If I run the regression:
$y = \beta_o + \beta_1 * Black + \beta_2 Black*X + \eta$
where Black = 1 if the individual is black, and x is a continuous variable, and I am omitting x separately as a regressor, and $\eta$ is the error term. What exactly is the interpretation of $\beta_2$?
The way I see it there are two different ways to look at it:
- $\frac{dE[y|x,black=1]}{dx}$ = $\beta_2$
which is then the marginal effect of x on y for blacks, but also:
- Taking mean differnces: $E[y|x,black=1]-E[y|x,black=0]$ = $\beta_1 + \beta_2 x$,and then:
$\frac{d(E[y|x,black=1]-E[y|x,black=0])}{dx}$ = $\beta_2$
, which is now saying the mean difference between black and non black is changing with x. These seem like quite different interpretations. Is one of these logically incorrect?
