by Parental Income and Race Circumstance for the years 1960,
1980 and 2000 are reported in Table 5. As may be seen EO is likewise ubiquitously and
significantly lower for those in the lowest income circumstance quartile than for those
in the highest income quartile for all three observation years. Again the gaps are much
narrower in 2000 as compared to 1960. Generally differences between blacks and whites
had been greatly reduced by the year 2000.
Table 6 presents the results for the measure of the QEO imperative. Since in this case
the just outcome objective is a variable weighted sum of the EO outcome and the status
quo outcome, with weight on the status quo increasing with family income, it will naturally
overlap with the status quo better than would the pure EO model, so higher measures
18These results are similar to those for Canada (Anderson et al. 2014).
16
of Justice are to be expected than for the pure EO case. Specifically, the hypothesized
QEO is that represented in equation (5) using the three variables of parental income and
education, and the educational attainment of the child, with the qualification at three
differing cutoffs. Indeed for the highest quartile, the overlaps are very close so they were
not reported, however the lower quartiles are of interest, and record improvements in
social justice for all quartiles over all categories during the period. Worthy of note is the
fact that gains for males were made steadily throughout the period, whereas the gains for
females were primarily made by the 1980s.
17
Table 3: Overall Measures of Different Notions of Social Justice
Six Variable OV*
Pure EO QEO
1960 OV Unbiased 0.8229 0.8416
S.E. (0.0007) (0.0007)
# Obs. 11978 11978
1980 OV Unbiased 0.8874 0.9015
S.E. (0.0005) (0.0005)
# Obs. 22386 22386
2000 OV Unbiased 0.8802 0.9009
S.E. (0.0006) (0.0006)
# Obs. 17055 17055
“t” Statistic of
Improvement in EO
1980−1960 72.60 67.48
[1.00] [1.00]
2000−1980 -8.92 -0.76
[0.00] [0.22]
2000−1960 60.94 63.06
[1.00] [1.00]
* The six variables are child education (effort variable),
and five circumstance variables, gender of child, race of
child, family structure, parental income and parental
education.
Note: 1. Standard Errors in Parenthesis, and Pr(T < t)
are in brackets. 2. Ouyang et al. (2006) smoothing of
discrete variables was not employed in these calculations
since over-smoothing led to a lack of discrimination with
large numbers of observations(see appendix A.1).
18
Table 4: Gender Equality of Opportunity Indices by Parental Income Quartile & Year
Panel A: Equality of Opportunity by Gender & Year
Income Males Females
Quartile 1960 1980 2000 1960 1980 2000
1st Quartile OV 0.8388 0.8684 0.9285 0.8717 0.9014 0.8996
Standard Error (0.0005) (0.0004) (0.0006) (0.0006) (0.0005) (0.0006)
# of Observations 1714 3126 2439 1298 2477 2050
2nd Quartile OV 0.8915 0.9316 0.9486 0.9094 0.9637 0.9531
Standard Error (0.0005) (0.0006) (0.0006) (0.0006) (0.0006) (0.0007)
# of Observations 1668 2998 2473 1308 2592 2012
3rd Quartile OV 0.9176 0.9512 0.9474 0.9423 0.9582 0.9482
Standard Error (0.0006) (0.0006) (0.0007) (0.0007) (0.0007) (0.0007)
# of Observations 1586 2989 2267 1409 2607 1891
4th Quartile OV 0.9462 0.9471 0.9355 0.9272 0.9611 0.9157
Standard Error (0.0006) (0.0006) (0.0008) (0.0008) (0.0007) (0.0009)
# of Observations 1608 3022 2179 1387 2575 1744
Panel B: Difference Between Quartiles (tStatistics & [Pr(T < t)])
1st - 2nd Quartile -73.55 -91.56 -23.20 -41.87 -77.69 -57.58
[0.00] [0.00] [0.00] [0.00] [0.00] [0.00]
1st - 3rd Quartile -99.45 -116.67 -21.04 -75.76 -69.99 -51.44
[0.00] [0.00] [0.00] [0.00] [0.00] [0.00]
1st - 4th Quartile -131.10 -110.14 -6.95 -56.61 -72.80 -14.87
[0.00] [0.00] [0.00] [0.00] [0.00] [0.00]
Note: Standard errors are in parenthesis.
