Reflection paper

Multidimensional Racial Inequality in the United States

Nicholas Rohde • Ross Guest

Accepted: 25 September 2012 / Published online: 5 October 2012 � Springer Science+Business Media Dordrecht 2012

Abstract This paper measures racial inequalities in the US using a multidimensional ‘wellbeing’ approach that simultaneously considers the distributions of income, health and

education. The primary objective is to examine trends in US wellbeing inequality with an

emphasis on changes in racial composition. Data is taken from 1990 to 2007 and we

observe increases in income inequality, a decline in education inequality and unchanged

health inequality over the period. Taken together, these results show a slight increase in the

dispersion in multidimensional wellbeing. Stratifying by racial groups shows that this

increase is due to widening intra-racial inequalities while inter-racial differences remained

unchanged. The method is also used to evaluate wellbeing across groups and we estimate

black wellbeing to average around 76 % of whites, while persons from other races average

approximately 93 %. Some other changes in composition occur through time and the

results are shown to be robust to a number of changes in parametric weightings.

Keywords Decomposition � Inequality � Race � Wellbeing

1 Introduction

Studies of economic inequality usually focus on the distribution of income, wealth or some

other monetary variable that is taken as a proxy for overall wellbeing. It is unclear however

that the use of such a narrowly defined variable can adequately capture the ‘true’ inequality

between individuals or households as there are many other factors that may influence

wellbeing such as health, job satisfaction, leisure time or education. If wellbeing is defined

more broadly, there is a possibility that results will emerge that conflict with established

notions about the levels, trends and composition of inequality based purely upon material

factors. For example, a highly unequal distribution of income may become acceptable if

low income earners are found to be compensated with greater quantities of some other

desirable characteristic, such as better health or more leisure time. Alternatively a low

N. Rohde (&) � R. Guest Griffith Business School, Gold Coast, QLD, Australia e-mail: n.rohde@griffith.edu.au

123

Soc Indic Res (2013) 114:591–605 DOI 10.1007/s11205-012-0163-0

degree of income inequality might mask a high degree of wellbeing inequality if other

important attributes are distributed more unequally. For this reason it is useful to support

standard univariate inequality measures with multivariate indices that account for the

distribution of several variables simultaneously.

There are a few different econometric methods for studying inequality over multiple

variables. These include the one-at-a-time approach employed and critiqued by Justino

(2005), generalizations of various dominance criteria into a multidimensional framework

(Atkinson and Bourguignon 1982, 1987; Kolm 1977; Trannoy 2005; Muller and Trannoy

2011a, b; Duclos et al. 2011) and explicitly normative methods (List 1999; Weymark

2006) which are related to the univariate Atkinson (1970) methodology.

This paper adopts the multidimensional ‘index’ approach of Maasoumi (1986) and

Maasoumi and Nicklesburg (1988) which is attractive as it allows for simple cardinal

interpretations of wellbeing inequality. The method involves specifying an explicit func-

tion that summarizes an individual’s circumstances by aggregating over multiple dimen-

sions and examining the distribution of the resultant index. For example if income, wealth,

health and leisure are to be considered, the scores of an individual across these criteria are

projected onto a single variable indicative of wellbeing, which can then be analysed and

decomposed using standard univariate techniques.

This aggregative approach has some advantages and disadvantages for studying

inequality. Firstly it is clear that such a process involves a simplification of the notion of

wellbeing, as even in a multidimensional context there are many factors that are likely to

be important but cannot be included in a formal analysis. Nevertheless the technique is

considerably broader than a casual interpretation might suggest, as a large number of the

unobservable characteristics that are indicative of wellbeing are likely to be highly cor-

related with the dimensions used. For example job satisfaction is likely to be correlated

with income and education (Albert and Davia 2005), a sense of financial security should be

highly dependent upon wealth (see theoretical work by Bossert et al. 2011) and the quality

of personal relationships is likely to be related to (mental) health (Bertera 2005) and

perhaps leisure time. For this reason a combination of several important dimensions should

capture most of the underlying drivers of wellbeing and form an effective (but imperfect)

indicator of overall wellbeing. 1

A second complication with the approach is that there is no obviously natural way to

determine an appropriate aggregation function to combine the various dimensions, and

hence the results are sensitive to the method employed by the analyst. That is, it can be

hard to unambiguously determine quite how important one dimension is relative to another,

and how much a surplus on one criterion should be used to compensate for a shortfall in

another. Practically this can be solved by involving a large number of different weighting

specifications, however their use places some caveats on any single finding as the choice of

functional specification for the index is hard to justify, but must be accepted for the results

to be meaningful.

Despite these drawbacks the technique has been used recently with success by Justino

(2012) who also provides a comprehensive overview of other multidimensional methods,

Lugo (2005) who studies the case of Argentina, Brandolini (2008) who compares various

European countries and also Nilsson (2010) who produces a similar study of Zambia. In

1 While it may be possible to increase the number of indicators used to generate a more well-rounded

measure, the more variables that are used, the more dependent the results become on certain aggregation assumptions. Thus the optimal way to proceed is to attempt to strike the right balance between parsimony and breadth of indicators.

