The decomposition of wage gaps in the
Netherlands with sample selection
adjustments
Jim Albrecht∗ Aico van Vuuren† Susan Vroman
‡
November 2003, Preliminary
Abstract
In this paper, we use quantile regression methods to analyze the gender
gap in the Netherlands. The decomposition technique we use is the Machado
and Mata (2000) method, as applied in Albrecht, Bj¨orklund and Vroman
(2003). There is strong evidence of a glass ceiling effect in the Netherlands;
i.e., the gender log wage gap is larger for higher quantiles. Because parttime work is common among women in the Netherlands and because the
female participation rate is relatively low, sample selection is a serious issue.
We apply Buchinsky’s technique for quantile regression with selectivity bias
correction and estimate a series of quantile regressions to find the marginal
contributions of individual characteristics to log wages for men and for women
at various quantiles in their respective wage distributions. We then use the
Machado/Mata technique amended to deal with sample selection to construct
a counterfactual distribution, namely, the distribution of wages that would
prevail among women were women to work full-time to the same extent as
men do. This allows us to decompose the gender gap at different quantiles
taking account of sample selection and to determine how much of the gap is
due to differences in the labor market characteristics of men and women and
how much is due to gender differences in rewards to these characteristics.
∗
Georgetown University, Department of Economics, Washington, DC 20057, USA.
Erasmus University Rotterdam, Department of Economics H7-19, Burgermeester Oudlaan 50,
3062PA Rotterdam, The Netherlands, email:[email protected]
‡
Georgetown University, Department of Economics, Washington, DC 20057, USA.
†
Keywords: Gender, labor economics, selection models. JEL-codes: C24, J22, J31, J71. We
thank participants of the ESPE meeting in New York for useful comments.
1
Introduction
In this paper, we use quantile regression methods to analyze the gender gap in the
Netherlands. Studies of other European countries have found significant differences
in the gender gap at different quantiles of the wage distribution. (See Albrecht,
Bj¨orklund, and Vroman (2003) for Sweden, Bonjour and Gerfin (2001) for Switzerland, Datta Gupta and Smith (2002) for Denmark, Fitzenberger and Wunderlich
(2001) and Blundell, Gosling, Ichimura and Meghir (2002) for the UK and Fitzenberger and Wunderlich (2002) for Germany). The Netherlands is a particularly
interesting case to examine from the point of view of understanding how differences
between the genders in employment affect the gender wage gap. While the female
employment rate in the Netherlands is similar to that of other Western European
countries, the rate of part-time work among women is by far the highest among the
OECD countries.
In terms of gender and employment, the Netherlands has changed dramatically
over the past 20 years.1 In 1980, the gender gap in employment in the Netherlands
(about 40%) was similar to that of Spain, Greece, Italy, and Ireland. By 2000,
this gap had fallen to a level (about 18%) more like that of most other Western
European countries, albeit still above the levels observed in the U.S., the U.K., and
the Scandinavian countries. Most of this change was due to an expansion of parttime work for women.2 The Netherlands has the highest rate of part-time work
(defined as fewer than 30 hours per week) among women in all the OECD countries.
About 56% of employment among women aged 25-54 in the Netherlands is part time.
This is considerably higher than the corresponding rates for Germany and Belgium
(35%), France (23%), the U.K. (38%) and the U.S. (14%). Only Switzerland (47%)
is remotely comparable. This difference relative to other countries is mainly caused
by high part-time employment rates for mothers. For example, 83% of working
mothers aged 25-54 with 2 or more children work part time in the Netherlands. The
comparable figure for the U.S. is 24%. The part-time employment rate of men in
the same age group in the Netherlands (6%) is not substantially different from the
corresponding rates in other countries, especially when we look at fathers.
The gender wage gap at the mean in the Netherlands is among the highest in the
OECD countries (20%). It is only a little higher in the U.S., while it is about the
same in the U.K. The fraction of this gap that can be explained by gender differences
in labor market characteristics is relatively small in the Netherlands, indicating that
1
The figures in the next two paragraphs are from the July 2002 OECD Employment Outlook.
The other main source for the change was due to the lower participation rates of men over the
age of 55. We abstract from this issue in this paper focusing only on individuals in the age group
25 to 55.
2
1
the difference in returns between the genders is particularly large in the Netherlands.
Of course, this conclusion is based on comparing means only and it is important to
state that the OECD studies do not correct for any selection issue in the data. Our
aim in this paper is to look at the gender wage gap in the Netherlands not only at
the mean, but also at other points in the wage distribution. In addition, we will
take account of selection into employment.
In this paper we focus on the gender gap for those men and women who are
working full time. We estimate quantile regressions by gender to see whether returns to labor market characteristics differ by gender at different points in the two
distributions. As noted above, the labor market behavior of Dutch women implies
that we need to adjust for sample selection in estimating these returns for full-time
working women.
There has been a lot of attention to sample selection models in the literature (Gallant and Nychka, 1987, Newey, 1988, Buchinsky, 1998a, Blundell, Gosling, Ichimura
and Meghir, 2002, Das, Newey and Vella, 2003. A review article is by Vella, 1998).
Many of these models try to extent the classical model by Heckman taking account of
non-normality. For our paper, it is important to emphasize that we cannot restrict
ourselves to a normal distribution to explain the gap over the wage distribution.
More in particular, estimation of the mean and variance of both distributions is
enough to determine the wage gaps over the whole distribution. In that sense, our
exercise would be nothing else than estimation at the mean. In this paper, we apply
the method introduced by Buchinsky (1998a) to correct for sample selection in a
semiparametric model.
It should be emphasized that there are other techniques that we could have used
to correct for sample selection. We would like to mention two of them. First, there is
the method introduced by Gallant and Nychka (1987). These authors use a flexible
specification for the distribution of the selection and outcome equation. Although in
any practical application, the researcher needs to rely on a parametric specification,
Gallant and Nychka show that eventually a large class of distributions is kept within
their framework. One of the drawbacks of this model is that we have to restrict
the way in which the quantiles and regressors are related. Although the method
is essentially more flexible, we obtain the same problems with the assumption of
normality. The second method we would like to emphasize is by Blundell, Gosling,
Ichimura and Meghir (2002). These authors use the idea of Manski (1994) using
bounds on the possible outcomes of selection. The idea of this method is simple:
although nothing is known about the wages of the non-workers, the bounds can be
obtained by assuming that either all non-workers earn higher wages than the workers
(resulting in the upper bound) or that all non-workers are less productive than the
workers (resulting in the lower bound). Although the method of Blundell, Gosling,
2
Ichimura and Meghir is more flexible than the method that we adopt in our paper,
it has the disadvantage of being less precise. In addition, it is much more useful to
analyze whether the wage gap changed over time than to analyze the wage gap in a
particular moment in time. The latter is the purpose of our paper.
We devote a lot of attention to the choice of instruments used for applying
Buchinsky’s method. Apart from the usual instruments used in the literature like
the presence of children in the household and husbands’ income, we use attitude
variables to correct for selection. More in particular, these attitude variables are
derived from questions about the respondents’ view on the working behavior of
women with families. Besides this, we use some information about the respondents’
view of the importance of working. We are aware of one earlier study by Vella (1994)
that also used this type of information in the context of selection models.3 Following
this study we deal with the problem that this type of information is not only able
to explain the working behavior of women, but can be suspected to be endogenous
as well. We also investigate and correct for the possible endogeneity of education.
After applying Buchinsky’s method, we decompose the difference between the
male and female wage distributions, as was done by Albrecht, Bj¨orklund, and Vroman (2003) for the Swedish gender gap, using the Machado and Mata (2000) quantile
regression decomposition technique. This decomposition allows us to determine how
much of the difference between the distributions at various quantiles is due to gender
differences in characteristics and how much is due to gender differences in returns
to these characteristics. Again we take account of the low rate of participation in
full-time work by Dutch women and seek to determine the answer to the following
question. If Dutch women participated in full-time work to the same extent as Dutch
men do, what would the gender gap be across the distribution? Further, would the
decomposition of this counterfactual gender gap give similar results? Accordingly,
we extend the Machado and Mata (2000) technique to account for sample selection.
In addition to the decomposition method, we prove that our method is consistent
and we derive the asymptotic covariance matrix of the estimators. To our knowledge
this is the first econometric analysis of the Machado-Mata technique.
The remainder of the paper is set up as follows. In the next section, we present
the data used in our paper. Some preliminary issues on the estimation method are
in section 3. Section 4 presents our results on the estimation of quantile regressions.
The decompositions of the gender wage gap are in section 5. Conclusions follow in
section 6.
3
In a follow-up study Das, Newey and Vella (2003) use the same data set.
3
2
Data
We use data from the OSA (Dutch Institute for Labor Studies) Labor Supply Panel.4
This panel dates from 1985 and is based on biannual (in 1986, 1988, etc.) interviews of a representative sample of about 2,000 households. All members of these
households between the ages of 16 and 65 who were not in (daytime) school and who
could potentially work were interviewed. The survey focuses on respondents’ labor
market experiences. A detailed description is given in Van den Berg and Ridder
(1998).
