Understanding Sources of Wage Inequality:

Additive Decomposition of the Gini Coefficient Using

Quantile Regression

Carlos Hurtado∗

January 31, 2017

Abstract

The constant increase in wage inequality in the United States from the early 1980s up to the

present has been well documented. The development of methodologies to model the complete

wage distribution has led to the hope of a better understanding of the factors that affect the

distribution. Comprehending how measurements of inequality vary as functions of sociode-

mographic characteristics and individual endowments, as well as the wage returns to those

characteristics and endowments, is an alternative approach to understand the disparity of

the wage distribution. This paper uses the relation between the Gini index and conditional

quantile functions to develop a new methodology for the measurement of the impact of var-

ious factors in the disparity of a distribution. Starting with a linear model to explain the

conditional quantile function in terms of covariates, the proposed procedure uses polynomial

approximations of the estimates of the quantile regression coefficients to additively decom-

pose the factors and characteristics that contribute most to the inequality of the distribution.

Moreover, using counterfactual scenarios, this paper proposes a technique to disentangle the

temporal changes in the distribution. The empirical application uses data on US hourly wages

from the Ongoing Rotation Group of the Current Population Survey for the years 1986 and

1995.

Keywords: Gini Index, Wage Structure, Inequality, Quantile Regression

JEL Classification Numbers: C13, C21, C43, J31, D63

∗Department of Economics, University of Illinois at Urbana-Champaign, 214 David Kinley Hall, 1407 West Gregory Drive,

Urbana, IL 61801, USA. Email: hrtdmrt2@illinois.edu. The author would like to thank Dr. Roger Koenker for his advise and

guidance throughout this research project. The valuable comments and suggestions from Dr. George Deltas, Dr. Elizabeth

Powers, Dr. Mark Borgschulte and Dr. Sergio Firpo improved substantially the quality of the paper. Data and code of this

paper are available at http://www.econ.uiuc.edu/∼hrtdmrt2/InequalityData&Code/

1 Introduction

It is well documented that during the decade of the 1970s, wage differences by education

and occupation narrowed in the United States. This was followed by a constant increase in

wage inequality beginning in early 1980s up to the present1. The sociodemographic charac-

teristics and individual endowments, as well as the returns (also referred as prices) of those

characteristics and endowments, are the factors that influence the final wage distribution.

Clearly, those factors are not independent of each other and is difficult to measure their

impact on wage inequality. Using statistical techniques to understand the distribution of

wages has been important to measure the impact of factors affecting the inequality of the

distribution.

Starting with the methods proposed by Oaxaca (1973) and Blinder (1973), there has

been an increasing number of approaches trying to disentangle the factors that contribute

to the differences in wages2. The classical model of labor supply and demand with homo-

geneous agents implicitly assumes that there is a unique wage that clears the market, as

opposed to a distribution of wages when there are heterogeneous agents; given the classical

model, part of the literature has focused on the average wage (controlling for individual

and institutional characteristics3).

Going beyond the analysis of mean wage differences has gained an increasing interest

in recent years. For example, using quantile regression, Buchinsky (1994) shows that the

returns to education are bigger for individuals in the upper tail of the wage distribution

than the returns of those at the bottom of the distribution. Angrist et al. (2006) finds a

similar result for a more contemporary subsample of the US population. In Arellano and

Bonhomme (2017) the authors show similar findings for the UK. These results suggest that

the increasing number of more educated workers contributes towards more inequality in

the distribution of wages.

The development of methodologies to model the complete wage distribution has led

to the hope of a better understanding of the factors that affect the distribution per se4.

Recently, Machado and Mata (2005) developed a counterfactual decomposition technique

using quantile regression. Exploiting the probability integral transformation theorem, the

authors estimate marginal (log) wage distributions consistent with a conditional distribu-

1See Levy and Murnane (1992), Katz (1999) and Autor et al. (2008) for a review of the literature.2For a review on many of the decomposition methods please refer to Fortin et al. (2011).3See for example Katz and Murphy (1992), Bound and Johnson (1992), Blau and Kahn (1996), Card

and Lemieux (2001)4To my knowledge, the first method that models the distribution of wages is in DiNardo et al. (1996).

The authors developed an estimation procedure to analyze counterfactual (log) wage distributions usingkernel density methods to appropriately weighted samples.

1

tion estimated by quantile regression. The authors perform counterfactual investigations by

comparing the implied marginal distributions for different distributions of covariates. They

apply this methodology to Portuguese data and also find that the increase in educational

levels contributes to higher inequality in the wage distribution for that country.

Although considerable research has been devoted to developing techniques to identify

the sources of wage differentials by estimating the density of the distribution, there has

not been much attention on understanding how inequality measures vary as function of the

factors that influence the distribution. The visual evidence presented by kernel estimates,

or the analysis of some quantiles of the distribution may be hard to interpret or may miss

information that a measurement of inequality provides. In fact, a researcher can only

report a few statistics in a table before a reader gets completely lost in numbers. To

link the factors that influence the (log) wage distribution with an inequality measure, this

paper exploits the link between the conditional quantile function and the Gini index. The

previous relationship makes it possible to measure the effect of the factors that impact

the wage distribution without modeling the density, but directly measuring its inequality

through the Gini coefficient.

This paper develops a method to measure the contribution of various factors to the dis-

parity of the distribution of (log) wages5. Employing the relationship between the Lorenz

curve and the conditional quantile function it is possible to additively decompose the Gini

index using quantile regression. Starting with a linear model to explain the conditional

quantile function in terms of covariates in a given year, I propose a method that uses poly-

nomial approximations of the estimates of the quantile regression coefficients to determine

the factors that contribute most to the inequality of the distribution. Using the estimates

of the Gini coefficient for different years, I also propose a decomposition of the changes on

the wage distribution using counterfactual scenarios.

To preview the findings, an empirical application of the procedure is developed using

the Ongoing Rotation Group (ORG) of the Current Population Survey (CPS) for 1986

and 1995. The estimates computed with the proposed method show that the impacts

of various characteristics have changed over time. Additionally, the proposed approach

shows that only the upper tail of the distribution contributes to a significant increment

in wage inequality. More interestingly, when comparing the proposed technique and the

Machado and Mata (2005) algorithm, both procedures reach similar conclusions. However,

the proposed method finds that changes in the proportion of workers with high school and

associate degree are significantly related to reductions in inequality of the wage distribution

5The method is developed in a general framework that would allow a researcher to use it for anotherpositive variable of interest, e.g. Income.

2

in the US during the period of analysis.

The paper proceeds as follows. Section 2 presents the details of the data on hourly

wages for the US. Section 3 develops the theoretical link between the Lorenz curve and

the Gini index, explaining the additive decomposition of the Gini coefficient as well as the

temporal changes in the distribution. Section 4 presents the proposed estimation procedure.

In section 5 the empirical application is developed. Finally, section 6 concludes.

2 Hourly Wage Series From the CPS

This paper uses data from the CPS to analyze changes in the distribution of wages in the

US from 1980 to 2015. Starting in 1979, workers in the ORG of the CPS are asked detailed

questions related to earnings from work. A major advantage of using the ORG is that

these detailed questions contain information that can be used to estimate hourly wages.

Using the answers to hourly earnings, or weekly earnings divided by usual hours work per

week, it is possible to compute hourly wages as a good measure of the price of labor. This

measure of hourly wages is closely related to the economic theory of wage determination

based on supply and demand.

A difficulty of using the ORG is that the CPS classifies and processes differently the

earnings of hourly paid and non-hourly paid workers throughout the years of analysis. To

create a consistent series of hourly wages it is necessary to adjust for changes in the top-

coding of weekly earnings, changes in the classification of overtime, tips and commissions

for hourly paid workers, and the change of the response of ’usual weekly hours’ for some of

the years. The previous differences demand particular attention to the changes in the sur-

vey for more than three decades. In an effort to construct consistent wages using the ORG

of the CPS, the Center for Economic and Policy Research (CEPR) has developed publicly

available code that uses the National Bureau of Economic Research (NBER) Annual Earn-

ings Files6 as well as the CPS basic monthly files7. All programs used by the CEPR are

available under the GNU General Public License from their web page. I downloaded and

modified those programs to create a consistent hourly wage series form 1980 to 2015. The

data and codes are available from the web page of this article8.

Several manipulations were performed to the data to get a consistent hourly wage series.

First, every wage was updated to constant dollars of 2015 using the Consumer Price Index

6These files are also known as Merged Outgoing Rotation Groups (MORG) and can be downloadedfrom: http://www.nber.org/morg/annual/.

7These files are known as the Basic Monthly CPS, and the Bureau of Labor Statistics maintains thosefiles available at: http://thedataweb.rm.census.gov/ftp/cps ftp.html#cpsbasic

8To download data and code go to: http://www.econ.uiuc.edu/∼hrtdmrt2/InequalityData&Code/

3

reported by the Bureau of Labor Statistics9. Second, for the series I only kept workers

reporting an hourly wage between $1 and $100 (in 1979 dollars) and with ages between 16

and 65 years. Third, I computed a potential experience variable using individuals ages, and

subtracting individuals years of education and also discounting five years before elementary

school. Fourth, the series included an indicator variable for female and non-white workers.

