Decomposition of gender wage inequalities through calibration: Application to the Swiss structure of earnings survey
Section 4. The weighted DFL method

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4.1  The method

The method proposed by DiNardo et al. (1996) uses a reweighting function by which women’s distribution of characteristics is rendered similar to men’s distribution of characteristics. The reweighted distribution is the women’s counterfactual distribution of characteristics. The DFL method is presented through the use of survey weights in order to take the sampling design into account.

The reweighting function is equal to

$$\psi \left(x_k\right)\mathrm{<mo>=</mo>}\frac{\mathrm{Pr}\left(D_{Mk}\mathrm{<mo>=</mo>}1|x_k\right)/\mathrm{Pr}\left(D_{Mk}\mathrm{<mo>=</mo>\; 1}\right)}{\mathrm{Pr}\left(D_{Mk}\mathrm{<mo>=</mo>}0|x_k\right)/\mathrm{Pr}\left(D_{Mk}\mathrm{<mo>=</mo>\; 0}\right)}\mathrm{,}$$

where $D_{Mk}\mathrm{<mo>=</mo>\; 1}$ if individual $k$ is a man and $D_{Mk}\mathrm{<mo>=</mo>\; 0}$ otherwise and $x_k$ is the vector of observed characteristics for individual $k.$ Obviously, $\mathrm{Pr}\left(D_{Mk}\mathrm{<mo>=</mo>}1|x_k\right)$ and $\mathrm{Pr}\left(D_{Mk}\mathrm{<mo>=</mo>}0|x_k\right)$ must be estimated. For this type of estimation, DiNardo et al. (1996) suggested the use of a logit or a probit model. Using the information from the sample,

$$\widehat{\psi }\left(x_k\right)\mathrm{<mo><mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext></mo>}\frac{\widehat{\mathrm{Pr}}\left(D_{Mk}\mathrm{<mo>=</mo>}1|x_k\right)/\widehat{\mathrm{Pr}}\left(D_{Mk}\mathrm{<mo>=</mo>\; 1}\right)}{\widehat{\mathrm{Pr}}\left(D_{Mk}\mathrm{<mo>=</mo>}0|x_k\right)/\widehat{\mathrm{Pr}}\left(D_{Mk}\mathrm{<mo>=</mo>\; 0}\right)}.(4.1)$$

Using the reweighting factor, women’s counterfactual wage mean is estimated by

$${\widehat{\overline{Y}}}_{F|M}^{\text{DFL}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k\widehat{\psi }\left(x_k\right)y_k}}{{\displaystyle {\sum }_{k\in S_F}{d}_k\widehat{\psi }\left(x_k\right)}}\mathrm{,}(4.2)$$

and women’s counterfactual means of characteristics by

$${\widehat{\overline{X}}}_{F|M}^{\text{DFL}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k\widehat{\psi }\left(x_k\right)x_k}}{{\displaystyle {\sum }_{k\in S_F}{d}_k\widehat{\psi }\left(x_k\right)}}.(4.3)$$

The estimated reweighting factor defined in equation (4.1) will be equal to

$$\widehat{\psi }\left(x_k\right)\mathrm{<mo><mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext></mo>}\widehat{a}\text{exp}\left(x_k^{ㄒ}\widehat{\gamma }\right)\mathrm{,}$$

where $\widehat{\gamma }$ is the estimation of $\gamma$ from the sample using empirical likelihood and $\widehat{a}$ is the ratio of estimated proportions of women and men. It is given by:

$$\widehat{a}\mathrm{<mo>=</mo>}\frac{\widehat{\mathrm{Pr}}\left(D_{Mk}\mathrm{<mo>=</mo>\; 0}\right)}{\widehat{\mathrm{Pr}}\left(D_{Mk}\mathrm{<mo>=</mo>\; 1}\right)}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k}}{{\displaystyle {\sum }_{k\in S_M}{d}_k}}\mathrm{.}$$

Since the DFL method is presented taking the survey weights into account, the reweighting factor ${\psi }_k$ will be multiplied by $d_k\mathrm{,}k\in S_F.$ This resulting factor will be termed “weighted DFL factor”. Women’s estimated counterfactual wage mean can be re-expressed as

$${\widehat{\overline{Y}}}_{F|M}^{\text{DFL}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k\widehat{\psi }\left(x_k\right)y_k}}{{\displaystyle {\sum }_{k\in S_F}{d}_k\widehat{\psi }\left(x_k\right)}}=\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k\exp \left(x^{ㄒ}\widehat{\gamma }\right)y_k}}{{\displaystyle {\sum }_{k\in S_F}{d}_k\exp \left(x^{ㄒ}\widehat{\gamma }\right)}}.(4.4)$$

Women’s counterfactual means of characteristics are estimated as

$${\widehat{\overline{X}}}_{F|M}^{\text{DFL}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k\widehat{\psi }\left(x_k\right)x_k}}{{\displaystyle {\sum }_{k\in S_F}{d}_k\widehat{\psi }\left(x_k\right)}}=\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k\exp \left(x^{ㄒ}\widehat{\gamma }\right)x_k}}{{\displaystyle {\sum }_{k\in S_F}{d}_k\exp \left(x^{ㄒ}\widehat{\gamma }\right)}}.$$

