Silviase/jgov_v1.4_full_ann huggingface dataset:
10 million in 1992 as the analysis sample. Third, we assigned 15 million earnings for $ p_{>15} $ proportion of people who earned more than 10 million in 1987 based on the predicted probability to earn more than 15 million, using the probit coefficient of 1992 for both sexes. Hourly wage is calculated based on these imputed annual earnings.

Full-time workers in this study are defined as those who work without a term contract (Joyo Rodosha) and work regular hours for an employer (Ippan Rodosha).

Table 2 tabulates the full-time employment rate by sex, age, and education groups. The overall, full-time employment rate did not change much between 1987 and 2002 for either sex. However, the age-specific, full-time employment rate has changed substantially. In particular, among males, youths' full-time employment rate declined drastically in the last 15 years, as articulated by Genda [2001]. In contrast, the full-time employment rate has increased among the elderly, as articulated by Chuma [1997]. Thus, those who presumably have more human capital are more likely to be employed full-time in terms of the age structure among males.

With respect to years of education, the employment rate has fallen for almost all educational and age categories. However, the rate of drop is smaller among those with more years of education. This tendency is stronger among women.

Table 3 reports the estimation results of the linear regression of the full-time employment dummy on education and age, using the male out-of-school
sample. We again can confirm that males over 50 years old were less likely to retire from full-time employment in 2002. The estimation results for females reported in Table 4 indicate that females with junior- or 4-year-college education were more likely to be full-time workers. Also those females aged 25-29, 30-34, and 35-39 were less likely to drop off from full-time employment in 2002.

These figures suggest that the average human capital level among full-time workers increased for both males and females from 1987 to 2002, but the degree of change seems larger for females. Thus the convergence of the observed wage distribution can be attributed to the convergence in the unobserved wage determinants, if the observed wage determinants and the unobserved wage determinants are positively correlated.

## 4 Results

To address the possibility that the gender wage convergence occurred due to a compositional change in full-time employment, this section implements the bound estimation of the gender wage convergence, allowing for a change in the distribution of unobserved characteristics. Table 5 reports the descriptive statistics of the analysis sample used in the wage regression analysis.

Table 6 reports the estimation results of the log wage equations for 1987, 1992, 1997, and 2002. The adjusted gender wage differential narrowed from 33 to 27 percent during this period; and the gender wage convergence is estimated to be about 6 percentage points. The estimated coefficients for
education, potential job experience, and tenure are similar across years. This stable wage structure already has been pointed out by Ohtake [2005].

As discussed previously, about 7 percentage points of the convergence in the gender wage differential could be a product of the convergence in the distribution of unobserved wage determinants. To allow for the change in the distribution of unobserved wage determinants, we now estimate the bound of the true gender wage convergence following the empirical strategy discussed in the previous sections. This strategy trims some observations so that the full-time employment ratios are identical between 1987 and 2002, conditional on the subset of workers' education and potential years of experience.

We obtain $ \overline{w} $ and $ \underline{w} $ by trimming the bottom or top tail of the wage distribution for each sex and year.

The results of the trimming estimations using the 1987-1992 sample are reported in Table 7. Column (1) reports the OLS regression result applied to the untrimmed log (wage) that replicates the results reported in Table 6. We do not observe a convergence of the gender wage differentials during this period among working people. The second column reports the OLS regression coefficients applied to $ \overline{w} $ . The number of observations is dropped from the number used for column (1) due to the trimming. This estimate is based on the assumption that the observed, stable gender wage differential is observed regardless of the divergence of the unobserved wage determinant between the sexes. Thus, the underlying gender wage gap could have converged up to 21 percentage points. The second column reports the OLS regression result
applied to w. This estimate attributes the observed wage convergence to the convergence of the unobserved wage determinant between the sexes at the maximum possible strength. This estimate implies that the offered wage gap for the sexes could have been diverged by 22 percentage points, while a stable gender wage gap is observed.

Table 8 reports the results of the trimmed estimation applied to the 1987 and 1997 data. We do not observe a convergence of wages for this period among working people (the gender wage convergence was only 2 percentage points). However, the underlying offered wage might have converged by 26 percentage points during this period as the upper bound. In contrast, the lower-bound estimate implies a possible 20-percentage-point widening gender wage gap during the period. Again, we cannot rule out a wide range of stories underlying the observed 2-percentage-point gender wage convergence between 1987 and 1997.