We use data from the 1992 survey for all of our analysis. This year is particularly
rich in terms of variables that can be used to explain participation in full-time
work. The total number of individuals in the survey in that year equals 4536. In
order to focus on those who are most likely to be working full-time, we restrict our
sample to individuals between 25 and 55 years of age. This resulted in deleting 1167
individuals. There are relatively many individuals with missing numbers for the
years of work experience when they report not to work at the survey date. We set
the number of years of work experience for these individuals equal to 0 when they
did not work for the past 2 years. The other individuals are deleted from the data
set. These are 96 individuals.
In Table 1, we give some descriptive statistics for women and we list the same
descriptive statistics for men in Table 2. The deletion of individuals as described
above results in a sample of 3185 individuals with 1617 females and 1568 males.
We distinguish between women working part time and women working full time.
Working full time is defined as working at least 21 hours per week. Although this
seems a rather low level of hours for such a definition, there are not that many
women with working hours in the range between 21 and 32. As expected, almost
all men between the ages of 25 and 55 (about 95%) work, and among those who are
working, 90% work full time. Among women, the situation is quite different. Slightly
less than 50% of the women work, and among those who work, only 55% work full
time. In terms of the variables that we can use to explain variation in wages, there
are some important differences between the genders. Men who work full time are
on average almost 4 years older than women who work full time, and years of work
experience are much higher for men than for women. Men have on average achieved
higher levels of education than women have, but if we compare men and women
who work full time, this pattern is reversed. Women are more likely to be married
than men are. This is a direct result of our restriction to individuals between 25
and 55 years of age and the empirical finding that women marry men that are on
average older then they are. Among the men and women who work full time, men
4
For more information about these data, see http://www.uvt.nl/osa.
4
are much more likely to be married. For the group of married women, we find the
husbands’ wage to be a little less than 13 thousand euro. It is somewhat higher for
those women who work part-time and somewhat lower for women working full-time.
Finally, there is a dummy variable indicating whether an individual lives in a city
which has at least 50 thousand inhabitants. Men and women are approximately
equally likely to live in cities, but women who work full time are much more likely
to do so than are men who work full time.
We also have attitude variables that we can use to help explain the propensity
to work full time. Specifically, we use responses to the statements (i) “Working is a
duty to society”, (ii) ”People neglect their duty when they do not work while still
below the age of 65”, (iii) “There are more important things in life than work”,
(iv) “It is not possible to require that women should give up their job in order to
spend more time at home”, (v) “Women with families should stay at home”, (vi)
“Women with children younger than 12 years of age should not have a paid job”,
(vii) “Parents should be willing to reduce working hours for childcare”. Respondents
were able to state that they totally disagreed, disagreed, are indifferent, agreed or
completely agreed. For all statements we take those individuals who either agreed
or totally agreed as the individuals that agree with the statement. Males have on
average more traditional views than females about the working before of women
with children. Interestingly, women who work full time have somewhat different
attitudes, in particular, these women are more likely than both men and other
women to agree that there are more important things in life than work and less
traditional in their views with respect to working women. Finally, we have data on
children. We use a dummy variable whether there are children living at home and
we have a set of dummy variables whether the youngest child is either (i) below 5
years of age, (ii) between 5 and 12 years of age, (iii) between 12 and 18 years of age,
or (iv) 18 years or older. We find that relatively few women who are working full
time have children living at home (42%); relatively many women working part time
do. This phenomenon is even more apparent when we look at the lowest age groups
of the youngest child. Only 8 percent of the full-time working women have children
living at home who are below 5 years of age, while among all women this figure is
equal to 18 percent. The same holds true for the second and third age group, but
the magnitude is smaller. It does not hold for the eldest age group.
We use four different regions: (1) the northern region, (2) the eastern region, (3)
the western region and the (4) southern region. The western region is by far the
most important region in terms of population size and economic activity, while the
northern region is far less densely populated and has much higher unemployment
rates.
Our attitude variables are able to act as instruments in the selection model.
5
However, they are unlikely to be exogenous themselves. Following Vella (1994), we
use religion and parents’ education as possible explanations for traditional behavior.
Around 60 percent of the respondents stated that they were related with a particular
religion. Catholism is the biggest religion in the Netherlands and the other two major
religions are two different branches of Calvinism. Full-time working women are less
likely to be religious, but the magnitude of the differences is small. There is no
difference in fathers’ and mothers’ education between women and men. Note that
due to the relatively many missing numbers, mothers’ education levels do not sum
up to a number that is close to 1. Interestingly, fathers’ education levels are higher
for full-time working women.
Concluding, full time working women are higher educated, less likely to be married, less likely to have (young) children at home and have different attitudes towards
working than women who do not work full time.
In order to obtain the wage data, we delete individuals without a reported wage.
These can be individuals that did not work at the time of the survey, but there are
also some non-responses. In addition, we delete individuals with missing contractual working hours and those who reported contractual working weeks exceeding 60
hours. The latter is likely to be due to measurement error. In total this resulted
in deleting 94 individuals. Column four of Table 1 and column three of Table 2 list
descriptive statistics for the data set that remains after the deletion of these individuals. We do not find any important differences between individuals that report
a wage and the total data set for full time workers. Table 1 lists descriptive statistics of the wages for men and women in the year 1992. As expected, men’s wages
(measured in euros per hour) are on average higher than women’s wages. The average wage among women working full time is slightly higher than the corresponding
average among women who work part time.
We next look at the unconditional distributions of wages for men and women.
In Figure 1, the estimated kernel densities of men’s and women’s wages are plotted.
In this figure, we use all wages including the individuals who work part-time. The
gender gap for all working women, i.e., the difference in log wages between the male
and female wage at each quantile of their respective distributions, is plotted in Figure
2. This figure can be derived directly from the kernel density plots in Figure 1. It
shows the gap for all working men versus all working women, i.e., both the male and
female distributions include part-time workers. Figure 3 shows the gap for full-time
men versus full-time women and comparing this figure with Figure 2, we find very
different patterns for the gender gaps across the distribution. When all working
women are considered (Figure 2), the gender gap is U-shaped, but when only fulltime working women are considered (Figure 3) the gender gap is larger at higher
quantiles. That is, as compared to men, full-time working women do relatively well
6
Figure 1: Kernel density estimates of wage data in OSA labor
supply panel survey.
males
females
.45
.4
.35
density
.3
.25
.2
.15
.1
.05
0
0
5
10
wage
15
20
at the bottom of the wage distribution but not so well at the top. This pattern is
presented in a different form in Figure 4, which plots the gap between women’s fulltime and part-time wages. Since there is likely a quite difficult selection procedure
underneath this difference in pictures, we focus on the pattern in Figure 3. That is,
we focus on the participation into full time employment of women.
In order to interpret the results in this paper, the relevant question is whether
women voluntarily work part time or whether there is a demand driven reason for
women to work part time. Or to phrase our question in another way: is it reasonable
to assume that the number of working hours is a choice rather than the result of
discrimination? From our investigations not presented here we find that most women
are rather satisfied with the number of hours that they work and that it is even the
case that some women would prefer to work even fewers hours.
7
Figure 2: Gender wage gap from raw data of the labor supply
panel survey (95 % confidence intervals).
0.28
0.26
0.24
wage gap
0.22
0.2
0.18
0.16
0.14
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
percentile
Figure 3: Gender wage gap between full-time working women
and full-time working men from raw data of the labor supply
panel survey (95 % confidence intervals).
0.35
0.3
wage gap
0.25
0.2
0.15
0.1
0
0.1
0.2
0.3
0.4
0.5
percentile
8
0.6
0.7
0.8
0.9
1
Figure 4: Wage gap between full-time and part-time wages of
women.
0.3
0.2
0.1
wage gap
0
-0.1
-0.2
-0.3
-0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
percentile
3
Preliminary issues in the empirical implementation
Before reporting the main results of our paper, we present the estimation of our instrument equations. We start with the attitude variables. As stated in the previous
section, there are 4 attitude variables related to the working behavior of working
women (see statements (iv) to (vii)). Although we could have included these attitude variables as different dummies into the regression equations, we follow Vella
(1994) by constructing a qualitative variable of attitudes. For every statement individuals are able to give 5 different answers to the question. We add 5 points
to those individuals giving the least traditional answer while individuals with the
most traditional answer obtain only 1 point. This procedure results in a qualitative
variable that can take any discrete number between 5 and 20. Although a correct
empirical procedure should take account of the restrictions imposed on this qualitative variable, we estimate a regression by ordinary least squares. It is not very
restrictive since the range of the discrete variable is rather large and can hence be
treated as if it were continuous. The results are listed in Table 3. Note that a
positive sign implies that the individual is less traditional. Not surprisingly, older
individuals are more traditional. In addition, we find the fathers’ education level
to have a highly positive impact and it is more important for women than for men.
9
The mothers’ education level is positive and significant as well. Finally, the different
religions show up to be highly significant with both branches of calvinism and the
other christians to be the most traditional.
