Blinder-Oaxaca decomposition
Decomposition of gender wage gap using using R software
TA: Yaya DIALLO
1. Facts.
We plot the distributon of hourly wage by gende and age group. We observed that wage differences between men and women accross age group. Is this graph illustrates discrimination against women in the Canadian Labor Market ?
The goal of this tutorial is to provides a technique to disentagle discrimination from other relevant factors in the labor market.
1.1 Taste discrimination
Following becker(1957), the discrimination may arise from the fact that some employers feel a desinclination to hire workers belonging to certain groups. To illustrate the theory of taste discrimination, let us consider a labor market composed of workers who each produce quantity \(y\) and who belong to two groups A and B, All employers have an aversion for workers of group A, even they have the same productivity as workers in group B.
- Gain of employing a worker from group B : \(y-w_A -u\)
- Gain of employing a worker from group A : \(y-w_A\)
where
\(w_A\) is the wage received by workers in group A
\(u>0\) is a parameter that represents the aversion employers feel toward workers of group A
Wage gap .
Under perfect competition, if the economy is composed solely of employers having an aversion toward workers of group A, the members of this group obtain \(w_A= y-u\) and members of group B receives \(w_A= y\). Thus, individuals in the nondiscriminated group, i.e belonging in B, obtain a higher wage, equal to their productivity.
Employement gap .
In the same setting, if the labor supply of both groups rises with wages, individuals in the discriminated group are, all other thing being equal, in employment less often.
Gap will vanishes in long run under perfect competition.
If there exist employers who experiences no aversion for indfividuals in group A and if these employers can freely enter this labor market, the wage difference between the two groups will disapears. The null profit entails, on one hand, that the workers of group A and B obtain the same wage equal to their productivity (\(w_A=w_B\)). At the end, the employers experiencing an aversion towards workers of group A refrain form hiring these workers.
According to the theory advocated by Becker(1957), employer and employee discrimination resulting in persistent wage differences cannot occur in perfect competitive markets, in which by definition all workers are paid according to their marginal productivity. Hence, discriminaton occurs under imperfect competition.
2. Measuring wage discrimination.
Discrimnation is rarely observed directly. In most cases, it is impossible to disentagle what factor in wage differences among people holding the same job reflects individual differences or deliberate discriminatory decisions by the employer. In this part, we will present two methods used by economist to quantify discrimination.
Estimations of wage equations : Including controls variables.
Decomposition methods
2.1 Estimation of discrimination: The case of gender wage gap in Swiss
The standard approach consists of estimating an equation in which the logaithm of income is explained by a set of factors like the duration and quality of schooling, experience, and region and by dummy variables representing etchnic origin.
\[\Large \ln w_{i}=x_{i}\beta + \alpha F_i + \gamma Z_i + \varepsilon_i\]
Where
\(x_i\) is the vector of individual characteristics (age,schooling,experience)
\(F_i\) is a dummy variable, which take 1 if the worker is female
\(Z_i\) represents relevant factor in the labor market: tenure, experience and level of education.
In this equation, if the set of variables explaining wage is sufficiently rich, the value of \(\alpha\) should represent the discrimination against women.
Illustration with Swiss Labor Market Survey
The Swiss labor survey data frame contains information about the demographic characteristics and labor market outcomes of 1647 employed. I listed the variables below including the log of wage (lnwage).
## [1] "lnwage" "educ" "exper" "tenure" "isco" "female"
## [7] "lfp" "age" "agesq" "single" "married" "divorced"
## [13] "kids6" "kids714" "wt"Wage_equation=lm(lnwage ~ educ + exper + tenure + female, data=LFSSwiss)
coef=round(summary(Wage_equation)$coefficients,3)
kable(coef, align = "c", caption="Table 1: Estimation of wage equations")| Estimate | Std. Error | t value | Pr(>|t|) | |
|---|---|---|---|---|
| (Intercept) | 2.213 | 0.068 | 32.384 | 0.000 |
| educ | 0.085 | 0.005 | 16.337 | 0.000 |
| exper | 0.011 | 0.002 | 7.220 | 0.000 |
| tenure | 0.008 | 0.002 | 4.101 | 0.000 |
| female | -0.084 | 0.025 | -3.345 | 0.001 |
The negative value on Female indicates an unexplained component of gender wage. Thus, we suspect discrimination against women in Swiss Labor Market(the value is signicantly different to zero).
