Silviase/jgov_v1.4_full_ann huggingface dataset:
are out of school was stable, around 27% between 1987 and 2002, while the corresponding figure for men dropped slightly from 64% to 61% (Table 1). Second, the effect of education on full-time participation became stronger and women were less likely to drop off from full-time employment as they aged, as we will see later. Thus, those women who presumably had higher human capital became more likely to be employed full-time over the last 15 years.

From the strengthened, positive self-selection mechanism based on observed characteristics, we could speculate whether the same selection rule would work based on unobserved characteristics. If so, the convergence of the gender wage gap among the employed population, conditioned on changes in observed characteristics, might have occurred merely because of changes in the composition of the full-time employed population’s unobserved characteristics.

This paper aims to identify the change in the mean, offered wage gap of males and females for the whole population, including those who do not work. A similar attempt was made using US data by Mulligan and Rubinstein [2004] and Mulligan and Rubinstein [2005]. Their straightforward strategy for dealing with the sample-selection issue was to look at the gender wage gap of those with advanced degrees because this group's labor force participation rate was almost 100 percent for the entire sample period. Thus, the sample-selection issue can be essentially ignored. In addition, they also employed the Heckman sample-selection correction method, using spouse's
characteristics as an excluded variable from the wage equation. They concluded that the observed male-female wage convergence in the US was largely due to the labor force’s compositional change for both sexes. Blundell et al. [2004] also attempted to identify the change in the offered wage gap using British data. Their analysis was based on Manski [1994]'s bound estimator, which imposes a moderately weak assumption on the distribution of the non-participant’s offered wage. Exploiting predictions from economic theory, they attempted to tighten the bounds by imposing several plausible assumptions on the distribution of non-participants’ offered wage in relation with the distribution of the participants’ actual wage. They rejected the hypothesis that the observed gender wage convergence was due to only composition effects.

Closely related research has been conducted on the convergence of the wage gap between African-Americans and whites in the US. Butler and Heckman [1977] first pointed out that dropping out of the labor force could be a source of the wage convergence between these two groups. Brown [1984] suggested a method for calculating the median wage convergence under an assumption that the median offered wages for non-working people are lower than the median offered wages for working people. Johnson et al. [2000] pursued this idea using micro data with a wage imputation for non-working people. Chandra [2000] conducted a variation of this study. Most of these studies focused on the median wage convergence rather than the mean wage convergence because the median value is not directly affected by the imputed
value to the extent that the employment rate exceeds 50% and labor market drop out occurs from the bottom of the wage distribution. This method is not applicable to our analysis because, in many groups, women's full-time employment rate is below 50%.

The empirical strategy in this paper is straightforward. To eliminate the change in the observed wage gap due to sample selection, we trim some observations from either the 1987 or the 2002 wage distribution so that the full-time employment rates are identical for these two years. For example, if the employment rate of female college graduates with 5 potential years of experience is increased by 5 percentage points in 2002 from 1987, we trim the top or bottom 5 percent of the wage distribution of the same group in 2002. Whether we trim from the top or from the bottom determines whether we estimate the upper bound or the lower bound of the gender wage convergence. We apply the same trimming for all groups, and this trimming artificially creates a sample whose employment rates are identical for 1987 and 2002 for all groups. A regression applied to this trimmed data estimates the upper and lower bounds for the true gender wage convergence. This approach enables us to estimate sharp bounds without imposing any assumption on the non-participants' wage distribution. Instead, we essentially impose an assumption on the distribution of unobserved characteristics among the employed population.

The contribution of this paper to the literature is two-fold. First, we suggest a sharp bound estimator to estimate the change in the mean wage across
genders when the composition of workers for each group is changing, without relying on a strong assumption. Second, we credibly identify the change in the gender wage gap for Japan where the gender wage gaps narrowed when the composition of male and female workers changed simultaneously.

It is worth noting that the purpose of this paper is not to identify a causal mechanism of wage determination, such as the return to education. This is principally due to the lack of a good instrument for potentially endogenous variables and thus is a limit of this study. Assuming the exogeneity of explanatory variables in the whole population, we focus our discussion on the issues of biases due to sample selection.

