Question about interpretting an interaction term in a regression 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?
 A: Given the model you have set up none of them are logically incorrect.
The model $$Y = \beta_0 + \beta_1 Black + \beta_2 X\cdot Black + u,$$ has what I will call a weird restriction
$$\mathbb E[y\lvert x , black=0] = \mathbb E[y \lvert black=0] = \beta_0.$$
The restrictions says that while for black individuals $X$ is important for the outcome variable $Y$ this is not the case for non-black individuals. If you really could come up with such a variable - and I am not arguing it is impossible - then it should come to you as no surprise that
$$(1) \ \frac{\partial \mathbb E[Y \lvert black=1, x]}{\partial x} = \beta_2$$
and
$$(2) \ \frac{\partial (\mathbb E[Y \lvert black=1, x] - \mathbb E[Y \lvert black=0, x]}{\partial x} = \beta_2$$
If however, you really believe that (1) and (2) should not be the same - you say they express different things - then you should simply use a more flexible model
$$Y = \beta_0 + \beta_1 Black + \beta_2 X\cdot Black + \beta_3 X + u$$
for which
$$(1) \ \frac{\partial \mathbb E[Y \lvert black=1, x]}{\partial x} = \beta_2 + \beta_3$$
and
$$(2) \ \frac{\partial (\mathbb E[Y \lvert black=1, x] - \mathbb E[Y \lvert black=0, x]}{\partial x} = (\beta_2 + \beta_3) - \beta_3 = \beta_2$$
In any case the more flexible model is probably preferred because it allows you to test whether the "weird" restriction is "true".
