Decomposition of gender wage inequalities through calibration: Application to the Swiss structure of earnings survey
Section 3. The weighted BO decomposition
3.1 The decomposition
Using
the setup in Section 2, the findings of Blinder (1973) and Oaxaca (1973) are
summarized in the context of sampling theory, namely by using sampling weights.
Assume that in each sample, a linear relationship is suitable between the
characteristics that are available and the
logarithm of the wage. A regression is done separately in each subpopulation
At the subpopulation level, the values of the
regression coefficients are given by
They can be
estimated from the sample by
where
are the
sampling weights. The regression coefficients
are called the
group wage structure or the returns on characteristics and they represent the
contribution of each characteristic to the wage.
Result 1 A sufficient
condition to obtain the following equalities
is that there exists a vector
such that
for all
Since
it is assumed that
for all
with
the equality is always fulfilled. The proof of
Result 1 can be found in Appendix A. Putting together the result above, equations (2.1)
and (3.1), the average difference between the wages of two groups can be
written as
The difference between average
wages of the groups contains two elements: an explained part, also called the composition
effect
and an unexplained part, or the structure
effect
The former encompasses differences in
characteristics between the two groups. The latter is the difference in the
returns on characteristics between the two groups, the part that is not
attributable to objective factors (Oaxaca, 1973; Blinder, 1973). It is obtained
using characteristics as a proxy for productivity. The estimation of the structure
effect is the central element of this paper. Equation (3.2) has the
same elements as the one proposed by Oaxaca (1973) and Blinder (1973). The
methodology applied to obtain the estimated average values and coefficients
differs from the traditional regression technique. The BO method uses the
estimated regression coefficients obtained through ordinary least squares (OLS)
and the vectors of average values of the observed explanatory variables. The
proposed approach takes into account the survey weights. However, the weighted
BO method is the same as the original BO method if the sampling weights are all
equal to 1.
3.2 A note on the structure effect
The
two elements in equation 3.2 have different names across the literature.
The first one, whose denomination we retained as composition effect is
also termed endowments effect. The second one, which we call structure
effect is also found in the literature as unexplained residual, price
effect, sex effect, calculated effect or unequal treatment (Weichselbaumer
and Winter-Ebmer, 2006). Using the BO method, the structure effect is an
estimation of the discrimination level. However, discrimination is an intricate
phenomenon that might not be always fully observed. Unobserved variables,
selection bias or some mechanisms on the labour market can help to increase the
explained part of the wage difference. Moreover, Weichselbaumer and
Winter-Ebmer (2005) note two potential issues regarding the chosen model.
First, if the characteristics chosen in the linear model are themselves subject
to discrimination, then the resulting structure effect will be over-estimated.
Second, if the characteristics are not a proper measure of the productivity,
then again, the structure effect might be under- or over-estimated. Weichselbaumer
and Winter-Ebmer (2006) warn about the legitimacy of the characteristics as
productivity indicators, since “wages may also be determined by bargaining
power, compensating differentials or efficiency wages”. However, for
simplicity, in what follows, we will assume that there are no such issues and
that the estimated structure effect is the result of discrimination on the
labor market. Moreover, we do not examine sample selection bias or other
mechanisms underlying the distribution of men and women in certain jobs.
3.3 The counterfactual wage distribution
In
general, the counterfactual wage distribution is an artificial distribution
obtained by using the characteristics of a group to estimate the wages of
another group (see, for instance Bourguignon, Ferreira, and Leite, 2002). Examples
of counterfactual distributions are found in DiNardo et al. (1996) or
DiNardo (2002). The term
that appears in equation (3.2) is called
the women’s counterfactual average wage. It is interpreted as the estimated
average wage of women if they had the same average characteristics as men and
if their return on characteristics remained unchanged. Women’s counterfactual
wage distribution is obtained by using the characteristics of men
and the wage structure of women
In terms of interpretation, it is the wage
distribution of women, if they had the same characteristics as men.
Using
Result 1 from the previous section, women’s counterfactual mean wage
equals
and is estimated
from the sample by
where
are estimated
in equation (2.1) and
are the
coefficients estimated by means of equation (3.1). With this notation, the
BO decomposition given in (3.2) is re-expressed as
3.4 Using the counterfactual distribution to estimate the composition
and the structure effects
Building
the counterfactual average wage allows for the estimation of the two effects
that make up the wage difference at the average levels. From equation (3.3),
the composition effect is equal to
The composition
effect can be interpreted as the difference between what women would earn on
average if they had the characteristics of men and what they actually earn.
Thus, it reflects the inequality due to the differences in characteristics. The
structure effect in equation (3.3) is equal to
The structure effect
is the difference between the actual average wage of men and what women would
earn if they had the average characteristics of men and their wage own
structure. The equations above express the composition and structure effects at
the average levels, since this is the limitation of the BO method. The next
section presents a method that allows for the construction of the entire
counterfactual distribution. This in turn results in the ability of estimating
the composition and structure effects along the entire wage distribution.
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