<p>Estimate the model 
<span class="math-container">$$W_i=a_1+a_2E_i+a_3G_i+a_4E_iG_i+\epsilon_i,$$</span>
where <span class="math-container">$W,E,G$</span> denote wage, education, and an indicator for gender (e.g. 1 if male, 0 if female), respectively. The term <span class="math-container">$EG$</span> is an interaction term obtained by multiplying the gender dummy by education. A simple t-test of the hypothesis <span class="math-container">$H_0: a_4=0$</span> will tell you if the effects of additional education on wage are significantly different for different genders. </p>

CC BY-SA 4.0
2019-03-18T03:45:44.497Z
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
CC BY-SA 4.0
2019-03-18T03:48:38.527Z
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
CC BY-SA 4.0
2019-03-18T18:31:07.827Z
gotcha. it's an intercept test. but I wonder if the intercept coefficient should be in there when testing for different slopes because the intercept inclusion could maybe "effect" the slope effect ? Just a thought. Thanks for great explanation.
mark leeds
CC BY-SA 4.0
2019-03-18T18:47:28.930Z
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
CC BY-SA 4.0
2019-03-17T21:41:16.233Z
2019-03-17T21:41:16.233Z
dlnB
<p>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?</p>

<p>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.</p>

<statistics><regression>
Multivariate linear regression: how to test for whether the slopes are the same?
rchng
# Multivariate linear regression: how to test for whether the slopes are the same?

(Asked by: username_0 on 2019-03-17)

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.

**Tags:** statistics, regression

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## 1 answers

### Answer by username_1 on 2019-03-17. Score: 2 | Accepted answer

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.

> **Comments:**
> **username_2**: 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.  
> **username_1**: 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.  
> **username_2**: gotcha. it's an intercept test. but I wonder if the intercept coefficient should be in there when testing for different slopes because the intercept inclusion could maybe "effect" the slope effect ? Just a thought. Thanks for great explanation.  
> **username_1**: 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  $G_i$.  

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