understanding sources of wage inequality: additive...
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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: [email protected]. 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
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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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24
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