592 N. Rohde, R. Guest

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this paper we follow these authors by applying the technique to recent US data obtained

indirectly from the PSID. In particular we explore changes in the levels and composition of

wellbeing inequality over time and determine whether the racial disparities that are

commonly observed using univariate approaches are reflective of broader disparities in

wellbeing.

We use the method to focus on the impact of racial stratifications on inequality (as

opposed to other potential variables such as sex, age or geographic location) for two

reasons. Firstly substantial economic differentials are regularly observed over racial groups

(which makes these differences an interesting vehicle for analysis) and secondly racial

groups are fixed while other factors such as age or location can or will change over time. 2

As a consequence racial stratifications are commonly cited as drivers of economic

inequality, which in turn relate to important issues such as poverty, segregation and social

mobility.

The three variables we employ to compare racial wellbeing are (i) equivalized house-

hold income after taxes and transfers, (ii) a general self-reported health variable and (iii)

the number of years of formal education an individual receives. 3

While there are other

variables that may contribute significantly to wellbeing we are constrained by practical

limitations on the type of data available. There has been a number of studies on racial

inequality in the US that have focussed on these variables independently, 4

including

various on material factors (Darity and Myers 1998; Hoover and Yaya 2011; Western and

Pettit 2005; D’Ambrosio and Wolff 2006; Gradin 2010; and results evident in the US

census 5 ) education, (Kao and Thompson 2003) and health (Krieger et al. 2005; Crimmins

and Saito 2001). These and other works have highlighted substantial differences between

racial groups, and generally show that the disparities are persistent over time. By studying

inequality in the distributions of these variables simultaneously, we find that the

inequalities observed in income or wealth alone are probably greater than the racial dif-

ferences in overall wellbeing. However the racial disparities we observe are still notable

and our results suggest that there has been little change in the degree of racial disparity

from 1990 to 2007.

The rest of the paper is structured as follows. Section 2 covers the empirical approach

and highlights some of the difficulties and subjectivities involved in measuring inequality

of more than one variable. Section 3 introduces the data and Section 4 presents results of

univariate inequality estimates. Section 5 gives the corresponding multidimensional esti-

mates while Section 6 summarizes and gives some concluding comments.

2 Methodology

To measure multidimensional inequality we define Xif as the amount of attribute

f = 1 … M accruing to individual i = 1 … N. X is thus an N 9 M matrix of attributes and Xi = (Xi1, Xi2, … XiM) is a vector of individual level characteristics. We standardize each

2 While gender differences may be treated as fixed, the sharing of resources within households dilutes this

inequality and complicates measurement issues. 3

These three indicators represent a fairly typical set e.g. see the recent work by Bossert et al. (2011). 4

The multidimensional work of the National Urban League (Jones, 2008, 2010) is a notable exception. 5

Census data on median family incomes for racial groups can be found at http://www.census.gov/ compendia/statab/2011/tables/11s0696.pdf.

Multidimensional Racial Inequality 593

123

observation relative to its mean value and denote the standardized values with lower case

script e.g. xi1 = Xi1/l1 where l1 ¼ð1=nÞ Pn

i¼1 Xi1. A general index is then calculated for each individual to summarize the wellbeing as

aggregated over the M attributes. Following Maasoumi (1986) we adopt the Constant

Elasticity of Substitution (CES) form

wiðxiÞ¼ a1xbi1 þ a2x b i2 þ�� �þ aM x

b iM

h i1 b

b 2 ð�1; 0Þ ð1Þ

where wi is the estimated wellbeing of individual i and a1 … aM and b are subjective weighting parameters.

As highlighted above, the choices made for both the dimension weights (a1 … aM) and the substitution weight b are important in specifying the index. Different values will imply different sensitivities of wellbeing to changes in attributes which in turn will affect

inequality estimates. Although it is difficult to settle unambiguously on parameter values, a

theoretical guidance of the various weights and rationales for their selection is given by

Decancq and Lugo (2012). Following these authors we first consider the set of weights a1 … aM which determine the relative value of each criterion to the aggregate index, where a larger value for a particular attribute corresponds to a greater degree of influence. Without

loss of generality, if parameter a1 is set equal to one the index will be entirely dominated

by that facet of wellbeing, provided weights are normalized such that PM

f¼1 af ¼ 1. Conversely the use of more equal weightings causes the index to consider wellbeing as an

average across multiple criteria. As there is no explicit guidance as to how these param-

eters should be chosen however, a parsimonious method is to use an equal scheme, which

imposes af = 1/M for all f. This specification is one of two employed in the paper and this vector of equal weights is denoted aE. Justification for employing aE can come from an agnostic viewpoint, as one may not want to impose that any criterion be more important

than any other. Decancq and Lugo (2012) however argue that while this approach appears

non-committal, it still implies a specific set of tradeoffs that may not be reasonable, and

instead these authors advocate more analytical approaches.