19
Table 4 (Continued): Gender Equality of Opportunity Indices by Parental Income Quartile & Year
Panel C: Difference Between Years
(tStatistics & [Pr(T < t)])
Panel D: Difference Between Genders
(tStatistics & [Pr(T < t)])
Males Females 1960 1980 2000
1960-1980 1960-2000 1980-2000 1960-1980 1960-2000 1980-2000 Males-Females Males-Females Males-Females
1st Quartile -45.63 -114.20 -83.93 -37.66 -31.57 2.33 -40.79 -52.77 33.52
[0.00] [0.00] [0.00] [0.00] [0.00] [0.99] [0.00] [0.00] [1.00]
2nd Quartile -53.18 -71.26 -20.13 -59.61 -46.21 11.32 -21.84 -37.66 -4.90
[0.00] [0.00] [0.00] [0.00] [0.00] [1.00] [0.00] [0.00] [0.00]
3rd Quartile -39.83 -32.99 4.26 -16.73 -5.93 10.41 -26.88 -8.03 -0.84
[0.00] [0.00] [1.00] [0.00] [0.00] [1.00] [0.00] [0.00] [0.20]
4th Quartile -1.03 10.39 11.58 -33.75 9.86 41.05 19.22 -15.71 16.52
[0.15] [1.00] [1.00] [0.00] [1.00] [1.00] [1.00] [0.00] [1.00]
20
Table 5: Racial Equality of Opportunity Indices by Parental Income Quartile & Year
Panel A: Equality of Opportunity by Race & Year
Income Whites Blacks
Quartile 1960 1980 2000 1960 1980 2000
1st Quartile OV 0.8376 0.8764 0.9147 0.8867 0.9014 0.9270
Standard Error (0.0004) (0.0004) (0.0005) (0.0008) (0.0005) (0.0009)
# of Observations 2240 4006 3580 772 1597 909
2nd Quartile OV 0.8945 0.9433 0.9461 0.8827 0.8958 0.9386
Standard Error (0.0004) (0.0005) (0.0005) (0.0012) (0.0010) (0.0015)
# of Observations 2722 4990 4077 254 600 408
3rd Quartile OV 0.9341 0.9591 0.9507 0.9537 0.8979 0.9349
Standard Error (0.0005) (0.0005) (0.0005) (0.0016) (0.0015) (0.0018)
# of Observations 2908 5249 3890 87 347 268
4th Quartile OV 0.9408 0.9564 0.9208 0.9102 0.9029 0.9557
Standard Error (0.0005) (0.0004) (0.0006) (0.0030) (0.0017) (0.0022)
# of Observations 2962 5375 3750 33 222 173
Panel B: Difference Between Quartiles (tStatistics & [Pr(T < t)])
1st - 2nd Quartile -91.79 -111.34 -44.24 2.72 4.99 -6.83
[0.00] [0.00] [0.00] [1.00] [1.00] [0.00]
1st - 3rd Quartile -147.88 -137.31 -50.15 -38.12 2.22 -4.03
[0.00] [0.00] [0.00] [0.00] [0.99] [0.00]
1st - 4th Quartile -154.65 -135.81 -7.83 -7.54 -0.81 -12.17
[0.00] [0.00] [0.00] [0.00] [0.21] [0.00]
Note: Standard errors are in parenthesis.
