These estimates indicate that part of the wage difference cannot be explained by the productivity difference. The estimation that takes into account the correlated productivity and demand shocks suggests the robustness of the results. This article attempts to complement Kawaguchi (2007) by obtaining structural parameters, and therefore sheds light more directly on the mechanism of the gender wage gap. 2005) and Sano (2005) also found that companies with a higher share of female employees achieve higher profits. 2002), we use panel data to estimate the production function and the wage equation.

These estimates show that part of the wage gap cannot be explained by the productivity gap. We used the Basic Survey of Firm Activity collected by the Japanese government's Ministry of Economy, Trade and Industry (METI) to examine the relative productivity of women to men. From the data sets, we extracted each firm's total sales, cost of sales, overhead cost, data on the firm's employees, such as the number of employees by gender, the book value of its fixed assets, the year of origin of the firm . , and a three-digit code indicating the industry in which the firm operates.

The mode of the distribution is around 0.2 and the distribution is skewed to the left. However, if the relative wage of women relative to men is lower than the relative productivity of women relative to men, a higher proportion of women results in a higher volume of sales relative to the mass of wages and other inputs. The average firm-level residual from this regression is regressed on the average proportion of women at the firm level.

To estimate the marginal product of the labor of male and female workers, we need to specify the functional form of the production function.

## Pooled Cross-Sectional Estimation

One way to allow for the time dependence of the error term is to not impose assumptions on the error structure and to estimate E(x0iuiu0ixi) directly. The GMM estimator using this matrix as the weighting matrix results in the efficient estimator under the assumption of very flexible error structure.

## Fixed Effects Estimation

Strict exogeneity means that the idiosyncratic shock of the current period is uncorrelated with the explanatory variable in other periods. Under this strict exogeneity assumption, the composite error term ujit is unrelated to the form of the mean deviation of the explanatory variable xjit. The choice of the weighting matrix Ξ depends on the assumption of the covariance structure of the error as in the cross-sectional estimator.

The variance of the estimator is obtained by replacing xi with zi in formula (16).

## Proxy Variable Estimation

The first part of the v1 error term is the productivity hit that can be related to qlitandmit. Following Levinsohn and Petrin (2003), we assume that this productivity shock is recovered by the firm's choice of intermediate input quantity, given the level of capital which is a state variable (i.e. for simplicity of analysis, we assume that the productivity shock follows a specific process that is E(v1it|v1it−1) = νv1it−1 or equivalently, f(·) =ν·(·).

All the explanatory variables are exogenous, but the problem is that part of f1(qlit, kit, mit;θ1) cannot be identified due to the multicollinearity with g(mit, kit). To overcome this difficulty in identification, we exploit the Markov property of the productivity shock. We use mit−2 as well as qlit−2 and mit−2 as an additional instrumental variable to achieve identification.

## Cross-sectional and fixed effects estimation

If we take the point estimates seriously, of the 70 percentage points of wage differentials observed in the data, 15 percentage points cannot be explained in the productivity gap between men and women. The pooled least squares estimator is a consistent estimator when the time-constant heterogeneity of each firm is unrelated to the firm's input mix. If the technological heterogeneity of each firm is related to the inputs, the pooled least squares estimator is unstable.

To circumvent this potential endogeneity issue, the production function and wage equations are estimated through an instrumental variable estimation using the mean deviation of the explanatory variables from each firm's mean. To ensure that the idiosyncratic error term of (4) is exogenous to each firm's mean of independent variables, the strict exogeneity of the error term, denoted as (??), is required. This firm-level heterogeneity may arise due to the heterogeneity of workers' quality across firms.

If the quality of workers in a given company is constant over time, then the effect of heterogeneity in the quality of workers is captured by di. However, the IV estimator, which uses the average deviation of the independent variables from each firm's mean, is a consistent estimator. The result in column (4) shows that the productivity of female workers compared to male workers is 54 percent.

Compared to the cross-sectional estimate in column (1) of table 3, this number is 10 percentage points higher, implying that these are ai and llf. Under the assumption of homoscedasticity for the idiosyncratic error term, the nonlinear least squares estimator is an efficient estimator under the zero value of no correlation between ai and lf itl. At first glance, these results seem to suggest that the relative wages of female workers compared to that of male workers reflect their relative productivity when working for the same company.

However, we should note that the fixed-effects estimator is poor when the time shock and the proportion of women are strongly correlated because the identification is based entirely on the variation in the proportion of women. As Houseman and Abraham (1993) report, Japanese firms tend to adjust their female workforce more quickly than their male workforce in response to demand or technological shocks. If the share of women increases in response to a positive demand shock due to fairly rapid labor adjustment, the fixed effects estimator may be highly biased upward.

## Proxy variable estimation

The estimate of the relative productivity of female workers relative to male workers falls slightly, but relative wages do not change in a meaningful way. The coefficients for labor and material increase and for capital decrease; the results come closer to the results without proxy variables. Overall, we assess the results based on the delayed moment for the production function and the current moment in the wage equation is most preferable.

In an attempt to explain the wage gap between men and women, we estimated relative marginal productivity and the relative wage of female workers compared to male workers using panel data from Japanese firms. According to the null hypothesis that the difference in wages between men and women represents a difference in productivity, the share of women in a company should not affect production, because the total mass of wages fully covers the quality-adjusted labor input. Based on the results of the reduced form approach, we continued with the structural assessment of relative productivity and pay between male and female workers with a joint estimate of the production function and the wage equation.

Cross-sectional estimates showed that the marginal productivity of female workers was 44 percent of that of male workers, while female wages were 31 percent of male workers' wages. However, the IV estimates, which allowed for firm-level fixed effects, indicated that both female workers. Although these fixed effects results appear to be consistent with the absence of discrimination, relative female productivity may be highly biased due to the temporal correlation between the female share and demand/productivity shocks in both the production function and the wage equation.

The most preferred estimates based basically on Levinsohn and Petrin (2003) indicate that the level estimates were not heavily biased due to the demand/productivity shock and the relative female productivity is 47 percent. The results corresponding to the employer's taste discrimination against women are consistent with the American evidence provided by Hellerstein et al. In the Japanese context, the implied employer discrimination against women is consistent with the robust findings that firms with a higher proportion of female workers earn higher profits (Kodama et al.

The research was carried out within the project of the Research Institute for Economy, Trade and Industry. We thank the Ministry of Economy, Trade and Industry and the Ministry of Public Administration, Internal Affairs, Posts and Telecommunications of the Japanese government for releasing the micro data used in this study. We thank Hiroyuki Chuma, Gigi Foster, Hidehiko Ichimura, Hisako Ishii, Sadao Nagaoka, Kazuhiko Odaki, Kei Sakata, Tetsushi Sonobe, Kotaro Tsuru, Futoshi Yamuchi, Jeffrey Wooldridge, and seminar participants at Hitotsubashi University, University of South Australia, Japan Conference on of Labor Economics, the FASID Hakone Conference, the International Conference on Panel Data, the Australian Meeting of the Econometric Society, and the Economic Trade Industry Research Institute for their comments.

Estimation of the production function and wage equation with heterogeneous labor: Evidence from a new matched employer-employee data set. Note: The hourly rate is a weighted average of the hourly rate of full-time workers (ippan rodosha, in Japanese), which is calculated as a fixed monthly wage (Shoteinai Kyuyo Gaku in Japanese) divided by a fixed monthly working hours (Shoteinai.