Finally, Table 9 reports the results of the trimming estimations applied to the 1987 and 2002 sample. We observe a 6-percentage-point convergence in the gender wage gap in these 15 years among full-time employees. However, the upper-bound estimate implies a possible 33-percentage-point convergence of the mean offered wage between genders. However, the lower-bound estimate indicates a possible gender divergence of 18 percentage points.

Overall, the bound estimates based on the worst-case scenario tend to indicate a wide bound, and we cannot informatively infer the sexes' mean wage convergence. However, at least the results reported in this section
warn us to infer the gender wage convergence at the mean only for full-time, employed workers. We must carefully consider the effects of sample selection.

## 5 Tightening the Bounds Using the Excluded Variable Assumption

We now consider how to exploit more economic information from the data to tighten the bound. We assume that there is an excluded variable that determines full-time employment status, but not offered wage. We use spousal income as the excluded variable. The analysis sample for this section includes only married couples to use spousal income as an excluded variable.

The structural equations for the offered wage are the following, as before.

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

The wage w is observed only among full-time workers s = 1. We assume a binary response model for full-time participation, and the conditional probability of being a full-time worker is given as:

$$ s=1(x\theta_{0}+f x\theta_{1}+x t\theta_{2}+f x t\theta_{3}+z\theta_{4}+f z\theta_{5}+v\geq0), $$

where z is spousal income. We assume that leisure is a normal good, i.e. $ \partial P(s = 1|x, f, z, t)/\partial z > 0 $ for all $ (x, f, t) $ . We also assume that the coefficients for spousal income are constant over time, and this is a crucial assumption for identifying the upper and lower bounds.

The error terms of two equations are assumed to have a positive correlation. This is natural because both the wage equation error term u and the
participation equation error term v should include the unobserved wage determinants. Among full-time workers, those who have high, non-own labor, household income should have a high, unobserved earning ability v and thus a higher u, conditional on x, f, and t.

We now assume that $ (u,v) $ are stationary, conditional on x, f, z, and t in the wage and selection equations. We exploit the information given by the positive correlation between u and v. Because u and v are positively correlated, trimming observations from the higher value of v (the higher value of non-labor income among full-time workers) is equivalent to trimming the observations from the higher value of u.

The following upper- and lower-bound estimators implement this idea:

$$ \begin{array}{r c l}{\beta_{3\{x,z\}}^{u}}&{=}&{1(p_{x}^{f02}>p_{x}^{f87})\{E[w_{x}^{f02}|s=1,z\geq z_{\frac{p_{x}^{f02}-p_{x}^{f87}}{p_{x}^{f02}}}]-E[w_{x}^{f87}|s=1]\}}\\ {}&{+}&{1(p_{x}^{f02}\leq p_{x}^{f87})\{E[w_{x}^{f02}|s=1]-E[w_{x}^{f87}|s=1,zp_{x}^{m87})\{E[w_{x}^{m02}|s=1,z\leq z_{1-\frac{p_{x}^{f02}-p_{x}^{f87}}{p_{x}^{f02}}}]-E[w_{x}^{m87}|s=1]\}}\\ {}&{-}&{1(p_{m}^{f02}\leq p_{x}^{m87})\{E[w_{x}^{m02}|s=1]-E[w_{x}^{m87}|s=1,z>z_{\frac{p_{x}^{f87}-p_{x}^{f02}}{p_{x}^{f87}}}\}\end{array} $$
$$ \begin{array}{r c l}{\beta_{3\{x,z\}}^{l}}&{=}&{1(p_{x}^{f02}>p_{x}^{f87})\{E[w_{x}^{f02}|s=1,zp_{x}^{f87})\{E[w_{x}^{f02}|s=1]-E[w_{x}^{f87}|s=1,z\geq z_{\frac{p_{x}^{f87}-p_{x}^{f02}}{p_{x}^{f87}}}\}]\}}\\ {}&{-}&{1(p_{x}^{m02}>p_{x}^{m87})\{E[w_{x}^{m02}|s=1,z>z_{\frac{p_{x}^{m02}-p_{x}^{m87}}{p_{x}^{m02}}}]-E[w_{x}^{m87}|s=1]\}}\\ {}&{-}&{1(p_{x}^{m02}\leq p_{x}^{m87})\{E[w_{x}^{m02}|s=1]-E[w_{x}^{m87}|s=1,z\leq z_{1-\frac{p_{x}^{m87}-p_{x}^{m02}}{p_{x}^{m87}}}]\}}\\ \end{array} $$