The second variable that we instrument is education. Instead of using dummy
variables we use a qualitative variable for the education level. It is equal to 1
if the individual did not have any educational qualifications. The numbers 2 to
5 are used for elementary schooling, lower secondary education, upper secondary
education, and bachelors/masters. In order to estimate the education equation,
we use a single-index method (Ichimura, 1993). The flexibility of this approach
captures the problem that the variable can only take discrete values between 1 and
5 (for example see Dustmann and Van Soest, 2000). Table 4 reports the results. We
make the identifying assumption that the coefficient of age is equal to -1. We use
the minus sign in order to obtain an increasing link function since age has a negative
impact on the education level. This implies that a positive sign in Table 4 results
in a positive impact on the education level. More in particular, the coefficient 7.55
for living in a city for women implies that the same education level is predicted for
a woman living in a city as would have been predicted for the same woman had she
been 7.55 years younger and not living in a city.
As mentioned, living in a city has a positive impact on the level of education for
women, but no impact on the education level of men. Having children before the
age of 23 is significantly negative for both women and men, but surprisingly it has a
larger impact on the education level of men. Interestingly, the attitude variable has
no impact on the educational attainment. The education level of the father has a
large significant impact on the education level. It is the largest for men. There are
no significant differences between the education levels of women in different regions,
but we find significantly lower levels for the northern region and the southern region
for men. Finally, individuals who agree with the statement that not working below
the age of 65 is neglecting duty are likely to have lower levels of education.
4
Quantile regressions
In this section, we present log wage quantile regressions for women and for men
who work full time. We regress log wage on the basic human capital variables
experience, measured as years of work experience, and education as defined in the
previous section and region.
In comparing the regressions for men and women who work full time, we need
to account for the selection of women into full time work. We do this using the
sample selection adjustment technique for quantile regressions that is described in
Buchinsky (1998a). We make this adjustment only for women. Before reporting the
10
results of the quantile regressions, we present the estimations for choice of full-time
work. These are presented in Table 5. The first column reports the Probit results,
while the second column contains the results using a single-index technique5 . Table 5
indicates that older and married women are less likely to work full time. This is also
true for those having (young) children at home. Women with a relatively high level
of work experience are more likely to work full-time. Again, the attitude variable
is not significant but it has the right sign. It is surprising that women who state
that there are more important things in life than working tend to be more likely to
work full-time. Finally, women from the northern region are less likely to work fulltime. Comparing the results of Probit with our single-index method, we find these
to be roughly the same. Still, the Hausman statistic testing normality is equal to
48.5. Under the null hypothesis this test statistic is χ2 -distributed with 18 degrees
of freedom. This implies that the test on normality rejects the null-hypothesis that
the stochastic part of our selection equation is normally distributed.
Table 6 presents the uncorrected quantile regressions results for full-time working
women. We take account of the endogeneity of education in Table 7, while Table
8 presents the corresponding results with the Buchinsky correction. The quantile
regression results for men are in Table 9 and 10. In the uncorrected estimates for
women, the basic human capital variables are the most important and have the
anticipated effects. The wages in the southern region are lower for most of the
quantiles estimated, while the wages in the eastern region are lower for some of
the quantiles around the median. The other variables have insignificant effects at
almost all the quantiles. Comparing the Tables without correcting for endogeneity of
education with those where we take account of this problem, we find the coefficient
for education to become larger. However, the standard deviation also increases.
Finally, comparing Tables 6 and 8 indicates that the coefficients at the median
and below are affected by the correction, but not those in the upper quantiles.
Specifically, the returns to experience and education fall in the lower quantiles and
the constant terms in the regressions become much smaller.
Table 9 indicates that men receive a positive return to education which does not
vary much across the quantiles. The years of experience have negative coefficients,
but note that age has a strongly positive effect that rises across the distribution.
Interestingly, there is a strong marriage premium for men in the Netherlands and it
is relatively constant across the distribution.
5
Details of the estimation are given in the Appendix.
11
5
Machado Mata decompositions
In this section, we present decompositions of the gender wage gap using the method
of Machado and Mata (2000). We use this method to construct distributions of
full-time men’s and women’s wages and a counterfactual distribution, namely, the
distribution of wages that women would earn if they were paid like men, but retained
the female distribution of characteristics. In addition, we construct the counterfactual distribution of wages that women would receive if they were paid like women
but obtain the distribution of characteristics of men. These decompositions are
related to the famous Oaxaca-Blinder decompositions.
In order to do so, we start with a more general setup. We consider two different
groups called group A and B, with characteristics given by the stochastic vectors
XA for group A and XB for group B. Without loss of generality we assume that
both XA and XB have dimension k and that the distribution function of XA is given
by GXA . Likewise we use the notation GXB . The endogenous variable is given by
YA for group A and YB for group B. Let the regression quantiles be given by β A (u)
for group A and β B (u) for group B. Hence, Quantu (YA |XA ) = XA β A (u); u ∈ [0, 1]
and likewise for YB . Finally, we assume that for both groups the outcomes as well
as the characteristics are observed. The sample sizes of the data sets are equal to
NA for group A and NB for group B.
Consider a random variable YAB , which has the properties that its quantiles
conditional on xA are given by
Quantu (yAB |xA ) = xA β B (u)
u ∈ [0, 1]
where xA is a k-dimensional random vector (i.e. characteristics of the A-group
members). The method of Machado and Mata is a method to obtain a sample from
the unconditional distribution of this random variable and can be described by the
following steps
1. Sample ui from a uniform distribution with support on the interval [0, 1].
2. Compute βbB (ui ), being the regression quantile of yjB on Xj,B , j = 1, . . . NB .
bX (using
3. Sample x
bi from the empirical distribution (i.e. bootstrapping) G
A
observations Xj,A j = 1, . . . NA ).
4. Compute ybi = xi βbB (ui )
5. Repeat steps 1 to 4 M times.
12
Of course, ybi ; i = 1 . . . M is not a real sample from the stochastic variable yAB
since it is based on estimates rather than the real parameters of the distribution.
This implies that estimators (like sample means, medians etc.) based on the sample
obtained by the method cannot be interpreted as estimates based on the population
YAB . However, one may expect that when NA and NB become large this problem
may not be that important anymore. In the remainder of this section, we make this
more precise using the sample quantile obtained from the method and comparing it
with the population quantile of YAB .
Let θ0 (q) be the q − th quantile of the distribution of YAB
θ0 (q) = FY−1
(q)
AB
R
b
where FYAB (yA ) = supp(XA ) FB (yA |XA = xA )dGXA (xA ). Let θ(q)
be its estimate
obtained from the sample using the Machado-Mata technique. In the appendix we
prove the following.
Theorem 1 Assume that GXA and GXB are both continuous on their support, then
P
b →
θ(q)
θ0 (q), when M, NA , NB → ∞, while M/NA → IA < ∞, M/NB → IB < ∞
The most important step in the proof is relatively simple and based on the
famous inverse transformation method (see for example Law and Kelton). However,
this method is based on the assumption that the underlying population distributions
of both data sets are known. In the Machado-Mata approach, these distributions
are estimated. Hence, most of the proof is devoted to show that when the sample
sizes of both data sets are large, the impact of the estimation method is negligible.
Finally, note that although M needs to become large as well, it should not increase
at a faster rate than both NA and NB .
Apart from consistency, it is important to show asymptotic normality. This was
not so much of an issue in Albrecht, Bj¨orklund and Vroman, since their sample sizes
of the data sets were extremely large. This is not the case for our application and
hence, we should investigate this a little further before we proceed. In the appendix,
the we prove following.
Theorem 2 Assume that limM,NB →∞
√
M, NA , NB → ∞, with
Ω=
M
NB
= IB < ∞, NMB = IA < ∞. Then,
b − θ(q)) Ã N (0, Ω)
M (θ(q)
´
³
´
i
³
¡ T¢
q(1 − q) h
AB
T
AB
(0|x
)x
V
(q)E
f
+
I
E
x
V
E
(x
)
(0|x
)x
1
+
I
E
f
A
A
A
G
A
A
B
uq
A
A
uq
fy2AB (θ)
13
´−1
´−1
³
³
T
B
T
T
(q(1 − q)E(xB xB )) EfuBq (0|x )xB xB
V (q) = EfuBq (0|xB )xB xB



VG = 

GA1 (x1 )(1 − G(x1 ))
−GA1 (x1 )GA2 (x2 )
...
−GA1 (x1 )GA2 (x2 ) GA2 (x2 )(1 − GA2 (x2 )) . . .
..
...
.
−GA1 (x1 )GAk (xk )
...
d
−GA1 (x1 )GAk (xk )
. . . GAk (xk )(1 − GAk (xk ))





d
B
B
B
where uAB
q |xA = yAB − xA β (q), uq |xB = yB − xB β (q).
Up till now, the presentation of the Machado-Mata approach was rather general.
Note that for our applications, the most interesting case is obtained when group A is
women and group B is men. However, the approach can also be used to decompose
differences in the distribution between time periods as in Machado and Mata (2000).
