The decomposition of wage gaps in the Netherlands with sample selection adjustments

Transcription

1 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, Björklund 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, vuuren@tinbergen.nl 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.

2 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, Björklund, 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 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 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. 2 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. 1

3 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, 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

4 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, Björklund, 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

5 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 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 4

6 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

7 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 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

8 Figure 1: Kernel density estimates of wage data in OSA labor supply panel survey. males females density wage 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

9 Figure 2: Gender wage gap from raw data of the labor supply panel survey (95 % confidence intervals) wage gap 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) wage gap percentile 8

10 Figure 4: Wage gap between full-time and part-time wages of women wage gap 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

11 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

12 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 technique 5. 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 Under the null hypothesis this test statistic is χ 2 -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

13 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 X A for group A and X B for group B. Without loss of generality we assume that both X A and X B have dimension k and that the distribution function of X A is given by G XA. Likewise we use the notation G XB. The endogenous variable is given by Y A for group A and Y B for group B. Let the regression quantiles be given by β A (u) for group A and β B (u) for group B. Hence, Quant u (Y A X A ) = X A β A (u);u [0, 1] and likewise for Y B. 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 N A for group A and N B for group B. Consider a random variable Y AB, which has the properties that its quantiles conditional on x A are given by Quant u (y AB x A ) = x A β B (u) u [0, 1] where x A 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 u i from a uniform distribution with support on the interval [0, 1]. 2. Compute β B (u i ), being the regression quantile of y B j on X j,b, j = 1,...N B. 3. Sample x i from the empirical distribution (i.e. bootstrapping) ĜX A (using observations X j,a j = 1,...N A ). 4. Compute ŷ i = x i βb (u i ) 5. Repeat steps 1 to 4 M times. 12

14 Of course, ŷ i ;i = 1...M is not a real sample from the stochastic variable y AB 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 Y AB. However, one may expect that when N A and N B 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 Y AB. Let θ 0 (q) be the q th quantile of the distribution of Y AB θ 0 (q) = F 1 Y AB (q) where F YAB (y A ) = supp(x A ) F B(y A X A = x A )dg XA (x A ). 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 G XA and G XB are both continuous on their support, then θ(q) P θ 0 (q), when M,N A,N B, while M/N A I A <, M/N B I B < 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 N A and N B. Apart from consistency, it is important to show asymptotic normality. This was not so much of an issue in Albrecht, Björklund 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 lim M,NB M N B = I B <, M N B = I A <. Then, M( θ(q) θ(q)) N(0, Ω) M,N A,N B, with Ω = q(1 q) f 2 y AB (θ) [ ( 1 + I B E f AB u q (0 x A )x T A ) ( ) V (q)e fu AB q (0 x A )x A + I A E ( ] xa) T VG E (x A ) 13

15 ( ) 1 ( ) 1 V (q) = Ef u B q (0 x B )x B x T B (q(1 q)e(xb x T B)) Ef u B q (0 x B )x B x T B V G = G A1 (x 1 )(1 G(x 1 )) GA 1 (x 1 )GA 2 (x 2 )... GA 1 (x 1 )GA k (x k ) G A1 (x 1 )G A2 (x 2 ) GA 2 (x 2 )(1 GA 2 (x 2 )) G A1 (x 1 )G Ak (x k ) GA k (x k )(1 G Ak (x k )) where u AB q x d A = y AB x A β B (q), u B q x d B = y B x B β B (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 I B grow larger and larger (i.e. sampling more and more), then ultimately Ω/M becomes approximately equal to E ( x T) V (q)e (x)/n. This is exactly the variance that we would obtain using a direct method without simulation. Third, if I B becomes smaller, then ultimately Ω becomes equal to q(1 q)/f u AB q (θ). This is exactly the variance of an estimated quantile of a distribution using M observations. This makes sense as well because if N B 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 Î B = M N B ( ) ( ) Ê f u AB q (0 x A )x A = 1 M y ABi x K βb Ai (q) Mh M h i=1 M ( ) ( ) Ê f u AB q (0 x A ) = 1 M y ABi x K βb Ai (q) Mh M h M i=1 14 x Ai

16 Figure 5: Gender gap over the distribution taking acount of differences in observed characteristics using the Machado Mata approach wage gap percentile where h M is the bandwidth and K is the kernel function. We use regression cross validation techniques to find the right levels of h M (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

17 Figure 6: Difference between men s wages and women s wages when paid like men wage gap percentile 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 while Quant u (Y A x A ) = x A β A (u) u [0, 1] 16

18 Figure 7: Gender wage gap between men s wages and women s wages when having characteristics of men wage gap percentile Figure 8: Difference between men s wages and women s wages when having the characteristics of men wage gap percentile 17

19 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 wage gap wage gap percentile Education percentile Region wage gap wage gap percentile City percentile Experience wage gap wage gap percentile percentile 18

