# Linear Hypothesis with insignificant coefficient in linear regression

In the following linear regression:

$$\text{Wage}=\beta_0+\beta_1 \, \text{Female} +\beta_2 \, \text{Time}+\beta_3 \, \text{Female}\cdot \text{Time} + e$$ $\beta_2 = 0.5$ and $\beta_3 = -1$ are significant but $\beta_1 =-0.2$ is insignificant. The $F$-test result show that $\beta_1+\beta_3 = 0$ has $(p\geq0.1)$ due to the high std. error of $\beta_1$.

Question: Based on negative and significant coefficient $\beta_3$, can I conclude that the wage gap between male and female increases over time? My concern is the main effect $\beta_1$ is not significant and the F-test of $(H_0: \beta_1 + \beta_3 = 0)$ result shows that $H_0$ is not rejected.

• How are you saying F-test result means b1+b3=0? – Bach Mar 3 '16 at 4:31
• I conducted F-test (H0: b1+b3=0) and couldn't reject the null hypothesis- – user3509199 Mar 3 '16 at 4:38
• I do not think you can combine both to reach at your hypothesis conclusion. But you can use the info at your hand, even if b1 is not significant, the sign of other two significant parameters can help you answer your question – Bach Mar 3 '16 at 4:40

You are interested in $\frac{\partial Wage}{\partial Time} = \beta_2 + \beta_3 Female$. This answers the question, "what happens when we increase time by 1?". Since $\beta_3 < \beta_2$, the wage will raise for men but not for women - since the raise of time time if completely offset for women.

Also, IMHO, the relevant F-test is $H_0: \beta_2 + \beta_3 = 0$, given the above.

Now, your next question should be: "Do I trust my model?". I, for one, would not trust your model. You do not control for race, education, occupation and hours worked. So omittted variable bias could be a huge problem for this model.

• Thanks for the quick answer. Above model (Wage, gender) is hypothetical. I use the hypothetical example to communicate easily. My model is different from the model in the question- Thanks! – user3509199 Mar 3 '16 at 14:39

I think what you are after is the gender gap, $\frac{\delta Wage}{\delta Female} = \beta_1 + \beta_3 Time$. Notice that in this case, the gender gap depends on time. Notice also that the gender gap does not equal $\beta_1 + \beta_3$. What your F-test tests is whether there is no gender gap when time = 1. Maybe this is something you want to test but this could also be nonsensical for instance if your time variable goes from 1990 to 2010.

Your confusion comes from the fact that you think you have tested with the F-test whether there is a gender gap in your entire dataset. So you think you have the following results

1. There is no gender gap (based on the F-test)
2. The gender gap increases over time (based on $\beta_3$)

which sounds weird. However, this is not true. What you have is

1. There is no gender gap at time = 0 ($\beta_1$)
2. There is no gender gap at time = 1 ($\beta_1 + \beta_3$)
3. The gender gap increases over time ($\beta_3$)

This seems to be plausible for instance if there is a significant gender gap at time = 10 ($\beta_1 + 10 \beta_3$). If you want to test whether there is a gender gap on average over time, you can just run a regression without the time variable and investigate the coefficient of Female.

All this is of course predicated on what your actual question is. If you just want to say that women, get lower wages on average, this is fine. If you want to say something along the line of "Women get underpaid for the same work and because they are women", you will need a more complicated model with at least more control variables.