We can show that $ \beta_{3\{x,z\}}^{u}\geq\beta_{3} $ and $ \beta_{3\{x,z\}}^{l}\leq\beta_{3} $ . To see this, $ \beta_{3\{x,z\}}^{u} $ is decomposed into the true parameter and the bias term:

$$ \begin{aligned}\beta_{3\{x,z\}}^{u}\ &=\beta_{3}\\&+\quad1(p_{x}^{f02}>p_{x}^{f87})\{E[u_{x}^{f02}|s=1,z\geq z_{\frac{p_{x}^{f02}-p_{x}^{f87}}{p_{x}^{f02}}}]-E[u_{x}^{f87}|s=1]\}\\&+\quad1(p_{x}^{f02}\leq p_{x}^{f87})\{E[u_{x}^{f02}|s=1]-E[u_{x}^{f87}|s=1,zp_{x}^{m87})\{E[u_{x}^{m02}|s=1,z\leq z_{1-\frac{p_{x}^{f02}-p_{x}^{f87}}{p_{x}^{f02}}}]-E[u_{x}^{m87}|s=1]\}\\&-\quad1(p_{m}^{f02}\leq p_{x}^{m87})\{E[u_{x}^{m02}|s=1]-E[u_{x}^{m87}|s=1,z>z_{\frac{p_{x}^{f87}-p_{x}^{f02}}{p_{x}^{f87}}}].\}\\ \end{aligned} $$

This bias term is positive if the error terms of the wage equation and the selection equation are positively correlated in the way that $ u = \rho v + e $ , where $ E[e|x, z] = 0 $ .

The following discussion heuristically shows that the bias terms are positive. Given an increase in the full-time employment rate among females for
group x, (i.e. $ p_{x}^{f02} > p_{x}^{f87} > 0 $ ), the first two terms become

$$ \rho\{E[v_{x}^{f02}|s=1,z\geq z_{\frac{p_{x}^{f02}-p_{x}^{f87}}{p_{x}^{f02}}}]-E[v_{x}^{f87}|s=1]\}. $$

The term $ E[v_{x}^{f02}|s = 1, z \geq z_{\frac{p_{x}^{f02}-p_{x}^{f87}}{p_{x}^{f02}}}] - E[v_{x}^{f87}|s = 1] $ is positive because $ E[v_{x}^{f02}|v \geq -x(\theta_{0} + \theta_{1} + \theta_{2} + \theta_{3}) - z(\theta_{4} + \theta_{5}), z \geq z_{\frac{p_{x}^{f02}-p_{x}^{f87}}{p_{x}^{f02}}}] - E[v_{x}^{f87}|v \geq -x(\theta_{0} + \theta_{1} + \theta_{2} + \theta_{3}) - z(\theta_{4} + \theta_{5}), v \geq -x(\theta_{0} + \theta_{1}) - z(\theta_{4} + \theta_{5})] $ is positive. Given $ v \geq -x(\theta_{0} + \theta_{1} + \theta_{2} + \theta_{3}) - z(\theta_{4} + \theta_{5}) $ is satisfied, the proportion $ (p_{x}^{f02} - p_{x}^{f87})/p_{x}^{f02} $ should be trimmed from the 2002 distribution to balance the full-time employment for 1987 and 2002. Once the value of x is fixed, the largest value of $ -x(\theta_{0} + \theta_{1} + \theta_{2} + \theta_{3}) - z(\theta_{4} + \theta_{5}) $ is attained by trimming the bottom $ (p_{x}^{f02} - p_{x}^{f87})/p_{x}^{f02} $ proportion of z because $ (\theta_{4} + \theta_{5}) < 0 $ under the assumption that leisure is a normal good. Because $ -x(\theta_{0} + \theta_{1} + \theta_{2} + \theta_{3}) - z(\theta_{4} + \theta_{5}) $ attains highest value under $ z_{\frac{p_{x}^{f02}-p_{x}^{f87}}{p_{x}^{f02}}} $ after fixing the value of x, $ E[v_{x}^{f02}|v \geq -x(\theta_{0} + \theta_{1} + \theta_{2} + \theta_{3}) - z(\theta_{4} + \theta_{5}), z \geq z_{\frac{p_{x}^{f02}-p_{x}^{f87}}{p_{x}^{f02}}}] - E[v_{x}^{f87}|v \geq -x(\theta_{0} + \theta_{1} + \theta_{2} + \theta_{3}) - z(\theta_{4} + \theta_{5}), v \geq -x(\theta_{0} + \theta_{1}) - z(\theta_{4} + \theta_{5})] $ is positive.