Fifth, the sample included a consistent classification for twenty industries and four regions

in all the years of analysis. Finally, I created a consistent classification of years of education

as non-school or dropouts (between zero and eleven years of education), high school (exactly

twelve years of education), associate degree (between thirteen and fifteen years of education)

and college degree (sixteen or more years of education).

Relatively large samples of workers are available to estimate changes in the wage dis-

tribution; the sample sizes are on average 165,000 workers per year from 1980 to 2015. To

acknowledge the gender gap, table 1 presents summary statistics of the CPS samples10 for

men and women separately. While real wages remain close to the average of 3.11 ($22.3)

for men over the period of analysis, they systematically increased for women. Although the

gender gap was reduced between 1980 and 2015, there still are wage differences by gender.

Potential experience (age-years of education-5) exhibits, in the late 1980s, the complete

entrance into the labor force of the baby boom generation as well as the retirement of some

of the baby boomers after 2010. Table 1 also shows that there is an increase in educational

attainment for men and women, with higher average years of education for women than

that for men. Finally, the rate of unionization exhibits a precipitously decrease during the

time of analysis whereas the rate of participation of nonwhite workers presents a constant

increase.

To visually understand the inequality in the wage distribution within and between gen-

ders, figures 1a and 1b present weighted kernel density estimates11 of hourly wages of men

and women from 1980 to 2015. The vertical line in each figure indicates the correspond-

ing real minimum wage, presented in table 1, as a reference of the bunching of the wage

distributions in the lower tail. From these figures it is evident that, for recent years, the

upper tail of the distributions is heavier than the upper tail of preceding years. Moreover,

it is also clear that for both, men and women, the distribution of hourly wages exhibits

9Specifically, using the seasonally adjusted index for all items based on US city average (Series Id:CUSR0000SA0).

10The summary statistics an all the estimates reported in this paper are weighted by the CPS sampleweights.

11These figures are similar to those presented in DiNardo et al. (1996), with the difference that here Iam using the CPS sample weights whereas they use hours-weighted kernel estimates. Similarly to DiNardoet al. (1996), the choice of the bandwidth for this estimation uses the Sheather and Jones (1991) method.

4

wider spreads over time with respect to the mean, particularly higher spreads for women.

These visual evidence is in concordance with the results previously exposed by Levy and

Murnane (1992), DiNardo et al. (1996), Katz (1999) and Autor et al. (2008).

3 The Lorenz Curve and the Gini Coefficient

The Lorenz curve is a compelling tool to describe the inequality of the distribution of a

positive random variable. For example, to measure inequality in wages, the curve relates

the cumulative share of wages earned by the cumulative share of people from the lowest to

the highest wages. Generally, following Koenker (2005), the Lorenz curve is defined as

L(τ) =

´ τ0QY (t)dt´ 1

0QY (t)dt

=1

µ

τˆ

0

QY (t)dt, (1)

where Y is a continuous and positive random variable, with cumulative density function

FY (y), quantile function denoted by QY (t) = inf y : FY (y) ≥ t = F−1Y (t), yτ = QY (τ),

and mean 0 < µ < ∞. As explained in appendix A, using the properties of the quantile

function, a monotone transformation, h(·), such that h(Y ) ≥ 0 and 0 < µh < ∞, with

µh = E [h(y)], lead us to a Lorenz curve of the transformed variable given by

Lh(τ) =1

µh

τˆ

0

Qh(Y )(t)dt =τE [h(y)|h(y) ≤ h(yτ )]

µh. (2)

Using the fact that 0 ≤ E [h(y)|h(y) ≤ h(yτ )] ≤ E [h(y)] = µh, and given τ ∈ (0, 1), it is

clear that the Lorenz curve of the transformed variable is also between zero and one.

Let Qh(Y )(t|x), with t ∈ (0, 1), denote the t-th conditional quantile of the distribution

of h(Y ), given a vector of covariates, x ∈ RP . Let us assume that we can model this

conditional quantile function as a linear combination of the covariates:

Qh(Y )(t|x) = xTβ (t) =P∑j=1

xjβj(t), (3)

where each βj(t) is the coefficient corresponding to the covariate j at the t-th quantile. Let

λh(τ) ∈ RP be the vector whose j-th component is defined as λj,h(τ) = 1τ

´ τ0βj(t)dt, which

roughly speaking is the mean of the j-th coefficient in the interval (0, τ)12. From equations

12Note that´ τ

0βj(t)dt = τλj,h(τ), and λh(τ) = 1

τ

(´ τ0β1(t)dt, · · · ,

´ τ0βP (t)dt

)= 1

τ

´ τ0β(t)dt.

5

(2) and (3) the conditional Lorenz curve of the transformed variable is reduced to

Lh(τ |x) =1

µh

τˆ

0

Qh(Y )(t|x)dt =1

µh

P∑j=1

xj

τˆ

0

βj(t)dt =τxTλh(τ)

µh. (4)

By comparing equations (2) and (4), note that E [h(y)|x ∧ (h(y) ≤ h(yτ ))] = xTλh(τ). By

taking the limit when τ goes to one we have that E [h(y)|x] = xTλh(1) = xT´ 1

0β(t)dt,

provided that the integral exists for each characteristic j. This is interesting because I am

linking an expression that involves the integral of the coefficients of the quantile regression

whit the conditional expectation of h(y), which can be also linked with the estimates of the

OLS method. This relation suggest that perhaps, under certain additional conditions, it is

possible to assert that βOLS =´ 1

0β(t)dt. This opens the possibility for further research to

explore this relation.

Based on the Lorenz curve, the Gini coefficient has a widespread use to summarize the

disparity of the distribution of a positive random variable. The relationship between the

coefficient and the curve is given by

G = 1− 2

1ˆ

0

L(τ)dτ, (5)

where G is the value of the Gini index. The index simply measures how much the Lorenz

curve of a given random variable deviates from the line of perfect equitability13. The

conditional Gini coefficient given a vector of covariates can be computed by using the

conditional Lorenz curve, equation (4), into the definition of the Gini index:

Gh (x) = 1− 2

1ˆ

0

Lh(τ |x)dτ

= 1− 1

µh

P∑j=1

xj

1ˆ

0

τˆ

0

2βj(t)dtdτ, (6)

where x ∈ RP . Equation (6) is an additive decomposition of the Gini index which can be

used to investigate the evolution of changes in the distribution of h(Y ) as a function of the

factor endowments and sociodemographic characteristics, xj, as well as the returns (prices)

of these endowments and characteristics, 1µh

´ 1

0

´ τ0

2βj(t)dtdτ .

13The line of perfect equitability is the Lorenz curve of a degenerate random variable δµ, which onlytakes the single value µ.

6

It is interesting to note that one can rewrite the coefficient by dividing the interval (0, 1)

into n equally spaced sub-intervals as

G = 1−n−1∑i=0

2

τi+1ˆ

τi

L(τ)dτ, (7)

with τi = in, for i = 0, · · · , n − 1. By noting that τi+1 − τi = 1

n, and noting that the area

under the line of perfect equitability can be written in terms of rectangles and triangles, it

is clear that

G = 2n−1∑i=0

i

n2+

1

2n2−

τi+1ˆ

τi

L(τ)dτ. (8)

What is interesting about equation (8) is that it allows us to identify the sub-intervals

that contribute most to the Gini index, that is, the quantiles that contribute most in order

to increase the inequality of the distribution. At last, combining equations (8) and (6) we

can re-express the coefficient, given a vector of covariates, as

Gh (x) = 1−n−1∑i=0

P∑j=1

xj1

µh

τi+1ˆ

τi

τˆ

0

2βj(t)dtdτ

= 2n−1∑i=0

i

n2+

1

2n2−

n−1∑i=0

P∑j=1

xj1

µh

τi+1ˆ

τi

τˆ

0

2βj(t)dtdτ (9)

Equation (9) takes into account the additive decomposition of the coefficient in terms of

the endowments and characteristics, xj, as well as the sub-intervals that contribute most

to increase the inequality of the distribution.

3.1 Impact of Individual Characteristics on the Gini Index

We can use equation (6) to compute the change in the Gini index, given a small positive

change in a characteristic j from xj to x′j, as

∆Gh

∆xj=Gh(x

′j, x−j)−Gh(xj, x−j)

x′j − xj= − 1

µh

1ˆ

0

τˆ

0

2βj(t)dtdτ ≡ −Πj

µh, (10)

7

where, x = (xj, x−j) = (x1, · · · , xj, · · · , xp) ∈ RP . By assumption µh > 0, and the sign of

the change in the Gini coefficient depends solely on the sign of Πj =´ 1

0

´ τ0

2βj(t)dtdτ . A

negative sign of Πj implies that a small positive change in covariate j is associated with

an increase in the Gini index, which implies more inequality in the distribution of h(Y ).

Alternatively, given a small positive change in covariate j, a positive sign of Πj is associated

with a reduction in the inequality of the distribution of h(Y ).