21
Table 5 (Continued): Racial Equality of Opportunity Indices by Parental Income Quartile & Year
Panel C: Difference Between Years
(tStatistics & [Pr(T < t)])
Panel D: Difference Between Races
(tStatistics & [Pr(T < t)])
Whites Blacks 1960 1980 2000
1960-1980 1960-2000 1980-2000 1960-1980 1960-2000 1980-2000 Whites-Blacks Whites-Blacks Whites-Blacks
1st Quartile -65.46 -115.21 -60.94 -15.62 -34.14 -25.39 -53.65 -39.33 -12.23
[0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [0.00]
2nd Quartile -77.55 -77.65 -4.14 -8.34 -29.44 -24.10 9.23 43.04 4.83
[0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [1.00] [1.00] [1.00]
3rd Quartile -37.74 -23.54 12.20 25.85 7.93 -16.02 -11.94 39.42 8.53
[0.00] [0.00] [1.00] [1.00] [1.00] [0.00] [0.00] [1.00] [1.00]
4th Quartile -23.50 25.61 47.44 2.12 -12.17 -18.88 9.97 29.90 -15.32
[0.00] [1.00] [1.00] [0.98] [0.00] [0.00] [1.00] [1.00] [0.00]
22
Table 6: Qualified Equality of Opportunity Hypothesis by Year, Gender & Race
Overall Males Females
1960 1980 2000 1960 1980 2000 1960 1980 2000
25th Percentile
& Below
Independent
OV 0.9625 0.9704 0.9797 0.9587 0.9655 0.9818 0.9672 0.9756 0.9742
s.e. (0.0002) (0.0002) (0.0002) (0.0003) (0.0003) (0.0003) (0.0003) (0.0003) (0.0004)
# of Obs. 11978 22386 17055 6576 12135 9358 5402 10251 7697
50th Percentile
& Below
Independent
OV 0.9305 0.9496 0.9657 0.9216 0.9402 0.9670 0.9403 0.9608 0.9604
s.e. (0.0002) (0.0002) (0.0002) (0.0003) (0.0003) (0.0003) (0.0003) (0.0003) (0.0004)
# of Obs. 11978 22386 17055 6576 12135 9358 5402 10251 7697
75th Percentile
& Below
Independent
OV 0.9010 0.9346 0.9797 0.8890 0.9225 0.9492 0.9155 0.9494 0.9474
s.e. (0.0002) (0.0002) (0.0002) (0.0002) (0.0003) (0.0003) (0.0003) (0.0003) (0.0003)
# of Obs. 11978 22386 17055 6576 12135 9358 5402 10251 7697
Whites Blacks
1960 1980 2000 1960 1980 2000
25th Percentile
& Below
Independent
OV 0.9603 0.9716 0.9799 0.9653 0.9666 0.9722
s.e. (0.0002) (0.0002) (0.0003) (0.0006) (0.0004) (0.0007)
# of Obs. 10832 19620 15297 1146 2766 1758
50th Percentile
& Below
Independent
OV 0.9331 0.9561 0.9648 0.9535 0.9437 0.9555
s.e. (0.0002) (0.0002) (0.0003) (0.0006) (0.0004) (0.0007)
# of Obs. 10832 19620 15297 1146 2766 1758
75th Percentile
& Below
Independent
OV 0.9076 0.9464 0.9496 0.9311 0.9255 0.9530
s.e. (0.0002) (0.0002) (0.0003) (0.0006) (0.0004) (0.0007)
# of Obs. 10832 19620 15297 1146 2766 1758
Note: Standard errors are in parenthesis, and Pr(T < t) are in brackets.
23
Table 7 reports the degree of overlap between the empirical density against the hypo-
thetical EO density conditional on different family structures. It is clear from panel A of
Table 7 that in the 1960s, neither boys or girls in any family structure exhibit a great deal
of EO, however with the exception of girls from divorced/separated families, there is sig-
nificant evidence of increased EO between 1960 and 2000. Panel B reports the significance
tests of within year differences between boys and girls, while panel C reports the tests of
cross year differences for each gender, derived from the indices of panel A. With regard
to the gender comparisons, in 1960s girls enjoyed significantly more EO than boys in all
family structures. However, by 2000 this pattern is no longer true with the exception of
children from widowed parent families. In fact, for children in divorced/separated parent
families, boys enjoy significantly more EO than girls. The changes within each gender
across the four decades as reflected in panel B are that although all children experienced
significant increases in EO, the improvements are larger for boys than they are for girls.
The extension of the examination to cross family structure comparisons by gender and
year in panel C, finds that with the exception of the divorced/separated versus intact
comparisons for boys, in 1960 and 2000, children of single parent families exhibited sig-
nificantly less EO than their intact counterparts. On the other hand, with the exception
of boys of widowed parents, the gaps had significantly diminished by 2000 reflecting the
impact of family law legislation that took place over the preceding period.
The growing similarity in EO says little explicitly about differences in the distribu-
tions of academic attainments across race or gender. Table 8 reports the corresponding
stochastic dominance tests19, utilizing similar ideas in the mixture method of the density
comparisons above, estimating the continuous variable using kernel density estimation
methods (Linton et al. 2005). Notice that despite the improvements among boys in terms
of EO, the educational outcomes of 18 year old girls continue to first order stochastically
dominate boys regardless of their familial background. In other words, all things equal
girls have a higher probability of performing better in school relative to boys. The com-
parisons pertaining to race on the other hand affirms the improvements that the civil
rights policies have afforded African Americans, since whereas in 1960, whites first order
stochastically dominated blacks, there is no longer any dominance relationship in 2000
for all family structures.
19This test corresponds to those presented by Lefranc et al. (2008, 2009), though in this case the
respective circumstance classes are race and gender.