Repeating a similar argument for all the terms, we can confirm that all the bias terms are non-negative. We can show $ \beta_{3}^{l,\{x,z\}} \leq \beta_{3} $ by the same argument.

The actual implementation of the estimation is identical to the trimming estimator based on wage trimming. Instead of trimming certain observations, we trimmed the observation based on the non-labor income within 185 cells based on education $ \times $ experience. We balanced the full-time employment rate for two years for all 185 cells.
The analysis sample is restricted to those who are married and who are the household head or his/her spouse. This sample restriction is applied because spousal income is identified for these individuals. We dropped the individuals whose spouse is employed, but for whom annual income is not reported. Because the analysis sample is restricted to married individuals different from the previous analysis, we repeated the trimming estimation without the excluded variable assumption for this analysis sample.

The results of the analysis appear in Table 7. The first column reports the gender wage convergence between 1987 and 1992, 1997, or 2002. Among married people, we do not observe a mean wage convergence between 1987 and 1992, but a 3-percentage-point convergence is observed between 1987 and 1997. Between 1987 and 2002, the gender wage gap converged by 11 percentage points. This number is even larger than the gender wage convergence observed among the entire working population reported in the previous table.

The upper- and lower-bound estimates for the wage convergence under the agnostic assumption are reported in Columns (2) and (3), respectively. For all sample years, the upper bound is about 20 percentage points higher than the point estimate and the lower bound is about 20 percentage points lower than the point estimate. The bound is so wide that we cannot rule out the possibility that the observed gender wage convergence occurred due to a compositional change of full-time workers.

Columns (4) and (5) report the upper- and lower-bound estimates based
on the trimming by spousal annual income. The excluded variable assumption turns out to add much information, and the bounds for the gender wage convergence become very narrow. For the period between 1987 and 1992, the convergence was bounded between -1 and 1 percentage points where the untrimmed estimate indicates a -1 percentage point convergence (in fact, a 1 percentage point divergence). The estimation results for the 1987 - 1997 period are a bit problematic because the untrimmed estimate does not reside in the range of lower- and upper-bound estimates. The point estimate is 0.03 and the bounds are 0.04 and 0.05. However, this could occur due to sampling error.

The highlight of the bound estimation based on the excluded variable assumption is the results between 1987 and 2002. We observe an 11-percentage-point convergence of the gender wage gap for this 15-year period among all full-time workers. The bound estimates based on spousal income trimming render a 10-percentage-point convergence as the lower bound and a 15-percentage-point convergence as the upper bound. This rather narrow bound clearly excludes the possibility that the observed gender wage convergence among full-time workers is an illusion due to a change in employment composition.

## 6 Conclusion

This paper discusses the identification and estimation of the gender wage convergence among full-time, regular workers. The identification of the gender
wage convergence is generally difficult because the composition of full-time workers tend to change contemporaneously. The identification strategy is essentially based on trimming the wage distribution, and this strategy contrasts with the identification strategy proposed by previous studies that have been based on imputing the non-employed population’s offered wage.

The adjusted gender wage differentials narrowed from 33 percent to 27 percent during the corresponding period. Thus, among the working population, a 6-percentage-point gender wage convergence was observed. The upper and lower bounds of the gender wage convergence based on agnostic assumption was 33 percentage points and -18 percentage points, based on an agnostic assumption.

