Decomposition of gender wage inequalities through calibration: Application to the Swiss structure of earnings survey
Section 7. Conclusion

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The phenomenon of discrimination has multiple facets and there are many mechanisms that can generate it. However, this paper only examines its estimation from a methodological point of view. The two calibration cases taken into consideration represent a generalization of two existing decomposition methods, the technique of Blinder (1973) and Oaxaca (1973) and the semi-parametric method of DiNardo et al. (1996), both expressed using sampling weights. The original methods can also be obtained, if all the sampling weights are considered to be equal to 1. The linear case yields the same result as the BO method. However, since the resulting weights are unbounded, negative values might be observed. Just as the DFL method, the calibration approach allows for the decomposition of wage differences at other points other than the mean, such as quantiles. However, the raking-ratio calibration is an improvement of the DFL method, in that the estimation of the structure effect will always include a residual effect equal to 0. Therefore, the structure effect will only be composed of the pure effect. Decomposing wage differences along quantiles enables the conclusion that in low-paying jobs, the inequalities are due solely to discrimination. In this article, the emphasis was placed on the generalization of two well-established decomposition methods through the calibration approach.

Acknowledgements

The authors are grateful to the Swiss federal statistical office for the financial support and the LOHN department for providing the data. However, the opinions expressed in this paper do not necessarily reflect those of the Swiss federal statistical office.

Appendix A

Proof of Result 1

$$\begin{array}{ll}{\widehat{X}}_h{\widehat{\beta }}_h\hfill & \mathrm{<mo>=<mtext>\hspace{0.17em}</mtext></mo>}{\widehat{X}}_h{\left({\displaystyle \sum _{k\in S_h}{d}_k{x}_k{x}_k^{ㄒ}}\right)}^{-1}{\displaystyle \sum _{l\in S_h}{d}_l{x}_l{y}_l}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{j\in S_h}{d}_j{x}_j^{ㄒ}}{\left({\displaystyle \sum _{k\in S_h}{d}_k{x}_k{x}_k^{ㄒ}}\right)}^{-1}{\displaystyle \sum _{l\in S_h}{d}_l{x}_l{y}_l}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\left({\displaystyle \sum _{j\in S_h}{\varsigma }^{ㄒ}{d}_j{x}_j{x}_j^{ㄒ}}\right){\left({\displaystyle \sum _{k\in S_h}{d}_k{x}_k{x}_k^{ㄒ}}\right)}^{-1}{\displaystyle \sum _{l\in S_h}{d}_l{x}_l{y}_l}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{l\in S_h}{\varsigma }^{ㄒ}{d}_l{x}_l{y}_l}\mathrm{<mo>=</mo>}{\displaystyle \sum _{l\in S_h}{d}_l{y}_l}\mathrm{<mo>=</mo>}{\widehat{Y}}_h\mathrm{.}\hfill \end{array}$$

By dividing this equation by ${\displaystyle {\sum }_{k\in S_h}{d}_k}\mathrm{,}$ Result 1 is obtained.

Appendix B

B.1 Linearization of the means

In order to compute the variance of the average means and of the counterfactual means we have used the linearization method proposed by Graf (2011). The author proposes to compute the partial derivative of the estimator with respect to the sample indicator. This derivative provides the linearized variable that can be plugged in the variance estimator. The average means are defined by:

$${\widehat{\overline{Y}}}_F=\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k{y}_k}}{{\displaystyle {\sum }_{k\in S_F}{d}_k}},$$

and

$${\widehat{\overline{Y}}}_M=\frac{{\displaystyle {\sum }_{l\in S_M}{d}_l{y}_l}}{{\displaystyle {\sum }_{l\in S_M}{d}_l}}.$$

For the two average wages, we obtain the linearized variables:

$$\frac{\partial {\widehat{\overline{Y}}}_F}{\partial I_j}=\{\begin{array}{ll}\frac{d_j\left(y_j-{\widehat{\overline{Y}}}_F\right)}{{\displaystyle {\sum }_{k\in S_F}{d}_k}}\hfill & j\in S_F\hfill \\ 0\hfill & j\in S_M\hfill \end{array},$$

and

$$\frac{\partial {\widehat{\overline{Y}}}_M}{\partial I_j}=\{\begin{array}{ll}\frac{d_j\left(y_j-{\widehat{\overline{Y}}}_M\right)}{{\displaystyle {\sum }_{l\in S_M}{d}_l}}\hfill & j\in S_M\hfill \\ 0\hfill & j\in S_F\hfill \end{array}.$$

B.2 Linearization of the counterfactual

In order to compute the counterfactual mean, we compute the weights $v_k$ defined by the system

$$A={\displaystyle \sum _{k\in S_F}{v}_k{d}_k{x}_k=\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k}}{{\displaystyle {\sum }_{l\in S_M}{d}_l}}{\displaystyle \sum _{l\in S_M}{d}_l{x}_l}}<mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext>{\widehat{\overline{X}}}_M{\displaystyle \sum _{k\in S_F}{d}_k,}$$

with

$$v_k=F\left(x_k^{ㄒ}\lambda \right).$$

For the linearized variables, we have to consider two cases:

-     If $j\in S_F$

$$\frac{\partial A}{\partial I_j}=v_j{d}_j{x}_j+\left[{\displaystyle \sum _{k\in S_F}{F}^{\prime }\left(x_k^{ㄒ}\lambda \right)d_k{x}_k{x}_k^{ㄒ}}\right]\frac{\partial \lambda }{\partial I_j}=d_j{\widehat{\overline{X}}}_M.$$

