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 1


$b_3$ is the difference between white females and the sum of $a+b_1+b_2$. That is, the difference between white females and the sum of non-white males plus the difference between non-white females and non-white males plus the difference between white males and non-white males.
\begin{align} b_3 = \bar x_\text{white female} - \big[&\ \ \bar x_\text{non-white male}\quad\quad\quad\quad\quad\quad\quad\ \ + \\ &(\bar x_\text{non-white female} - \bar x_\text{non-white male}) + \\ &(\bar x_\text{white male}\quad\quad\! - \bar x_\text{non-white male})\quad\ \big] \end{align}
Honestly, it's a bit of a mess to interpret in this way. More typically, we interpret the test of $b_3$ as a test of the additivity of the effects of ${\rm white}$ and ${\rm female}$. (The expression within the square brackets $[]$ is the additive effect of ${\rm white}$ and ${\rm female}$.) Then we make more substantive interpretations only of simple effects (i.e., the effect of one factor within a pre-specified level of the other factor). People rarely try to interpret the interaction effect / coefficient in isolation.

It may also help you to read my answer here: Interpretation of betas when there are multiple categorical variables, which covers an analogous, but simpler, situation without the interaction.

  • $\begingroup$ ok can I interpret b3 as earnings of female white is different by b3 from earnings of females or white $\endgroup$
    – user59740
    Commented Nov 1, 2014 at 4:48
  • $\begingroup$ @ahmed, b3 is not white female - non-white female, nor is it white female - white male. I'll try to make my answer clearer. $\endgroup$ Commented Nov 1, 2014 at 14:31
  • $\begingroup$ 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 can we answer how to know whether there is a gender pay gap among whites only? $\endgroup$
    – user59740
    Commented Nov 1, 2014 at 18:36
  • 1
    $\begingroup$ @ahmed, the way you have this set up is that non-white males are the reference group. The other coefficients are mostly being tested against that (w/ the interaction term as specified above). If you want to test males vs females w/i white, make white males (or white females) the reference group & the test of female (male) will do it. Alternatively, you can do a t-test on those 2 subgroups. $\endgroup$ Commented Nov 2, 2014 at 3:11
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    $\begingroup$ The intercept isn't a % change, it's just a constant. You can exponentiate that value to get a value on the original scale for that cell, if you want. The other coefficients represent % changes. $\endgroup$ Commented Nov 3, 2014 at 3:47

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