24
Table 7: Equal Opportunity Measures by Family Structure & Gender
Panel A: Overlap Measure & Difference Between Family Structures by Year & Gender
1960 1980 2000
Intact Divorced +
Separated
Widowed Intact Divorced +
Separated
Widowed Intact Divorced +
Separated
Widowed
Male OV Unbiased 0.8505 0.8870 0.7145 0.9152 0.8787 0.8631 0.9156 0.9273 0.8176
Standard Error (0.0010) (0.0043) (0.0038) (0.0008) (0.0020) (0.0032) (0.0009) (0.0018) (0.0048)
# Observations 5819 334 423 9943 1581 611 7312 1780 266
Female OV Unbiased 0.8816 0.8525 0.8435 0.9385 0.9043 0.9161 0.9180 0.9028 0.9149
Standard Error (0.0011) (0.0049) (0.0041) (0.0009) (0.0022) (0.0033) (0.0010) (0.0021) (0.0054)
# Observations 4766 261 375 8392 1309 550 6052 1448 197
Panel B: Difference Between Genders by Year & Family Structure (tStatistics & [Pr(T < t)])
Males −Females -20.25 5.31 -23.18 -20.05 -8.74 -11.51 -1.77 8.84 -13.43
[0.00] [1.00] [0.00] [0.00] [0.00] [0.00] [0.04] [1.00] [0.00]
Note: Standard errors are in parenthesis.
25
Table 7 (Continued): Equal Opportunity Measures by Family Structure & Gender
Panel C: Difference Between Family Structures by Gender & Year (tStatistics & [Pr(T < t)])
Intact −
(Divorced +
Separated)
Intact −
Widowed
(Divorced +
Separated)
−Widowed
Intact −
(Divorced +
Separated)
Intact −
Widowed
(Divorced +
Separated)
−Widowed
Intact −
(Divorced +
Separated)
Intact −
Widowed
(Divorced +
Separated)
−Widowed
Male -8.26 34.43 30.03 17.20 15.95 4.18 -5.66 20.16 21.44
[0.00] [1.00] [1.00] [1.00] [1.00] [1.00] [0.00] [1.00] [1.00]
Female 5.82 9.05 1.43 14.68 6.49 -2.97 6.60 0.57 -2.07
[1.00] [1.00] [0.92] [1.00] [1.00] [0.00] [1.00] [0.72] [0.02]
Panel D: Difference Between Years by Gender & Family Structure (tStatistics & [Pr(T < t)])
1960−1980 1960−2000 1980−2000
Male -49.97 1.74 -29.96 -46.85 -8.62 -16.90 -0.30 -17.97 7.94
[0.00] [0.96] [0.00] [0.00] [0.00] [0.00] [0.38] [0.00] [1.00]
Female -40.02 -9.73 -13.83 -23.78 -9.54 -10.52 15.30 0.49 0.20
[0.00] [0.00] [0.00] [0.00] [0.00] [0.00] [1.00] [0.69] [0.58]
26
Table 8: Stochastic Dominance Test by Family Type & Race
1960 2000
Panel A: Stochastic Dominance Test by Gender
Family Structure\Hypothesis Boys iGirls Girls iBoys Boys iGirls Girls iBoys
Intact 6.3076 -0.0676 3.3651 0.0896
[1.0000] [0.0000] [1.0000] [0.0159]
Result Girls 1Boys Girls 1Boys
Single
Parent
3.4246 -0.0106 1.7016 0.0318
[1.0000] [0.0000] [0.9969] [0.0020]
Result Girls 1Boys Girls 1Boys
Panel B: Stochastic Dominance Test by Race
Family Structure\Hypothesis White iBlack Black iWhite White iBlack Black iWhite
Intact 0.0000 21.8600 2.1075 1.8672
[0.0000] [1.0000] [<0.90] [<0.90]
Result White 1Black No Dominance at 3rd Order
Single
Parent
0.0000 11.9580 0.4736 3.0115
[0.0000] [1.0000] [<0.90] [<0.90]
Result White 1Black No Dominance
Note: idenotes ith order stochastic dominance. The test performed tests up to third order of
dominance, and if no dominance relationship is revealed, it is concluded that there is “No
Dominance”. The upper tail probabilities of the test statistic are in brackets for first order dominance
comparisons. Comparisons at the second and third order are based on comparisons against upper tail
critical values at the 0.9, 0.95 and 0.99 level of significance, as suggested by Linton et al. (2005).