2.2 Decomposition of gender wage gap.
2.2.1 Decomposition : methods and formulas.
Another approach to the measurement of discrimination is to estimate separately wages equations for female and male. Then, identify what part of the wage gap stems from individuals characteristics that vary across groups (female and male for our case) and what part is unexplained. When two groups are considered, the method used to estimate to decompose the wage gap is Blinder-Oaxaca Method (BO) as seen in class.
Let consider two groups male (m)and female(f) and the outcome is the wage \(w\) in the labor market. The gender wage gap can expressed as following \[\Delta w= \mathbb{E}(w_m)-\mathbb{E}(w_f)\] where E() denotes the expected value of wage.
The wage depend on individuals characteristics of the workers, X, and assuming the wage equation is linear then the wage equation of individual i belonging in group \(g \in \{ m, f \}\) can be expressed as follows
\[w_{i,g} = X_{i,g} \beta_m + \varepsilon_{i,g}\] where \(\beta\) is the vector of coefficients : slopes and the constant. \(\varepsilon\) is the error terms.
- The expected wage (or average wage) for men
\[ \large \mathbb{E}(w_{m}) = \mathbb{E}(X_{m}) \beta_m + \underbrace{\mathbb{E}(\varepsilon_{m})}_{=0} \] - The expected wage (or average wage) for women
\[ \large \mathbb{E}(w_{f}) = \mathbb{E}(X_{f}) \beta_f + \underbrace{\mathbb{E}(\varepsilon_{f})}_{=0} \]
- Let simplify the notation and assume \(\mathbb{E}(Y)=\bar{Y}\). The gender wage gap is :
\[ \large \Delta \bar{w}= \bar{w_m}-\bar{w_f}= \bar{X_m}\beta_m -\bar{X_f}\beta_f \] - To identify the contribution of group differences in predictors to the overall gender wage gap, the equation of \(\Delta \bar{w}\) can be rearranged as presented in class.
\[\begin{align} \large \Delta \bar{w} & = \underbrace{\left(\bar{X_m}-\bar{X_f} \right)\beta_m }_{\text{Explained}}+ \underbrace{\left(\beta_m-\beta_f \right)\bar{X_f} }_{\text{Unexplained(coefficients)}} \end{align}\]a.Threefold decomposition
Usually, economist decompose the first term in “endowment effect” and “interactions component” such that the gender wage gap becomes:
\[\begin{align} \large \Delta \bar{w} & = \underbrace{\left(\bar{X_m}-\bar{X_f} \right)\beta_f }_{\text{Explained }}+ \underbrace{\left(\beta_m-\beta_f \right)\bar{X_f} }_{\text{Unexplained(coefficients)}} + \underbrace{\left(\bar{X_m}-\bar{X_f} \right)\left(\beta_m-\beta_f \right)}_{\text{Interactions}} \label{qd} \end{align}\]The difference was expressed from the viewpoint of women but the inveserve is also possible.
b.Twofold decomposition
An alternative decomposition is to consider a reference where there is a nondiscriminatory coefficient that can be used to determine the contribution of the differences. Let assume the reference coefficient is \(\beta_R\). The coefficient can be interpreted as non-discriminatory. The gender wage gap can be expressed in two terms.
\[\begin{align} \large \Delta \bar{w} & = \underbrace{\left(\bar{X_m}-\bar{X_f} \right)\beta_R }_{\text{Explained }} + \underbrace{\underbrace{\left(\beta_R - \beta_f \right)\bar{X_f} }_{\text{Unexplained Female }} + \underbrace{\left(\beta_m - \beta_R \right)\bar{X_m} }_{\text{Unexplained Male}} }_{ \text{Unexplained}} \end{align}\]The “unexplained part” is now decomposed two parts
- “Unexplained Male” : discrimination in favor of men.
- “Unexplained Female” : discrimination againts women.
2.2.2 Application of decomposition in R : The case of gender wage gap in Swiss.
There is available package of oaxaca. First, you have to install the package. FOr the estimation, use the command oaxaca which performs the Blinder-Oaxaca decompositions.
- Command Oaxaca to decompose the gender wage gap. I save my results in results.