At this point, it is worthwhile to make some additional notes about our findings in Theorem 2. First, if groups A and B are equal, then the Machado and
Mata approach just replicates the distribution of A. Of course, in that case the
Machado-Mata approach is not so interesting but all of the results above still apply.
Second, if groups A and B are equal and if we let IB grow larger and larger (i.e.
sampling more and more), then ultimately Ω/M becomes approximately equal to
¡ ¢
E xT V (q)E (x) /N . This is exactly the variance that we would obtain using a
direct method without simulation. Third, if IB becomes smaller, then ultimately
Ω becomes equal to q(1 − q)/fuAB
(θ). This is exactly the variance of an estimated
q
quantile of a distribution using M observations. This makes sense as well because if
NB becomes larger and larger for fixed M , we can interpret the estimated regression
quantiles as if they were known in advance.
We use the kernel density method to estimate the covariance matrix V (q) (see
Buchinsky, 1998b). In addition we use the following estimates to complete the
calculation of Ω
³
´
b
E fuAB
(0|xA )xA =
q
³
´
b
E fuAB
(0|xA ) =
q
M
IbB =
NB
1
M hM
M
X
K
i=1
M
1 X
K
M hM i=1
14
Ã
yAB i
Ã
− xAi βbB (q)
hM
!
yAB i − xAi βbB (q)
hM
x Ai
!
Figure 5: Gender gap over the distribution taking acount of
differences in observed characteristics using the Machado Mata
approach.
0.26
0.24
0.22
wage gap
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
percentile
where hM is the bandwidth and K is the kernel function. We use regression cross
validation techniques to find the right levels of hM (see for example Silverman, 1988).
Figure 5 plots the wage gap taking account of the differences in observed characteristics between men an women. In addition, Figure 6 plots the difference between
the wages of men and the wages of women had they been paid like men. We find
the gap to be smaller over the whole distribution. However, the difference is only 2
percent with the only exception of levels around 3.5 percent in between the second
and third decile. The differences are smaller at the top of the distribution. Hence,
taking account of the differences in observed characteristics makes the glass ceiling
even more apparent. Most of the results are not significant as can be seen from
Figure 6.
Instead of calculating the women’s wages when paid like men it is also possible to
correct for the differences in characteristics between men and women by calculation
of the wages that women would have earned would they be paid like women but
having the characteristics of men. This is done in figure 7. The impact on the wage
gap is illustrated in figure 8. In this figure a positive level implies that the gender
wage gap becomes smaller. This exercise has the advantage of being able to further
disentangle the wage gap as described in Machado and Mata (2000). The impact on
the gender wage gap by changing the characteristics of women in just one particular
15
Figure 6: Difference between men’s wages and women’s wages
when paid like men.
0.08
0.06
0.04
wage gap
0.02
0
-0.02
-0.04
-0.06
0
0.2
0.4
0.6
percentile
0.8
1
variable are illustrated in figure 9. In this figure a positive number implies that
if we would assume the particular characteristic to have the same distribution of
men, then the wage gap will drop. A negative level implies a rise in the wage gap.
Taking this into account, we find that the wage gap decreases when we take age
and experience into account. The wage gap increases when we take education into
account. The latter is as expected since full time working women are in general
higher educated.
From the above we can conclude that the wage gap decreases somewhat when
we take account of the differences in observed characteristics between women and
men. In addition, the wage gap is explained mostly by the differences in age and
experience and that these characteristics are more important than the differences in
education. However, the general result is that we cannot explain most of the wage
gap as well as the glass ceiling.
Next, we investigate the effect of sample selection on the women’s log wage distribution and on the counterfactual distribution. Consider the model by Buchinsky
(1998a), where we have that
Quantu (YA |xA ) = xA β A (u)
while
16
u ∈ [0, 1]
Figure 7: Gender wage gap between men’s wages and women’s
wages when having characteristics of men.
0.28
0.26
0.24
wage gap
0.22
0.2
0.18
0.16
0.14
0.12
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
percentile
Figure 8: Difference between men’s wages and women’s wages
when having the characteristics of men.
0.06
0.05
0.04
0.03
wage gap
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
0
0.1
0.2
0.3
0.4
0.5
percentile
17
0.6
0.7
0.8
0.9
1
Figure 9: Decomposition of difference between wages of women
when having the characteristics of men and women’s wages
(full-time working women and full-time working men).
Age
Married
0.03
-0.002
0.025
-0.004
0.02
-0.006
0.015
-0.008
wage gap
wage gap
0.01
0.005
-0.01
0
-0.012
-0.005
-0.014
-0.01
-0.016
-0.015
-0.02
-0.018
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
percentile
Education
0.5
0.6
0.7
0.8
0.9
percentile
Region
0
0.01
0.008
-0.005
0.006
-0.01
wage gap
wage gap
0.004
-0.015
0.002
-0.02
0
-0.025
-0.002
-0.03
-0.004
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
percentile
City
0.4
0.5
0.6
0.7
0.8
0.9
0.7
0.8
0.9
percentile
Experience
0.001
0.04
0
0.03
-0.001
0.02
0.01
-0.003
wage gap
wage gap
-0.002
-0.004
0
-0.005
-0.01
-0.006
-0.02
-0.007
-0.008
-0.03
0.1
0.2
0.3
0.4
0.5
0.6
percentile
0.7
0.8
0.9
0.1
0.2
0.3
0.4
0.5
percentile
18
0.6
1
Quantu (YA |xA ; d = 1) = xA β A (u) + λ(zA γ A )
u ∈ [0, 1]
The variable y is only observed if d = 1. The Machado-Mata algorithm is
changed as follows
1. Sample u from a uniform distribution with support on the interval [0, 1].
2. Compute β A (u) and λ using the technique by Buchinsky.
3. Draw an xAi from the data of the population A (taking the complete sample).
4. Compute ybAi = xAi β A (u).
This procedure is repeated M times. Note that in (3) sampling is from the
complete sample and not from the data set of those who have an uncensored observations from y. When we would sample from the latter data set, we would not
obtain the distribution of y for the whole population. Instead, we would obtain
the distribution that would result when those who have a censored observation (for
example non-participating women) would have the same observed characteristics
as those who have an uncensored observation of y. This implies that we would not
take the difference between the observed characteristics between the two groups into
account. Although this technique does not provide the right distribution, we show
later on that it may give us some additional information on the selection bias.
We can extend theorem 2 by replacing V (q) by the covariance matrix that is
computed using the technique of Buchinsky (1998a). Figure 10 gives an illustration
comparing the women’s wages with the corrected women’s wages
Quantq (log (wF ) |d = 1) − Quantq (log ((wF ))
A positive value implies positive selection bias (i.e. the full-time working women
have relatively high (full-time) wages). Figure 10 illustrates the interesting result
that the overall selection effect is positive and significant. This implies that full-time
working women are the women with the highest earning potential. We find this to
be rather constant over the distribution.
Figure 11 plots the wage gap that results from the selection correction. This is
the gap between the men’s wages and the corrected women’s wages. As should be
expected from the results above, we find the glass ceiling to be even more apparent
then before.
As explained above, the results taking the selection effects into account are based
on sampling the characteristics of women from the data set of all women. If we would
have sampled from the data set of the full-time working women, then we would
19
Figure 10: Comparison of women’s wages before and after
correcting for measurement error.
0.09
0.08
0.07
0.06
wage gap
0.05
0.04
0.03
0.02
0.01
0
-0.01
-0.02
0.2
0
0.4
0.6
0.8
1
percentile
Figure 11: Gender wage gap taking account of selection effects
of women.
0.32
0.3
0.28
wage gap
0.26
0.24
0.22
0.2
0.18
0.16
0.14
0
0.1
0.2
0.3
0.4
0.5
percentile
20
0.6
0.7
0.8
0.9
1
Figure 12: Sample selection bias based on observed characteristics.
0.015
0.01
0.005
wage gap
0
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
0
0.2
0.4
0.6
0.8
1
percentile
have ignored the impact of the difference between the observed characteristics of
women who do and do not work full time. Although this would have resulted in
the wrong distribution and hence the wrong wage gaps, it may help us to further
disentangle the total sample selection effect. As mentioned, the distribution that
results from sampling from the full time working women is the distribution that
would have resulted when the women who do not work full time would have had
the same characteristics as those who do work full time. Hence comparing the right
distribution with this distribution, results in the selection effect due to observed
characteristics. The remainder of the sample selection effect is that part of the
selection effect due to unobserved characteristics. This is the difference between the
distribution obtained from sampling from the data set of full-time working women
and the distribution of observed women’s wages.
Figures 12 and 13 present the observed and unobserved part of the selection
effect. We find the observed part to be relatively small. It is negative over the
whole distribution but only significantly different from zero at the top. This negative
effect most probably results from the fact that full-time working women are in
general younger than the other women. In addition, we find the unobserved part
of the sample selection effect to be positive and significant over almost the whole
distribution.
As a final exercise we compare the wages of all women when paid like men with
21
Figure 13: Sample selection bias based on unobserved characteristics.
0.1
0.08
wage gap
0.06
0.04
0.02
0
-0.02
0
0.2
0.4
0.6
0.8
1
percentile
the wages of all women when corrected for sample selection (as described above).