Moreover, µh is just a positive scaling parameter that normalizes the Lorenz curve and

the Gini coefficient to be between zero and one. Hence, the magnitude of Πj also reveals

information about the magnitude of the change in the Gini index, given a small positive

change in the covariate j. Bigger values of Πj, in absolute terms, are associated with bigger

changes in the Gini coefficient, in absolute terms. I will refer to the absolute magnitude

ofΠjµh

as the impact of covariate j on the distribution of h(Y ). Using this notion, some

covariates will have bigger impact that others on the distribution of h(Y ).

3.2 Temporal changes in the distribution of h(Y )

Understanding the inequality in the distribution of h(Y ) allow us to decompose the effect

of several factors on the change of the distribution over a period of time. This has practical

implications, because it allow us to distinguish between the effect on the change of the

distribution of h(Y ) driven by changes in the individuals’ characteristics and changes in the

returns to those characteristics. Previous decomposition methods have also developed this

type of analysis; DiNardo et al. (1996) apply kernel density methods to reweighted samples

to analyze counterfactual wage distributions. In a method more closely related to the

one presented here, Machado and Mata (2005) developed a counterfactual decomposition

technique using quantile regression. In both cases, as also the case for this study, the

decomposition is a generalization of the Oaxaca (1973) method, which was developed to

analyze counterfactual differences in mean earnings. In Chernozhukov et al. (2013), Rothe

(2013) and Melly (2006), the authors go deeper on counterfactual decomposition methods,

but I leave the use of this techniques for future research.

Assume that we would like to analyze the changes in the distribution during two years

Ψ ∈ 0, 1. There are two types of counterfactual scenarios that we would like to investi-

gate; on the one hand, we would like to estimate the inequality in the distribution of h(Y )

in year Ψ = 1, corresponding to the distribution of the covariates in year Ψ = 0. On the

other hand, we would like to estimate the inequality in the distribution of h(Y ) in year

Ψ = 1 if only one covariate is distributed as in year Ψ = 0. Using these counterfactuals it is

possible to understand the effect on the distribution of h(Y ) given changes in the covariates

8

as well as changes in the returns of those covariates.

Let us model the conditional quantile function in year Ψ as

Qh(Y )(t|x; Ψ) = xTβΨ (t) , (11)

and denote by X(Ψ) the NΨ×P matrix of data on covariates. Denote by Xj(Ψ) the average

of column j of the matrix X(Ψ). Using the additive decomposition of the Gini coefficient,

equation (6), an estimate for the Gini index in year Ψ can be computed as

GΨh = 1−

P∑j=1

Xj(Ψ)ΠΨj

µΨh

, (12)

where µΨh and ΠΨ

j are estimates, in year Ψ, for µh and Πj =´ 1

0

´ τ0

2βj(t)dtdτ respectively.

Although there are general equilibrium effects given changes in the distribution of the

covariates because those changes will also affect the returns to the characteristics, yet let

me assume for simplicity that the changes in the covariates do not modify the returns of

those characteristics14. Under this assumption, an estimate of the Gini index in year Ψ = 1

if all covariates had been distributed as in year Ψ = 0 can be easily computed as

G1h (X(0)) = 1−

P∑j=1

Xj(0)Π1j

µ1h

. (13)

Let GΨh denote the Gini index computed from an observed sample of h(Y ) in year Ψ.

The changes in the distribution of h(Y ) can be capture by changes in the Gini coefficient:

G1h −G0

h = G1h − G0

h + residual

= G1h − G1

h (X(0))︸ ︷︷ ︸change in covariates

+ G1h (X(0))− G0

h︸ ︷︷ ︸change in returns

+ residual. (14)

The effect on the change of the distribution of h(Y ) driven by changes in the returns to

individual characteristics is capture by G1h (X(0))− G0

h, where covariates are held constant

and only returns are changing. The effect on the change of the distribution associated with

changes in the individual characteristics is measured by G1h− G1

h (X(0)), where returns are

constant and only covariates change.

Let X1−j(0) =

(X1(1), · · · , Xj(0), · · · , XP (1)

)be the vector in RP with j-th entrance

14This is an inherent assumption of the Oaxaca (1973) decomposition that is also present in DiNardo etal. (1996) and Machado and Mata (2005).

9

defined as the average of characteristic j in year Ψ = 0, and all other entrances defined by

the average of the covariates in year Ψ = 1. To isolate the effect of having only one covariate

distributed as in year Ψ = 0, let us define the effect on the change of the distribution of

h(Y ) given the change on an individual covariate as

G1h − G1

h

(X1−j(0)

)= −

(Xj(1)− Xj(0)

) Π1j

µ1h

. (15)

This analysis further assumes that the changes from Ψ = 0 to Ψ = 1 take place in

this particular order, which is arbitrary. It is also interesting to understand the changes

in the distribution at Ψ = 0 if all covariates had been distributed as in Ψ = 1. That

counterfactual gives us alternative measurements of the contribution of the changes in the

returns of the covariates as well as the changes in the covariates.

3.2.1 Alternative Method

An alternative approach to understand the changes in the distribution of h(Y ) is the pro-

posed method of Machado and Mata (2005), which actually estimates the entire distribution

to isolate the factors that contribute to the changes. The authors introduced a decomposi-

tion technique to understand the changes in the distribution of h(Y ) over a period of time

using quantile regression. Their method is based on the probability integral transformation

theorem: if U is a uniform random variable on [0, 1], then F−1 (U) has distribution F . They

model the conditional quantile of h(Y ) in year Ψ as in equation (11)

Qh(Y )(t|x; Ψ) = xTβΨ (t) ,

and then estimate the marginal densities implied by the conditional model as follows:

1. Generate a random sample of size m from a uniform random variable on [0, 1]:

u1, · · · , um

2. Estimate Qh(Y )(ui|x; Ψ) yielding m estimates βΨ(ui).

3. Generate a random sample of size m with replacement from X(Ψ), the NΨ×P matrix

of data on covariates, denoted by x∗i (Ψ)mi=1.

4. Generate a random sample of h(Y ) that is consistent with the conditional distribution

defined by the model:η∗i (Ψ) ≡ x∗i (Ψ)T βΨ(ui)

mi=1

.

10

To generate a random sample from the marginal distribution of h(Y ) that would have

prevailed in Ψ = 1 if all covariates had been distributed as in Ψ = 0, assuming that

changes in the covariates do not modify the returns of those covariates, one can use X(0)

in step 3 described above.

A counterfactual scenario where only one covariate, xi(1), is distributed as in year

Ψ = 0, requires an additional procedure. In Machado and Mata (2005), the authors

defined a partition of the covariate xi(1) in J classes, Cj(1), with relative frequencies fj(·),for j = 1, · · · , J , and propose the following procedure:

1. Generate η∗i (1)mi=1, a random sample of h(Y ), with size m, that is consistent with

the conditional distribution defined by the model.

2. Take the first class, C1(1), and select all elements of η∗i (1)mi=1 that are generated

using this class, I1 = i|xi(1) ∈ C1(1), that is η∗i (1)i∈I1 . Generate a random sample

of size m× f1(0) with replacement from η∗i (1)i∈I1

3. Repeat step 2 for j = 2, · · · , J .

With the previous two procedures it is possible to generate random samples of counterfac-

tual scenarios; using these random samples it is possible to decompose the changes in the

density of h(Y ).

Let f(η(Ψ)) denote an estimator of the marginal density of an observed sample of h(Y )

in year Ψ and f(η∗(Ψ)) an estimator of the density of h(Y ) based on generated sample

η∗i (Ψ)mi=1. Denote by f(η∗(1);X(0)) an estimate of the counterfactual density in Ψ = 1 if

covariates had been distributed as in Ψ = 0 and f(η∗(1);xi(0)) an estimate of the density

in Ψ = 1 if only covariate xi is distributed as Ψ = 0. If α(·) denotes a summary statistics

(e.g. quantile or scale measure), the decomposition of the changes in α is

α(f(η(1))

)− α

(f(η(0))

)= α

(f(η∗(1))

)− α

(f(η∗(1);X(0))

)︸ ︷︷ ︸

change in covariates

(16)

+α(f(η∗(1);X(0))

)− α

(f(η∗(0))

)︸ ︷︷ ︸

change in returns

(17)

+ residual. (18)

In the same way, the contribution of an individual covariate is

α(f(η∗(1))

)− α

(f(η∗(1);xi(0))

). (19)

11

Notice that this alternative method estimates the entire distribution to isolate the factors

that contribute to the changes of the distribution, as opposite as the method proposed

using the Gini index.

3.3 Inequality in the distribution of Y

The linear decomposition of the Gini index in equation (6) is computed for the trans-

formed variable, h(Y ), but it would be interesting to estimate the impact of the individual

characteristics on the distribution of the positive random variable, Y . The study of the

transformed variable instead of the variable in original scale may create a conflict between

the statistical objective and the economic objective of study. However, given the assumed

properties the transformation h(·), and using the properties of the quantile function,

QY (t|x) = h−1(Qh(Y )(t|x)

)= h−1

(xTβ (t)

), (20)

which implies

L(τ |x) =1

µ

τˆ

0

h−1(xTβ (t)

)dt (21)

and

G (x) = 1− 2

µ

1ˆ

0

τˆ

0

h−1(xTβ (t)

)dtdτ. (22)

The previous relation is not necessarily linear because h−1(·) is not necessarily linear, but

equation (22) links the Gini coefficient of Y with a transformation of a linear combination

of the quantile regression coefficients. Further research is required to explore this link.

