Volume 44, Issue 1
Oaxaca-Blinder decomposition of changes in means and inequality: A
simultaneous approach
Arthur Charpentier
UQAM

Emmanuel Flachaire
Aix-Marseille University, AMSE

Abstract
In this paper, we show that a decomposition of changes in inequality, with the mean log deviation index, can beobtained directly from the Oaxaca-Blinder decompositions of changes in means of incomes and log-incomes. It allows
practitioners to conduct simultaneously empirical analyses to explain which factors account for changes in means andin inequality indices between two distributions with strictly positive values.

Paper #4 in Special issue “In memory of Pr. Michel Terraza” This work was supported by the Natural Sciences and Engineering ResearchCouncil of Canada Grant NSERC-2019-07077, and by the French government under the “France 2030” investment plan managed by the
French National Research Agency (reference: ANR-17-EURE-0020, ANR-17-CE41-0007, ANR-19-FRAL-0006) and from Excellence
Initiative of Aix-Marseille University - A*MIDEX.
Citation: Arthur Charpentier and Emmanuel Flachaire, (2024) ''Oaxaca-Blinder decomposition of changes in means and inequality: A
simultaneous approach'', Economics Bulletin, Volume 44, Issue 1, pages 308-320
Contact: Arthur Charpentier - charpentier.arthur@uqam.ca, Emmanuel Flachaire - emmanuel.flachaire@univ-amu.fr.
Submitted: May 24, 2023. Published: March 30, 2024.

Picture credits: Virginie Terraza

Special issue “In memory of Professor Michel Terraza”

1

Introduction

The Oaxaca-Blinder decomposition is widely used in empirical studies toanalyze the sources of changes in income distributions (Oaxaca 1973, Blinder
1973). It separates changes due to diﬀerences in characteristics between two
groups, from other eﬀects. Initially developped to explain the male-female
wage diﬀerence in means (gender gap), this method has been extended tovarious distributional measures (DiNardo et al. 1996, Fortin et al. 2011).
This paper focuses on the Oaxaca-Blinder decomposition of changes ininequality indices. When comparing inequality between two years or two
countries, diﬀerence in inequality may be mainly driven by diﬀerences incharacteristics between two groups (age, education, etc.), while there would
be no diﬀerence if individuals from the two groups share, on average, thesame characteristics. With an Oaxaca-Blinder decomposition, we are thenconcerned by measuring the share of the diﬀerence in inequality which is notdue to diﬀerences in characteristics between individuals in the two groups.
Such decomposition is usually done with regression based on the RecenteredInﬂuence Function (RIF), when applied to inequality indices (Ferreira et al.
2017, Firpo et al. 2018, Rios-Avila 2020, Gradı́n 2020).
In this paper, we show that the use of RIF regression is not required
to obtain a decomposition of changes in inequality indices. To decomposechanges in inequality, we consider the mean log deviation (MLD) measure,also known as the second Theil index or Generalized Entropy index withparameter zero, which is deﬁned for strictly positive values. The MLD
index has an attractive property compared to other inequality indices. It
is the only relative inequality measure which respects both the principle oftransfer (a transfer from an individual to a poorer individual cannot increase
inequality) and the principle of monotonicity in distance (when a rich getricher inequality cannot decrease), see Cowell and Flachaire (2018). Here,
we show another attractive feature of the MLD index. An Oaxaca-Blinder
decomposition of changes in MLD inequality indices can be obtained directlyfrom the results of a standard decomposition of changes in means of incomes
and log-incomes. In other words, standard Oaxaca-Blinder decomposition
analysis can be used to simultaneously explain which factors account forchanges in means and in inequality indices between two distributions.
In section 2, we present the standard Oaxaca-Blinder decomposition ofchanges in means. In section 3, we show that an Oaxaca-Blinder decomposition of changes in MLD indices can be derived from the results obtained on
the decomposition of changes in means. Finally, an application is presented
in section 4.
1

2

Decomposing difference in means

The Oaxaca-Blinder decomposition is mainly used to study changes in means
between two groups A and B of individuals. Let us consider the following
linear regression models:
log yA = XA βA + εA