An alternative to the use of equal weights is to construct an artificial set of weights

based upon the dataset. This is the second approach we employ in the paper, where we

determine data-driven weights to account for observed correlations between the variables.

If two dimensions of wellbeing are strongly correlated there is the possibility of over-

emphasizing or double counting these variables under an equal weighting scheme. As such

it can be more appropriate to give proportionally less weighting to correlated variables.

There are a number of ad hoc methods for doing this. Our method is to average the

correlation coefficients between the dimensions for each period and select the weighting

for attribute f to be proportional to the inverse of the product of the correlation between f

and all the other variables. This set of data driven weights is denoted aD. As well as specifying a vector of dimension weights the single parameter b must be

chosen. This parameter gives the elasticity of substitution between the variables and

therefore may be interpreted as the degree that one variable (e.g. income) can offset a low

score for another (e.g. health). This variable can be shown to be equal to 1 -1/r, where r is the elasticity which is constant across all variables. As particular values for b yield special cases of Eq. (1) it is instructive to examine particular parametric specifications. If

b !�1, it follows that r ! 0, which causes the index to lose all substitutability between attributes. In this instance the wellbeing index approaches a Leontief function wi ¼ minf¼1...MðxfÞ where total wellbeing is equal to the magnitude of the poorest performing

594 N. Rohde, R. Guest

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characteristic. Thus if an individual scores poorly on one variable (e.g. health) then no

degree of income, education, leisure, wealth or other factor can compensate appropriately

under this specification. Alternatively if b ! 0, the wellbeing index approaches the geo- metric mean of the dimension scores (wi ¼

QM f¼1 xf when criteria weights are equal)

implying unit elasticity of substitution. In this case an individual would be unaffected by a

percentage increase in one variable that coincided with a percentage decrease in any other

variable. Other specifications include setting b = 1 which gives the perfectly substitutable weighted arithmetic mean wi ¼

Pm f¼1 af xf , and b = -1 which preserves the functional

form in Eq. (1) with an implicit elasticity of 0.5. Although virtually any values for b are likely to be reasonable we use integer values (b ? 0 and b = -1) for the sake of con- venience, and hence we are imposing differing degrees of imperfect substitution in both

cases. 6

Given that it is difficult to justify any particular choice of weightings relative to others,

it is possible to add robustness to results by determining inequality measurements across

several different parametric specifications. Results that hold over multiple sets of weights

can be considered more reliable than results that depend on a particular parametric

specification and accordingly more emphasis is placed on less sensitive findings. We have

chosen two alternative specifications for the two types of weighting parameters and hence

there are four combinations of weights used in the paper. These are (b = 0, a = aE), (b = -1, aE), (b = 0, a = aD) and (b = -1, a = aD) and are henceforth referred to in shorthand as Specifications 1–4, respectively.

Once the weights have been determined the index specified in Eq. (1) can be applied to

each individual. The inequality between values of the index may then be estimated using

the Theil inequality measure (Theil 1967). This index measures the information-theoretic

divergence between the wellbeing shares of the sample and the shares that would exist

under perfect equality. It can be written as

T ¼ 1

n

Xn

i¼1

wi

�w ln

wi

�w ð2Þ

where �w represents the average wellbeing score. The Theil index functions well as an inequality measure (Cowell and Kuga 1981; Foster

1983), satisfying the fundamental Pigou-Dalton transfer principle (Pigou 1912). This

principle dictates that a progressive transfer (i.e. a small transfer of wellbeing from an

individual with a high score to a person with a low score) has the property of always

reducing the inequality measurement. This property ensures that the statistic gives larger

values for more unequal distributions, while a score of zero occurs under perfect equality

(where each individual has identical wellbeing).

An advantage of Theil’s measure (as a member of the Generalized Entropy class of

index) is that it can be additively decomposed into the inequality that occurs between the

racial groups and a weighted sum of the inequalities within each group. If wellbeing is

stratified according to racial groups j = 1 … k such that the number of persons within subgroup j is nj and

P nj ¼ n then Theil’s index can be written as

6 In principle b could vary among attributes, reflecting different degrees of flexibility in substituting one

attribute for another within a given index. This is left for future work.

Multidimensional Racial Inequality 595

123

T ¼ Xk

j¼1 w�j Tj þ

1

k

Xk

j¼1

�wj �w

ln �wj �w

ð3Þ

Here w�j is the proportional share of wellbeing scores for individuals within j, Tj is the Theil

estimate of inequality within the subgroup relative to �wj which denotes the subgroup average. The second term on the right is thus a measure of the inequality between racial

wellbeing means and is denoted TR. Like the overall applications of Theil’s measure, larger

estimates of TR indicate large discrepancies between racial groups while smaller values

imply more equal values. An estimate of zero occurs only when all three racial groups have

the exact same average score on the welfare criterion.

3 Data

Data for the study comes from the Panel Study of Income Dynamics (PSID) component of

the Cross National Equivalence File (CNEF). The specific variables used are household

post-government income (coded I11113_X), number of years of education (D11109_X)

and a self-rated health variable (M11126_X) and are considered in detail below. Cross

sectional individual weights are also used throughout the paper, while missing and negative

observations are dropped. The sample sizes vary slightly from year to year but generally

consist of around 8,000 individual observations for each wave.