4 Estimation Procedure

The measurement of the impacts in the previous section, as well as the decomposition of the

temporal changes in the distribution of h(Y ), relies on estimating Πj =´ 1

0

´ τ0

2βj(t)dtdτ

in equation (6). The natural approximation is Πj =´ 1

0

´ τ0

2βj(t)dtdτ , where βj is the

estimated quantile regression coefficient. To fix ideas, let Qh(Y )(t|x) for t ∈ (0, 1) be the

t-th conditional quantile function of h(Y ), given a vector of covariates, x ∈ RP . Assume

12

that the conditional quantile function can be modeled as

Qh(Y )(t|x) = xTβ (t) , (23)

where β(t) is a vector in RP whose entries are the quantile regression coefficients. Following

Koenker and Bassett (1978), for a given t ∈ (0, 1), β(t) can be estimated by solving

minb∈RP

N∑i=1

ρt(h(yi)− xTi b

), (24)

with

ρt(u) = u (t− I (u < 0)) , (25)

where N is the number of observations and I(·) is the indicator function.

Denote by β(t) the solution to the optimization problem in equation (24), and let βj(t)

denote the j-th component of the estimated vector, which is a function of the quantile t.

The objective is to find

Πj =

1ˆ

0

τˆ

0

2βj(t)dtdτ =n−1∑i=0

τi+1ˆ

τi

τˆ

0

2βj(t)dtdτ, (26)

and one option would be to numerically find the double integrals in equation (26), given a

grid of point evaluations of βj(t). The problem of extending the one-dimensional methods of

integration to multiple dimensions is the increasing required number of function evaluations

(“curse of dimensionality”). If we take m points of evaluation, a numerical approximation

of the double integrals would be proportional to m2 number of functional evaluations.

An alternative approach would be to find a smooth approximation of βj(t) using a known

functional form with known antiderivative, and then compute Πj analytical using the known

functional form of the approximation. An advantage of this approach is that it simplifies

the computations and keeps the number of functional evaluations equal to m, the number

of points of evaluation.

There are many approximation procedures to smooth a continuous function. Perhaps

the most familiar procedure is the use of splines, which approximates a function using a

piecewise continuous polynomial. A disadvantage of the use of splines to estimate Πj is that

it requires the definition of the knots15, and it also demands multiple piecewise integration

that depends on the number of knots. Another procedure available for smoothing is the

15The knots, in the jargon used for splines, are the places where the polynomial pieces connect.

13

use of orthogonal polynomials to approximate any continuous function16. This procedure

generates a unique polynomial of order K that minimizes the square of the error between

the smoothing polynomial and the observed values of the function. A disadvantage of

the use of orthogonal polynomials to estimate Πj is that it requires the definition of the

polynomial order, K. An advantage of using this method is that the computation of the

estimate of Πj can be easily obtain by finding the double integrals of equation (26) in one

step, as opposite as the piecewise integration necessary when using splines.

Assuming that βj(t) is continuous in [0, 1], we can approximate the function using any

family of orthogonal polynomials on that interval, as explained in Judd (1998). Under

the assumptions, the Weierstrass Approximation Theorem guarantees that βj(t) can be

uniformly approximated on [0, 1] by polynomials to any degree of accuracy. There are

many families of orthogonal polynomials, e.g. Legendre, Chebyshev, Laguerre or Hermite;

the main difference between the families is the weighting functions and the domain of

the polynomials. For a function with bounded domain the simplest weighting function is

w(x) = 1, which corresponds to the Legendre polynomials. In the interest of simplicity, I

use the Legendre polynomials to approximate βj(t) on [0, 1].

The domain of the Legendre polynomials is [−1, 1], but we would like to approximate

βj(t) on [0, 1]. To compute the approximation, it is therefore necessary to reshape the

orthogonal Legendre polynomials to [0, 1] as explained in appendix B17. With the reshaped

polynomials, it is possible to find the least-square approximation of βj(t) with a polynomial

of order K. Specifically, let βj,K(t) denote a polynomial of the form

βj,K(t) = α0,jp0(t) + α1,jp1(t) + · · ·+ αK,jpK(t)

where pkKk=0 are the first K + 1 Legendre polynomials on [0, 1]. The objective is to

minimize the sum of the squared errors between βj(t) and βj,K(t) defined by

E(α0,j, · · · , αK,j) =

1ˆ

0

[βj(t)− βj,K(t)

]2

dt.

16A weighting function, w(x), on [a, b] is any function that is positive almost everywhere and has afinite integral on [a, b]. Given a weighting function, the inner product between the polynomials f and g is

defined as 〈f, g〉 =´ baf(x)g(x)w(x)dx. A family of polynomials pn(x) is orthogonal with respect to the

weighting function w(x) if and only if 〈pm, pn〉 = 0 for all m 6= n.17Reshaping the Legendre orthogonal polynomials into the interval [0, 1] requires a simple linear substi-

tution that affects the limits of the integral.

14

For a given K, let˜βj,K(t) = arg min

α0,j ,··· ,αK,jE(α0,j, · · · , αK,j);

the polynomial ˜βj,K(t) is a smoothed approximation of βj(t) based on K + 1 known poly-

nomials on [0, 1]. This polynomial approximation has the advantage of having close form

antiderivatives that are easy to compute. An estimate of the impact of the j-th covariate

to the inequality of the distribution of h(Y ), equation (26), can be easily obtained using

the smoothed approximation.

To understand the effect of the order K of the polynomial approximation, assume for

a moment that a researcher specifies a model for the conditional quantile function of the

logarithm of wages (e.g. h(·) = ln(·)). Moreover, assume that the researcher estimates

a quantile regression coefficient, βj(t), for a grid of m = 69 quantiles. Figure 2 presents

an example of the smooth least-square approximation of this hypothetical estimate of the

quantile regression coefficient, βj(t). For each panel in the figure the dashed line corresponds

to the smoothed polynomial approximation, ˜βj,K(t). Panel (a) exhibits the results using

a polynomial of degree 2; if we only take into account the point estimate of the quantile

regression, a polynomial of degree 2 apparently does a poor job of approximation. If we

consider the 95% confidence interval, it is clear that a polynomial of degree 2 may do a

decent job of approximation. Panel (b) shows an approximation using a polynomial of

degree 6. Here it is evident that increasing the degree of the polynomial improves the

approximation to the point estimate of the quantile regression coefficient, which may be

desire.

From the panels in figure 2 we learn that the approximation to the point estimate of

the quantile regression is better when we use higher degree for the polynomial, but this

approximation does not smooth some jumps of the estimate that may appear presumably

due to lack of data of the corresponding quantile (in the case of wages this lack of data is

generally on the top or bottom 1% of the distribution). It is evident that there is a trade-off

when choosing the degree of the polynomial for the approximation. A bigger degree of the

polynomial increases the accuracy to the point estimate, which may be desirable but is not

necessarily the objective of the approximation. Moreover, from figure 2 it is also evident

that the approximation may be ’good enough’ as long as the smoothing polynomial is inside

the confidence bands.

Bootstrapping is used to find whether the estimate differs significantly from zero and

compute confidence intervals for Πj. To clarify the procedure, let N be the sample size and

R the number of repetitions for the bootstrap. To construct confidence intervals, in each

repetition re-sample N observations with replacement; with the re-sample data estimate

15

the quantile regression coefficients β(t), and find the smooth polynomial approximations˜βj,K(t). Using the smooth approximation, for each repetition calculate an estimate of

equation (26). The point estimate for the impacts Πj is the average of the R previous

estimations, and the 95% bootstrap confidence intervals can be constructed using the 2.5-

th and 97.5-th quantiles of the R repetitions.

The accuracy of the estimation procedure relies on the accuracy of the model for the

conditional quantile function Qh(Y )(t|x), e.g. the linearity of the quantile regression model.

Given the consistency of the quantile regression estimate under regularity conditions (see

Bantli and Hallin (1999) for details), it is reasonable to expect that the estimate of Πj

is accurate. Particularly, suppose for a moment that we know the quantile process β(t)

and hence the true impact estimate Πj; for a large sample, under conditions explained in

Koenker (2005), we know that βn(t) → β(t) and then we expect Πj ≈ Πj. Appendix C

develops exactly this idea and shows that the estimation procedure accurately measures

the impact of the j-th covariate.

5 Sources of Wage Inequality in US

Using the link between the Gini coefficient and the quantile regression explained in the

previous sections, it is possible to implement an empirical application using the rich data

on hourly wages from the ORG of the CPS. To execute the procedure, set h(·) = ln(·).Moreover, given that the constant increase in wage inequality in the US started in the

early 1980s, I use the years 1986 (Ψ = 0) and 1995 (Ψ = 1) to cover the fist decade of this

period of wage disparity18. Further, assume that the conditional quantile function of the

logarithm of wages can be modeled as

Qln(w)(t|x; Ψ) = xTβΨ (t) , (27)

where x is a vector that contains individual characteristics on unionization status, poten-

tial experience and its square; an indicator variable for each classification of schooling19;

nonwhite, women, part time and marital status dummy variables; controls for region and

industry; and a constant term.