# The decomposition is represented by female
results <- oaxaca(formula = lnwage ~ educ + exper| female, data = LFSSwiss)
# Main outputs of the command
names(results)## [1] "beta" "call" "n" "R" "reg" "threefold"
## [7] "twofold" "x" "y"- The gender wage gap.
wA=round(results$y$y.A,4)
wB=round(results$y$y.B,4)
wd=round(results$y$y.diff,4)
wagegap=cbind(wA,wB,wd)
colnames(wagegap)=c("Male","Female","Gender wage gap")
rownames(wagegap)=c(" ")
kable(wagegap, align = "c", caption="Table 1: Wage differences")| Male | Female | Gender wage gap | |
|---|---|---|---|
| 3.4402 | 3.2668 | 0.1735 |
In our sample, the mean of log wages (lnwage) is 3.44 for men and 3.27 for women, yielding a wage gap of 0.17. We can decompose the wage to look at which part is due to discrimination. The following table shows the wage gap is divided into three parts.
- Results of the estimation of wage equations by gender.
betam= round(results$beta$beta.A,4)
betaf=round(results$beta$beta.B,4)
betadiff=round(results$beta$beta.diff,4)
tabeta=cbind(betam,betaf,betadiff)
colnames(tabeta)=c("Male","Female","Differences")
kable(tabeta, align = "c", caption="Table 2: Coefficients differences")| Male | Female | Differences | |
|---|---|---|---|
| (Intercept) | 2.2399 | 2.1002 | 0.1397 |
| educ | 0.0840 | 0.0880 | -0.0040 |
| exper | 0.0148 | 0.0147 | 0.0001 |
- Difference in endowments.
xm= round(results$x$x.mean.A,4)
xf=round(results$x$x.mean.B,4)
xdiff=round(results$x$x.mean.diff,4)
tabx=cbind(xm,xf,xdiff)
colnames(tabx)=c("Male","Female","Differences")
kable(tabx, align = "c", caption="Table 3: Endowments differences")| Male | Female | Differences | |
|---|---|---|---|
| (Intercept) | 1.0000 | 1.0000 | 0.0000 |
| educ | 11.8142 | 11.2321 | 0.5822 |
| exper | 14.0768 | 12.1377 | 1.9392 |
a. Threefold decomposition
rowtab=c("Endowments", "Coefficients", "Interactions")
coltab=c("Coef", "se")
tab1=results$threefold$overall[1:2]
tab2=results$threefold$overall[3:4]
tab3=results$threefold$overall[5:6]
tab=rbind(tab1,tab2,tab3)
colnames(tab) <- coltab
rownames(tab)<-rowtab
kable(tab, align = "c", caption="Table 4: Decomposition of wage gap")| Coef | se | |
|---|---|---|
| Endowments | 0.0797073 | 0.0155901 |
| Coefficients | 0.0959280 | 0.0255305 |
| Interactions | -0.0021746 | 0.0101077 |
b.Twofold decomposition
We have to define the reference coefficient to be able to decompose. In this package, the author consider many cases: \(\beta_R=\alpha \beta_m + (1-\alpha) \beta_f\) where \(\alpha\) is the weight. The default option of oaxaca command displays for \(\alpha \in \{0, 1, 0.5, -1, -2 \}\). Let take \(\alpha = 1\) i.e \(\beta_R=\beta_m\).
rowtab=c("Explained","Unexplained (Male+Female)", "Unexplained Male", "Unexplained Female")
coltab=c("Coefficients", "se")
tab1=c(results$twofold$overall[2,2],results$twofold$overall[2,3])
tab2=c(results$twofold$overall[2,4],results$twofold$overall[2,5])
tab3=c(results$twofold$overall[2,6],results$twofold$overall[2,7])
tab4=c(results$twofold$overall[2,8],results$twofold$overall[2,9])
tabtwofold=rbind(tab1,tab2,tab3,tab4)
colnames(tabtwofold) <- coltab
rownames(tabtwofold)<-rowtab
kable(tabtwofold, align = "c", caption="Table 5: Decomposition of wage gap")| Coefficients | se | |
|---|---|---|
| Explained | 0.0775327 | 0.0119196 |
| Unexplained (Male+Female) | 0.0959280 | 0.0255305 |
| Unexplained Male | 0.0000000 | 0.0000000 |
| Unexplained Female | 0.0959280 | 0.0255305 |
References : - g - b