Post-government household income is used in the analysis to provide a general measure

of monetary wellbeing. Although the unit of analysis is the individual, household incomes

provide a better measure of wellbeing due to the sharing of incomes within families. To

account for household size, each income is equivalized by dividing by the square root of

the household size to approximate the amount accruing to the individual. While direct data

on post-government household incomes are not available (due to the PSID stopping the

recording of this variable in 1992) a simulated series generated using the TAXSIM pro-

gram (Feenberg and Coutts 1993) is employed in its place. The variable is recommended

for this purpose in the PSID codebook and estimates the sum of total family income from

labour earnings, asset flows, private transfers, private pensions, public transfers, and social

security pensions minus total household taxes.

The education of each individual is measured in terms of number of years of formal

schooling received and is top-coded at 17 years. As a result of this top-coding, estimates of

inequality in education are likely to understate the true value as education levels reaching

this point are not particularly uncommon. Further, as average education levels increased

over the time span there is the possibility of a slight downward bias in inequality estimates

later relative to earlier in the time period. 7

This is due to the point at which the variable is

censored (i.e. 17 years) becoming lower relative to the mean of the distribution over time.

The third variable, self-rated health, poses some problems for the analysis as it is quoted

as an ordinal qualitative variable rather than numerical. Respondents are asked to rate their

health from ‘Excellent’, ‘Very Good’, ‘Good’, ‘Fair’ or ‘Poor’. As we wish to make

cardinal inequality measurements it is necessary to apply some type of numerical values to

represent the relative wellbeing of each category. This is done by applying a nominal scale

where the category of ‘Poor’ receives a score of one and ‘Excellent’ receives a score of

7 For example 8.6 % of observations hit the educational top-code of 17 years in 2007 while only 6.9 % of

observations reached this value in 1990.

596 N. Rohde, R. Guest

123

five. While such an assignment is unattractive as it imposes linearity between the cate-

gories, the problem is not too great in practice for two reasons. Firstly, the measurement

technique subjects the variables to three different sets of weightings, none of which have

especially compelling natural choices. Thus all variables, even those with clearly defined

units of measurement such as income, will take on a somewhat arbitrary scaling. Secondly,

we are more interested in proportional relationships between various sets of estimates than

their absolute values and this type of comparison only requires consistency with respect to

scaling and weighting. 8

Lastly racial stratifications are determined using the variable D11112LL. While there

are a number of racial groups with which an individual may identify, the majority (over

90 %) of persons either identify as ‘‘White’’ or ‘‘Black’’. 9

All other observations are

grouped into a generic ‘Other Race’ variable in order to ensure that inequality estimates for

some racial groups are not based upon unduly small samples.

4 Income, Education and Health Inequalities in Isolation

A preliminary guide to trends and racial differences in the income, education and health

variables can be gained from Table 1 which reports estimates for the mean of each variable

in each year from 1990 to 2007. It is evident that in almost all cases whites had the highest

greater equivalized household incomes, 10

levels of education and health scores, while

blacks always scored the lowest on these three criteria. It is also apparent that average

income and education levels had risen for all three groups, which has probably driven an

increase in wellbeing over the period. There is however no strong sign of increase in the

self-reported health scores, with estimates for whites and blacks roughly constant, although

estimates for the ‘other race’ category declined slightly. This lack of trend does not imply

that health has declined or failed to improve however as (i) there is the possibility that

individuals report health scores relative to a benchmark that has increased through time,

and (ii) we have noted the variable has been subjected to a linear scale which makes such

cardinal comparisons difficult to interpret.

To measure inequality in these variables we first consider each variable separately using

the one-at-a-time approach. 11

Estimates of Theil’s T metric are given in Table 2 for both

the inequality in the sample as a whole (column 1) and the inequality between the means

for each racial group (column 2). These provide a general summary and serve as a

benchmark for comparisons with the true multidimensional estimates given in Tables 5

and 6. The results show considerable income inequality relative to education and health

scores which are much more evenly distributed.

To search for changes in both the aggregate levels of inequality and the inequality that

occurs between the races, each set of estimates is regressed against a time trend using the

general equation yt = k0 ? k1t ? et where y denotes the dependent variable at t is a linear

8 The assessment of trends in perceived health is valid provided that (i) the subjective scores are appropriate

indicators of true health status, and (ii) the self assessments are consistent across both time and across the racial groups. 9

Persons of Latino origin are classified as white. 10

The racial group ‘Other’ reports a higher average income than whites for 1991 and 1994. In all other instances whites outscore this group over all three criteria. 11

It should be noted that combining univariate inequality measurements is not a true ‘multidimensional’ technique as it misses the interaction that occurs between dimensions.

