10.2 Slope dummy variable
While an intercept dummy variable is a very powerful tool, it makes one glaring assumption. Consider the regression results above, namely the estimated slope coefficient with respect to tenure
\[\hat{\beta}_3 = 0.14\]
The interpretation of this slope coefficient is as follows:
Holding education, experience, and gender constant, an individual will receive $0.14 more in wages for every additional year of tenure, on average.
In particular, holding gender constant implies that a female receives the same annual raise than a male. This is an explicit assumption of the model, because the model is incapable of differentiating the annual wage with respect to gender. Do men and women get significantly different average annual raises? We can extend the model to test this assumption with the use of a slope dummy variable.
A slope dummy variable is an example of an interaction term. In other words, it is a new variable that arises from taking the product of two existing variables. In this case, in order for us to examine the gender difference of tenure, we consider the product between female and tenure.
\[wage_i = \beta_0+\beta_1\;educ_i+\beta_2\;exper_i+\beta_3\;tenure_i+...\] \[\beta_4\;female_i+\beta_5\;(tenure_i*female_i)+\varepsilon_i\]
Like our illustration of an intercept dummy, we can see what this PRF looks like for males and females.
\[Male: wage_i=\beta_0+\beta_1educ_i+\beta_2exper_i+\beta_3tenure_i+\varepsilon_i\] \[Female: wage_i=(\beta_0+\beta_4)+\beta_1educ_i+\beta_2exper_i+(\beta_3+\beta_5)tenure_i+\varepsilon_i\]
For males, the PRF looks exactly as it does when we only considered an intercept dummy because both \(\beta_4\) and \(\beta_5\) drop out when \(female_i = 0\). For females, we can see the potential change in the intercept (as before), but we can now see a potential change in the slope with respect to tenure.
## (Intercept) educ exper tenure female
## -2.00229568 0.58279061 0.02834532 0.17780235 -1.17787884
## tenure:female
## -0.14359567Our extended model now gives a better picture of the gender impact on wages. \[\hat{\beta}_4 = -1.18\]
Holding all else constant, a female earns $1.18 less than a male on average.
When considering the impact of tenure on wages, we could show the difference explicitly:
\[Males: \frac{\Delta wage}{\Delta tenure} = \hat{\beta}_3=0.18\] \[Females: \frac{\Delta wage}{\Delta tenure} = \hat{\beta}_3+\hat{\beta}_5=0.18-0.14=0.04\]
The regression states that males receive an $0.18 increase in wages on average for every additional year in tenure (holding all else constant), while females receive only a $0.04 increase in wages on average.
Note that we could also consider slope dummy variables with respect to education as well as experience. You can analyze those on your own.