(1)

log yB = XB βB + εB(2)where E(εA |XA ) = E(εB |XB ) = 0. From an OLS estimation, we have
log y A = X̄A β̂A(3)

log y B = X̄B β̂B(4)where log y A , log y B , X̄A and X̄B are the means of the variables for observations in group A and group B, including a constant vector. By substracting
these two equations, the diﬀerence of log-incomes means between group A
and B is then deﬁned as follows:
log y A − log y B = X̄A β̂A − X̄B β̂B(5)

By adding and substracting X̄A β̂B , we obtain:∆µ = log y A − log y B = X̄A (β̂A − β̂B ) + (X̄A − X̄B )β̂B
|
{z
} |
{z
}
∆S
µ (unexplained)(6)∆Xµ (explained)

It is a standard Oaxaca-Blinder aggregated decomposition, into explained
and unexplained eﬀects:• The explained eﬀect reﬂects the diﬀerences in (average) covariates between A and B. For instance, it is diﬀerent from zero when the averagelevel of education is not the same in A and B.
• The unexplained eﬀect reﬂects the diﬀerences of the ”eﬀects” of the covariates between A and B. For instance, education may explain income
diﬀerences among individuals in A more strongly than in B.
The additive property of linear regression models allows us to decompose the explained and unexplained eﬀects in greater details. Indeed, thecontributions of the kth covariate to the explained and unexplained eﬀects
in (6) are, respectively, equal to:∆Xµ,k = (X̄A,k − X̄B,k )β̂B,k(7)∆Sµ,k = X̄A,k (β̂A,k − β̂B,k )(8)2

However, the detailed decomposition analysis of the unexplained eﬀect produces arbitrary results. It is known to have an identiﬁcation problem: thecoeﬃcients eﬀect of sets of dummy variables and the intercept are sensitive
to the choice of reference groups (Oaxaca and Ransom 1999, Yun 2005).
One solution is to adjust the coeﬃcients (Gardeazabal and Ugidos 2004).
The shares of the explained and unexplained eﬀects are sensitive to thedecomposition method. Indeed, another decomposition can be obtained byadding and substracting X̄B β̂A in (5):∆µ = log y A − log y B = X̄B (β̂A − β̂B ) + (X̄A − X̄B )β̂A
{z
} |
{z
}
|
∆S
µ (unexplained)(9)∆Xµ (explained)

The diﬀerence comes from the reference group used in the decomposition:• decomposition in (6) is based on X̄A β̂B , that is, what would be thelog-income mean of individuals in A if they would be living in B,• decomposition in (9) is based on X̄B β̂A , that is, what would be thelog-income mean of individuals in B if they would be living in A.
The choice of the reference group is arbitrary and the decomposition
can be viewed as arbitrary as well. There is no general guidance to thischoice (Fortin et al. 2011). Nevertheless, the reference group has some
economic meaning in the decomposition of changes in inequality. Indeed, theunexplained component in (6) can be interpreted as an average treatmenteﬀect on individuals in group A:∆Sµ = X̄A (β̂A − β̂B ) = log y A − X̄A β̂B(10)

The term X̄A β̂B is a counterfactual, it represents the mean of log-incomes
induced by individuals in A if they would be living in B. Similarly, theunexplained component in (9) can be interpreted as an average treatmenteﬀect on individuals in group B.

3

Decomposing difference in inequality indicesIn this section, we show that the Oaxaca-Blinder decomposition of the difference of MLD inequality indices can be obtained from the decompositionof the diﬀerence of log-incomes means from the previous section.
From a sample of incomes {y1 , y2 , . . . , yn }, the MLD index is computed
as follows:
✓ ◆
n
1X
yi
ν=−
(11)
= log ȳ − log y
log
n
ȳ
i=1

3

It is the diﬀerence between the log of incomes mean and the mean of logincomes. The diﬀerence of the MLD indices between individuals from groups
A and B is then equal to:νA − νB = log(ȳA /ȳB ) + log y B − log y A(12)

In other words, the diﬀerence of the MLD inequality indices between two
groups can be rewritten as a function of the diﬀerence of the mean of logincomes deﬁned in (6) and (9):∆ν = log(ȳA /ȳB ) − ∆µ(13)

