suppose I have this simple model
$$ log(wage_i) = \beta_0 + \beta_1 educ_i + \beta_2 white + \epsilon_i$$
where $white$ is a dummy equal to 1 if the individual is white. I want to see if the impact of education on wages are different for whites and non-whites. Maybe I think that for some reason, for the white group, education should have no impact at all or something. My question is: Interacting the dummy with $educ$ is the correct way to see if the impact is different?
Also, if I divide my sample in whites and non-whites, and regress $ log(wage_i) = \beta_0 + \beta_1 educ_i + \epsilon_i$ for both samples, and for whites I see no statistical significance, while for non-whites I see highly significant, can I interpret this as education having different impacts on wages?
I think that another way to put this question is, take the below model:
$$ log(wage_i) = \beta_0 + \beta_1 educ_i + \beta_2 white + \gamma educ_i \times white + \epsilon_i$$
Should $\beta_0 + \beta_1 educ_i + \beta_2 + \gamma educ_i$ and $\beta_0^w + \beta_1^w educ_i^w$ have similar interpretations?