The results of the analysis are summarized as follows. First, we find gender wage convergence among full-time workers, even after adjusting for their observed characteristics. The regression-adjusted gender wage gap narrowed from 31 percent to 26 percent between 1987 and 2002. Second, the bound estimation based on very agnostic assumptions render a wide range of bounds, and thus we cannot rule out the possibility that the observed gender wage convergence is due to a change in the workers' composition. Third, the bound estimation under an excluded variable assumption that spousal income affects reservation wage, but not offered wage renders very narrow bounds. Thus, if we believe in the excluded variable assumption, we could conclude that the observed gender wage convergence during the period was not an illusion.

The paper is organized as follows. Section 2 lays out a simple Roy model
to illustrate changes in the selection pattern into employment and discusses the empirical strategy. Section 3 describes the Japanese micro data used in this study and describes the change in the gender wage gap and the workforce composition in terms of males and females. Section 4 reports the results of the trimming regressions. Section 5 discusses the trimming estimation under an excluded variable assumption and reports the results. The last section concludes.

## 2 Methodology

### 2.1 Bound Estimators

This section discusses the identification of bounds on the change in the underlying gender wage gap, allowing for a change in the unobserved distribution of human capital. Instead of identifying the exact parameter value of the wage convergence, the discussion here attempts to identify the bounds of the parameter value. The identification of the sharp upper- and lower-bounds of the wage gap convergence is based on the discussion by Lee [2005] in the context of program evaluation under non-ignorable sample attrition.

To discuss the convergence of the gender wage gap, we assume that the equilibrium offered wage is determined by the following equation.

$$ w^{o}=x\beta_{0}+f\beta_{1}+t x\beta_{2}+t f\beta_{3}+u, $$

where $ w^{o} $ is the log of hourly offered wage and x is the vector of education, potential experience, and job tenure. The dummy variable f takes 1 if the
individual is female, and t is the time dummy that takes 1 if the year is 2002. We do not allow for different coefficients for education experience and tenure between sexes. The same potential experience most likely corresponds to the different actual experiences of males and females. Thus, the estimated coefficient for potential experience should be interpreted as the coefficient for the age conditioning variable, rather than the return to general human capital. By specifying the wage equation in this way, the coefficient $ \beta_{3} $ corresponds to the gender wage convergence.

We assume the exogeneity of the error term $ E[u|x,f,t]=0 $ . This is a strict assumption that is likely to be violated. Thus, we cannot interpret the estimated coefficient as causal. The focus of this study is limited to examining the bias due to sample selection. To save space, we denote $ [w|x,f=0,t=0]=w_{x}^{m87} $ , $ [w|x,f=1,t=0]=w_{x}^{f87} $ , $ [w|x,f=0,t=1]=w_{x}^{m02} $ and $ [w|x,f=1,t=1]=w_{x}^{f02} $ . We also denote the conditional distribution of u following the same convention.

The reservation wage for each individual is given as

$$ w^{r}=x_{1}\alpha_{0}+f x_{1}\alpha_{1}+t x_{1}\alpha_{2}+t f x_{1}\alpha_{3}+e, $$

where $ x_{1} $ is a part of x that includes education and potential experience variables. The reservation wage could be higher among women with high human capital because of assortative mating and the income effect due to husband's income. This implies a positive $ \alpha_{1} $ . If the degree of positive assortative mating has changed between 1987 and 2002, we expect $ \alpha_{3} \neq 0 $ .
Each individual is a full-time worker when the offered wage exceeds the reservation wage. Thus, the reduced-form equation for full-time employment is defined as follows:

$$ s=1[x\theta_{0}+f x\theta_{1}+t x\theta_{2}+t f x\theta_{3}+v\geq0], $$

The log hourly wage w is observed when s = 1. We denote $ P[s = 1 | x, f = 0, t = 0] = p_{x}^{m87} $ , $ P[s = 1 | x, f = 0, t = 1] = p_{x}^{m02} $ , $ P[s = 1 | x, f = 1, t = 0] = p_{x}^{f87} $ , and $ P[s = 1 | x, f = 1, t = 1] = p_{x}^{f02} $ .

