Decomposition of gender wage inequalities through calibration: Application to the Swiss structure of earnings survey
Section 5. The calibration approach

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5.1  The calibration method

The calibration method was introduced by Deville and Särndal (1992). The idea behind the technique is to make use of the information known at the population level on some auxiliary variables to estimate a function of a variable of interest. Usually, the auxiliary variables and the variable of interest are correlated. The resulting estimates are consistent and efficient.

Assuming that the sampling weights $d_k$ are available and that the totals of auxiliary information at the population level given by

$$X\mathrm{<mo><mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext></mo>}{\displaystyle \sum _{k\in U}{x}_k}\mathrm{,}$$

are known, new weights $w_k\mathrm{,}k\in S$ should be constructed, such that the following constraint (or calibration equation) is respected

$${\displaystyle \sum _{k\in S}{w}_k{x}_k}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in U}{x}_k}\mathrm{.}(5.1)$$

The weights are determined by solving in $\lambda$ the calibration equations that become

$${\displaystyle \sum _{k\in S}{w}_k{x}_k}\mathrm{<mo><mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext></mo>}{\displaystyle \sum _{k\in S}{d}_k{F}_k\left(x_k^{ㄒ}\lambda \right)x_k}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in U}{x}_k}\mathrm{,}$$

where $F_k\left(x_k^{ㄒ}\lambda \right)$ is the calibration function. The resulting calibration estimation of $Y$ is

$$\widehat{Y}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S}{d}_k{y}_k{F}_k\left(x_k^{ㄒ}\lambda \right)}.(5.2)$$

In what follows, we will use the linear case, where the pseudo-distance function is the chi-square distance and the calibration function is given by $F_k\left(x_k^{ㄒ}\lambda \right)\mathrm{<mo>=</mo>\; 1}+x_k^{ㄒ}\lambda .$ In the second case, we will use the raking-ratio, which uses the Entropy pseudo-distance and where the calibration function is given by $F_k\left(x_k^{ㄒ}\lambda \right)\mathrm{<mo>=</mo>}\exp \left(x_k^{ㄒ}\lambda \right).$

5.2  Calibration of women’s characteristics on the men’s characteristics

Suppose that for all the units of the sample, there is a given sampling weight $d_k.$ In the current context, the auxiliary variables that are used in the calibration process are some selected characteristics measured for every individual. The aim is to ‘divert’ the calibration technique in order to compute a weighting system that adjusts the totals of the auxiliary variables of women on the totals of men. The variable of interest is the logarithm of the wage.

In the women sample, new weights $w_k$ close to $d_k$ are computed, such that ${\displaystyle {\sum }_{k\in S_F}}G\left(w_k\mathrm{,}{d}_k\right)$ is minimized. The following calibration equation is satisfied

$${\displaystyle \sum _{k\in S_F}{w}_k{x}_k}\mathrm{<mo><mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext></mo>}{\widehat{\tilde{X}}}_M\mathrm{,}(5.3)$$

where the vector ${\widehat{\tilde{X}}}_M$ stores the totals of men’s characteristics adjusted on the total of the weights of the women over the total of the weights of the men.

$${\widehat{\tilde{X}}}_M\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_F}{d}_k}}{{\displaystyle {\sum }_{k\in S_M}{d}_k}}{\displaystyle \sum _{k\in S_M}{d}_k{x}_k}\mathrm{.}$$

Dividing the calibration equation (5.3) by ${\displaystyle {\sum }_{k\in S_F}{d}_k}$ yields

$$\frac{{\displaystyle {\sum }_{k\in S_F}{w}_k{x}_k}}{{\displaystyle {\sum }_{k\in S_F}{d}_k}}\mathrm{<mo>=</mo>}\frac{{\widehat{\tilde{X}}}_M}{{\displaystyle {\sum }_{k\in S_F}{d}_k}}\mathrm{<mo><mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext></mo>}{\widehat{\overline{X}}}_M\mathrm{.}(5.4)$$

So with the new weights $w_k,$ the new women’s means of characteristics are equal to those of men. Another interesting equality is

$${\displaystyle \sum _{k\in S_F}{w}_k}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_F}{d}_k}\mathrm{,}(5.5)$$

which holds because $x_{k1}\mathrm{<mo>=</mo>\; 1,}k\in S_M$ and calibration is performed on it. If

$${\widehat{\overline{\tilde{X}}}}_M\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_F}{w}_k{x}_k}}{{\displaystyle {\sum }_{k\in S_F}{w}_k}}\mathrm{,}$$

by putting together equations (5.4) and (5.5), this means that

$${\widehat{\overline{\tilde{X}}}}_M\mathrm{<mo>=</mo>}{\widehat{\overline{X}}}_M\mathrm{.}(5.6)$$

Women’s counterfactual wage mean estimator is thus

$${\widehat{\overline{Y}}}_{F|M}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_F}{w}_k{y}_k}}{{\displaystyle {\sum }_{k\in S_F}{d}_k}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_F}{w}_k{y}_k}}{{\displaystyle {\sum }_{k\in S_F}{w}_k}}\mathrm{.}$$

5.3  Linear calibration

Result 2 Women’s counterfactual wage mean obtained using linear calibration is equal to the counterfactual wage mean obtained using the weighted BO method, i.e., ${\widehat{\overline{Y}}}_{F|M}\mathrm{<mo>=</mo>}{\widehat{\overline{X}}}^{ㄒ}{\widehat{\beta }}_F\mathrm{.}$