20 Quant u (Y A x A ;d = 1) = x A β A (u) + λ(z A 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 x Ai from the data of the population A (taking the complete sample). 4. Compute ŷ Ai = x Ai 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 Quant q (log (w F ) d = 1) Quant q (log ((w F )) 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

21 Figure 10: Comparison of women s wages before and after correcting for measurement error wage gap percentile Figure 11: Gender wage gap taking account of selection effects of women wage gap percentile 20

22 Figure 12: Sample selection bias based on observed characteristics wage gap 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

23 Figure 13: Sample selection bias based on unobserved characteristics wage gap 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, Björklund 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

24 Figure 14: Relative difference between wages of women when participating and paid like men with the actual women s wages wage gap percentile 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 (x 1, u) be the reservation wage of an individual based on his or her observed characteristics x 1 and unobserved characteristics u, E(u x 1 ) = 0. In addition, let y (x 2, v) be the wage of that individual based on observed characteristics x 2 and unobserved characteristics v, E(v x 2 ) = 0. We assume that the variables contained in x 2 is a subset of those contained in x 1. 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 = x 1 α + u 23

25 and y = x 2 β + v From this, it directly follows that for u [0, 1] Quant u (y x 2, d = 1) = x 2 β + Quant u (v x 1, d = 1) x 2 β Hence, a quantile regression of y on x 2 results in biased estimates. Define h(x 1 ) = E(v x 1, d = 1) and let y = x 2 β + h(x 1 ) + v 1 By definition v 1 = v h(x 1 ). And hence Quant u (v 1 x 1, v) = 0. We have that h(x 1 ) = Quant u (v y y R ) = Quant u (v v u x 1 γ)dv h 1 (z) where γ = β α and z = x 1 γ. β is equal to β for all elements included in both x 1 and x 2 and equal to zero if excluded from x 2. Hence, h 1 is only a function of z and that h 1 < 0. The exact function of h 1 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 following series estimator ĥ1(z) = i=0 δ iλ(z) i, where λ is the inverse Mill s ratio. The function ĥ1(z) is a power series of h 1 (z) and for appropriate values of δ, ĥ1(z) h 1 (z) when the number of terms goes to infinity. Before estimation of h 1 is feasible z = x 1 γ must be estimated. We estimate γ from a semi-parametric single-index regression of x 1 and d using the procedure as described in Ichimura (1993). This implies that γ is estimated by where γ = argmin 1 n Ĝ (z) = n i=1 ( ) 2 d i Ĝ(x 1iγ) ) n (z x i=1 d ik 1i γ h n ) n (z x i=1 K 1i γ h n where K is the kernel function and h n 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

26 intercept as well as δ 0 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 We first consider the estimator θ, which is the sample median obtained from the sample ỹ i ; i = 1...M. This sample is obtained as follows 1. Sample u i from a uniform distribution with support on the interval [0, 1] 2. Sample x i from the distribution G XA. 3. Compute ỹ i = x i 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 ỸAB. The distribution function of this variable is equal to FỸAB (y) = P(ỸAB y) = 1 0 P(ỸAB y U = u)du 1 = P(F 1 (U) y U = u; X A = x A )dg XA (x A )du 0 supp(x A ) 1 = P(U F YB (y X A = x A ))dg XA (x)du 0 supp(x A ) = F YB (y X A = x A )dg XA (x) = F YAB (y) supp(x A ) Hence, ỸAB = d Y AB. The most important step in this derivation is in the third line and is derived from the observation that given x A and u, Ỹ must be the u-th quantile of the Y B. This implies that the observations from the sampling method are sampled from the population distribution Y AB. For the remainder of this proof we use v i as a k-dimensional vector that is used to sample from the distribution G XA and ĜX A. The elements of v i are sampled 25

27 from the standard uniform distribution. 6 Hence X A,i = X A (v i ) = G 1 X A (v i ) and likewise for X A,i Let Ψ M (θ(q)) = 1 M i m q(ỹ AB,i, θ(q)) and Ψ M (θ(q)) = 1 M i m q(ŷ AB,i, θ(q), where q(y AB θ(q)) if y AB > θ(q) m(y AB, θ(q)) = (1 q)(θ(q) y AB ) if y AB θ(q) In addition, define Ψ(θ(q)) as Ψ(θ(q)) = (1 q) θ(q) F yab dy + θ F yab (y)dy It is obvious that Ψ(θ(q)) is maximized whenever θ(q) = θ 0 (q), which is the identification condition for quantiles. It remains to show that Ψ M (θ(q)) Ψ(θ(q)) = o P (1) By the triangular inequality sup θ(q) sup θ(q) Ψ M (θ(q)) Ψ(θ(q)) sup Ψ M (θ(q)) Ψ M (θ(q)) + sup Ψ M (θ(q)) Ψ(θ(q)) θ(q) θ(q) The last term on the last line is o P (1) by the law of large numbers. It remains to show that the first term is o P (1) as well. We have that (dropping the subscripts AB for ŷ and ỹ 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