Multidimensional Racial Inequality 597

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Table 1 Racial means of income, education and health variables 1990–2007

White Black Other

Income Education Health Income Education Health Income Education Health

1990 21,551 12.81 3.61 12,412 11.64 3.15 19,913 12.67 3.60

1991 21,183 12.83 3.62 12,500 11.68 3.18 21,411 12.72 3.55

1992 21,479 12.88 3.62 12,650 11.75 3.16 20,526 12.77 3.56

1993 22,062 12.99 3.65 12,980 11.92 3.16 20,657 12.82 3.61

1994 21,723 13.02 3.63 11,998 11.98 3.15 23,793 12.73 3.61

1995 21,831 13.03 3.66 12,697 12.00 3.19 20,935 12.62 3.56

1996 22,023 13.04 3.63 12,920 12.00 3.17 21,434 12.64 3.48

1997 23,709 13.06 3.64 13,864 12.05 3.26 18,335 12.10 3.49

1999 25,247 13.17 3.66 13,986 12.05 3.25 19,899 12.18 3.44

2001 26,642 13.19 3.66 16,098 12.14 3.25 23,814 12.21 3.44

2003 25,803 13.22 3.67 15,233 12.19 3.24 24,075 12.30 3.44

2005 26,972 13.40 3.56 14,792 12.45 3.16 22,954 12.56 3.38

2007 26,749 13.16 3.59 14,615 12.25 3.24 24,240 12.38 3.44

Source Authors’ own calculations

Note Income is quoted in equivalized household income deflated relative to 1990 dollars. Health means are subject to the nominal linear scaling mentioned in Section 2 and hence are best interpreted with caution. Population shares are not quoted but average approximately 83 % White, 11 % Black and 6 % ‘Other Race’ throughout the period

Table 2 Sample inequality estimates for income, education and health 1990–2007

Years Income Education Health

T TR T TR T TR

1990 0.24598 0.01143 0.02534 0.00043 0.05493 0.00087

1991 0.21994 0.01073 0.02461 0.00043 0.05336 0.00078

1992 0.23897 0.01063 0.02396 0.00041 0.05279 0.00086

1993 0.23585 0.01080 0.02147 0.00035 0.04877 0.00096

1994 0.29044 0.01366 0.02117 0.00035 0.05107 0.00097

1995 0.27225 0.01129 0.02080 0.00034 0.04951 0.00090

1996 0.25279 0.01086 0.02085 0.00034 0.05036 0.00086

1997 0.25302 0.01177 0.02318 0.00041 0.04983 0.00059

1999 0.34734 0.01348 0.02309 0.00049 0.04858 0.00073

2001 0.29227 0.00953 0.02246 0.00046 0.04819 0.00072

2003 0.27754 0.01038 0.02180 0.00043 0.04760 0.00081

2005 0.37457 0.01406 0.02105 0.00038 0.05264 0.00075

2007 0.37631 0.01398 0.02139 0.00034 0.05146 0.00054

Source Authors’ own calculations

Note The left columns give estimates of Theil’s T index for the overall inequality within the sample for each variable as per Eq. (2) while the right columns give the T index between the means of the racial groups as per the second term in Eq. (3)

598 N. Rohde, R. Guest

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time-trend. 12

The well documented increase in income inequality over this period is evi-

dent in the first column which shows estimates of T increasing from 0.246 in 1990 to 0.376

in 2007. While the time-series is fairly volatile, a regression of the T index for overall

incomes on the relevant trend reveals a statistically significant upward trend 13

(p = 0.0003). Similar regressions for education and health inequalities show a marginal

downward trend for education (p = 0.0065) and a negative but insignificant trend for

health (p = 0.195). Thus the strong increase in income inequality appears to have coin-

cided with smaller reductions in inequalities in these other dimensions.

Trends in the composition of inequality can also be identified using information drawn

from Table 2. This is done with respect to racial groupings by examining the inequality

estimates that are determined between the racial means. Time-trends for between-race

inequalities are insignificant for income and education (income: p = 0.166, education:

p = 0.95) indicating that the inequality in these variables that can be explained by racial

stratifications has been relatively unchanged. Conversely there has been a reduction in the

racial inequality of the health index with a significant negative trend (p = 0.018) over the

period.

5 Multidimensional Inequality Measurements

Given that we have observed: (i) an increase in income inequality, (ii) a decrease in

education inequality and (iii) unchanged inequality between racial groups for two of our

three variables, it is difficult to judge how inequality in wellbeing, particularly with respect

to racial divisions, has changed over time. To address this issue we turn to the multidi-

mensional wellbeing index that captures not only their distributions but the relationships

between them. Indices of the form illustrated in Eq. (1) are estimated over all four para-

metric specifications to depict the overall wellbeing of each racial group. As the indices

have no interpretation in terms of units of measurement they are standardized relative to

the wellbeing of whites, which is normalized at unity in every period. Hence there is no

capacity to evaluate trends in the absolute wellbeing from the quoted figures, only their

relative magnitudes. 14

Results are reported in Tables 3 and 4 for the four sets of parametric

weights.