However, the knowledge that leisure is a normal good and spousal income affects the reservation wage, but not offered wage, helps to tighten the bounds dramatically. Among married men and women, the gender wage differential converged by 11 percentage points between 1987 and 2002. The lower- and upper-bound estimates under the assumption that spousal income is excludable were 10 and 13 percentage points, respectively. Based on this tight bound, we exclude the possibility that the observed wage convergence among full-time workers is spurious due to a change in workers' composition.

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Table 1: Trend in Full-time Population Ratio and Gender Wage Differential Sample: Age 15-64, Out of School

Full-time/Population Ratio	Full-time Hourly Wage (Yen, Year 2000 Value)	Female/Male Wage Ratio
Male	Female	Male	Female
1987	0.64	0.27	2788	1690	0.61
1992	0.66	0.31	3290	1995	0.60
1997	0.67	0.31	3562	2291	0.64
2002	0.61	0.27	3467	2413	0.70
Table 2: Full-Time Employment Population Ratio by Gender, Education and Age.

Male	Female
1987	1992	1997	2002	1987	1992	1997	2002
Total	0.64	0.66	0.67	0.61	0.27	0.28	0.31	0.27

Educ=9
15-19	0.47	0.51	0.37	0.21	0.33	0.29	0.17	0.05
20-24	0.62	0.65	0.61	0.41	0.29	0.30	0.23	0.11
25-29	0.65	0.66	0.62	0.50	0.21	0.22	0.18	0.13
30-34	0.64	0.65	0.61	0.53	0.20	0.21	0.16	0.13
35-39	0.62	0.63	0.60	0.52	0.22	0.27	0.21	0.14
40-44	0.62	0.61	0.61	0.51	0.24	0.27	0.26	0.16
45-49	0.56	0.60	0.60	0.52	0.25	0.29	0.26	0.21
50-54	0.50	0.57	0.60	0.52	0.22	0.26	0.25	0.20
55-59	0.39	0.48	0.53	0.48	0.13	0.18	0.20	0.17
60-64	0.15	0.21	0.22	0.17	0.05	0.07	0.07	0.06

Educ=12
15-19	0.59	0.64	0.57	0.46	0.72	0.74	0.54	0.38
20-24	0.80	0.82	0.77	0.61	0.65	0.68	0.59	0.41
25-29	0.82	0.84	0.81	0.73	0.33	0.40	0.40	0.35
30-34	0.79	0.81	0.81	0.75	0.23	0.26	0.27	0.25
35-39	0.74	0.76	0.78	0.75	0.23	0.27	0.26	0.23
40-44	0.72	0.72	0.75	0.72	0.25	0.29	0.28	0.25
45-49	0.68	0.70	0.70	0.68	0.25	0.29	0.29	0.26
50-54	0.62	0.66	0.68	0.63	0.23	0.25	0.26	0.23
55-59	0.46	0.56	0.61	0.56	0.15	0.18	0.21	0.19
60-64	0.59	0.21	0.21	0.16	0.72	0.08	0.07	0.07

Educ=14
20-24	0.82	0.86	0.81	0.66	0.73	0.81	0.72	0.60
25-29	0.82	0.84	0.84	0.77	0.43	0.51	0.53	0.49
30-34	0.79	0.80	0.82	0.78	0.31	0.32	0.33	0.34
35-39	0.74	0.75	0.77	0.73	0.29	0.32	0.30	0.29
40-44	0.73	0.70	0.70	0.69	0.29	0.31	0.32	0.30
45-49	0.67	0.67	0.68	0.68	0.31	0.29	0.33	0.32
50-54	0.73	0.66	0.67	0.60	0.31	0.29	0.30	0.30
55-59	0.58	0.62	0.61	0.55	0.19	0.21	0.26	0.22
60-64	0.18	0.17	0.18	0.19	0.06	0.07	0.09	0.09
Educ=16
20-24	0.86	0.91	0.82	0.66	0.70	0.81	0.69	0.60
25-29	0.89	0.91	0.89	0.81	0.51	0.63	0.61	0.55
30-34	0.86	0.88	0.89	0.85	0.36	0.41	0.45	0.43
35-39	0.81	0.82	0.86	0.84	0.36	0.38	0.38	0.39
40-44	0.77	0.77	0.80	0.81	0.34	0.35	0.37	0.37
45-49	0.75	0.72	0.75	0.75	0.41	0.33	0.35	0.36
50-54	0.68	0.71	0.70	0.67	0.37	0.35	0.36	0.34
55-59	0.57	0.59	0.61	0.57	0.27	0.28	0.28	0.28
60-64	0.24	0.25	0.23	0.19	0.13	0.09	0.11	0.10
N	305994	361858	334095	315915	325956	357262	345187	328547
Table 3: The OLS regression of full-time employment Dependent variable: Full-time=1; Sample: Male