The convergence of the wage gap that we want to estimate is

$$ (E w_{x}^{m87}-E w_{x}^{f87})-(E w_{x}^{m02}-E w_{x}^{f02})=\beta_{3}. $$

We, however, cannot estimate this because we observe w only for s = 1. Instead, we can estimate

$$ \begin{align*}\big(E[w_{x}^{m87}|s=1]-E[w_{x}^{f87}|s=1]\big)-\big(E[w_{x}^{m02}|s=1]-E[w_{x}^{f02}|s=1]\big)\\=\beta_{3}+\big(E[u_{x}^{m87}|s=1]-E[u_{x}^{f87}|s=1]\big)-\big(E[u_{x}^{m02}|s=1]-E[u_{x}^{f02}|s=1]\big).\end{align*} $$

The last bias terms are likely to be non-zero due to the change in the sample-selection structure. The sample-selection structure could have changed because of changes in the offered wage structure or the reservation wage structure. Instead of specifying the sample-selection terms, we estimate the bound for these bias terms and for $ \beta_{3} $ .

We attempt to identify the sharp upper and lower bounds of $ \beta_{3} $ , which critically depends on the following assumption.
Assumption 1 The joint distribution of $ (u,v) $ , conditional on x, f and t, is invariant over time.

It is important to note that this assumption still allows for changes in the offered-wage structure and the reservation-wage structure because the coefficients can change over time. This assumption implies that under the same selection rules for 1987 and 2002, the conditional means of the unobserved wage determinants among employed populations are identical. This is to say

$$ E[u_{x}^{g87}|v\geq-(x\theta_{0}+fx\theta_{1}+tx\theta_{2}+txf\theta_{3})]=E[u_{x}^{g02}|v\geq-(x\theta_{0}+fx\theta_{1}+tx\theta_{2}+txf\theta_{3})] $$

for all $ g = (m, f) $ , $ t = (0, 1) $ , and $ f = (0, 1) $ . Note that $ u_{x}^{m87} = [u | x, f = 0, t = 0] $ for example.

Claim 1 Under assumption 1, the sharp upper- and lower- bounds

of the gender wage convergence parameter, $ \beta_{3} $ , is given as

$$ \begin{aligned}\beta_{3}^{u}&=1(p_{x}^{f02}>p_{x}^{f87})\{E[w_{x}^{f02}|s=1,u>u_{(p_{x}^{f02}-p_{x}^{f87})/p_{x}^{f02}}]-E[w_{x}^{f87}|s=1]\}\\&+1(p_{x}^{f02}\leq p_{x}^{f87})\{E[w_{x}^{f02}|s=1]-E[w_{x}^{f87}|s=1,u\leq u_{1-(p_{x}^{f87}-p_{x}^{f02})/p_{x}^{f87}}]\}\\&-1(p_{x}^{m02}>p_{x}^{m87})\{E[w_{x}^{m02}|s=1,up_{x}^{f87})\{E[w_{x}^{f02}|s=1,up_{x}^{m87})\{E[w_{x}^{m02}|s=1,u>u_{(p_{x}^{m02}-p_{x}^{m87})/p_{x}^{m02}}]-E[w_{x}^{m87}|s=1]\}\\&-1(p_{x}^{m02}\leq p_{x}^{m87})\{E[w_{x}^{m02}|s=1]-E[w_{x}^{m87}|s=1,u\leq u_{1-(p_{x}^{m87}-p_{x}^{m02})/p_{x}^{m87}}]\},\\ \end{aligned} $$
where, $ 1(\cdot) $ is the indicator function. The notation $ u_{p} $ stands for the pth percentile of the distribution of u. The condition $ u > u_{p} $ implies that the very bottom p percent is trimmed from the wage distribution, and $ u \leq u_{1-p} $ implies that the very top p percent is trimmed from the wage distribution.

The reasoning behind $ \beta_{3}^{u} $ is the following. When the employment rate of women with characteristics x increases by $ p_{x}^{f02} - p_{x}^{f87} $ , we obtain the upper bound of women's average wage growth by trimming the bottom $ (p_{x}^{f02} - p_{x}^{f87})/p_{x}^{f02} $ proportion of the 2002 women's wage distribution. After this trimming, the artificial labor force participation rate of group x becomes $ p_{x}^{f87} $ , because $ (p_{x}^{f02} - p_{x}^{f02}\frac{(p_{x}^{f02} - p_{x}^{f87})}{p_{x}^{f02}}) = p_{x}^{f87} $ . In contrast, when the labor force participation rate declines by $ p_{x}^{f87} - p_{x}^{f02} $ , we obtain the upper bound of women's average wage growth by trimming the top $ (p_{x}^{f87} - p_{x}^{f02})/p_{x}^{f87} $ proportion of the women's 1987 wage distribution.

