Gender Wage Gap Accounting: The Role of Selection Bias

Abstract

Mulligan and Rubinstein (2008) (MR) argued that changing selection of working females on unobservable characteristics, from negative in the 1970s to positive in the 1990s, accounted for nearly the entire closing of the gender wage gap. We argue that their female wage equation estimates are inconsistent. Correcting this error substantially weakens the role of the rising selection bias (39 % versus 78 %) and strengthens the contribution of declining discrimination (42 % versus 7 %). Our findings resonate better with related literature. We also explain why our finding of positive selection in the 1970s provides additional support for MR’s main hypothesis that an exogenous rise in the market value of unobservable characteristics contributed to the closing of the gender gap.

Appendix: Derivation of Partial Derivatives in Eqs. (10) and (11)

Recall that σ _v and ρ _{u v} can be expressed in terms of parameters underlying the GHR model, ${\upsigma }_{v}=\left [ {\upsigma }_{u}^{2}+{\upsigma }_{\upvarepsilon }^{2}-2{\uprho }_{u\upvarepsilon } {\upsigma }_{u}{\upsigma }_{\upvarepsilon } \right ]^{1/2}$and ${\uprho }_{uv}=\frac {{\upsigma }_{u}-{\uprho }_{u\upvarepsilon } {\upsigma }_{\upvarepsilon } }{{\upsigma }_{v}}$.

First, consider the effect of σ _u on σ _v :

$$\begin{array}{@{}rcl@{}} \frac{\partial {\upsigma}_{v}}{\partial {\upsigma}_{u}} &=&\frac{\partial} { \partial {\upsigma}_{u}}\left[ {{\upsigma}_{u}^{2}}+{\upsigma}_{\upvarepsilon}^{2}-2{\uprho}_{u\upvarepsilon} {\upsigma}_{u}{\upsigma}_{\upvarepsilon} \right]^{1/2}=\frac{1}{2} \left[ {{\upsigma}_{u}^{2}}+{\upsigma}_{\upvarepsilon}^{2}-2{\uprho}_{u\upvarepsilon} {\upsigma}_{u}{\upsigma}_{\upvarepsilon} \right]^{-\frac{1}{2}}\left( 2{\upsigma}_{u}-2{\uprho}_{u\upvarepsilon} {\upsigma}_{\upvarepsilon} \right)\\ &=&\frac{{\upsigma}_{u}-{\uprho}_{u\upvarepsilon} {\upsigma}_{\upvarepsilon} }{\left[ {{\upsigma}_{u}^{2}}+{\upsigma}_{\upvarepsilon} ^{2}-2{\uprho}_{u\upvarepsilon} {\upsigma}_{u}{\upsigma}_{\upvarepsilon} \right]^{\frac{1}{2}}}=\frac{{\upsigma}_{u}-{\uprho}_{u\upvarepsilon} {\upsigma}_{\upvarepsilon} }{{\upsigma}_{v}}={\uprho}_{uv}. \end{array} $$

(14)

Using this in the derivation that follows obtains the first desired result:

$$\begin{array}{@{}rcl@{}} \frac{\partial} {\partial {\upsigma}_{u}}\Pr \left( L^{\ast} >0|Z,I_{h}\right) &=&\frac{\partial} {\partial {\upsigma}_{u}}\Pr \left( \frac{v}{{\upsigma}_{v}} \leq \frac{Z\upgamma -\upalpha I_{h}}{{\upsigma}_{v}}\right) \\ &=&\frac{\partial} {\partial {\upsigma}_{u}}{\Phi} \left( \frac{Z\upgamma -\upalpha I_{h}}{{\upsigma}_{v}}\right) \\ &=&\upphi \left( \frac{Z\upgamma -\upalpha I_{h}}{{\upsigma}_{v}}\right) \left[ -\left( \frac{Z\upgamma -\upalpha I_{h}}{{{\upsigma}_{v}^{2}}}\right) \right] {\uprho}_{uv}. \end{array} $$

Next, consider the effects of σ _u on ρ _{u v} , ρ _{u v} σ _u , and λ.

