# Dummy variable interaction regression [duplicate]

I have a model: $$\ln({\rm earnings}) = a+b_1{\rm female}+b_2{\rm white}+b_3{\rm female}\times{\rm white}$$ ${\rm female}$ and ${\rm white}$ are dummy variables.

I have interpreted $b_1$ and $b_2$:

• $b_1$ = change in female earnings comparing to male given you are non white
• $b_2$ = change in white earnings comparing to non white given you are male

But I am unable to interpret the coefficient of the interaction term ($b_3$). Please help me with this.

Let me make it more clear what I need out of this regression $$\ln({\rm earnings}) = 2.618656-.0899657{\rm female}+.382019{\rm white}-.2754126 {\rm female}\times{\rm white}$$ Now i know there is gender pay difference with b1, I also know there is race pay difference with b2. Now with b3 i need to know is their a gender pay gap for whites only. How can I figure that out with regression above and without test.

## 1 Answer

Your interpretation of the first two coefficients leads you a bit astray. If you look at b1 only, this tells you the average difference in earnings between men and women in percent (given that the regression is log linear), holding skin color fixed. Using your regression results women earn on average 9% less than men. Then b2 tells you the average difference in earnings between white and non-white regardless of gender. Again from your regression you see that white people earn on average 38% more than noon-whites. What you are looking for is a combination of these two effects.

The best thing to do is to make a table of average earnings like this

          male  female
white
non-white


and calculate the differences between these cells and then the difference of those differences. You should get results similar to your regression coefficients. Think about whose earnings you get from your regression when you "switch on/off" your two dummy variables female and white, i.e. change them from 0 to 1 or vice versa. Remember there that you have an interaction term which will eventually end up in the partial effect of b1 and b2, e.g. the partial effect for female is the regression equation differentiated with respect to female. But from the table and changing the regression equation you will surely figure out what b3 does in this context.

• I still didn't get how I will be able to get to know if there is gender pay gap among white individuals? – user59740 Nov 1 '14 at 9:38
• When you differentiate your regression equation with respect to female $$\frac{\partial \ln (\text{earnings})}{\partial \text{female}} = b1 + b3\cdot \text{white}$$ see what happens when you turn the white dummy from 0 to 1. How do you interpret b3? – Andy Nov 1 '14 at 10:24
• t= female=-1.65 white=8.86 femalewhite=-4.61 cons=66.07 p>|t|= female=(0.100) white=0.000 femalewhite=0.000 cons=0.000 std error= female=0.0546456 white=0.043098 female*white=0.059699 cons=0.0396351 95% confidence interval. Now i know there is gender pay difference with b1 (9%), I also know there is race pay difference with b2(38%). Now i need to know is their a gender pay gap for whites only. How can I figure that out regression above. – user59740 Nov 1 '14 at 18:33