Figure 3 presents some of the estimated quantile regression coefficients of equation (27)

for a grid of 69 equally spaced points over the interval (0, 1)20. In each panel, the solid line

18Refer to table 1 for summary statistics. For example, the number of observation in 1986 is 175,533and in 1995 is 162,383.

19The included variables are high school, associate degree and college degree.20The 69 equally spaced points create a grid that has constant step of around 1.4%.

16

corresponds to the estimates in 1986 whereas the dashed line shows the estimates in 1995.

In each case, the shaded regions around the lines correspond to the 95% confidence interval

obtained by computing a Huber sandwich estimate using a local estimate of the sparsity.

In figure 3, the plot corresponding to the race variable (nonwhite) shows that nonwhite

workers earn less than white workers (negative coefficients). Moreover, the estimated coef-

ficients in both years overlap their confidence bands almost over all the wage distribution,

showing that the racial wage gap in the US has remain in similar levels during the years of

analysis. The plot corresponding to the unionization status reveals that unionized workers

earn more than nonunionized workers, and this difference is reduced as we move up through

the wage distribution. Moreover, this wage difference has been reduced for the bottom half

of the wage distribution between 1986 and 1995. The plot of potential experience in figure 3

exhibits that workers with more training earn more, especially at relatively high-paid jobs.

Moreover, more experience increased its returns on the upper quartile of the distribution

of wages for the period of analysis.

As expected, wages increase with education, and it is true across the whole distribution

and all the classifications of schooling. In figure 3, the plots corresponding to high school,

associate degree and college degree show that this effect is more important at the highest

quantiles of the distribution of wages. Moreover, the plots reveal that the returns to

education differ across educational attainment; a higher degree is associated with higher

returns to education. Notably, obtaining a high school degree in the US increased its

returns on the upper quartile of the wage distribution for the period of analysis. With a

different pattern than high school and college degree, the returns to an associate degree

were reduced for the bottom half of the wage distribution and increased on the upper decile

between 1986 and 1995. The biggest change in the returns to schooling happens for college-

educated workers during the years of analysis, where the returns increased almost across

the whole distribution. Finally, the plot corresponding to the gender variable (women) in

figure 3 shows that female workers earn less that male, and this gender gap increases as we

move up through the wage distribution. Furthermore, the gender gap has been reduced in

almost all the distribution or wages, except the higher 95-th percentile.

5.1 Impact of individual characteristics

I choose the order of the polynomial approximation to be 9 for year Ψ = 0 (1986) and Ψ = 1

(1995), with the restriction of using a smoothing approximation inside the confidence bands

for each repetition. Additionally, I set the number of repetitions for the estimation proce-

dure explained in section 4 as R = 1, 000. For each year and repetition, µln is computed as

17

the weighted average of the logarithm of real wages in 2015 dollars. Using this setup, it is

possible to estimate Πj in equation (26) for the years of analysis and compute the impact

estimateΠjµln

.

The results of the estimation for selected covariates are presented in table 2. As dis-

cussed in section 3.1, given a small positive change in covariate j, a positive sign of Πj is

associated with a reduction in the inequality of the distribution of (log) wages. The first

entry of each cell in table 2 presents the impact estimation whereas the second reports the

95% bootstrap confidence interval; each column exhibits the results for the corresponding

year. All covariates presented in table 2, except (potential) experience, are binary vari-

ables21, which is relevant when thinking about small positive changes. It is worth noting

that the impact estimates for gender and race are negative in both years. A negative sign

for the impact estimate is associated with an increase in the Gini index and implies more

inequality in the distribution of (log) wages. In other words, for both years of analysis,

if everything else is held constant, an increment in the proportion of female or nonwhite

workers would increase the inequality in the distribution of hourly wages. It is also in-

teresting to note that the negative impact of gender has been reduced from 1986 to 1995

whereas the negative impact of race remains similar for both years.

Experience, education attainment and unionization have positive impact estimates for

both years of analysis in table 2. A positive impact estimate is associated with a reduction

in the Gini coefficient and a reduction in the inequality of the wage distribution. Experience

has the smallest effect reducing the inequality in the distribution. Moreover, the impact of

experience is similar for both years of analysis. Higher educational attainment has higher

impact on the equality of the wage distribution. Everything else held constant, an increase

in the proportion of college educated workers is associated with a reduction of the Gini

index that is almost four times bigger that a equivalent increase in the proportion of high

school educated workers in both yeas of analysis. Finally, unionization reduces inequality,

with a smaller effect in 1995 than in 1986.

5.2 Changes in the distribution of (log) wages

It is interesting to compare the results of the my methodology in section 3.2 with the

existing algorithm of Machado and Mata (2005) in order to learn about the advantages

of using the proposed method. To implement the procedure of section 3.2, let us divide

the interval (0, 1) into 4 equally spaced sub-intervals (quartiles Q1, Q2, Q3, and Q4), keep

21Gender: dummy for female. Unionization: dummy for union status. Race: dummy for nonwhite. HighSchool, Associate Degree and College are self explanatory.

18

the order of the polynomial approximation to be 9 for years 1986 and 1995, and keep the

number of repetitions in 1, 000 as before. Moreover, to execute the Machado and Mata

(2005) method (MM), algorithm explained in subsection 3.2.1, let us set m = 4, 500 and

let α(·) be the quantile statistic.

Table 3a presents the results when the MM decomposition is implemented. The first

two columns of table 3a present the estimates of selected quantiles of the distribution

of (log) wages for the years of analysis. The third column shows the point estimates

for the overall changes between 1986 and 1995. In addition to the point estimate, this

column reports the 95% bootstrap confidence intervals for the estimate using 1000 bootstrap

samples, computed using the 2.5-th and 97.5-th quantiles of the bootstrap distribution of

the summary statistic. The third column in table 3a shows that wages decreased from the

10-th to the 90-th quantile, and the biggest reduction occurred in the median (a reduction

of 4.1%). The 1-st and 99-th quantiles in the column of changes show that wages increased

in the upper and bottom tail of the distribution, with a significant 12.8% increase in the

99-th quantile. Notice, however, that we can not conclude about the behavior of the

distribution between the presented quantiles. It may be the case that wages are increasing

for some quantiles between the 25-th and the 90-th, but we can not know because not

all the quantiles are reported. In fact, one can realistically only report few editorializing

statistics in a table before a reader gets completely lost in numbers. Indeed, it is word

noting that the last row in table 3a shows that there is no significant change in the Gini

coefficient of (log) wages, presumably because there are opposite movements of the wage

distribution during the years of analysis.

Columns (4) to (6) in table 3a decompose the total changes in the wage distribution

into the part due to changes in covariates (equation (16)), changes due to the changes in

the returns (equation (17)), and the residual (equation (18)). The first entry in each cell

of columns (4) to (6) in table 3a is the point estimate of the change of the distribution for

the given quantile, explained by the indicated factor. The second entry of each cell also

presents the 95% bootstrap confidence interval, as before. The third entry in each cell is

the proportion of the total change explained by the indicated factor. It is interesting to

note that using the MM method none of the analyzed quantiles significantly change due to

the covariates (all confidence intervals include zero). Moreover, the aggregate contribution

to the change in wages, explained by the change in the returns, is negative and significant

from the 25-th to the 90-th quantiles; that is, there is a significant reduction in the wages

of the mentioned quantiles that is explained by the change in the returns of the character-

istics. Finally, columns (7) to (13) in table 3a present the estimates of the change in the

distribution due to the changes in the specific characteristics. It is interesting to note that

19

the algorithm in MM does not find significant effects of any covariate; it is also the case

when summarizing the inequality of the distribution using the Gini index (last row on the

table).

Table 3b presents the decomposition of the changes in the (log) wage distribution when

using the additive decomposition of the Gini index, equation (14). The structure of the

table is similar to the structure of table 3a. In other words, in each cell of the table, the

first entry corresponds to the point estimate, the second entry reports the 95% bootstrap

confidence intervals for the estimate using 1000 bootstrap samples and the third entry

reports the proportion of the total change explained by the indicated factor.

A negative point estimate in table 3b implies a reduction in the Gini coefficient, and

the negative sign is associated here with a reduction in the inequality of the distribution.

First two columns in table 3b present the estimates of the Gini index for the corresponding

quartile in the indicated year; adding up the first four rows of the panel ( rows Q1, Q2,

Q3 and Q4) recovers the point estimate of the last row (Total). As before, third column

in table 3b shows the point estimates for the overall changes, Notice that, as in the MM

decomposition, there is no significant change in the Gini index of (log) wages for the

overall distribution (the 95-th bootstrap confidence interval in column (3) includes zero

in the row Total), but the only positive and significant change in the participation of the

Gini coefficient occurs in the forth quartile (row Q4) , that is, only the upper tail of the

distribution contributes to a significant increase in the inequality of wages. The proposed

method has the benefit of being able to capture this effect, something that we could not

assert using the MM algorithm.