$$\begin{array}{@{}rcl@{}} \frac{\partial {\uprho}_{uv}}{\partial {\upsigma}_{u}} &=&\frac{\partial} { \partial {\upsigma}_{u}}\left( \frac{{\upsigma}_{u}-{\uprho}_{u\upvarepsilon} {\upsigma}_{\upvarepsilon} }{{\upsigma}_{v}}\right) =\frac{{\upsigma}_{v}-\left( {\upsigma}_{u}-{\uprho}_{u\upvarepsilon} {\upsigma}_{\upvarepsilon} \right) \frac{\partial {\upsigma}_{v}}{\partial {\upsigma}_{u}}}{{{\upsigma}_{v}^{2}}} \\ &=&\frac{1-\left( \frac{{\upsigma}_{u}-{\uprho}_{u\upvarepsilon} {\upsigma}_{\upvarepsilon} }{{\upsigma}_{v}}\right) {\uprho}_{uv}}{{\upsigma}_{v}}=\frac{1-{\uprho}_{uv}^{2}}{{\upsigma}_{v}}>0. \end{array} $$

$$\frac{\partial \left( {\uprho}_{uv}{\upsigma}_{u}\right)} {\partial {\upsigma}_{u}}= \frac{\partial {\uprho}_{uv}}{\partial {\upsigma}_{u}}{\upsigma}_{u}+{\uprho}_{uv}=\left( \frac{1-{\uprho}_{uv}^{2}}{{\upsigma}_{v}}\right) {\upsigma}_{u}+{\uprho}_{uv}. $$

Finally,

$$\begin{array}{@{}rcl@{}} \frac{\partial \uplambda} {\partial {\upsigma}_{u}} &=&{\uplambda}^{\prime} \left( \frac{Z\upgamma -\upalpha I_{h}}{{\upsigma}_{v}}\right) \left[ \frac{\partial} { \partial {\upsigma}_{u}}\left( \frac{Z\upgamma -\upalpha I_{h}}{{\upsigma}_{v}}\right) \right] \\ &=&{\uplambda}^{\prime} \left( \frac{Z\upgamma -\upalpha I_{h}}{{\upsigma}_{v}}\right) \left[ -\left( \frac{Z\upgamma -\upalpha I_{h}}{{{\upsigma}_{v}^{2}}}\right) \frac{ \partial {\upsigma}_{v}}{\partial {\upsigma}_{u}}\right] \\ &=&-{\uplambda}^{\prime} \left( \frac{Z\upgamma -\upalpha I_{h}}{{\upsigma}_{v}} \right) \left[ \left( \frac{Z\upgamma -\upalpha I_{h}}{{{\upsigma}_{v}^{2}}}\right) {\uprho}_{uv}\right], \end{array} $$

where we used the definition $\uplambda \left (\frac {Z\upgamma -\upalpha I_{h}}{{\upsigma }_{v}}\right ) =\frac {\upphi \left (\frac {Z\upgamma -\upalpha I_{h}}{ {\upsigma }_{v}}\right )} {\Phi \left (\frac {Z\upgamma -\upalpha I_{h}}{{\upsigma }_{v}} \right )}$ and the above result shown in Eq. (14).

Differentiating the bias B = ρ _{u v} σ _u λ gives the second desired result:

$$\begin{array}{@{}rcl@{}} \frac{\partial B}{\partial {\upsigma}_{u}} &=&\frac{\partial \left( {\uprho}_{uv}{\upsigma}_{u}\right)} {\partial {\upsigma}_{u}}\uplambda +\frac{\partial \uplambda} {\partial {\upsigma}_{u}}{\uprho}_{uv}{\upsigma}_{u} \\ &=&\left[ \left( \frac{1-{\uprho}_{uv}^{2}}{{\upsigma}_{v}}\right) {\upsigma}_{u}+{\uprho}_{uv}\right] \uplambda +\left[-{\uplambda}^{\prime} \left( \frac{ Z\upgamma -\upalpha I_{h}}{{\upsigma}_{v}}\right) \right] \left[ \left( \frac{ Z\upgamma -\upalpha I_{h}}{{{\upsigma}_{v}^{2}}}\right) {\uprho}_{uv}\right] {\uprho}_{uv}{\upsigma}_{u}. \end{array} $$

Table 3 Summary statistics

Table 4 Coefficients and standard errors in the estimation of the benchmark and MR models

Table 5 Estimation of the simple benchmark and simple MR models for the 1970s