We are interested in the industry wage differential of retail services relative to the aggregate economy. The raw sector differential can partly be explained by specific characteristics of workers in this sector compared to the aggregate and partly by the fact that workers (observed) characteristics are priced differently in different sectors of the economy. In order to shed some light on the relative magnitude of these alternative explanatory factors a decomposition technique in the spirit of the well-known Blinder (1973) and Oaxaca (1973) approach is employed. In what follows, the raw retail differential is spread into three components: a rewards, a characteristics and an interaction effect.
Let the usual wage equation model be described for a specific sector of the economy as
′ (1)
= +
y x β ε
and correspondingly for the aggregate as . (2)
= ′ + Y X B E Then define
∆ = −βˆ: β Bˆ ˆ and ∆ = −x: x X, (3)
where the vectors x and X contain average values of the regressor in the sector and in the aggregate, respectively. A sensible decomposition of the raw earnings differential is given by
rewards effect characteristics interaction
To illustrate the decomposition consider the simple case, where earnings are explained by a qualification dummy variable only, which takes a zero value for low skilled and one for skilled workers. Assume that remuneration of low-skilled workers is identical across sectors so that the regression constant can be disregarded for the ease of exposition. Assume further that, compared to the aggregate, the average worker in the sector is more qualified and the skill premium is higher in the sector, hence >x X > 0 and β >B> 0, respectively. It is evident that under these circumstances the wage differential is favourable to the sector. The situation is depicted in figure 1.
The total wage bonus of the sector is equivalent to the sum of the shaded areas. The total effect can be divided into
• the rewards effect (area A) or X⋅ ∆
β
; given our assumptions the rewards effect here indicates a pay bonus for the sector even if the share of qualified workers were the same as in the aggregate economy;• the characteristics effect (area B) or ∆ ⋅x B; even if the group of qualified workers were paid the same as in the aggregate, there would be an advantage to the sector because of its favourable skill structure.
• the interaction effect (area C) or ∆ ⋅ ∆x
β
; in the context here, this effect indicates an earnings advantage for the sector because there are relatively more high-paid qualified workers in the sector and these workers are doing especially well compared to the aggregate.In contrast to the example given here, the negative rewards and/ or the characteristics effect could lead to a negative wage differential, or a “wage penalty” for a specific sector. This is what can be expected for retail services.
Note that the sign of the rewards (characteristics) effect is determined by ∆
β
(or ∆x,respectively) given that X >0 and which has been assumed here. If the rewards and characteristics effects have equal signs the interaction effect is positive. Consider the case where both are negative, hence the sector has a relatively low educated work force and the skill premium is less than in the aggregate. Clearly this would lead to a wage penalty for the sector. The interaction effect, however, would be positive, since the sector penalty is
“reduced” by the fact that the sector has a relatively low share of workers in the category with a negative differential compared to the aggregate. In other words, the interaction effects corrects for a double counting of the sector’s disadvantage that would otherwise occur.
>0 B
B
Figure 1. Decomposition of the raw sectoral differential
Application of the decomposition technique in case of standard OLS (or TOBIT) estimation is straightforward. In the context of quantile regressions, however, the approach requires some further clarifications.
Consider the quantile regression model
, (5) coefficients and εθ and Eθdenote error terms.34 Replacing the vector of explanatory variables in eq. (5) by the corresponding sample means gives the expectation for the θ -quantile of dependent variable yi conditional to average characteristics
( ) ( )
As an example consider the case with log wages as dependent variable and years of schooling as the only regressor. Assume that the average duration of schooling is x in the sector and Xin the aggregate. Inserting these values into eq. (5) yields the expected median for workers with average years of schooling as xβˆ0.5 in the sector and XBˆ0.5in the aggregate economy. Note that in general this expectation is not identical to the unconditional median:
( ) ( ) ( ) ( )
Qθ yi ≠Qθ yi |x = xi and Qθ Yi ≠Qθ Yi |X = Xi .
By contrast, for standard (mean) regression the following relationship holds:
( )
i(
i |)
ˆy =E y =E y x =x β′
& OLS (and similar for the aggregate).
To summarize these considerations, the decomposition approach can be applied to quantile estimation results as well. However, one has to stress the fact that the results are conditional to average characteristics, a restriction that is not necessary in the context of standard estimation like OLS.35
Let us now define the conditional θ -th quantile wages in logs as
( ) ( )
:=Q i i and :=Q i i
yθ θ y x =x Yθ θ Y X =X .
Then one can calculate the conditional sector wage differential at quantile θ as CWDθ =yθ−Yθ. For example, for θ =0.5 the indicator compares median wages of workers with typical (average) characteristics in the sector to median wages of a group of workers with typical characteristics in the aggregate.
CWD0.5
By applying the same decomposition techniques as for the mean wage differential one obtains the breakdown that is used in our empirical study:
rewards effect characteristics interaction
Blinder, Alan S. (1973): "Wage Discrimination: Reduced Form and Structural Estimates", Journal of Human Resources, 8, pp.436-455.
Oaxaca, Ronald (1973): "Male-Female Wage Differentials in Urban Labor Markets", International Economic Review, 14, pp.673-709.
35 In case of only minor differences between the conditional and unconditional quantiles, however, the specific restraints in the interpretation of the quantile regression approach can be neglected.