Similarly, if the employment rate of men with characteristics x increases by $ p_{x}^{m02} - p_{x}^{m87} $ , we obtain the lower bound of men's average wage growth by trimming the bottom $ (p_{x}^{m02} - p_{x}^{m87})/p_{x}^{m87} $ proportion from the 1987 men's wage distribution. In contrast, when the employment rate decreases by $ p_{x}^{m87} - p_{x}^{m02} $ we obtain the lower bound of men's wage growth by trimming the top $ (p_{x}^{m87} - p_{x}^{m02})/p_{x}^{m02} $ proportion of the men's 2002 wage distribution.

The upper bound of women’s wage growth minus the lower bound of men’s wage growth defines the upper bound of the gender wage convergence, $ \beta_{3}^{u} $ . In addition, the lower bound of women’s wage growth minus the upper
bound of men’s wage growth defines the lower bound of the gender wage convergence, $ \beta_{3}^{l} $ .

Now we give the proof for Claim 1.

## Proof

The $ \beta_{3}^{u} $ is decomposed into the true convergence part and the bias terms as follows:

$$ \begin{aligned}\beta_{3}^{u}\ &=\beta_{3}\\&+1(p_{x}^{f02}>p_{x}^{f87})\{E[u_{x}^{f02}|s=1,u>u_{(p_{x}^{f02}-p_{x}^{f87})/p_{x}^{f02}}]-E[u_{x}^{f87}|s=1]\}\\&+1(p_{x}^{f02}\leq p_{x}^{f87})\{E[u_{x}^{f02}|s=1]-E[u_{x}^{f87}|s=1,u\leq u_{1-(p_{x}^{f87}-p_{x}^{f02})/p_{x}^{f87}}]\}\\&-1(p_{x}^{m02}>p_{x}^{m87})\{E[u_{x}^{m02}|s=1,u p_{x}^{f87} $ , from the selection equation $ s = 1[x\alpha_{0} + fx\alpha_{1} + tx\alpha_{2} + tf x\alpha_{3} + v \geq 0] $ , we know that $ x\alpha_{2} + fx\alpha_{3} > 0 $ . Then, $ \{E[u_{x}^{f02}|s = 1, u > u_{(p_{x}^{f02} - p_{x}^{f87})/p_{x}^{f02}}] - E[u_{x}^{f87}|s = 1]\} $ is positive. To see this, $ E[u_{x}^{f02}|s = 1, u > u_{(p_{x}^{f02} - p_{x}^{f87})/p_{x}^{f02}}] = E[u_{x}^{f02}|v \geq -x\alpha_{0} - x\alpha_{1} - x\alpha_{2} - x\alpha_{3}, u > u_{(p_{x}^{f02} - p_{x}^{f87})/p_{x}^{f02}} $ is obtained by removing the very bottom $ (p_{x}^{f02} - p_{x}^{f87})/p_{x}^{f02} $ proportion of $ u_{x}^{f02} $ from the conditional distribution of $ u_{x}^{f02} $ on $ v \geq -x\alpha_{0} - x\alpha_{1} - x\alpha_{2} - x\alpha_{3} $ . However, $ E[u_{x}^{f87}|s = 1] = E[u_{x}^{f87}|v \geq -x\alpha_{0} - x\alpha_{1}] = E[u_{x}^{f87}|v \geq -x\alpha_{0} - x\alpha_{1}, v \geq -x\alpha_{0} - x\alpha_{1} - x\alpha_{2} - x\alpha_{3}] = E[u_{x}^{f02}|v \geq -x\alpha_{0} - x\alpha_{1}, v \geq -x\alpha_{0} - x\alpha_{1} - x\alpha_{2} - x\alpha_{3}] $ is obtained by removing
$ (p_{x}^{f02}-p_{x}^{f87})/p_{x}^{f02} $ proportion of $ u_{x}^{f02} $ from the conditional distribution of $ u_{x}^{f02} $ on $ v \geq -x\alpha_{0} - x\alpha_{1} - x\alpha_{2} - x\alpha_{3} $ . The former expectation is obtained by removing $ (p_{x}^{f02}-p_{x}^{f87})/p_{x}^{f02} $ proportion from the bottom, and the latter expectation is obtained by removing $ (p_{x}^{f02}-p_{x}^{f87})/p_{x}^{f02} $ proportion that does not satisfy $ v \geq -x\alpha_{0} - x\alpha_{1} $ . Thus, the former expectation should be larger than or equal to the latter expectation.