Proof

In order to determine the vector $\lambda$ in the case when the chi-squared pseudo-distance is used, the following equation must be solved

$$\begin{array}{ll}{\widehat{\tilde{X}}}_M\hfill & ={\displaystyle \sum _{k\in S_F}{d}_k{x}_{k}F\left(x_k^{ㄒ}\lambda \right)}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_F}{d}_k{x}_k\left(1+x_k^{ㄒ}\lambda \right)}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_F}{d}_k{x}_k}+\left({\displaystyle \sum _{k\in S_F}{d}_k{x}_k{x}_k^{ㄒ}}\right)\lambda \mathrm{.}\hfill \end{array}$$

Thus,

$$\lambda \mathrm{<mo><mtext>\hspace{0.17em}</mtext>=<mtext>\hspace{0.17em}</mtext></mo>}{\left({\displaystyle \sum _{k\in S_F}{d}_k{x}_k{x}_k^{ㄒ}}\right)}^{-1}\left({\widehat{\tilde{X}}}_M-{\displaystyle \sum _{k\in S_F}{d}_k{x}_k}\right)\mathrm{<mo>=</mo>}{T}^{-1}\left({\widehat{\tilde{X}}}_M-{\widehat{X}}_F\right)\mathrm{,}$$

where

$$T\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_F}{d}_k{x}_k{x}_k^{ㄒ}}\mathrm{.}$$

Thus

$$w_k\mathrm{<mo>=</mo>}{d}_{k}F\left(x_k^{ㄒ}\lambda \right)\mathrm{<mo>=</mo>}{d}_k\left\{1+x_k^{ㄒ}{T}^{-1}\left({\widehat{\tilde{X}}}_M-{\widehat{X}}_F\right)\right\}\mathrm{.}$$

Using the result from the previous equation, the numerator of expression (5.2) becomes

$$\begin{array}{ll}{\widehat{Y}}_{F|M}^{\text{LC}}\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_F}{d}_{k}F\left(x_k^{ㄒ}\lambda \right)y_k}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_F}{d}_k{y}_k}+{\left({\widehat{\tilde{X}}}_M-{\widehat{X}}_F\right)}^{ㄒ}{T}^{-1}{\displaystyle \sum _{k\in S_F}{d}_k{x}_k{y}_k}\mathrm{,}(5.7)\hfill \end{array}$$

where ${\widehat{Y}}_{F|M}^{\text{LC}}$ denotes the total of the logarithm of the wage in the women sample, when the total is constructed using the chi-squared pseudo-distance. Let

$${\widehat{\beta }}_F\mathrm{<mo>=</mo>}{T}^{-1}{\displaystyle \sum _{k\in S_F}{d}_k{x}_k{y}_k}\mathrm{.}$$

Vector ${\widehat{\beta }}_F$ has already been defined in the same way in equation (3.1) for the weighted BO method. Equation (5.7) is rewritten as

$$\begin{array}{ll}{\widehat{Y}}_{F|M}^{\text{LC}}\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_F}{d}_k{y}_k}+{\left({\widehat{\tilde{X}}}_M-{\widehat{X}}_F\right)}^{ㄒ}{\widehat{\beta }}_F\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\widehat{Y}}_F+{\left({\widehat{\tilde{X}}}_M-{\widehat{X}}_F\right)}^{ㄒ}{\widehat{\beta }}_F\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\widehat{\tilde{X}}}_M^{ㄒ}{\widehat{\beta }}_F\mathrm{,}(5.8)\hfill \end{array}$$

because under the condition of Result 1, ${\widehat{X}}_F^{ㄒ}{\widehat{\beta }}_F\mathrm{<mo>=</mo>}{\widehat{Y}}_F\mathrm{.}$ By dividing (5.8) by ${\displaystyle {\sum }_{k\in S_F}{w}_k}\mathrm{,}$ Result 2 is obtained.

Using the chi-squared pseudo-distance, the resulting weights have no bounds. This means that the calibration weights might be negative. Even though this calibration instance yields the same results as the BO method for average wages, we advocate for the use of an instance that gives nonnegative weights.

5.4  Raking-ratio calibration

The second instance of calibration uses the entropy pseudo-distance. It is also known as “raking-ratio” calibration. Using the entropy pseudo-distance, equation (5.3) becomes

$${\widehat{\tilde{X}}}_M\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_F}{d}_k{x}_{k}F\left(x_k^{ㄒ}\lambda \right)}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_F}{d}_k{x}_k\text{exp}\left(x_k^{ㄒ}\lambda \right)}\mathrm{.}(5.9)$$

This resulting system of equations cannot be solved analytically. However, the value of $\lambda$ can be found through the Newton-Raphson algorithm.

The equation (5.2) can be now written as

$${\widehat{Y}}_{F|M}^{\text{RRC}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_F}{d}_k\text{exp}\left(x_k^{ㄒ}\lambda \right)y_k}\mathrm{,}$$

where ${\widehat{Y}}_{F|M}^{\text{RRC}}$ denotes the total of the logarithm of the wage in the women sample, when the total is constructed using the raking-ratio calibration. The counterfactual wage mean of women is written as

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

The equation above is very similar to equation (4.4). The only difference lies in the estimation of the parameters $\lambda$ and $\gamma .$ The vector $\lambda$ contains the Lagrangian multipliers solving equation 5.9 under constraint (5.1), while the vector $\gamma$ is found through maximum likelihood.

After computing the calibration weights $w_k$ defined in (5.3) and by using the information in equation (5.6), it results that

$${\widehat{\overline{X}}}_M\mathrm{<mo>=</mo>}{\widehat{\overline{X}}}_{F|M}^{\text{RRC}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_F}{x}_k{w}_k}}{{\displaystyle {\sum }_{k\in S_F}{w}_k}}\mathrm{,}$$

which ensures that the residual part of the structure effect defined in equation (4.8) will equal 0. This is a solution to the problem shown in Section 4.3. This instance of calibration also remedies the issue of the negative weights that may arise when using the chi-squared pseudo-distance.

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