Columns (4) to (6) in table 3b decompose the total changes in the distribution due to

the changes in the covariates, changes in the returns, and a residual (i.e., the components

of equation (14)). The point estimates in column (4), in table 3b, shows that the overall

changes in the covariates, if anything, have reduced the inequality in the wage distribution,

although none of the computed effects is significantly different from zero. Column (5) of the

same table exhibits that all quartiles, and the entire wage distribution, have significantly

increased their inequality due to changes in the returns of the coefficients. The aggregate

changes in the returns of the covariates are consistent with significant increments in the

inequality of the distribution of wages, particularly higher for the upper quartiles of the

distribution. The findings of the previously discussed columns are consistent with the

results of the MM technique, although the change in the upper and lower quartiles of the

distribution is more clear when using the proposed method.

Finally, columns (7) to (13) in table 3b show the changes in the wage distribution

due to changes in some specific characteristics. As in the case of the implemented MM

20

algorithm, most of the changes in the individual covariates have no significant effect in

the change of the distribution. However, when using the proposed method, the changes in

the proportion of workers with high school and associate degree are significantly related to

changes in the wage distribution. The proportion of workers with a high school degree in

1986 was 39.5%. This fell to 32.8% in 1995. On the contrary, the fraction of workers with

an associate degree in 1986 was 23.1% whereas it was 29.8% in 1995. Given the positive

impact of having a high school degree on reducing inequality (refer to table 2 for the

impact estimates), the contraction in the proportion of workers with a high school degree

is associated with an increase in the inequality of the distribution for all the quartiles and

the overall distribution. On the other hand, having an associate’s degree has a positive

impact on the wage distribution, impact that is bigger than that for high school (refer to

table 2). As a result, the increase in the proportion of workers with an associate’s degree is

associated with a reduction in the inequality of the distribution. Within all quartiles and

for the entire distribution.

In summary, the increase in inequality of (log) wages distribution between 1986 and

1995, is significantly accounted for the changes in the returns (coefficients) of the charac-

teristics. Using the MM algorithm and the proposed method the conclusions are similar,

but the proposed method allow us to identify a significant increment in the inequality of

the wage distribution for all the quartiles, as opposite to few summary statistics. Moreover,

the proposed methodology isolates the effect towards equality in the wage distribution due

to the change in the proportion of workers with associate degree, something that may be

interesting from a policy perspective, if the reduction of disparity among the society is the

objective of the planers.

6 Conclusion

This project was undertaken to propose a methodology to evaluate the contribution of

various factors to the disparity of the (log) wage distribution in the US. Using the link

between the Lorenz curve and the quantile function, the paper explored the theoretical link

between the Gini index and the quantile regression. Using this relationship, I proposed an

estimation procedure to measure the impact of individual characteristics in the distribution

of wages. Moreover, using the estimates of the Gini index for the years 1986 and 1995, I

also decomposed the changes in the wage distribution due to changes in the covariates and

the returns to those covariates using counterfactual scenarios.

This work contributes to existing literature in decomposition methods by directly mea-

21

suring inequality in the (log) wage distribution through the Gini coefficient without model-

ing the density. Previous decomposition methods focused on the visual evidence of kernel

estimates or the analysis of selected quantiles without considering the information summa-

rized by an inequality measure. By assuming a linear quantile function of the (log) wages

conditional on covariates, the proposed method shows that the impact of various factors

has changed over time. More importantly, from the comparison between the proposed

method and the algorithm in Machado and Mata (2005), it is clear that both procedures

reach similar conclusion, but the proposed decomposition technique shows sources of wage

inequality that were not clearly isolated by the method in Machado and Mata (2005).

A key strength of the present study is that for the year of analysis, the proposed method

shows that only the upper tail of the distribution contributes to a significant increase in

wage inequality. Moreover, the proposed decomposition technique finds that changes in

the proportion of workers with high school and associate degree are significantly related

to changes in the wage distribution. These results extend our knowledge on the sources

of wage inequality. In fact, the proposed decomposition method isolates the effect towards

equality in wage distribution due to the change in the proportion of workers with associate

degrees. This may have interesting implications from a policy perspective.

A limitation of this study, as an analogue to Oaxaca (1973) decomposition, is the

presumption that changes in the characteristics do not modify the returns of those charac-

teristics. Moreover, the analysis only accounts for changes in the covariates from the year

1986 to the year 1995, but the proposed decomposition technique could have considered

counterfactual scenarios in the reverse order. More importantly, the linear decomposition

works for a particular transformation of wages for which the conditional quantile functions

are assumed to be linear in parameters (i.e., log wages), but this may not be a natural scale

to analyze the disparity of the distribution.

Further research could usefully explore how to account for the general equilibrium ef-

fects given changes in the distribution of the covariates, because those changes will also

affect the returns to the characteristics. Moreover, a future study investigating different

counterfactual scenarios and more recent years of analysis would be very interesting. A

natural progression of this work is to extend the proposed method to the untransformed

variable (i.e., wages) to address questions related to the inequality of the distribution of

the variable in levels.

22

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Figures

Figure 1a: Kernel density estimates of men’s real log wages 1980 - 2015 ($2015)

1

25

Figure 1b: Kernel density estimates of women’s real log wages 1980 - 2015 ($2015)

1

26

Figure 2: Example of approximation using Legendre polynomials

(a) Polynomial of degree 2 (b) Polynomial of degree 6

Panels (a) and (b) present a generic example of the least-square approximation for the coefficients from

a quantile regression using Legendre polynomials of various orders. In both panels a generic βj(t) ispresented with its corresponding 95% confidence interval. In each panel, the dashed line corresponds to

the polynomial approximation,˜βj,K(t).

27

Figure 3: Selected Coefficients Estimates From the Quantile Regression

Selected coefficients estimates from the quantile regression of equation (27). Each βj(t) is presented with itscorresponding 95% confidence interval. In each plot you find superimposed the corresponding coefficientsfor year 1986, (Ψ = 0) and year 1995 (Ψ = 1).

28

Tables

Table 1: Summary Statistics for the CPS 1980-1997

Men

Women

Yea

rLogRea

lMinim

um

WageA

LogRea

lW

age

UnionB

Nonwhite

Educa

tion

Experience

CNo.Obs.

LogRea

lW

age

UnionB

Nonwhite

Educa

tion

Experience

CNo.Obs.