We can repeat exactly the same argument for the other three terms and show that all four selection-bias terms are non-negative. Thus $ \beta_{3}^{u} \geq \beta_{3} $ . The same argument is used to show $ \beta_{3}^{l} \leq \beta_{3} $ . The bound is sharp because the bias term can be zero. Under Assumption 1, if the employment rate does not change for all groups, the exact value of the parameter is identified (i.e., $ \beta_{3}^{l} = \beta_{3} = \beta_{3}^{u} $ ).

### 2.2 Estimation Procedure

The estimation of the upper- and lower-bounds of the gender wage convergence discussed above is implemented by the following procedure.

1. Calculate the employment rate by gender and $ x_{1} $ (education and potential experience) for 1987 and 2002. The conditioning is implemented based on $ x_{1} $ instead of x (education, potential experience, tenure) because tenure is not observed among non-employed individuals. Call the predicted value $ \hat{p}_{x_{1}}^{jt} $ for t = 87, 02 and $ j = (\text{male}, \text{female}) $ .
2. Define the upper-bound variable, $ \overline{w} $ , by trimming the bottom $ \left(\hat{p}_{x_{1}}^{f02}-\hat{p}_{x_{1}}^{f87}\right)/\hat{p}_{x_{1}}^{f02} $ fraction of the distribution of $ w_{x_{1}}^{f02} $ if $ \hat{p}_{x_{1}}^{f02}>\hat{p}_{x_{1}}^{f87} $ , and by trimming the top $ \left(\hat{p}_{x_{1}}^{f87}-\hat{p}_{x_{1}}^{f02}\right)/\hat{p}_{x_{1}}^{f87} $ fraction of $ w_{x_{1}}^{f87} $ if $ \hat{p}_{x_{1}}^{f02}\leq\hat{p}_{x_{1}}^{f87} $ . Note that the trimming based on w is identical to the trimming based on u because $ x_{1} $ is conditioned.

3. Define the upper-bound variable, $ \overline{w} $ , by trimming the top $ \left(\hat{p}_{x_{1}}^{m02}-\hat{p}_{x_{1}}^{m87}\right)/\hat{p}_{x_{1}}^{m02} $ fraction of the distribution of $ w_{x_{1}}^{m02} $ , if $ \hat{p}_{x_{1}}^{m02}>\hat{p}_{x_{1}}^{m87} $ . If $ \hat{p}_{x_{1}}^{m02}\leq\hat{p}_{x_{1}}^{m87} $ , trim the bottom $ \left(\hat{p}_{x_{1}}^{m87}-\hat{p}_{x_{1}}^{m02}\right)/\hat{p}_{x_{1}}^{m87} $ fraction from the distribution of $ w_{x_{1}}^{m87} $ .

4. Define the lower-bound variable, $ \underline{w} $ , by trimming the top $ (\hat{p}_{x_{1}}^{f02}-\hat{p}_{x_{1}}^{f87})/\hat{p}_{x_{1}}^{f02} $ of the distribution of $ w_{x_{1}}^{f02} $ if $ \hat{p}_{x_{1}}^{f02} > \hat{p}_{x_{1}}^{f87} $ , and by trimming the bottom $ (\hat{p}_{x_{1}}^{f87}-\hat{p}_{x_{1}}^{f02})/\hat{p}_{x_{1}}^{f87} $ from $ w_{x_{1}}^{f87} $ if $ \hat{p}_{x_{1}}^{f02} \leq \hat{p}_{x_{1}}^{f87} $ .