       Thus

$$\frac{\partial \lambda }{\partial I_j}=-T^{-1}{d}_j\left(v_j{x}_j-{\widehat{\overline{X}}}_M\right),$$

       where

$$T={\displaystyle \sum _{k\in S_F}{F}^{\prime }}\left(x_k^{ㄒ}\lambda \right)d_k{x}_k{x}_k^{ㄒ}.$$

-     If $j\in S_M$

$$\frac{\partial A}{\partial I_j}=\left[{\displaystyle \sum _{k\in S_F}{F}^{\prime }}\left(x_k^{ㄒ}\lambda \right)d_k{x}_k{x}_k^{ㄒ}\right]\frac{\partial \lambda }{\partial I_j}=d_j\left(x_j-{\widehat{\overline{X}}}_M\right)\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k}}{{\displaystyle {\sum }_{l\in S_M}{d}_l}}.$$

       Thus

$$\frac{\partial \lambda }{\partial I_j}=T^{-1}{d}_j\left(x_j-{\widehat{\overline{X}}}_M\right)\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k}}{{\displaystyle {\sum }_{l\in S_M}{d}_l}}.$$

Since we have supposed that there exists a vector $\gamma$ such that ${\gamma }^{ㄒ}{x}_k=1$ for all $k\in U,$ then, we have

$${\gamma }^{ㄒ}A={\displaystyle \sum _{k\in S_F}{v}_k{d}_k}={\displaystyle \sum _{k\in S_F}{d}_k}.$$

Now consider

$${\widehat{\overline{Y}}}_{F|M}=\frac{{\displaystyle {\sum }_{k\in S_F}{v}_k{d}_k{y}_k}}{{\displaystyle {\sum }_{k\in S_F}{v}_k{d}_k}}=\frac{{\displaystyle {\sum }_{k\in S_F}{v}_k{d}_k{y}_k}}{{\displaystyle {\sum }_{k\in S_F}{d}_k}}.$$

Again, two cases must be considered:

-     If $j\in S_F$

$$\begin{array}{ll}\frac{{\widehat{\overline{Y}}}_{F|M}}{\partial I_j}\hfill & =\frac{d_j\left(v_j{y}_j-{\widehat{\overline{Y}}}_{F|M}\right)+\frac{\partial {\lambda }^{ㄒ}}{\partial I_j}\left[{\displaystyle {\sum }_{k\in S_F}{F}^{\prime }}\left(x_k^{ㄒ}\lambda \right)d_k{x}_k{y}_k\right]}{{\displaystyle {\sum }_{k\in S_F}{d}_k}}\hfill \\ \hfill & =\frac{d_j\left(v_j{y}_j-{\widehat{\overline{Y}}}_{F|M}\right)-d_j{\left(v_j{x}_j-{\widehat{\overline{X}}}_M\right)}^{ㄒ}{T}^{-1}{\displaystyle {\sum }_{k\in S_F}{F}^{\prime }}\left(x_k^{ㄒ}\lambda \right)d_k{x}_k{y}_k}{{\displaystyle {\sum }_{k\in S_F}{d}_k}}\hfill \\ \hfill & =\frac{d_j\left[v_j{y}_j-{\widehat{\overline{Y}}}_{F|M}-{\left(v_j{x}_j-{\widehat{\overline{X}}}_M\right)}^{ㄒ}{B}_F\right]}{{\displaystyle {\sum }_{k\in S_F}{d}_k}},\hfill \end{array}$$

       where

$$B_F=T^{-1}{\displaystyle \sum _{k\in S_F}{F}^{\prime }}\left(x_k^{ㄒ}\lambda \right)d_k{x}_k{y}_k.$$

-     If $j\in S_M$

$$\begin{array}{ll}\frac{{\widehat{\overline{Y}}}_{F|M}}{\partial I_j}\hfill & =\frac{\partial {\lambda }^{ㄒ}}{\partial I_j}\frac{{\displaystyle {\sum }_{k\in S_F}{F}^{\prime }}\left(x_k^{ㄒ}\lambda \right)d_k{x}_k{y}_k}{{\displaystyle {\sum }_{k\in S_F}{d}_k}}\hfill \\ \hfill & =d_j{\left(x_j-{\widehat{\overline{X}}}_M\right)}^{ㄒ}\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k}}{{\displaystyle {\sum }_{l\in S_M}{d}_l}}{T}^{-1}\frac{{\displaystyle {\sum }_{k\in S_F}{F}^{\prime }}\left(x_k^{ㄒ}\lambda \right)d_k{x}_k{y}_k}{{\displaystyle {\sum }_{k\in S_F}{d}_k}}\hfill \\ \hfill & =d_j{\left(x_j-{\widehat{\overline{X}}}_M\right)}^{ㄒ}\frac{1}{{\displaystyle {\sum }_{l\in S_M}{d}_l}}{B}_F.\hfill \end{array}$$

Thus the linearized variable is

$$z_k=\{\begin{array}{ll}\frac{d_j\left[v_j{y}_j-{\widehat{\overline{Y}}}_{F|M}-{\left(v_j{x}_j-{\widehat{\overline{X}}}_M\right)}^{ㄒ}{B}_F\right]}{{\displaystyle {\sum }_{k\in S_F}{d}_k}}\hfill & \text{if}j\in S_F\hfill \\ \frac{d_j{\left(x_j-{\widehat{\overline{X}}}_M\right)}^{ㄒ}{B}_F}{{\displaystyle {\sum }_{l\in S_M}{d}_l}}\hfill & \text{if}j\in S_M.\hfill \end{array}$$

The linearized variable must only be plugged in the variance estimator corresponding to the sampling design. Note that the variance of the counterfactual depends on the variance computed for the sample of men for the part that is explained by the regression and the variance computed for the sample of women for the part that remains unexplained.

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