Table 3 shows estimates of the relative wellbeing for blacks and those of other races

using specification sets 1–2. The results from the left panel (specification 1) suggest that

black wellbeing is approximately 75–79 % as high as white wellbeing, while the alter-

native weighting specification used in the right panel (specification 2) gives these values at

73–78 %. These results are broadly consistent with other estimates produced by the

National Urban League (see Jones 2008, 2010). Wellbeing of those of other races is much

higher, around 87–99 % of that of whites over both sets of weights. Corresponding figures

can be found in Table 4, which refers to similar calculations employing specifications 3

12 The time series available is too short to distinguish between stochastic and deterministic trends with the

use of unit root tests and hence the regressions are conducted to identify significant movements within the sample rather than to describe the data-generating process. For this reason we also ignore potential auto- correlation issues. 13

Two sided p-vales are quoted in parentheses for a hypothesis test on the significance of the trend coefficient. 14

An impression of the changes in overall wellbeing over time can be gained by examining the racial means for each variable in Table 1.

Multidimensional Racial Inequality 599

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Table 3 Average relative racial wellbeing scores: specifications 1 and 2

Years Specification 1 (b ? 0, a= aE) Specification 2 (b ? -1, a= aE)

White Black Other White Black Other

1990 1 0.7635 0.9732 1 0.7393 0.9706

1991 1 0.7697 0.9921 1 0.7471 0.9888

1992 1 0.7646 0.9820 1 0.7381 0.9835

1993 1 0.7625 0.9809 1 0.7351 0.9870

1994 1 0.7511 0.9936 1 0.7209 0.9788

1995 1 0.7688 0.9616 1 0.7464 0.9545

1996 1 0.7743 0.9630 1 0.7530 0.9560

1997 1 0.7764 0.8278 1 0.7540 0.7888

1999 1 0.7624 0.8811 1 0.7363 0.8716

2001 1 0.7911 0.9011 1 0.7744 0.8827

2003 1 0.7791 0.9151 1 0.7589 0.9052

2005 1 0.7698 0.9068 1 0.7458 0.8948

2007 1 0.7706 0.9222 1 0.7485 0.9106

Source Authors’ own calculations

Note As wellbeing scores operate on a nominal scale the averages for each racial group are given relative to whites, which are normalized at one such that time trends in white wellbeing are eliminated

Table 4 Average racial wellbeing scores: specifications 3 & 4

Years Specification 3 (b ? 0, a= aD) Specification 4 (b ? -1, a= aD)

White Black Other White Black Other

1990 1 0.7423 0.9705 1 0.7217 0.9691

1991 1 0.7494 0.9924 1 0.7303 0.9886

1992 1 0.7437 0.9800 1 0.7213 0.9820

1993 1 0.7407 0.9798 1 0.7177 0.9875

1994 1 0.7271 0.9989 1 0.7020 0.9818

1995 1 0.7469 0.9618 1 0.7287 0.9544

1996 1 0.7530 0.9632 1 0.7358 0.9561

1997 1 0.7555 0.8188 1 0.7372 0.7836

1999 1 0.7402 0.8753 1 0.7183 0.8670

2001 1 0.7712 0.8993 1 0.7576 0.8800

2003 1 0.7580 0.9144 1 0.7412 0.9025

2005 1 0.7463 0.9034 1 0.7269 0.8925

2007 1 0.7471 0.9207 1 0.7296 0.9085

Source Authors’ own calculations

Note As wellbeing scores operate on a nominal scale the averages for each racial group are given relative to the pooled sample average. Therefore a score of unity implies that the group has a wellbeing score in line with the sample average, while lower and higher scores indicate lower and higher relative wellbeing. The wellbeing scores are normalized relative to each period such that sample-wide trends are eliminated

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and 4. In these instances black wellbeing ranges from 74 to 78 % while others average

again around 88–98 %, depending on the time period and the specification employed.

In most cases estimates are similar from year to year; however there is some evidence of

changes in wellbeing over time. It is clear that for all four sets of estimates the wellbeing

of persons in the ‘Other Race’ category relative to whites has declined as regressions of

relative wellbeing scores on time trends all produce significant negative coefficients

(p1 = 0.028, p2 = 0.037, p3 = 0.035 and p4 = 0.037) 15

while corresponding estimates of

black relative wellbeing show little sign of increasing. Estimates from Table 3 show

positive coefficients (implying a degree of convergence between black and white well-

being) but these are not substantial even at 10 % significance (p1 = 0.012, p2 = 0.012,

p3 = 0.211 and p4 = 0.172). While these estimates point to a gradual increase in black

relative to white wellbeing, the rate of convergence is slow. If the observed linear trends

are to continue, the four specifications imply times to equalization in wellbeing ranging

from 271 to 358 years from 2007 levels. 16

As there is some slight evidence of changes in the racial composition of wellbeing over

the time period, this raises questions concerning overall wellbeing inequality. Tables 5 and

6 use Theil’s T index to measure the inequality: (i) in total, (ii) within the racial groups,

and (iii) between the racial groups under the parametric assumptions employed previously.

The first three columns in each table give estimates for the inequality of wellbeing within

each racial group while the fourth column provides the inequality between the groups, and

the fifth column gives the inequality for the entire sample.

