Blinder–Oaxaca decomposition

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, β_A and β_B are vectors of coefficients and μ is an error term.

Let b_A and b_B be respectively the regression estimates of β_A and β_B. 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.

The unexplained differential in wages for the same values of explanatory variables should not be interpreted as the amount of the difference in wages due only to discrimination. This is because other explanatory variables not included in the regression (e.g. because they are unobserved) may also account for wage differences. For example, David Card and Alan Krueger found in a paper entitled, "School Quality and Black-White Relative Earnings: A Direct Assessment"[5] that improvements in the quality of schools for Black men born in the Southern states of the United States between 1915 and 1966 increased the return to education for these men, leading to narrowing of the black-white earnings gap. In terms of wage regressions, the poor quality of schools for Black men had meant a lower value of the β coefficient on years of schooling for Black men than for White men. Thus, some of this lower β coefficient reflected a difference in the quality of education for Black workers which could have otherwise been interpreted as an effect of direct discrimination; differences in the quality of education for Black workers would reflect historical or 'indirect' discrimination against them.

• Standardization (demographics)

• Autor, David (January 30, 2015). "MIT Graduate Labor Economics 14.662 Spring 2015 Lecture Note 1: Wage Density Decompositions" (PDF). Massachusetts Institute of Technology: MIT OpenCourseWare.
• Borjas, George J. (2000). "Measuring Discrimination". Labor Economics (Second ed.). Boston: Irwin McGraw-Hill. pp. 362–366. ISBN 0-07-231198-3.
• Cahuc, Pierre; Carcillo, Stéphane; Zylberberg, André (2014). "Decomposition Methods: The Case of the Gender Wage Gap". Labor Economics. Cambridge: MIT Press. pp. 504–514. ISBN 978-0-262-02770-0.
• Fortin, Nicole; Lemieux, Thomas; Firpo, Sergio (2011). "Decomposition Methods in Economics". Handbook of Labor Economics. Elsevier. pp. 1–102. doi:10.1016/S0169-7218(11)00407-2. ISBN 978-0-444-53450-7.
• Kevin Guo & Guillaume Basse. 2021. "The Generalized Oaxaca-Blinder Estimator." Journal of the American Statistical Association.