We need to further decompose log(ȳA /ȳB ) into explained and unexplained components. Let us consider the following linear regression models:yA = X A δ A + η A(14)

yB = X B δ B + η B(15)

where E(ηA |XA ) = E(ηB |XB ) = 0. With OLS estimators δ̂A and δ̂B , and byadding and substracting X̄A δ̂B , an Oaxaca-Blinder decomposition is givenby(16)∆ξ = y A − y B = X̄A (δ̂A − δ̂B ) + (X̄A − X̄B )δ̂B
{z
} |
{z
}
|
∆S
ξ (unexplained)∆Xξ (explained)

By adding and substracting log(X̄A δ̂B ) to log(ȳA /ȳB ), we havelog(ȳA /ȳB ) = log ȳA − log(X̄A δ̂B ) + log(X̄A δ̂B ) − log ȳB= log(X̄A δ̂A ) − log(X̄A δ̂B ) + log(X̄A δ̂B ) − log(X̄B δ̂B )
|
{z
} |
{z
}
unexplained

(17)
(18)

explained

From (6), (13) and (18), the aggregate decomposition of changes in MLD
indices - into explained and unexplained eﬀects - is equal to:X∆Xν = log(X̄A δ̂B ) − log(X̄B δ̂B ) − ∆µ(explained)(19)∆Sν = log(X̄A δ̂A ) − log(X̄A δ̂B ) − ∆Sµ(unexplained)(20)

A detailed decomposition of the unexplained eﬀect can be obtained by using
the following approximation, which holds for small log diﬀerences:log(X̄A δ̂A ) − log(X̄A δ̂B ) ≈4X̄A (δ̂A − δ̂B )X̄A δ̂B=∆SξX̄A δ̂B(21)

Hence, from (8), (20) and (21) we have∆Sν,k ≈

X̄A,k (δ̂A,k − δ̂B,k )X̄A δ̂B− X̄A,k (β̂A,k − β̂B,k ) =∆Sξ,kX̄A δ̂B− ∆Sµ,k(22)

where ∆Sν,k , ∆Sξ,k and ∆Sµ,k are the contributions of the variable k to theunexplained eﬀect of changes in, respectively, inequality, income means andlog-income means. With a similar development and approximation, from
(7) and (19), the contribution of the variable k to the explained eﬀect ofchanges in inequality is given by
∆X
ν,k ≈

(X̄A,k − X̄B,k )δ̂B,kX̄B δ̂B− (X̄A,k − X̄B,k )β̂B,k =∆X
ξ,kX̄B δ̂B

− ∆X
µ,k

(23)

The unexplained component (20) can be interpreted as an average treatment eﬀect on individuals in group A:∆Sν = log(X̄A δ̂A ) − log(X̄A δ̂B ) − X̄A (β̂A − β̂B )(24)= log ȳA − log yA − log(X̄A δ̂B ) + X̄A β̂B(25)= νA − log(X̄A δ̂B ) + X̄A β̂B(26)

The term log(X̄A δ̂B ) − X̄A β̂B is a counterfactual, it represents the income
inequality induced by individuals in A if they would be living in B.
Standard errors and conﬁdence intervals can be computed by bootstraping, using the following procedure: [1] resample with replacement in the OLS
residuals of (1) to form a vector of the same size and generate new values of
log-incomes by adding it to (3) ; [2] resample with replacement in the OLS
residuals of (2) to form a vector of the same size and generate new values of
log-incomes by adding them to (4) ; [3] Perform a decomposition with the
new values of log-incomes ; [4] Repeat the previous steps R times ; [5] Thebootstrap standard error is the standard deviation of the R decomposition
estimates, and the bootstrap conﬁdence interval is obtained from the 0.025
and 0.975 sample quantiles of the R decomposition estimates.

4

Application

The data are obtained from the application in Hlavac (2018), on labor wages
and demographic characteristics of 712 employed hispanic workers in the
Chicago metropolitan area.