The first distribution is obtained in the same way as before but now drawing from
the sample of all women (and not just the full-time working women). The obtained
distribution can be interpreted as the wage gap that would result if women participated like men taking differences between observed characteristics into account.
The gender wage gap obtained with this procedure is in Figure 14. We find the
difference to be relatively low at the bottom and extremely high at the top.
6
Conclusions
In this paper, we used quantile regression methods to analyze the gender gap in
the Netherlands. The decomposition technique we used was the Machado and Mata
(2000) method, as applied in Albrecht, Bj¨orklund and Vroman (2003). We applied
Buchinsky’s technique for quantile regression with selectivity bias correction and
estimated a series of quantile regressions to find the marginal contributions of individual characteristics to log wages for men and for women at various quantiles
in their respective wage distributions. We then used the Machado/Mata technique
amended to deal with sample selection to construct a counterfactual distribution,
namely, the distribution of wages that would prevail among women were women to
work full-time to the same extent as men do. This allowed us to decompose the
22
Figure 14: Relative difference between wages of women when
participating and paid like men with the actual women’s wages.
0.45
0.4
0.35
wage gap
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
percentile
0.7
0.8
0.9
1
gender gap at different quantiles taking account of sample selection and to determine how much of the gap is due to differences in the labor market characteristics
of men and women and how much is due to gender differences in rewards to these
characteristics.
Appendix A: explanation of the estimation method
Let y R (x1 , u) be the reservation wage of an individual based on his or her observed characteristics x1 and unobserved characteristics u, E(u|x1 ) = 0. In addition, let y ∗ (x2 , v) be
the wage of that individual based on observed characteristics x2 and unobserved characteristics v, E(v|x2 ) = 0. We assume that the variables contained in x2 is a subset of those
contained in x1 .
The individual will choose to work whenever y ∗ ≥ y R . Now define d∗ = y − y R and let
d = 1 if and only if d∗ ≥ 0. The variable d is equal to zero in the event that d∗ < 0. In
addition, define y as the observed wage. This variable is observed and equal to y ∗ only if
d = 1. In the event that d = 0, we do not observe y, though y ∗ exists for any individual
in the population. Assume the following specifications of y R and y ∗
y R = x1 α + u
23
and
y ∗ = x2 β + v
From this, it directly follows that for u ∈ [0, 1]
Quantu (y|x2 , d = 1) = x2 β + Quantu (v|x1 , d = 1) 6= x2 β
Hence, a quantile regression of y on x2 results in biased estimates. Define h(x1 ) =
E(v|x1 , d = 1) and let
y = x2 β + h(x1 ) + v1
By definition v1 = v − h(x1 ). And hence Quantu (v1 |x1 , v) = 0. We have that
h(x1 ) = Quantu (v|y ∗ ≥ y R ) = Quantu (v|v − u ≥ x1 γ)dv ≡ h1 (z)
where γ = β ′ − α and z = −x1 γ. β ′ is equal to β for all elements included in both x1
and x2 and equal to zero if excluded from x2 . Hence, h1 is only a function of z and that
h′1 < 0. The exact function of h1 depends on the distribution of both v and u. Therefore,
in general it is impossible to find a representation of h. Newey (1999) suggests to use the
P
i
following series estimator b
h1 (z) = ∞
i=0 δi λ(z) , where λ is the inverse Mill’s ratio. The
function b
h1 (z) is a power series of h1 (z) and for appropriate values of δ, b
h1 (z) → h1 (z)
when the number of terms goes to infinity.
Before estimation of h1 is feasible z = −x1 γ must be estimated. We estimate γ from a
semi-parametric single-index regression of x1 and d using the procedure as described in
Ichimura (1993). This implies that γ is estimated by
´2
1 X³
b
γ = argmin
di − G(x1i γ)
n i=1
n
where
Pn
³
z−x1i γ
hn
´
di K
³
´
z−x1i γ
K
i=1
hn
b (z) = i=1
G
Pn
where K is the kernel function and hn is the kernel bandwidth. More details are in
Ichimura (1993) and Horowitz (1998). For all models that estimate a semiparametric
sample selection model in the way described above, the intercept in the wage equation
is not identified. This can be seen most easily by taking into account that both the
24
intercept as well as δ0 appear in the linear (quantile) regression equation. As is done in
Buchinsky (1998a) and Andrews and Schafgans (1998), we estimate this intercept through
an identification at infinity approach. This implies that we restrict ourselves to a sample
that has a relatively high probability of working full time to estimate the intercept. For
more details we refer to Andrews and Schafgans (1998).
Appendix B: Proofs of theorems
Proof of theorem 1
e which is the sample median obtained from the sample
We first consider the estimator θ,
yei ; i = 1 . . . M . This sample is obtained as follows
1. Sample ui from a uniform distribution with support on the interval [0, 1]
2. Sample xei from the distribution GXA .
3. Compute yei = xei β B
4. Repeat steps 1 to 3 M times.
The realizations of this method can itself be seen as realizations from a stochastic variable.
Denote this variable by YeAB . The distribution function of this variable is equal to
Z
1
P(YeAB ≤ y|U = u)du
FYeAB (y) = P(YeAB ≤ y) =
0
Z 1Z
=
P(F −1 (U ) ≤ y|U = u; XA = xA )dGXA (xA )du
0
supp(XA )
Z 1Z
P(U ≤ FYB (y|XA = xA ))dGXA (x)du
=
0
supp(XA )
Z
FYB (y|XA = xA )dGXA (x) = FYAB (y)
=
supp(XA )
d
Hence, YeAB = YAB . The most important step in this derivation is in the third line and is
derived from the observation that given xA and u, Ye must be the u-th quantile of the YB .
This implies that the observations from the sampling method are sampled from the population distribution YAB . For the remainder of this proof we use vi as a k-dimensional vector
b X . The elements of vi are sampled
that is used to sample from the distribution GXA and G
A
25
eA,i = X
eA (vi ) = G−1 (vi ) and likewise for
from the standard uniform distribution.6 Hence X
XA
b M (θ(q)) = 1 P mq (b
bA,i Let Ψ
e M (θ(q)) = 1 P mq (e
y
,
θ(q))
and
Ψ
yAB,i , θ(q), where
X
AB,i
i
i
M
M

if yAB > θ(q)
 q(yAB − θ(q))
m(yAB , θ(q)) =

(1 − q)(θ(q) − yAB ) if yAB ≤ θ(q)
In addition, define Ψ(θ(q)) as
Ψ(θ(q)) = (1 − q)
Z
θ(q)
FyAB dy +
Z
∞
F yAB (y)dy
θ
∞
It is obvious that Ψ(θ(q)) is maximized whenever θ(q) = θ0 (q), which is the identification
condition for quantiles. It remains to show that
¯
¯
¯b
¯
sup ¯ΨM (θ(q)) − Ψ(θ(q))¯ = oP (1)
θ(q)
By the triangular inequality
¯
¯
¯
¯
¯
¯
¯
¯e
¯
¯b
¯
¯b
e
sup ¯Ψ
M (θ(q)) − Ψ(θ(q))¯ ≤ sup ¯ΨM (θ(q)) − ΨM (θ(q))¯ + sup ¯ΨM (θ(q)) − Ψ(θ(q))¯
θ(q)
θ(q)
θ(q)
The last term on the last line is oP (1) by the law of large numbers. It remains to show
that the first term is oP (1) as well. We have that (dropping the subscripts AB for yb and
ye for the remainder of the proof)
6
It is always possible to sample from a k-dimensional distribution with a known distribution
function based on repetitive conditioning and a draw from a k-dimensional vector sampled from
univariate uniform distributions. Although it is also possible to do this using the empirical distribution function of a k-dimensional stochastic vector, bootstrapping from the data would be a
much easier way to obtain such a sample. The results are not affected by the sampling method.