5. Define the lower-bound variable, $ \underline{w} $ , by trimming the bottom $ (\hat{p}_{x_{1}}^{m02}-\hat{p}_{x_{1}}^{m87})/p_{x_{1}}^{m02} $ fraction of the distribution of $ w_{x_{1}}^{m02} $ , if $ \hat{p}_{x_{1}}^{m02}>\hat{p}_{x_{1}}^{m87} $ . If $ \hat{p}_{x_{1}}^{m02}\leq\hat{p}_{x_{1}}^{m87} $ , trim the top $ (\hat{p}_{x_{1}}^{m87}-\hat{p}_{x_{1}}^{m02})/\hat{p}_{x_{1}}^{m87} $ fraction from the distribution of $ w_{x_{1}}^{m87} $ .

6. Run the OLS regression of $ \overline{w} $ on x, f, tx, and tf where x is education, potential experience, its squared, its cubic, its quartic, tenure and its squared, its cubic, and its quartic. The dummy variable f is the female dummy, and t is 2002 dummy. The coefficient for tx is the estimate for $ \beta_{3}^{u} $ .
7. Similarly, run the OLS regression of $ \underline{w} $ on explanatory variables to obtain $ \beta_{3}^{l} $ .

## 3 Data and Descriptive Statistics

The Employment Status Survey (ESS, Shugyo Kozo Kihon Chosa) for the years 1987, 1992, 1997, and 2002 is used in this study. The ESS is conducted every 5 years on household members 15 years old or more in approximately 440,000 households dwelling in sampled units that cover the complete population. $ ^{1} $ The survey collects information on household members and each member's labor force status on October 1 of each survey year. This study utilizes micro data and extracts information on age, educational attainment, employment status, annual work days, weekly work hours, and annual income from the main job during the previous year. The file contains about 1 million individuals with a half-million males and a half-million females for each year that the survey was conducted. The analysis sample is restricted to those ages 15-64 and out of school. The sample is further restricted to observations with a valid age, educational background, and employment status. Those without job tenure conditioned on being employed are dropped from the wage regression sample. The sizes of the analysis sample are about 600,000 males 600,000 females for the following year combinations: 1987 and 1992, 1987 and 1997, and 1987 and 2002. Although the previous discussion
on identification and estimation was based on the 1987-2002 case, the same method is applied to the 1987-1992 and 1987-1997 cases.

Regarding the construction of variables, we transformed the highest educational attainment into a continuous variable. For junior high-school graduates and high-school graduates, 9 and 12 years of education are assigned, respectively. Twelve years of education is assigned for junior-college and technical-college graduates, and 16 years of education is assigned for 4-year college graduates and graduate school graduates. The survey records the annual earnings in ranges. $ ^{2} $ This range is transformed into a continuous variable by using the center value for each range. For the highest open bracket range, the lowest value for the range is assigned for each year. Weekly work hours and annual work days are also recorded by ranges, and we transformed them into continuous variables using the same rule $ ^{3} $ .

Constructing the hourly wage from these brackets could result in a biased estimate for the true hourly wage, but what we care about is the change in the
hourly wage over time rather than the wage level. The ranges of the brackets for work days are consistent over the years, as are the ranges for weekly work hours. Regarding the annual income range, the change in the range of the brackets in the middle of annual income distribution does not change the hourly wage if the annual income is uniformly distributed within the range. Although the uniform distribution assumption is likely to be violated, the bias due to this would be minimal.

The critical change of the brackets for our purpose is the increase in the largest range from 10 million or above to 15 million or above between 1987 and 1992. If a disproportionately large number of males earns more than 15 million, conditional on earning more than 10 million, $ ^{4} $ the change of this brackets increased the annual earnings of males in 1992 and after, even if the underlying annual earnings had not changed.

To avoid this issue, we imputed those who might have earned more than 15 million in 1987 by the following procedure. First, we calculated the proportion of people who earned more than 15 million conditional on earning more than 10 million, using the 1992 sample by sexes. Let us call this proportion $ p_{>15} $ . Second, we ran separate probit regressions for both sexes, whose dependent variable is the dummy variable indicating earning more than 15 million, and the independent variables, which are education, experience, its squared, tenure and its squared, using the individuals who earned more than