# Multivariate linear regression: how to test for whether the slopes are the same?

If I regress wages on education and the dummy variable gender using a linear conditional expectation function (wage = a + b(education) + c(gender)), how can I test that the slope b is the same for both males and females? i.e. the returns to education are the same?

Intuitively, I was thinking that one could conduct simple linear regressions of wage on education for males and females separately and then do a two-sample hypothesis test for whether the slope estimates are significantly different. Would this introduce issues with the omitted variable gender? Thank you.

Estimate the model $$W_i=a_1+a_2E_i+a_3G_i+a_4E_iG_i+\epsilon_i,$$ where $$W,E,G$$ denote wage, education, and an indicator for gender (e.g. 1 if male, 0 if female), respectively. The term $$EG$$ is an interaction term obtained by multiplying the gender dummy by education. A simple t-test of the hypothesis $$H_0: a_4=0$$ will tell you if the effects of additional education on wage are significantly different for different genders.
• I see what you're doing but wouldn't using the same model without the interaction term and testing for $a3 = 0$ test the same thing ? Thanks for insight. – mark leeds Mar 18 '19 at 3:45
• Good question. $a3$ and $a4$ are actually addressing different effects. In particular, $a_3$ is an intercept shifter for different genders, while $a_4$ is a slope shifter with respect to education level. So testing $a_3=0$ is testing whether, all else equal, wages are significantly different across genders. On the other hand, testing $a_4=0$ is testing, all else equal, whether an extra year of education has a significantly different impact on wages across genders. – dlnB Mar 18 '19 at 3:48
• Whether or not one should include the dummy $G_i$ should be motivated by institutional knowledge of the dataset. In particular, if you have a reason to believe that men earn more than women (or vice versa) independent from the effects of education, one should include $G_i$. – dlnB Mar 18 '19 at 18:47