26
¯
¯
¯
¯
¯1 X
1 X
¯b
¯
e
sup ¯ΨM (θ(q)) − ΨM (θ(q))¯ = sup ¯¯
q(b
yi − θ) −
q(e
yi − θ)+
M
θ(q) ¯ M yb <θ
θ(q)
yei <θ
i
¯
¯
X
X
¯
1
1
(1 − q)(θ − ybi ) −
(1 − q)(θ − yei )¯¯ ≤
M
M
¯
ybi ≥θ
yei ≥θ
¯
¯
¯
¯
¯
¯
¯
¯
¯1 X
¯1 X
¯
¯
¯
¯
¯
q sup ¯
(b
yi − yei )¯ + (1 − q) sup ¯
(e
yi − ybi )¯¯
θ(q) ¯ M ye ≥θ
θ(q) ¯ M ye <θ
¯
¯
i
i
¯
¯
¯
¯
¯
¯
¯
¯
¯1 X
¯
¯1 X
¯
¯
¯
¯
+q sup ¯
(b
yi − θ)¯ + q sup ¯
(b
yi − θ)¯¯
θ(q) ¯ M yb <θ≤e
θ(q) ¯ M ye <θ≤b
¯
¯
yi
yi
i
i
¯
¯
¯
¯
¯
¯
¯
¯
¯1 X
¯
¯
¯1 X
(θ − ybi )¯¯ + (1 − q) sup ¯¯
(θ − ybi )¯¯
+(1 − q) sup ¯¯
θ(q) ¯ M
θ(q) ¯ M
¯
¯
ybi <θ≤e
yi
yei <θ≤b
yi
Making use of the definition of ybi , yei , x
bi and xi and using the triangular inequality again,
we obtain
¯
¯
¯
¯
¯
¯
¯
´¯¯
³
¯1 X
¯
¯1 X
b i ) − β(ui ) ¯
(b
yi − yei )¯¯ ≤ sup ¯¯
x
bi (vi ) β(u
q sup ¯¯
¯
θ(q) ¯ M ye <θ
θ(q) ¯ M ye <θ
¯
¯
i
i
¯
¯
¯
¯
´
¯ 1 X ³ −1
¯
−1
¯
+¯
GXA (vi ) − GXA (vi ) β(ui )¯¯
¯ M yei <θ
¯
P
b i) →
We have that β(u
β(ui ); i = 1, . . . M for NB → ∞ (consistency of quantile regressions). Since M/NB → IB < ∞ when M, NB → ∞ the first term is oP (1). For the
P
b −1 (vi ) →
second term, we have that G
G−1
XA
XA (vi ) when n → ∞ due to a combination of
the Glivenko-Cantelli theorem, the continuous mapping theorem and the assumption that
GXA is continuous (and hence its inverse exists). Hence, the second is oP (1) as well (using
M/NA → IA < ∞ as M, NA → ∞). The prove that the others are op (1) as well is along
the same lines.
¯
¯
¯1
q sup ¯¯
θ(q) ¯ M
¯
¯
¯
(b
yi − θ)¯¯
¯
ybi <θ≤e
yi
¯
¯
M
¯
¯1 X
´
³
¯
¯
−1
b
b −1
(v
)
β(u
)
<
θ(q)
≤
G
(v
)β(u
)
(b
y
−
θ)
1 G
= q sup ¯
¯
i
i
i
i
i
n
¯
¯
M
θ(q)
i=1
27
X
Using the same reasoning as above, we have that the first term in the sum converges in
probability to zero when NA , NB → ∞. This implies that the whole term converges to
zero in probability since M/NA → I < ∞.
¥
Proof of theorem 2
b is equal to
The criterion function to obtain θ(q)
1
min
θ(q) M
where
7
X
ybAB i >θ(q)
qkb
yAB i − θk + (1 − q)
m(yAB , θ(q)) =
X
ybAB i ≤θ(q)
kb
yAB i − θ(q)k = min

 q(yAB − θ(q))

θ
M
X
m(b
yABi , θ(q))
i=1
if yAB > θ(q)
(1 − q)(θ(q) − yAB ) if yAB ≤ θ(q)
and ybAB i is the estimated level of yAB i (i.e. the i-th observation from the sample using
Machado and Mata’s technique). The first order condition is equal to
M
X
b
∂m(b
yAB i , θ(q))
=0
∂θ(q)
i=1
Taking a first order Taylor series expansion of this results in
n
M
1 X ∂ 2 m(b
1 X ∂m(b
yABi , θ(q)) √
yABi , θ(q))
b − θ(q)) + oP (1)
M (θ(q)
0= √
+
∂θ(q)
M
∂θ(q)2
M i=1
i=1
(1)
We have that (see for example Van der Vaart (1998)
n
1 X ∂ 2 m(b
yABi , θ(q)) P
→ fyABi (θ(q))
M
∂θ(q)2
i=1
The first part of the right-hand side of equation (1) can be further expanded using a Taylor
series expansion.
7
Note that the function presented is not differentiable everywhere. This is not sufficient for the
proofs presented below. In general it suffices to show that a Lipshitz condition holds. It is not
difficult to show that this condition is satisfied. See Van der Vaart (1998) for more details)
28
M
M
1 X ∂m(yAB i , θ(q))
yAB i , θ(q))
1 X ∂m(b
√
+
=√
b
∂θ(q)
M i=1
M i=1
∂ θ(q)
r
M
M 1 X ∂ 2 m(yAB i , θ(q)) p
NB (βb − β)+
NB M
∂θ(q)∂β
i=1
r
M
X
M 1
∂ 2 m(yAB i , θ(q)) p
b A (x) − GA (x)) + oP (1)
NB (G
NB M
∂θ(q)∂GA (x)
i=1
We have that (see for example Buchinsky (1998b))
p
NB (βbB (q) − β B (q)) Ã N (0, V (q))
We have that (see for example Van der Vaart (1998), chapter 19)
p
b A − GA ) Ã N (0, VG )
NA (G
Moreover, using the central limit theorem results in
M
1 X ∂m(yAB i , θ)
√
Ã N (0, q(1 − q))
∂θ
M i=1
and the law of large numbers results in
M
³
´
1 X ∂ 2 m(yAB i , θ(q))
→ −E fuAB
(0|x
)x
A
A
q
M
∂θ(q)∂β
i=1
and
M
1 X ∂ 2 m(yAB i , θ(q))
→ βE (xA )
M
∂θ(q)∂G(x)
i=1
Taking this together (and using that M/NB → IB ), the final result is obtained.
¥
29
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31
Table 1: Descriptive statistics for women in the OSA data
All
Part-time
Full-time
Full-time
wage obs.
39.3
(8.36)
0.87
0.73
10.9
(7.32)
12954
(7806)
0.32
0.20
–
39.94
(7.48)
0.95
0.85
12.1
(7.89)
14373
(6316)
0.32
0.22
6.33
(2.15)
36.07
(8.56)
0.76
0.42
12.8
(7.84)
11558
(7882)
0.39
0.11
–
35.97
(8.61)
0.76
0.41
12.7
(7.80)
9868
(8857)
0.41
0.11
6.36
(2.12)
0.64
0.10
0.89
0.53
0.18
0.22
0.65
0.66
0.11
0.92
0.64
0.09
0.11
0.64
0.61
0.06
0.93
0.68
0.08
0.09
0.55
0.61
0.06
0.93
0.68
0.07
0.09
0.57
0.12
0.44
0.30
0.14
0.10
0.49
0.25
0.16
0.06
0.30
0.40
0.23
0.06
0.30
0.41
0.23
0.18
0.20
0.19
0.10
0.19
0.23
0.26
0.17
0.08
0.13
0.11
0.10
0.08
0.11
0.12
0.10
Personal characteristics
Age
Married
Children living at home
Number of years of work experience
Husbands’ net yearly wage (if present)
Living in city
First child before the age of 23
Net hourly wage rate
Attitude variables
(i) Working is a duty for the society
(ii) Not working ≤ 65 y/a is neglecting duty
(iii) More important things in life than working
(iv) Impossible to require that women should give up working
(v) Women with children should not work
(vi) Women with children of ≤ 12 y/o should not have a paid job
(vii) Prefer reducing working hours for care
Education levels
Up to elementary school
Lower secondary education
Higher secondary education
Bachelors/masters
Age of youngest child living at home
Below 5 years
Between 5 and 11 years
Between 12 and 17 years of age
18 years or older
32
Table 1: Descriptive statistics for women in the OSA data
(continued)
All
Part-time
Full-time
Full-time
wage obs.
0.13
0.19
0.32
0.24
0.13
0.18
0.36
0.23
0.08
0.21
0.33
0.23
0.08
0.22
0.33
0.22
0.30
0.15
0.10
0.04
0.01
0.30
0.18
0.09
0.03
0.01
0.30
0.14
0.08
0.01
0.01
0.30
0.14
0.08
0.01
0.01
0.40
0.19
0.09
0.06
0.41
0.19
0.07
0.01
0.30
0.22
0.13
0.09
0.29
0.22
0.14
0.08
0.14
0.07
0.02
0.00
0.16
0.08
0.03
0.00
0.08
0.07
0.02
0.01
0.08
0.06
0.02
0.01
Employee
Self employed
0.73
0.20
0.76
0.19
0.75
0.17
0.75
0.17
Number of observations
1617
336
410
391
Regions
North
East
West
South
Religion
Catholic
Dutch calvinists (Nederlands hervormd )
Calvinists (Gereformeerd )
Other christians
Other religions
Fathers’ education
Up to elementary school
Lower secondary education
Higher secondary education
Bachelors/masters
Mothers’ education
Up to elementary school
Lower secondary education
Higher secondary education
Bachelors/masters
Fathers’ work characteristics
33
Table 2: Descriptive statistics for men in the OSA data
All
Full-time
Full-time
wage obs.