27
5 Conclusions
A general readily applicable method for quantifying the progress of Social Justice has
been presented, and applied to the Equality of Opportunity notion of Social Justice. The
method relies upon measuring a sense of the distance of the joint probability distribution
of agent characteristics from that which would be desired under a particular notion of
Social Justice. The method does not run into the problems confronted by regression and
transition matrix techniques which are commonly employed, and it is sufficiently flexible
to admit a variety of agent characteristics that may be either discretely or continuously
measured. The technique has been used to measure the progress of Equality of Opportu-
nity, and a similar notion of Qualified Equal Opportunity for 18 year old children in the
United States over the last 4 decades of the last century.
Those decades saw considerable efforts through various family, education, and civil
rights law and policy to equalize opportunity in the U.S., especially with regard to el-
evating outcomes of those who were disadvantaged in their circumstance whether it be
gender, race or family background based. While these efforts have been much lauded,
their success has been contested by some. Here a new measure that provides a metric for
the level of equality of opportunity has been provided, which has well defined statistical
properties facilitating inference and which can handle collections of circumstance, and
outcome variables that can be discrete or continuous. Using the measure to relate 18
year old school attainments to their circumstances in the form of their gender, race, fam-
ily background (intact, or divorced/separated, or widowed parents) and the educational
status and income of the family, it is possible to conclude that the efforts have met with
some qualified success. With the exception of one group, daughters in the lowest parental
income quartile (especially those daughters of a widowed parent), all groups in all family
types, and of both genders have experienced significant improvements in Equality of Op-
portunity over the period. Some have advanced more than others, though it should be
said that the genders started from different positions, with girls generally experiencing
greater equality of opportunity than boys in the 1960s era.
28
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34
A Appendix
A.1 Estimation of Mixed Overlap Measure
In instances such as the current one where at least one of the variables considered is
continuous, the measure proposed by Anderson et al. (2010) is subject to biases. However
Anderson et al. (2012) proposed a similar measure using Kernel estimation techniques,
and showed that the kernel estimator of OV := θ=Rmin{fa(x), fb(x)}dxis distributed
as,
√n(b
θ−θ)−αn−→ N(0, v) (A-1)
where
v=p0σ2
0+pa(1 −pa) + pb(1 −pb)
p0= Pr(X ∈ Cfa,fb); Cfa,fb={x∈
R
n:fa(x) = fb(x)>0}
pa= Pr(X ∈ Cfa); Cfa={x∈
R
n:fa(x)< fb(x)}
pb= Pr(X ∈ Cfb); Cfb={x∈
R
n:fa(x)> fb(x)}
and where αnand σ2
0are bias correction factors. Since this estimator does not depend upon
arbitrarily chosen points in the support, it will provide for consistent testing of hypotheses.
This means that the two techniques can be mixed without any serious implications for the
bias and asymptotic distributions of the results. In addition, in the current application,
we have utlized the cross-validation kernel smoothing method for the discrete variable as
well, as suggested by Ouyang et al. (2006). To illustrate the technique used consider the
simple bivariate case, leting x∈Xbe the discete variable, and y∈Ybe the continuous
variable. Then the overlap measure that compares two densities f(.) and g(.) is,
OV =X
x∈XZy∈Y
min{f(x, y), g(x, y)}dy(A-2)
Notice that to estimate this mixed overlap index, we need to sum over the estimated
density at each discrete realization, besides integrating over the support of the continuous
variable. Keep in mind that g(.) may refer to either the hypothesized density in question
or the density of another population under comparison. We will describe the comparison
of two mixed distributions here, and the examinations of more general equal oportunity
hypotheses can be easily adapted from the procedure given below.
35
1. First calculate the overlap index,
d
OV =
J
X
j=1 Zy∈Y
min{fn(xj, y), gn(xj, y)}dy (A-3)
where jindexes the Junique discrete realizations. The estimators for the densities
are,
fn(xj, y) = 1
nbd
n
X
i=1
Lλ(Xf
i, xj)K y−Yf
i
b!(A-4)
gn(xj, y) = 1
nbd
n
X
i=1
Lλ(Xg
i, xj)Ky−Yg
i
b(A-5)
where ddenotes the number of continuous variable dimensions (d= 1 in the current
case) in the observed variables {Xf
i}n
i=1,{Xg
i}n
i=1,{Yf
i}n
i=1 and {Yg
i}n
i=1. Further,
for fn(.) and gn(.),
Lλ(Xi, xj) =
J
Y
j=1 λ
J−1I(Xi6=xj)
(1 −λ)I(Xi=xj)(A-6)
where λis estimated via cross-validation method as prescribed in Ouyang et al.