Initially we examine trends in both total inequality and the inequality that may be

attributed to racial divisions. Regressing total inequality on time-trends for specifications

1-4 shows statistically significant upward trends for total wellbeing inequality (p1 = 0.004,

p2 = 0.004, p3 = 0.002 and p4 = 0.003, respectively). As with most of the univariate

inequality measures, there does not appear to be any significant trends in the inequalities

between the racial groups. These regressions yield negative slope parameters (implying a

reduction in racial wellbeing inequality over time) but with insignificant results in all cases

(p1 = 0.621, p2 = 0.454, p3 = 0.687 and p4 = 0.473). Thus while the data shows strong

evidence of an increase in the inequality of wellbeing, the inequality that occurs between

the races is virtually unchanged.

If the increase in wellbeing inequality is not explained by an increase occurring between

the racial groups, it must have occurred within these groups. Regressing the inequality

estimates within each race on the time-trends show significant upward movements for

whites in particular with significant positive coefficients in all cases (p1 = 0.005,

p2 = 0.003, p3 = 0.002, and p4 = 0.002). For the blacks there was no significant change

in within-group inequality using any of the four sets of weights (p1 = 0.489, p2 = 0.386,

p3 = 0.314 and p4 = 0.365) while for persons of other racial groups there was marginal

evidence of an increase in internal inequality at 10 % significance under two of the four

specifications (p1 = 0.062, p2 = 0.168, p3 = 0.064, and p4 = 0.183). Hence we conclude

that the increase of wellbeing inequality observed over time is strongly driven by the

upward trend in inequality amongst whites.

15 Here the subscripts on the p-values refer to the parametric weighting specification employed.

16 Parameter estimates for the time trend coefficient (k1) for the regressions wt = k1?k1t ? et are (1)

0.000801, (2) 0.00107, (3) 0.000715 and 0.000997 for the four specifications. Dividing the gaps between white and black wellbeing at 2007 levels by these estimates yields estimated times to convergence of (1) 286 years, (2) 234 years, (3) 358 years and (4) 271 years.

Multidimensional Racial Inequality 601

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Table 5 Racial inequalities in wellbeing using specifications 1 & 2

Years Specification 1 (b ? 0, a= aE) Specification 2 (b ? -1, a= aE)

White Black Other Between Total White Black Other Between Total

1990 0.0495 0.0569 0.0495 0.0032 0.0533 0.0550 0.0858 0.0602 0.0039 0.0617

1991 0.0476 0.0582 0.0483 0.0030 0.0516 0.0536 0.0888 0.0533 0.0037 0.0604

1992 0.0490 0.0630 0.0430 0.0031 0.0532 0.0550 0.0984 0.0492 0.0039 0.0625

1993 0.0480 0.0627 0.0494 0.0032 0.0526 0.0553 0.1004 0.0611 0.0041 0.0635

1994 0.0519 0.0656 0.0720 0.0036 0.0576 0.0592 0.1071 0.0766 0.0047 0.0688

1995 0.0510 0.0631 0.0577 0.0031 0.0555 0.0586 0.0982 0.0703 0.0038 0.0664

1996 0.0488 0.0584 0.0567 0.0029 0.0529 0.0553 0.0893 0.0713 0.0035 0.0625

1997 0.0503 0.0623 0.1112 0.0034 0.0573 0.0590 0.0978 0.1768 0.0043 0.0714

1999 0.0518 0.0619 0.0720 0.0034 0.0572 0.0592 0.0985 0.0925 0.0042 0.0686

2001 0.0506 0.0503 0.0722 0.0025 0.0544 0.0566 0.0755 0.0878 0.0031 0.0632

2003 0.0495 0.0574 0.0683 0.0028 0.0546 0.0589 0.0904 0.0828 0.0034 0.0666

2005 0.0556 0.0610 0.0741 0.0033 0.0606 0.0632 0.0987 0.0924 0.0040 0.0725

2007 0.0661 0.0734 0.0759 0.0032 0.0706 0.0781 0.1172 0.0925 0.0039 0.0866

Source Authors’ own calculations

Note The first three columns give inequality estimates within black, white and other racial groups employing weightings b ? 0 and aE. The fourth column gives the inequality between the subgroup means and the fifth column provides the inequality estimate for the entire sample. Columns on the right of the central divider have the same interpretation but are determined with b ? 1

Table 6 Racial inequalities in wellbeing using specifications 3 and 4

Years Specification 3 (b ? 0, a= aD) Specification 4 (b ? -1, a= aD)