39.26
(8.39)
0.86
0.66
19.53
(9.76)
0.34
0.05
–
38.90
(8.28)
0.87
0.67
19.61
(9.67)
0.32
0.06
–
38.75
(8.22)
0.87
0.67
19.47
(9.64)
0.33
0.05
7.72
(2.38)
0.76
0.10
0.91
0.53
0.21
0.26
0.60
0.77
0.09
0.91
0.54
0.21
0.25
0.61
0.77
0.09
0.91
0.54
0.20
0.25
0.61
0.10
0.36
0.33
0.21
0.10
0.36
0.33
0.21
0.10
0.36
0.33
0.21
0.19
0.19
0.17
0.12
0.19
0.19
0.17
0.11
0.20
0.19
0.17
0.11
Personal characteristics
Age
Married
Children living at home
Number of years of work experience
Living in city
First child before the age of 23
Net hourly wage
Attitude variables
(i) Working is a duty for the society
(ii) Not working ≤ 65 y/a is neglecting duty
(iii) More important things in life than working
(iv) Impossible to require that women should give up working
(v) Women with children should not work
(vi) Women with children ≤ 12 y/o should not have a paid job
(vii) Prefer reducing working hours for care
Education levels
Up to elementary school
Lower secondary education
Higher secondary education
Bachelors/masters
Age of youngest child living at home
Below 5 years
Between 5 and 11 years
Between 12 and 17 years of age
18 years or older
34
Table 2: Descriptive statistics for men in the OSA data (continued)
All
Full-time
Full-time
wage obs.
0.13
0.18
0.32
0.25
0.13
0.18
0.33
0.25
0.13
0.19
0.33
0.24
0.29
0.12
0.10
0.04
0.01
0.28
0.12
0.10
0.04
0.00
0.28
0.12
0.10
0.04
0.00
0.40
0.20
0.11
0.06
0.41
0.20
0.11
0.05
0.10
0.36
0.33
0.21
0.14
0.05
0.03
0.00
0.13
0.05
0.02
0.00
0.13
0.05
0.02
0.00
Employee
Self employed
0.74
0.20
0.76
0.18
0.76
0.18
Number of observations
1568
1312
1237
Regions
North
East
West
South
Religion
Catholic
Dutch calvinists (Nederlands hervormd )
Calvinists (Gereformeerd )
Other christians
Other religions
Fathers’ education
Up to elementary school
Lower secondary education
Higher secondary education
Bachelors/masters
Mothers’ education
Up to elementary school
Lower secondary education
Higher secondary education
Bachelors/masters
Fathers’ work characteristics
35
Table 3: Ordinary least squares estimates for the attitude variable
Constant
Age
Women
Men
15.46
( 0.41)
-0.05
( 0.01)
15.43
( 0.40)
-0.05
( 0.01)
-0.50
( 0.20)
0.55
( 0.24)
0.60
( 0.30)
1.13
( 0.36)
-0.69
( 0.20)
0.40
( 0.23)
0.22
( 0.28)
0.37
( 0.36)
0.52
( 0.27)
0.40
( 0.29)
0.12
( 0.25)
-0.15
( 0.22)
0.30
( 0.22)
0.01
( 0.25)
-0.24
( 0.21)
-0.18
( 0.22)
-0.57
( 0.21)
-0.86
( 0.24)
-1.10
( 0.28)
-1.46
( 0.40)
-0.16
( 0.95)
-0.29
( 0.21)
-1.42
( 0.25)
-0.71
( 0.27)
-1.54
( 0.40)
1.52
( 1.07)
0.08
0.09
Fathers’ education level
Elementary school
Lower secondary education
Upper secondary education
Bachelors/masters degree
Mothers’ education level
At least secondary education
Region
North
East
South
Religion
Catholic
Dutch calvinist
Calvinist
Other christian
Other
R2
36
Table 4: Single-index estimates of education levels
Age
Living in city
Having first child before the age of 23 y/o
Attitude
Women
Men
-1
–
7.55
( 3.89)
-16.60
( 3.89)
-4.35
( 2.34)
-1
–
-0.88
( 3.71)
-29.30
( 6.12)
-1.99
( 1.96)
0.57
( 7.82)
3.77
( 8.48)
-0.79
( 6.58)
6.48
( 7.47)
12.73
( 4.24)
18.43
( 5.68)
33.29
( 9.21)
32.68
( 9.02)
46.07
( 10.83)
65.06
( 12.43)
6.97
( 5.65)
3.90
( 4.30)
0.29
( 4.54)
-23.46
( 5.72)
4.74
( 3.82)
-9.74
( 4.41)
-10.69
( 5.54)
5.30
( 5.05)
-19.92
( 6.24)
3.97
( 5.95)
Characteristics of fathers’ job
Employee
Self employed
Fathers’ education level
Lower secondary education
Upper secondary education
Bachelors/masters degree
Region
North
East
South
Additional attitude variables
Not working ≤ 65 y/a is neglecting duty
More important things in life than working
37
Table 5: Estimates of the Incidence of Full-Time Work
Probit
Constant
(
Age
(
Education
(
Attitude
(
Married
(
Having children at home
(
Number of years of work experience
(
More important things
(
Living in city
(
-1.141
0.916)
-0.071
0.008)
0.632
0.197)
0.072
0.066)
-0.326
0.116)
-0.273
0.146)
0.064
0.006)
0.250
0.137)
-0.013
0.086)
Single-index
(
(
(
(
(
(
(
-1.141
(·)
-0.071
(·)
0.835
0.277)
0.086
0.082)
-0.182
0.147)
-0.425
0.193)
0.066
0.007)
0.219
0.165)
-0.047
0.110)
Age of youngest child
Younger than 5 years of age
Between 5 and 12 years of age
Between 12 and 18 years of age
-1.195
( 0.175)
-0.593
( 0.150)
-0.245
( 0.144)
-1.289
( 0.222)
-0.561
( 0.200)
-0.211
( 0.193)
North
Region
North
-0.458
( 0.140)
0.148
( 0.110)
0.044
( 0.102)
East
South
38
-0.310
( 0.164)
0.149
( 0.145)
0.110
( 0.133)
Table 6: Quantile regressions for women without corrections
for selectivity
Constant
Age
Married
Years of work experience
City
Education
10%
20%
30%
40%
50%
60%
70%
80%
90%
1.28
( 0.19)
-0.00
( 0.00)
0.01
( 0.05)
0.00
( 0.00)
-0.03
( 0.05)
0.08
( 0.03)
1.15
( 0.12)
0.00
( 0.00)
-0.00
( 0.04)
0.01
( 0.00)
-0.00
( 0.03)
0.11
( 0.02)
1.18
( 0.08)
-0.00
( 0.00)
0.01
( 0.02)
0.01
( 0.00)
0.01
( 0.02)
0.12
( 0.01)
1.19
( 0.09)
0.00
( 0.00)
0.02
( 0.03)
0.01
( 0.00)
0.02
( 0.03)
0.11
( 0.01)
1.21
( 0.09)
0.00
( 0.00)
0.00
( 0.03)
0.01
( 0.00)
-0.01
( 0.02)
0.12
( 0.01)
1.24
( 0.06)
0.00
( 0.00)
-0.00
( 0.02)
0.01
( 0.00)
0.02
( 0.02)
0.11
( 0.01)
1.19
( 0.13)
0.00
( 0.00)
-0.01
( 0.04)
0.01
( 0.00)
0.02
( 0.04)
0.13
( 0.02)
1.24
( 0.10)
0.00
( 0.00)
-0.02
( 0.03)
0.01
( 0.00)
0.00
( 0.03)
0.14
( 0.02)
1.42
( 0.13)
0.01
( 0.00)
-0.08
( 0.03)
0.01
( 0.00)
0.03
( 0.03)
0.11
( 0.02)
0.06
( 0.06)
-0.07
( 0.06)
-0.05
( 0.06)
0.07
( 0.05)
0.01
( 0.04)
-0.06
( 0.04)
0.03
( 0.04)
-0.03
( 0.03)
-0.09
( 0.03)
0.00
( 0.05)
-0.03
( 0.03)
-0.06
