Decomposition of gender wage inequalities through calibration: Application to the Swiss structure of earnings survey
Section 2. Problem and notation
The
question of interest is the estimation of the wage differences between women
and men, more specifically, how much of this difference is attributable to
discrimination. Assume there is a finite population
of size
that can be divided into two subpopulations,
women and men, denoted by
of size
Additionally, a random sample
is drawn from
which contains both women and men. Sample
is selected by means of a sampling design
for any
where
Sample
can be split into two subsamples,
women and men, such that
The variable of interest, denoted by
is in this case the logarithm of the wage. The
totals of the variable of interest in the two subpopulations are given by
where
is the
logarithm of the wage of the
individual.
Since not all units in the subpopulations are observed, the totals can be
estimated by
where
is a sampling weight
assigned to the
unit of the
sample. Sampling weights are obtained after several statistical treatments (for
example, adjustment for non-response).
The
population means of the logarithms of the wages are given by
and can be estimated
by
Moreover, assume
that for each
individual in
either of the two subsamples, there is a vector of
auxiliary
variables denoted by
This vector is
supposed to be known for each unit selected in the sample. The auxiliary
variables contain some characteristics of the individual, for instance the age,
the education level or the seniority level. They can be quantitative or qualitative
variables, thus
can be a
categorical variable or a quantity. Also assume that the first auxiliary
variable is a constant, i.e.,
for all
The
totals of these auxiliary variables at the subpopulation level are given by
Using the weights
defined above,
these two totals can be estimated by
Vectors
of average values can be analogously estimated. The average values at the
subpopulation levels are given by
and estimated by
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