1980

2.19

3.11

.0.17

12.75

18.26

106,936

2.70

.0.18

12.76

17.64

87,887

1981

2.17

3.09

.0.18

12.82

18.19

99,530

2.69

.0.18

12.80

17.68

83,220

1982

2.11

3.09

.0.18

12.92

18.23

92,249

2.71

.0.19

12.91

17.68

79,586

1983

2.08

3.09

0.28

0.18

12.99

18.11

91,049

2.73

0.18

0.19

12.99

17.59

79,013

1984

2.03

3.08

0.26

0.18

13.02

17.91

92,729

2.73

0.17

0.19

13.03

17.53

80,787

1985

2.00

3.09

0.25

0.20

13.03

18.02

93,763

2.75

0.16

0.20

13.07

17.58

82,817

1986

1.98

3.10

0.24

0.20

13.07

17.95

92,081

2.77

0.16

0.20

13.12

17.64

83,452

1987

1.94

3.09

0.23

0.21

13.08

18.00

92,005

2.78

0.15

0.21

13.15

17.69

84,676

1988

1.90

3.08

0.23

0.22

13.11

17.99

88,084

2.78

0.15

0.21

13.19

17.78

81,193

1989

1.86

3.10

0.22

0.22

13.14

18.09

89,459

2.79

0.15

0.22

13.23

18.01

82,979

1990

1.93

3.08

0.21

0.23

13.12

18.05

93,500

2.79

0.15

0.23

13.27

18.01

87,322

1991

2.00

3.07

0.21

0.24

13.18

18.25

90,127

2.80

0.15

0.23

13.33

18.25

85,313

1992

1.97

3.06

0.21

0.24

13.00

18.55

88,358

2.81

0.15

0.23

13.14

18.64

84,513

1993

1.94

3.05

0.20

0.24

13.06

18.59

86,804

2.82

0.15

0.23

13.20

18.78

83,902

1994

1.92

3.05

0.20

0.24

13.09

18.62

82,354

2.83

0.15

0.24

13.24

18.81

80,342

1995

1.89

3.05

0.19

0.24

13.12

18.74

82,510

2.81

0.14

0.24

13.26

18.96

79,873

1996

1.97

3.04

0.19

0.26

13.12

18.98

73,034

2.81

0.14

0.25

13.30

19.08

71,468

1997

2.03

3.05

0.18

0.27

13.10

19.09

74,576

2.83

0.14

0.26

13.30

19.27

72,763

1998

2.01

3.09

0.18

0.27

13.13

19.22

75,589

2.86

0.13

0.27

13.32

19.34

73,450

1999

1.99

3.12

0.18

0.27

13.18

19.34

76,746

2.88

0.13

0.27

13.35

19.45

74,432

2000

1.96

3.13

0.17

0.28

13.18

19.46

77,712

2.89

0.13

0.28

13.36

19.60

75,166

2001

1.93

3.14

0.17

0.28

13.23

19.73

82,348

2.92

0.13

0.28

13.41

19.85

80,038

2002

1.91

3.16

0.16

0.28

13.26

20.01

87,798

2.94

0.13

0.28

13.46

20.09

86,464

2003

1.89

3.15

0.16

0.31

13.23

20.16

85,502

2.95

0.13

0.30

13.50

20.46

85,298

2004

1.87

3.15

0.15

0.32

13.25

20.25

84,260

2.94

0.13

0.30

13.54

20.56

83,181

2005

1.83

3.14

0.15

0.32

13.24

20.47

84,803

2.94

0.13

0.31

13.58

20.69

83,569

2006

1.80

3.14

0.14

0.33

13.26

20.52

84,987

2.95

0.12

0.31

13.61

20.79

82,819

2007

1.90

3.14

0.14

0.33

13.31

20.60

83,717

2.95

0.13

0.32

13.67

20.87

82,302

2008

1.98

3.14

0.15

0.33

13.40

20.81

82,286

2.96

0.13

0.32

13.75

21.04

81,588

2009

2.08

3.17

0.15

0.33

13.46

21.15

78,864

2.98

0.13

0.32

13.80

21.31

79,945

2010

2.06

3.16

0.14

0.33

13.49

21.27

78,083

2.98

0.13

0.32

13.85

21.44

78,894

2011

2.03

3.13

0.14

0.34

13.52

21.19

77,769

2.97

0.13

0.32

13.89

21.50

77,611

2012

2.01

3.14

0.13

0.35

13.56

21.35

78,070

2.96

0.12

0.35

13.92

21.53

76,725

2013

2.00

3.13

0.13

0.36

13.58

21.35

78,030

2.97

0.12

0.35

13.99

21.45

76,301

2014

1.98

3.13

0.13

0.37

13.59

21.35

78,714

2.97

0.12

0.36

14.02

21.35

76,464

2015

1.98

3.15

0.13

0.37

13.63

21.30

77,740

2.99

0.12

0.37

14.06

21.29

75,537

A2015ConstantDollars

BUnion

statu

sofwork

ers

wasnotcollected

inth

eoutg

oing

rota

tion

gro

up

supplements

from

1980

to1982.

However,

using

theM

ay

pension

supplementit

may

bepossible

estim

ate

this

summary

statistic

forasu

bsa

mple

ofth

epopulation

CPotentialexperienceis

computed

asage-years

ofeducation

-5

29

Table 2: Impact Estimates of Selected Covariates

1986 1995

Gender -0.073 -0.056-0.074;-0.071 -0.058;-0.055

Unionization 0.078 0.0660.075; 0.080 0.064; 0.069

Race -0.038 -0.041-0.040;-0.036 -0.044;-0.039

High School 0.058 0.0610.056; 0.060 0.058; 0.063

Associate Degree 0.105 0.0960.103; 0.108 0.093; 0.098

College 0.202 0.2270.199; 0.205 0.224; 0.230

Experience 0.0104 0.01020.0102;0.0107 0.01;0.0105

Impact estimates for selected covariates from the model in equation (27) for year 1986, (Ψ = 0) and year

1995 (Ψ = 1). Each impact estimate,Πj

µln, is presented in the first entry of each cell on the table. The

second entry of each cell on table shows the 95% is bootstrap confidence interval.

30

Table 3a: Decomposition of the Changes in the Distribution of (log) Wages Using the Methodof Machado and Mata (2005)

Marginals

Aggregate

Con

tribution

sIndividual

Covariates

1986

1995

Chan

geCovariates

Returns

Residual

Gender

Unionization

Race

HighSchool

Associate

Degree

College

Experience

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

1stquan

t.1.469

1.501

0.032

0.036

0.012

-0.0151

0.021

0.004

-0.008

0.006

0.053

0.021

-0.028

-0.319;0.363

-0.201;0.212

-0.227;0.199

-0.228;0.287

-0.232;0.217

-0.260;0.164

-0.242;0.261

-0.182;0.280

-0.224;0.272

-0.248;0.236

1.109

0.356

-0.465

0.634

0.127

-0.255

0.174

1.640

0.632

-0.877

10th

quan

t.2.065

2.051

-0.014

0.003

-0.048

0.0308

0.008

-0.010

-0.015

-0.007

0.012

0.000

0.023

-0.065;0.028

-0.057;0.060

-0.111;0.015

-0.072;0.087

-0.079;0.068

-0.091;0.075

-0.084;0.075

-0.064;0.092

-0.081;0.078

-0.053;0.102

-0.229

3.428

-2.200

-0.543

0.694

1.052

0.529

-0.881

0.032

-1.606

25th

quan

t.2.380

2.362

-0.018

0.012

-0.078

0.0478

-0.012

-0.029

-0.037

-0.027

-0.018

-0.031

0.016

-0.094;0.008

-0.040;0.062

-0.131;-0.025

-0.074;0.056

-0.089;0.031

-0.106;0.029

-0.084;0.041

-0.077;0.047

-0.094;0.040

-0.056;0.082

-0.644

4.255

-2.611

0.648

1.574

2.005

1.464

0.991

1.717

-0.855

Median

2.785

2.744

-0.041

0.037

-0.111

0.0331

0.012

-0.006

-0.025

0.014

-0.006

-0.007

0.031

-0.113;0.008

-0.013;0.087

-0.157;-0.059

-0.060;0.073

-0.069;0.063

-0.093;0.038

-0.051;0.076

-0.076;0.065

-0.072;0.063

-0.031;0.092

-0.890

2.697

-0.807

-0.288

0.145

0.600

-0.351

0.145

0.161

-0.756

75th

quan

t.3.213

3.175

-0.038

0.056

-0.108

0.0135

0.015

-0.005

-0.017

0.030

-0.022

0.002

0.038

-0.106;0.036

-0.006;0.122

-0.161;-0.049

-0.059;0.104

-0.078;0.075

-0.089;0.057

-0.053;0.121

-0.097;0.056

-0.073;0.079

-0.034;0.128

-1.498

2.854

-0.357

-0.386

0.140

0.461

-0.795

0.571

-0.053

-1.004

90th

quan

t.3.543

3.536

-0.007

0.067

-0.088

0.0149

0.022

-0.028

-0.015

0.010

-0.006

0.022

0.041

-0.088;0.102

-0.007;0.149

-0.148;-0.015

-0.059;0.123

-0.123;0.077

-0.106;0.082

-0.084;0.117

-0.103;0.088

-0.082;0.111

-0.054;0.133

-9.511

12.642

-2.130

-3.193

3.966

2.151

-1.371

0.917

-3.193

-5.792

99th

quan

t.3.988

4.116

0.128

0.079

0.085

-0.0362

0.059

0.059

0.083

0.067

0.039

0.055

0.041

0.012;0.304

-0.098;0.245

-0.097;0.218

-0.123;0.251

-0.130;0.226

-0.115;0.250

-0.164;0.265

-0.190;0.224

-0.158;0.227

-0.220;0.250

0.617

0.666

-0.284

0.464

0.464

0.651

0.523

0.304

0.429

0.323

Giniof

logW

11.382

11.722

0.340

0.278

0.204

-0.1425

0.105

0.124

0.257

0.294

-0.149

0.163

0.064

-0.374;1.015

-0.284;0.846

-0.375;0.766

-0.620;0.867

-0.574;0.804

-0.446;0.963

-0.378;1.032

-0.885;0.580

-0.601;0.894

-0.679;0.838

0.820

0.600

-0.420

0.311

0.366

0.757

0.866

-0.437

0.480

0.188

Note

1:

The

firs

tentr

yin

each

cell

isth

ep

oin

test

imate

din

the

change

inth

eatt

ribute

of

the

densi

ty,

expla

ined

by

the

indic

ate

dfa

cto

r

Note

2:

The

second

entr

yis

the

95%

confi

dence

inte

rval

for

the

change

Note

3:

The

thir

dentr

yis

the

pro

port

ion

of

the

tota

lch

ange

expla

ined

by

the

indic

ate

dfa

cto

r

31

Table 3b: Decomposition of the Changes in the Distribution of (log) Wages Using theProposed Method

