vandanak
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- The Blinder–Oaxaca decomposition is a statistical method that explains the difference in the means of a dependent variable between two groups by decomposing the gap into that part that is due to differences in the mean values of the independent variable within the groups, on the one hand, and group differences in the effects of the independent variable, on the other hand. The method was introduced by sociologist and demographer Evelyn M. Kitagawa in 1955. I have confusion in understanding a term
The following three equations illustrate this decomposition. Estimate separate linear wage regressions for individuals i in groups A and B:
{\displaystyle {\begin{aligned}(1)\qquad \ln({\text{wages}}_{A_{i}})&=X_{A_{i}}\beta _{A}+\mu _{A_{i}}\\(2)\qquad \ln({\text{wages}}_{B_{i}})&=X_{B_{i}}\beta _{B}+\mu _{B_{i}}\end{aligned}}}
where Χ is a vector of explanatory variables such as education, experience, industry, and occupation, βA and βB are vectors of coefficients and μ is an error term.
Let bA and bB be respectively the regression estimates of βA and βB. Then, since the average value of residuals in a linear regression is zero, we have:
{\displaystyle {\begin{aligned}(3)\qquad &\operatorname {mean} (\ln({\text{wages}}_{A}))-\operatorname {mean} (\ln({\text{wages}}_{B}))\\[4pt]={}&b_{A}\operatorname {mean} (X_{A})-b_{B}\operatorname {mean} (X_{B})\\[4pt]={}&b_{A}(\operatorname {mean} (X_{A})-\operatorname {mean} (X_{B}))+\operatorname {mean} (X_{B})(b_{A}-b_{B})\end{aligned}}}
![{\displaystyle {\begin{aligned}(3)\qquad &\operatorname {mean} (\ln({\text{wages}}_{A}))-\operatorname {mean} (\ln({\text{wages}}_{B}))\\[4pt]={}&b_{A}\operatorname {mean} (X_{A})-b_{B}\operatorname {mean} (X_{B})\\[4pt]={}&b_{A}(\operatorname {mean} (X_{A})-\operatorname {mean} (X_{B}))+\operatorname {mean} (X_{B})(b_{A}-b_{B})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e50f13499df60059982b9094fbe754d31a22ec62 "{\displaystyle {\begin{aligned}(3)\qquad &\operatorname {mean} (\ln({\text{wages}}_{A}))-\operatorname {mean} (\ln({\text{wages}}_{B}))\\[4pt]={}&b_{A}\operatorname {mean} (X_{A})-b_{B}\operatorname {mean} (X_{B})\\[4pt]={}&b_{A}(\operatorname {mean} (X_{A})-\operatorname {mean} (X_{B}))+\operatorname {mean} (X_{B})(b_{A}-b_{B})\end{aligned}}}")
The first part of the last line of (3) is the impact of between-group differences in the explanatory variables X, evaluated using the coefficients for group A. The second part is the differential not explained by these differences in observed characteristics X.
I have confusion in last equation of equation 3. Please help I have kind of lost touch.
Thank you in advance
{\displaystyle {\begin{aligned}(1)\qquad \ln({\text{wages}}_{A_{i}})&=X_{A_{i}}\beta _{A}+\mu _{A_{i}}\\(2)\qquad \ln({\text{wages}}_{B_{i}})&=X_{B_{i}}\beta _{B}+\mu _{B_{i}}\end{aligned}}}
where Χ is a vector of explanatory variables such as education, experience, industry, and occupation, βA and βB are vectors of coefficients and μ is an error term.
Let bA and bB be respectively the regression estimates of βA and βB. Then, since the average value of residuals in a linear regression is zero, we have:
{\displaystyle {\begin{aligned}(3)\qquad &\operatorname {mean} (\ln({\text{wages}}_{A}))-\operatorname {mean} (\ln({\text{wages}}_{B}))\\[4pt]={}&b_{A}\operatorname {mean} (X_{A})-b_{B}\operatorname {mean} (X_{B})\\[4pt]={}&b_{A}(\operatorname {mean} (X_{A})-\operatorname {mean} (X_{B}))+\operatorname {mean} (X_{B})(b_{A}-b_{B})\end{aligned}}}
The first part of the last line of (3) is the impact of between-group differences in the explanatory variables X, evaluated using the coefficients for group A. The second part is the differential not explained by these differences in observed characteristics X.
I have confusion in last equation of equation 3. Please help I have kind of lost touch.
Thank you in advance