Through the use of the reweighting factor, the counterfactual coefficients in the women’s sample are given by

$${\beta }_F^{\text{DFL}}\mathrm{<mo><mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext></mo>}{\left({\displaystyle \sum _{k\in U_F}\psi \left(x_k\right)x_k{x}_k^{ㄒ}}\right)}^{-1}{\displaystyle \sum _{k\in U_F}\psi \left(x_k\right)x_k{y}_k}\mathrm{,}$$

and estimated by

$${\widehat{\beta }}_F^{\text{DFL}}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{k\in S_F}{d}_k\widehat{\psi }\left(x_k\right)x_k{x}_k^{ㄒ}}\right)}^{-1}{\displaystyle \sum _{k\in S_F}{d}_k\widehat{\psi }\left(x_k\right)x_k{y}_k}.(4.5)$$

The coefficients above have to be computed, because under the same condition as in Result 1, women’s counterfactual wage mean defined in (4.2) is given by

$${\widehat{\overline{Y}}}_{F|M}^{\text{DFL}}\mathrm{<mo>=</mo>}{\widehat{\overline{X}}}_{F|M}^{{\text{DFL}}^{ㄒ}}{\widehat{\beta }}_F^{\text{DFL}}\mathrm{.}$$

The BO decomposition formula can now be expressed as

$$\begin{array}{ll}{\widehat{\overline{Y}}}_M-{\widehat{\overline{Y}}}_F\hfill & \mathrm{<mo><mtext>\hspace{0.17em}</mtext>=</mo>}\left({\widehat{\overline{Y}}}_{F|M}^{\text{DFL}}-{\widehat{\overline{Y}}}_F\right)+\left({\widehat{\overline{Y}}}_M-{\widehat{\overline{Y}}}_{F|M}^{\text{DFL}}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\left({\widehat{\overline{X}}}_{F|M}^{{\text{DFL}}^{ㄒ}}{\widehat{\beta }}_F^{\text{DFL}}-{\widehat{\overline{X}}}_F^{ㄒ}{\widehat{\beta }}_F\right)+\left({\widehat{\overline{X}}}_M^{ㄒ}{\widehat{\beta }}_M-{\widehat{\overline{X}}}_{F|M}^{{\text{DFL}}^{ㄒ}}{\widehat{\beta }}_F^{\text{DFL}}\right)\mathrm{,}(4.6)\hfill \end{array}$$

where ${\widehat{\beta }}_M$ and ${\widehat{\beta }}_F$ are defined in (3.1). The first term of equation (4.6) is the composition effect and the second one the structure effect.

4.2  Further decomposition of the structure effect

As Fortin et al. (2011) note, the purpose of the DFL reweighting factor is to render the distribution of women’s characteristics identical to that of men. This implies that the means of the auxiliary variables in the two groups should be equal. However, with the DFL method, it is not the case. Indeed,

$${\widehat{\overline{X}}}_{F|M}^{\text{DFL}}<mtext>\hspace{0.17em}</mtext>\ne <mtext>\hspace{0.17em}</mtext>{\widehat{\overline{X}}}_M(4.7)$$

(see, for instance, Fortin et al. 2011; Donzé, 2013). The reweighting factor thus fails to match the two distributions perfectly.

The structure effect in equation (4.6) can be further divided in the following elements

$$\left({\widehat{\overline{X}}}_M^{ㄒ}{\widehat{\beta }}_M-{\widehat{\overline{X}}}_{F|M}^{{\text{DFL}}^{ㄒ}}{\widehat{\beta }}_F^{\text{DFL}}\right)\mathrm{<mo>=</mo>}{\widehat{\overline{X}}}_M^{ㄒ}\left({\widehat{\beta }}_M-{\widehat{\beta }}_F^{\text{DFL}}\right)+\left({\widehat{\overline{X}}}_M-{\widehat{\overline{X}}}_{F|M}^{\text{DFL}}\right){\widehat{\beta }}_F^{\text{DFL}}\mathrm{,}(4.8)$$

where ${\widehat{\overline{X}}}_{F|M}^{\text{DFL}}$ and ${\widehat{\beta }}_F^{\text{DFL}}$ are defined in equations (4.3) and (4.5), respectively (Fortin et al. 2011). The first element of the right-hand side of equation (4.8) is the pure effect and the second the residual effect or the total reweighting error (Fortin et al. 2011). The pure effect is the actual unexplained part of the wage difference. The residual effect contains the misfit of the model, in other words, what the reweighting factor fails to match between men’s and women’s distribution of characteristics. This method allows for the construction of a counterfactual wage distribution. This in turn allows for the comparison between this new distribution and the observed wage distributions of women and men. The drawback of the method is that it may happen that at least one characteristic is a good predictor of the gender (for instance, the economic sector). This implies that $\mathrm{Pr}\left(D_{Mk}\mathrm{<mo>=</mo>}1|x_k\right)$ may get close to 1 and that the reweighting factor will take take on a large value (Fortin et al. 2011). This obviously leads to a large variance of the factor. This will be shown in Section 8.

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