White Black Other Between Total White Black Other Between Total

1990 0.0590 0.0658 0.0585 0.0038 0.0634 0.0631 0.0951 0.0690 0.0045 0.0705

1991 0.0565 0.0676 0.0574 0.0036 0.0612 0.0617 0.0985 0.0617 0.0043 0.0691

1992 0.0586 0.0736 0.0510 0.0038 0.0634 0.0633 0.1090 0.0569 0.0045 0.0715

1993 0.0576 0.0747 0.0578 0.0039 0.0630 0.0637 0.1119 0.0692 0.0047 0.0727

1994 0.0628 0.0786 0.0890 0.0044 0.0697 0.0683 0.1197 0.0880 0.0054 0.0788

1995 0.0617 0.0753 0.0690 0.0037 0.0669 0.0673 0.1086 0.0800 0.0043 0.0758

1996 0.0588 0.0696 0.0665 0.0035 0.0636 0.0638 0.1002 0.0797 0.0041 0.0717

1997 0.0603 0.0748 0.1294 0.0040 0.0685 0.0673 0.1093 0.1910 0.0049 0.0808

1999 0.0632 0.0740 0.0811 0.0041 0.0692 0.0683 0.1095 0.1007 0.0048 0.0783

2001 0.0616 0.0604 0.0825 0.0030 0.0658 0.0654 0.0853 0.0961 0.0035 0.0725

2003 0.0602 0.0683 0.0833 0.0034 0.0659 0.0679 0.1004 0.0892 0.0039 0.0760

2005 0.0686 0.0735 0.0865 0.0040 0.0743 0.0734 0.1104 0.1029 0.0047 0.0834

2007 0.0814 0.0881 0.0902 0.0039 0.0865 0.0897 0.1294 0.1028 0.0045 0.0988

Source Authors’ own calculations

Note The first three columns give inequality estimates within black, white and other racial groups employing weightings b ? 0 and aD. The fourth column gives the inequality between the subgroup means and the fifth column provides the inequality estimate for the entire sample. Columns on the right of the central divider have the same interpretation but are determined with b = -1

602 N. Rohde, R. Guest

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Aside from the above time-trend regressions two further results can be obtained from

the inequality estimates. Firstly the figures in Tables 5 and 6 highlight the difficulties

involved in providing robust rankings of races with respect to within-group inequality. The

two sets of estimates determined using b = 0 appear highest for blacks and lowest for whites, while corresponding estimates when b = -1 give blacks the lowest within-group inequality. Given the dependence of the within-group estimates on the parametric speci-

fication it seems reasonable to conclude that no unambiguous relationships in the

inequality within the various groupings are evident.

Secondly a final insight into the role of race in explaining inequality can be gained from

the proportional share of between-race inequality relative to total inequality. Such shares

can be obtained by dividing the ‘between-race’ inequality estimate by the total inequality

measurement. This value may be interpreted as the proportion of total inequality that may

be explained by the racial divisions we have defined. For the income, education and health

variables it is shown that the between-race component of univariate inequality ranges from

averages of 1.6 % (health), 1.7 % (education) and 4.2 % (income) of total inequality. This

indicates that racial stratifications explain a greater proportion of income inequality than

for health or education. However the equivalent ratios for wellbeing inequality are higher,

with racial classifications accounting for 5–6 % of total inequality. This suggests that if

generalized to apply to wellbeing, the standard univariate measurements presented in

Table 2 understate the extent that racial differences drive wellbeing inequality.

6 Summary and Conclusion

This paper has studied racial disparities in income, education and health from several

angles. The primary goals have been to compare results from traditional measures of

inequality across races to results obtained using multidimensional ‘wellbeing’ indicators,

and to search for changes in the levels and composition of inequality over time. Initially

univariate applications of Theil’s T index are used to summarize changes in the distribution

of the three variables, which show an increase in income inequality occurring alongside a

decrease in education inequality. Wellbeing indices are then estimated and racial means are

compared, with blacks having wellbeing scores consistently averaging around 76 % of

whites, while persons from other races exhibiting equivalent scores of approximately

93 %. Results were generated over four different parametric weighting scenarios and

appear to be fairly robust to the choice of weights.

Inequality estimates are then determined for the wellbeing scores and changes in the

breakdown of inequality are examined using time-trend regressions. We observe a sig-

nificant increase in overall wellbeing inequality over time across four alternative para-

metric specifications; however the increase in inequality has not been mirrored in

increasing inequality between the racial groups. Rather the increase in wellbeing appears to

have occurred within the racial groups with a particularly strong increase for whites. We

also observed that the proportion of total inequality that can be explained by the racial

stratifications is higher for multidimensional inequality than for any of the three variables

independently. This suggests that a broad interpretation of univariate techniques may result

in an understatement of the role that race has in determining inequality.

Lastly we have focussed on the measurement of inequality but have not speculated on

the underlying forces that may have led to the changes we have observed. It is difficult to

come up with definitive explanations for various changes in distribution, however some

plausible reasons can be speculated upon. Two of the primary explanations for the increase

Multidimensional Racial Inequality 603

123

in income inequality (which is a primary driver of the multidimensional results we

observe) given in the literature are (i) changes in tax rates or other governmental policy

over time and (ii) increases in wage and salary returns to skilled works 17

which itself is

unequally distributed both within and between racial groups. However the means by which

these or other factors have driven the distributional changes we have observed is open for

debate. As inequality in health and education may be related to inequality in monetary

resources, it may also be partially explained by this type of phenomena, or may be driven

by other factors such as cultural differences.

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