Marginals

Aggregate

Con

tribution

sIndividual

Covariates

Gini1986

Gini1995

Chan

geCovariates

Returns

Residual

Gender

Unionization

Race

HighSchool

Associate

Degree

College

Experience

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

Q1

1.8800

1.8628

-0.0173

-0.0082

0.0858

-0.0949

0.0043

0.0134

0.0086

0.0208

-0.0300

-0.0140

-0.0450

-0.171;0.115

-0.078;0.061

0.074;0.100

-0.010;0.021

-0.003;0.029

-0.001;0.018

0.006;0.034

-0.048;-0.011

-0.055;0.030

-0.095;0.007

0.472

-4.964

5.492

-0.247

-0.777

-0.495

-1.203

1.738

0.812

2.604

Q2

3.8104

3.8529

0.0425

-0.0722

0.2326

-0.1180

0.0139

0.0416

0.0286

0.0683

-0.1048

-0.0528

-0.1686

-0.207;0.278

-0.289;0.150

0.203;0.268

-0.034;0.067

-0.010;0.091

-0.003;0.061

0.020;0.112

-0.169;-0.039

-0.205;0.113

-0.356;0.026

-1.698

5.473

-2.775

0.327

0.978

0.672

1.606

-2.465

-1.242

-3.967

Q3

3.8030

3.9578

0.1548

-0.1893

0.3778

-0.0338

0.0248

0.0662

0.0493

0.1249

-0.1988

-0.1010

-0.3283

-0.097;0.406

-0.575;0.213

0.329;0.431

-0.060;0.120

-0.015;0.145

-0.005;0.106

0.036;0.204

-0.320;-0.075

-0.393;0.216

-0.694;0.050

-1.223

2.441

-0.218

0.160

0.427

0.319

0.807

-1.284

-0.653

-2.121

Q4

1.8885

2.0463

0.1578

-0.3606

0.4981

0.0202

0.0366

0.0839

0.0681

0.1912

-0.3105

-0.1551

-0.5210

0.018;0.301

-0.913;0.221

0.429;0.570

-0.089;0.177

-0.019;0.184

-0.007;0.146

0.056;0.313

-0.500;-0.117

-0.603;0.331

-1.102;0.080

-2.285

3.157

0.128

0.232

0.532

0.432

1.212

-1.968

-0.983

-3.302

Total

11.382

11.722

0.340

-0.6302

1.1944

-0.2264

0.0796

0.2051

0.1546

0.4052

-0.6441

-0.3230

-1.0629

-0.374;1.015

-1.855;0.660

1.040;1.364

-0.192;0.385

-0.047;0.450

-0.017;0.331

0.118;0.663

-1.037;-0.242

-1.256;0.690

-2.247;0.162

-1.866

3.536

-0.670

0.236

0.607

0.458

1.200

-1.907

-0.956

-3.147

Note

1:

The

firs

tentr

yin

each

cell

isth

ep

oin

test

imate

din

the

change

inth

eatt

ribute

of

the

densi

ty,

expla

ined

by

the

indic

ate

dfa

cto

r

Note

2:

The

second

entr

yis

the

95%

confi

dence

inte

rval

for

the

change

Note

3:

The

thir

dentr

yis

the

pro

port

ion

of

the

tota

lch

ange

expla

ined

by

the

indic

ate

dfa

cto

r

32

Appendices

A The Lorenz Curve as an Expected Value

Let Z be a continuous random variable with support in R, with cumulative distribution

function FZ(z) and probability density function given by fZ(z). Assume that E [z|z ≤ a]

exist and is finite for all a ∈ R. Let τ ∈ (0, 1) and denote the quantile function by

QZ(t) = inf z : FZ(z) ≥ t = F−1Z (t). Also denote zτ = QZ(τ). Then,

τˆ

0

QZ(t)dt =

zτˆ

−∞

zfZ(z)dz

= τ

zτˆ

−∞

zfZ(z)

FZ(zτ )dz

= τ

zτˆ

−∞

zfZ<zτ (z)dz

= τE [z|z ≤ zτ ] . (28)

From equation (28) it is clear that

1ˆ

0

QZ(t)dt = E [z] . (29)

Moreover, ∀τ ∈ (0, 1)

E [z|z ≤ zτ ] =

zτˆ

−∞

zfZ(z)

FZ(zτ )dz ≤ zτ

zτˆ

−∞

fZ(z)

FZ(zτ )dz = zτ

and,

zτ = zτ

∞

zτ

fZ(z)

1− FZ(zτ )dz ≤

∞

zτ

zfZ(z)

1− FZ(zτ )dz = E [z|z ≥ zτ ] .

Then, ∀τ ∈ (0, 1)

0 ≤ (1− τ) (E [z|z ≥ zτ ]− E [z|z ≤ zτ ]) ,

33

which implies

E [z|z ≤ zτ ] ≤ τE [z|z ≤ zτ ] + (1− τ)E [z|z ≥ zτ ] = E [z] (30)

Let Y be a continuous and positive random variable, with cumulative density function

FY (y), quantile function denoted by QY (t) = inf y : FY (y) ≥ t = F−1Y (t), yτ = QY (τ),

and 0 < E [y] < ∞. Let h (·) be a continuous and monotone function. Define Z = h(Y )

and µh = E [h(y)] = E [z]. Assume that h(·) is such that h(Y ) ≥ 0 and 0 < µh < ∞. By

the properties of the quantile function, Qh(Y )(t) = h(QY (t)). Then, using equations (28)

and (29), the Lorenz curve of the transformed variable is given by

Lh(τ) =1

µh

τˆ

0

Qh(Y )(t)dt =τE [h(y)|h(y) ≤ h(yτ )]

E [h(y)].

Using the inequality in (30) it is clear that the transformed Lorenz curve takes values

between 0 and 1.

By the definition of the Gini coefficient we know that

Gh = 1− 2

1ˆ

0

Lh(τ)dτ

= 1− 2

µh

1ˆ

0

τE [h(y)|h(y) ≤ h(yτ )] dτ.

The Gini index, Gh, is always smaller or equal than 1 because h(Y ) ≥ 0. Moreover, using

inequality (30) we have

E [h(y)|h(y) ≤ h(yτ )] ≤ E [h(y)] ,

or,1ˆ

0

τE [h(y)|h(y) ≤ h(yτ )] dτ ≤1ˆ

0

τE [h(y)] dτ =1

2E [h(y)] .

In other words, the Gini index, Gh, is always positive.

34

B Integral Approximation

The Gini index can be written as

Gh (x) = 1− 1

µh

P∑j=1

xj

1ˆ

0

τˆ

0

2βj(t)dtdτ,

and we would like to approximate

βj(t) ≈ ˜βj,K(t) = α0p0(t) + · · ·+ αkpk(t) =K∑i=0

αipi(t),

a polynomial of degree K. It is easy to implement Least Squares Orthogonal Polynomial

Approximation when each pi(t) is a Legendre polynomial on [0, 1]. Denote by pi(u) the

Legendre polynomial on [−1, 1]. Using the transformation u = 2t − 1 it is clear that

pi(t) = pi(u).

Denote by I(u) the indefinite integral of pi(u), then

τˆ

0

pi(t)dt =1

2

2τ−1ˆ

−1

pi(u)du =1

2[I(2τ − 1)− I(−1)] .

Denote by II(w) the indefinite integral of I(w), then

bˆ

a

τˆ

0

pi(t)dtdτ =1

2

bˆ

a

[I(2τ − 1)− I(−1)] dτ.

By defining w = 2τ − 1 it follows

bˆ

a

τˆ

0

pi(t)dtdτ =1

4[II(2b− 1)− II(2a− 1)]− 1

2I(−1) [b− a] .

Note that, for each pi(u), the indefinite integral I(u) is another polynomial. The same

is true for II(w). The advantage of this procedure the easy implementation of the integral

of known polynomials.

35

C Accuracy of the Estimating Procedure

Set h(·) = ln(·) and assume that the conditional quantile function of the logarithm of

hourly wages can be modeled as

Qln(w)(t|x) = xTβ (t) = xageβ1(t) + x2ageβ2(t),

where x is a vector that contains age and its square. Suppose that we know the quantile

process β(t), defined by β1(t) = 0.2t + 0.05t2 and β2(t) = −0.0023t− 0.0003t2. Therefore,

the true values of Πj =´ 1

0

´ τ0

2βj(t)dtdτ are Π1 = 0.075 and Π2 = −0.00082.

For the purpose of this performance exercise, I create a random sample of N = 8, 000

observations with normally distributed ages, mean age 35 years and standard deviation 8

years. Using a normal error term with zero mean and variance proportional to the quantile

and employing the random sample generated before, I computed log wages consistent with

the assumed quantile process. Figure C.1 exhibits the sample as well as selected lines for

the conditional quantile functions. Notice that the volatility of the (low) wages increases for

higher quantiles; also note the concave shape of the conditional quantile functions, pattern

that is consistent with the findings in the literature.

Figure C.1: Random Sample and Selected Conditional Quantile Functions

36

Table C.1: Results of Performance Exercise

Πj Πj

(1) (2)

age 0.075 0.0720.068;0.077

age2 -0.00082 -0.00080-0.00091;-0.00068

Note 1: The first entry reports the average ofthe estimates in each of the 1,000 repetitions.

Note 2: The second entry reports the 95%confidence interval

To perform the proposed estimation, I set the number of repetitions for the bootstrap

to be R = 1, 000 and the order of the polynomial approximation to be K = 6. Using

the smooth approximation for each repetition, I calculate the point estimates for Πj and

report the average in column (2) of table C.1. The 95% bootstrap confidence intervals

are constructed using the 2.5-th and 97.5-th quantiles of the R repetitions. The second

entrance of each cell in column (2) of table C.1 reports those bootstrap confidence intervals.

As expected, the confidence intervals include the true values of Πj, reported on column (1) of

the same table. This performance exercise shows that the estimation procedure accurately

measures the impact of the j-th covariate.

37