5

group A group B diﬀerence∆µ (log y) 2.6967 2.5534
17.5828 14.5672
ξ (y)ν (mld) 0.1702 0.1254

0.14343.01560.0448explained∆Xs.e.

unexplained∆Ss.e

0.0769 (0.0213)
1.6165 (0.4076)
0.0283 (0.0114)

0.0665 (0.0395)
1.3991 (0.8021)
0.0165 (0.0256)

Table 1: Aggregated decomposition of changes in means of log-wages, wages,
and in MLD indices between native (A) and foreign-born (B) workers

4.1

Difference in means

Table 1 (ﬁrst line) shows the aggregated decomposition of changes in meansof log-wages between native (Group A) and foreign-born workers (GroupB), with native workers taken as reference group. The diﬀerence in means
between natives and foreign-born workers is equal to 0.1434, of which 0.0769
can be attributed to diﬀerences in characteristics (i.e. age, gender, education) and the remaining 0.0665 is unexplained.
Figure 1 shows the detailed decomposition of changes in means of logwages for each variable, computed from (7) and (8), with 95% bootstrap
conﬁdence intervals (R = 10,000). In the explained components, all variables are statistically signiﬁcant. It suggests that the explained part of thediﬀerence in means is driven by diﬀerences in characteristics between thetwo groups. In the unexplained components, only the variable age is clearlystatistically signiﬁcant. It suggests that the payoﬀ of an additional year of
age is greater for native workers.
Overall, this empirical analysis shows that the log-wage mean of native
hispanic workers is greater than that of foreign-born hispanic workers. In
addition, native workers have, on average, diﬀerent characteristics and they
enjoy greater returns to age. Similar conclusions can be obtained from thedecomposition of changes in means of wages, rather than log-wages (middle
line in Table 1 and Figure 2)

4.2

Difference in inequality indices

Table 1 (last line) shows the aggregated decomposition of changes in MLD
inequality indices, obtained from (13)–(20). The results show that there ismore inequality among native than foreign-born workers (0.1702 vs. 0.1254).
The diﬀerence in inequality between the two groups is equal to 0.0448. It
is decomposed into 0.0283, which can be attributed to diﬀerences in characteristics (i.e. age, gender, education), and the remaining 0.0165 which is6unexplained. These results suggest that if average characteristics of foreignborn and native workers were the same, the overall inequality diﬀerence
between the two groups would be smaller (0.0165 vs. 0.0448).
Figure 3 shows the approximated detailed decomposition of changes in
MLD indices for each variable, computed from (22) and (23), with 95%
bootstrap conﬁdence intervals. In the explained components, the variableadvanced degree is statistically signiﬁcant. It suggests that the explained
part of the diﬀerence in inequality is partly driven by the proportion ofhighly educated individuals, which is larger in native hispanic workers. In
the unexplained components, no variable is statistically signiﬁcant.
Overall, this empirical analysis shows that inequality diﬀerence between
native and foreign-born hispanic workers is mainly driven by more highly
educated native hispanic workers than foreign-born workers.

5

Conclusion

In this paper, it is shown that a decomposition of diﬀerence in inequality indices can be obtained directly from standard Oaxaca-Blinder decompositionof diﬀerence in log-income and income means. An application illustrates
how to investigate decomposition of diﬀerences in means and in inequality
simultaneously.

References
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Cowell, F. A. and E. Flachaire (2018). Inequality measures and the median: Why inequality increased more than we thought. PEP discussion
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wage decompositions. Review of Economics and Statistics, 4(10341036), 86. F
Gradı́n, C. (2020). Quantifying the contribution of a subpopulation toinequality an application to Mozambique. Journal of Economic Inequality, 18, 391419.
Hlavac, M. (2018). oaxaca: Blinder-Oaxaca decomposition in R. R package version 0.1.4. https://CRAN.R-project.org/package=oaxaca.
Oaxaca, R. L. (1973). Male-female wage diﬀerentials in urban labor markets. International Economic Review , 14, 693–709.
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decompositions. Review of Economics and Statistics, 81, 154–157.
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regression and RIF decomposition. Stata Journal , 20, 5194.
Yun, M.-S. (2005). A simple solution to the identiﬁcation problem indetailed wage decompositions. Economic Inquiry, 43, 766–772.

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Figure 1: Log-wage gap - The detailed explained and unexplained components of a Oaxaca-Blinder decomposition on the native vs. foreign-born
immigrant log-wage gap in means.

9Figure 2: Wage gap - The detailed explained and unexplained components
of a Oaxaca-Blinder decomposition on the native vs. foreign-born immigrantwage gap in means

10Figure 3: MLD inequality gap - The detailed explained and unexplained
components of a Oaxaca-Blinder decomposition on the native vs. foreignborn immigrant gap in mean-logarithmic deviation inequality indices.

11