(1)	(2)	(3)	(4)
1987	1992	1997	2002
Educ=12	0.11	0.09	0.10	0.12
(0.00)	(0.00)	(0.00)	(0.00)
Educ=14	0.12	0.09	0.10	0.13
(0.00)	(0.00)	(0.00)	(0.00)
Educ=16	0.18	0.15	0.15	0.18
(0.00)	(0.00)	(0.00)	(0.00)
20-24	0.21	0.19	0.23	0.19
(0.01)	(0.01)	(0.01)	(0.01)
25-29	0.23	0.21	0.27	0.31
(0.01)	(0.01)	(0.01)	(0.01)
30-34	0.20	0.18	0.27	0.33
(0.01)	(0.01)	(0.01)	(0.01)
35-39	0.15	0.13	0.24	0.33
(0.01)	(0.01)	(0.01)	(0.01)
40-44	0.13	0.08	0.20	0.30
(0.01)	(0.01)	(0.01)	(0.01)
45-49	0.09	0.06	0.16	0.26
(0.01)	(0.01)	(0.01)	(0.01)
50-54	0.03	0.03	0.14	0.21
(0.01)	(0.01)	(0.01)	(0.01)
55-59	-0.10	-0.07	0.07	0.15
(0.01)	(0.01)	(0.01)	(0.01)
60-64	-0.37	-0.38	-0.29	-0.21
(0.01)	(0.01)	(0.01)	(0.01)
Constant	0.48	0.54	0.44	0.30
(0.01)	(0.00)	(0.01)	(0.01)
Observations	305994	361858	334095	315915
$ R^{2} $	0.16	0.15	0.15	0.15

Note: Heteroskedasticity robust standard errors are reported in the parenthesis.
Table 4: The OLS regression of full-time employment Dependent variable: Full-time=1; Sample: Female

(1)	(2)	(3)	(4)
1987	1992	1997	2002
Educ=12	0.03	0.02	0.04	0.05
(0.00)	(0.00)	(0.00)	(0.00)
Educ=14	0.10	0.09	0.11	0.13
(0.00)	(0.00)	(0.00)	(0.00)
Educ=16	0.16	0.14	0.16	0.19
(0.00)	(0.00)	(0.00)	(0.00)
20-24	-0.02	0.03	0.12	0.13
(0.01)	(0.01)	(0.01)	(0.01)
25-29	-0.31	-0.23	-0.05	0.06
(0.01)	(0.01)	(0.01)	(0.01)
30-34	-0.41	-0.39	-0.20	-0.06
(0.01)	(0.01)	(0.01)	(0.01)
35-39	-0.41	-0.38	-0.22	-0.08
(0.01)	(0.01)	(0.01)	(0.01)
40-44	-0.39	-0.37	-0.20	-0.07
(0.01)	(0.01)	(0.01)	(0.01)
45-49	-0.38	-0.37	-0.19	-0.06
(0.01)	(0.01)	(0.01)	(0.01)
50-54	-0.41	-0.39	-0.21	-0.07
(0.01)	(0.01)	(0.01)	(0.01)
55-59	-0.50	-0.46	-0.25	-0.11
(0.01)	(0.01)	(0.01)	(0.01)
60-64	-0.58	-0.57	-0.38	-0.23
(0.01)	(0.01)	(0.01)	(0.01)
Constant	0.62	0.63	0.44	0.26
(0.01)	(0.01)	(0.01)	(0.01)
Observations	325956	383213	345187	328547
$ R^{2} $	0.12	0.13	0.10	0.07

Note: Heteroskedasticity robust standard errors are reported in the parenthesis.