( 0.03)
-0.00
( 0.04)
-0.06
( 0.03)
-0.08
( 0.03)
-0.02
( 0.03)
-0.06
( 0.02)
-0.04
( 0.02)
-0.00
( 0.06)
-0.03
( 0.04)
0.01
( 0.04)
-0.03
( 0.04)
-0.04
( 0.03)
-0.01
( 0.03)
-0.10
( 0.05)
-0.09
( 0.04)
-0.05
( 0.04)
Regions
North
East
South
39
Table 7: Quantile regressions for women without corrections
for selectivity taking account of endogeneity of education
Constant
Age
Married
Years of work experience
City
Education
10%
20%
30%
40%
50%
60%
70%
80%
90%
1.18
( 0.39)
0.00
( 0.00)
-0.01
( 0.05)
0.00
( 0.00)
-0.03
( 0.05)
0.10
( 0.09)
1.14
( 0.37)
0.00
( 0.00)
-0.01
( 0.05)
0.00
( 0.00)
-0.02
( 0.05)
0.13
( 0.09)
0.89
( 0.24)
0.00
( 0.00)
-0.03
( 0.03)
0.00
( 0.00)
-0.04
( 0.03)
0.19
( 0.06)
1.01
( 0.20)
0.00
( 0.00)
-0.03
( 0.03)
0.01
( 0.00)
-0.01
( 0.02)
0.18
( 0.05)
0.99
( 0.27)
0.00
( 0.00)
-0.03
( 0.04)
0.01
( 0.00)
-0.01
( 0.03)
0.20
( 0.07)
0.78
( 0.26)
0.00
( 0.00)
-0.04
( 0.04)
0.01
( 0.00)
-0.01
( 0.03)
0.27
( 0.06)
0.82
( 0.22)
0.00
( 0.00)
-0.02
( 0.03)
0.01
( 0.00)
-0.01
( 0.03)
0.26
( 0.05)
0.78
( 0.30)
0.01
( 0.00)
-0.04
( 0.04)
0.00
( 0.00)
0.02
( 0.04)
0.25
( 0.07)
1.05
( 0.43)
0.01
( 0.00)
-0.05
( 0.06)
0.01
( 0.00)
0.09
( 0.05)
0.19
( 0.10)
0.09
( 0.08)
-0.05
( 0.06)
-0.08
( 0.06)
0.09
( 0.08)
-0.01
( 0.05)
-0.08
( 0.06)
0.06
( 0.05)
-0.01
( 0.04)
-0.08
( 0.04)
0.01
( 0.04)
-0.04
( 0.03)
-0.09
( 0.03)
0.01
( 0.06)
-0.02
( 0.04)
-0.07
( 0.04)
-0.02
( 0.06)
-0.02
( 0.04)
-0.06
( 0.04)
-0.01
( 0.05)
-0.03
( 0.03)
-0.03
( 0.03)
-0.02
( 0.06)
-0.01
( 0.05)
0.06
( 0.04)
0.03
( 0.09)
-0.05
( 0.06)
0.05
( 0.06)
Regions
North
East
South
40
Table 8: Quantile Regressions for women with corrections for
selectivity
Constant
Age
Married
Years of work experience
City
Education
10%
20%
30%
40%
50%
60%
70%
80%
90%
1.16
(0.01)
-0.00
(0.01)
-0.01
(0.04)
0.00
(0.01)
-0.04
(0.04)
0.12
(0.09)
0.95
(0.01)
-0.00
(0.00)
-0.04
(0.04)
0.01
(0.00)
-0.03
(0.03)
0.22
(0.07)
0.73
(0.01)
-0.00
(0.00)
-0.04
(0.04)
0.01
(0.00)
-0.04
(0.03)
0.29
(0.07)
0.77
(0.01)
-0.00
(0.00)
-0.05
(0.04)
0.01
(0.00)
-0.02
(0.03)
0.28
(0.06)
0.87
(0.01)
-0.00
(0.00)
-0.06
(0.04)
0.01
(0.00)
-0.01
(0.03)
0.27
(0.06)
0.75
(0.01)
0.00
(0.00)
-0.08
(0.04)
0.01
(0.00)
-0.03
(0.03)
0.30
(0.07)
0.86
(0.01)
-0.00
(0.00)
-0.07
(0.04)
0.01
(0.00)
-0.03
(0.03)
0.28
(0.07)
0.81
(0.01)
0.01
(0.00)
-0.07
(0.05)
0.01
(0.00)
-0.01
(0.04)
0.26
(0.09)
1.24
(0.01)
0.01
(0.00)
-0.09
(0.06)
0.01
(0.00)
0.08
(0.04)
0.16
(0.10)
0.09
(0.05)
-0.06
(0.06)
-0.07
(0.05)
0.08
(0.05)
0.01
(0.04)
-0.07
(0.04)
0.02
(0.04)
-0.01
(0.04)
-0.08
(0.04)
0.00
(0.04)
-0.03
(0.03)
-0.08
(0.04)
-0.02
(0.04)
-0.02
(0.03)
-0.03
(0.04)
-0.05
(0.04)
-0.02
(0.04)
-0.03
(0.04)
-0.06
(0.04)
-0.01
(0.04)
-0.02
(0.04)
-0.05
(0.05)
-0.01
(0.04)
0.03
(0.05)
0.04
(0.08)
-0.04
(0.04)
0.08
(0.06)
Regions
North
East
South
41
Table 9: Estimates of the quantile regressions for men without
correction
Constant
Age
Married
Years of work experience
City
Education
10%
20%
30%
40%
50%
60%
70%
80%
90%
0.98
( 0.14)
0.01
( 0.00)
0.12
( 0.03)
-0.01
( 0.00)
-0.03
( 0.02)
0.09
( 0.04)
0.81
( 0.13)
0.02
( 0.00)
0.13
( 0.02)
-0.01
( 0.00)
-0.01
( 0.02)
0.12
( 0.03)
0.59
( 0.14)
0.02
( 0.00)
0.14
( 0.02)
-0.01
( 0.00)
-0.01
( 0.02)
0.16
( 0.03)
0.66
( 0.14)
0.03
( 0.00)
0.13
( 0.02)
-0.02
( 0.00)
-0.03
( 0.02)
0.12
( 0.03)
0.56
( 0.13)
0.03
( 0.00)
0.13
( 0.02)
-0.02
( 0.00)
-0.02
( 0.02)
0.13
( 0.03)
0.68
( 0.16)
0.04
( 0.00)
0.11
( 0.03)
-0.02
( 0.00)
-0.02
( 0.02)
0.10
( 0.04)
0.74
( 0.17)
0.04
( 0.00)
0.12
( 0.03)
-0.02
( 0.00)
-0.01
( 0.02)
0.08
( 0.04)
0.52
( 0.14)
0.04
( 0.00)
0.12
( 0.02)
-0.02
( 0.00)
0.00
( 0.02)
0.14
( 0.03)
0.61
( 0.27)
0.04
( 0.01)
0.08
( 0.05)
-0.03
( 0.00)
-0.03
( 0.03)
0.14
( 0.07)
-0.09
( 0.03)
-0.07
( 0.02)
-0.06
( 0.02)
-0.05
( 0.03)
-0.06
( 0.02)
-0.05
( 0.02)
-0.07
( 0.03)
-0.08
( 0.02)
-0.04
( 0.02)
-0.07
( 0.03)
-0.07
( 0.02)
-0.03
( 0.02)
-0.08
( 0.03)
-0.10
( 0.02)
-0.03
( 0.02)
-0.10
( 0.03)
-0.09
( 0.02)
-0.04
( 0.02)
-0.10
( 0.03)
-0.07
( 0.03)
-0.04
( 0.02)
-0.11
( 0.03)
-0.08
( 0.02)
-0.04
( 0.02)
-0.18
( 0.05)
-0.07
( 0.04)
-0.06
( 0.04)
Region
North
East
South
42
Table 10: Estimates of the quantile regressions for men without
correction and taking endogeneity of education into account
Constant
Age
Married
Years of work experience
City
Education
10%
20%
30%
40%
50%
60%
70%
80%
90%
0.98
( 0.14)
0.01
( 0.00)
0.12
( 0.03)
-0.01
( 0.00)
-0.03
( 0.02)
0.09
( 0.04)
0.81
( 0.13)
0.02
( 0.00)
0.13
( 0.02)
-0.01
( 0.00)
-0.01
( 0.02)
0.12
( 0.03)
0.59
( 0.14)
0.02
( 0.00)
0.14
( 0.02)
-0.01
( 0.00)
-0.01
( 0.02)
0.16
( 0.03)
0.66
( 0.14)
0.03
( 0.00)
0.13
( 0.02)
-0.02
( 0.00)
-0.03
( 0.02)
0.12
( 0.03)
0.56
( 0.13)
0.03
( 0.00)
0.13
( 0.02)
-0.02
( 0.00)
-0.02
( 0.02)
0.13
( 0.03)
0.68
( 0.16)
0.04
( 0.00)
0.11
( 0.03)
-0.02
( 0.00)
-0.02
( 0.02)
0.10
( 0.04)
0.74
( 0.17)
0.04
( 0.00)
0.12
( 0.03)
-0.02
( 0.00)
-0.01
( 0.02)
0.08
( 0.04)
0.52
( 0.14)
0.04
( 0.00)
0.12
( 0.02)
-0.02
( 0.00)
0.00
( 0.02)
0.14
( 0.03)
0.61
( 0.27)
0.04
( 0.01)
0.08
( 0.05)
-0.03
( 0.00)
-0.03
( 0.03)
0.14
( 0.07)
-0.09
( 0.03)
-0.07
( 0.02)
-0.06
( 0.02)
-0.05
( 0.03)
-0.06
( 0.02)
-0.05
( 0.02)
-0.07
( 0.03)
-0.08
( 0.02)
-0.04
( 0.02)
-0.07
( 0.03)
-0.07
( 0.02)
-0.03
( 0.02)
-0.08
( 0.03)
-0.10
( 0.02)
-0.03
( 0.02)
-0.10
( 0.03)
-0.09
( 0.02)
-0.04
( 0.02)
-0.10
( 0.03)
-0.07
( 0.03)
-0.04
( 0.02)
-0.11
( 0.03)
-0.08
( 0.02)
-0.04
( 0.02)
-0.18
( 0.05)
-0.07
( 0.04)
-0.06
( 0.04)
Region
North
East
South
43