(2006). While the kernel function, K(.), used is the Normal kernel as suggested
in Anderson et al. (2012), and the bandwidth used in estimating the overlap index
is the Silverman’s rule of thumb (bs= 1.84sn−1/5, where sis the sample standard
deviation).
2. However, the above measure is biased, due to the intersection between the density
functions, and consequently needs to be accounted for. To do so, we have to first
find the estimated contact set (where the two densities crosses each other) and its
complements,
b
Cf,g ={y∈
R
d:|fn(xj, y)−gn(xj, y)| ≤ cn, fn(xj, y)>0, gn(xj, y)>0}
b
Cf={y∈
R
d:fn(xj, y)−gn(xj, y)<−cn, fn(xj, y)>0, gn(xj, y)>0}
b
Cg={y∈
R
d:fn(xj, y)−gn(xj, y)> cn, fn(xj, y)>0, gn(xj, y)>0}
where the first equation above describes the contact set, while the others are its
complement, and cnis the tuning parameter which was set to b3/2
sif bs<1, and b2/3
s
otherwise. Further, note that this is to be performed for each j={1, . . . , J}.
36
3. The bias corrected overlap measure, and its variance are as follows,
d
OVbc =d
OV −bann−0.5(A-7)
bv=bp0σ2
0+bσ2
1(A-8)
where the calculations required to obtain these values, in sequence, are
(a) ||K||2
2and ||K||2are
||K||2
2=Z
R
dK2(u)du
⇒ ||K||2=sZ
R
dK2(u)du
Note that for the univariate uniform kernel function,
||K||2
2=Z∞
−∞
1(|u| ≤ 0.5)du
=Z0.5
−0.5
du = 1
⇒ ||K||2= 1
(b) banis the bias correction factor,
ban=Emin{Z1, Z2}||K||2
2bd
2
J
X
j=1 ZCf,g
f
1
2
n(xj, y)dy +ZCf,g
g
1
2
n(xj, y)dy!
where Emin{Z1, Z2}=−0.56, and Z1and Z2are independent standard normal
random variables if the sample sizes are the same for both densities under
consideration. But when they are different, letting the sample size for fn(.)
be n, and that for gn(.) be m, such that the ratio of the sample sizes are
m/n→τ∈(0,∞), Emin{Z1, Z2}=−1
√π=−0.5642 needs to be augmented
with Emin{Z1, Z2/τ}=−√1+1/τ
2q2
π. That is τis just the ratio of sample
sizes.
(c) The kernel constant, σ2
0, is defined as follows,
σ2
0=||K||2
2ZT0
cov min{Z1, Z2},min (ρ(t)Z1+p1−ρ(t)2Z3,
ρ(t)Z2+p1−ρ(t)2Z4)!dt = 0.6135
37
where T0={t∈
R
d:||t|| ≤ 1}and
ρ(t) = R
R
dK(u)K(u+t)du
||K||2
2
Note that the value for the kernel constant above is for case when the sample
sizes considered are the same. When this is not the case, the kernel constant
for that case can be calculated as,
σ2
0(τ) = 1 + τ−1
2σ2
0
(d) bp0in the calculation of the variance can is estimated by
bp0=1
2
J
X
j=1 Zb
Cf,g
fn(xj, y)dy +Zb
Cf,g
gn(xj, y)dy!
In other words, the estimator is the “average” of each of the density estimates
over the contact set.
(e) Finally, bσ2
1is estimated by
bσ2
1=bpf(1 −bpf) + bpg(1 −bpg)
where
bpf=
J
X
j=1 Zb
Cf
fn(xj, y)dy
bpg=
J
X
j=1 Zb
Cg
gn(xj, y)dy
In other words, bpfand bpgare simply the estimates of the respective density
over the complement of the contact set.
If the samples are not equal,
bσ2
1(τ) = bpf(1 −bpf)+[bpg(1 −bpg)/τ]
For a complete discussion of the asymptotic results, and intuition regarding the mea-
sure and the proofs, see Anderson et al. (2012).
When testing for mobility, the overlap index generated by this mixture of discrete and
continuous variables is,
d
OV =
J
X
j=1 Zy∈Y
min{fn(xj, y), g1
n(xj)g2
n(y)}dy (A-9)
where jindexes the Junique discrete realization as before.
38
