ECON 3412 - Introduction to Econometrics
Lecture 7 - Nonlinear Regression II: Interactions

(a) Interpretation of black coefficient:

The coefficient -0.198 represents the log salary difference between black and white players in a city with 0% black population. Since the interaction term is included, this coefficient alone doesn't tell the full story.

(b) Salary difference at 10% black population:

$\ln(salary)_{black} - \ln(salary)_{white} = -0.198 + 0.0125(10) = -0.198 + 0.125 = -0.073$

Black players earn approximately 7.3% less than white players in such cities.

(c) Equal salary point:

Set the difference to zero: $-0.198 + 0.0125 \times percblck = 0$

$percblck = 0.198 / 0.0125 = 15.84\%$

At 15.84% black population, there's no racial wage gap.

(d) Testing Hispanic wage difference:

This requires a joint test of $H_0: \beta_{hispan} = \beta_{hispan \times perchisp} = 0$

Individual t-tests may not be reliable due to multicollinearity between hispan and hispan×perchisp.

Need F-test for joint significance.

(a) Separate equations:

Males (female = 0):

$\ln(wage) = -2.27 + 0.626 \times educ + 0.026 \times exper$

Females (female = 1):

$\ln(wage) = (-2.27 - 0.060) + (0.626 - 0.140) \times educ + 0.026 \times exper$

$= -2.33 + 0.486 \times educ + 0.026 \times exper$

(b) Returns to education:

• Males: 62.6% per year of education
• Females: 48.6% per year of education
• Difference: 14.0 percentage points

(c) Test for equal education effects:

$H_0: \beta_4 = 0$ (female×educ coefficient = 0)

$t = -0.140 / 0.120 = -1.17$

$|t| = 1.17 < 1.96$ → Fail to reject at 5% level

We cannot conclude that education effects differ by gender.

(d) Wage penalty for female with 16 years education:

$\ln(wage)_{female} - \ln(wage)_{male} = -0.060 - 0.140(16) = -0.060 - 2.24 = -2.30$

This seems unreasonably large (230% lower), suggesting possible specification issues or the need to check the standard error for female (1.436 seems high).

(a) Number of regression lines:

With two binary variables, we have $2 \times 2 = 4$ groups:

• (HiEL = 0, LowInc = 0): High income, few English learners
• (HiEL = 1, LowInc = 0): High income, many English learners
• (HiEL = 0, LowInc = 1): Low income, few English learners
• (HiEL = 1, LowInc = 1): Low income, many English learners

(b) Marginal effects of STR:

(c) Economic meaning of $\beta_7 > 0$:

The triple interaction $\beta_7 > 0$ means that the combined effect of having both many English learners AND low income creates an additional impact beyond what we'd expect from just adding the two separate interaction effects.

This suggests synergy between the two disadvantages: schools facing both challenges benefit even more from smaller classes than the sum of the individual effects would suggest.

(d) Testing equal STR effects:

$H_0: \beta_4 = \beta_5 = \beta_7 = 0$

This is a joint F-test with 3 restrictions. If we fail to reject, then STR has the same effect ($\beta_1$) for all groups.

(a) Explaining the contradiction:

This is a classic case of multicollinearity. The variables bfriend and bfriend×skipped are highly correlated because:

• bfriend×skipped = 0 whenever bfriend = 0
• bfriend×skipped = skipped whenever bfriend = 1

This correlation inflates standard errors, making individual t-tests less powerful. However, the joint F-test can still detect that together these variables explain significant variation in GPA.

(b) Conclusion about boyfriend/girlfriend effect:

Yes, based on the significant F-test, we should conclude that having a boyfriend/girlfriend affects GPA, but the effect depends on class attendance behavior. We cannot rely on individual t-tests due to multicollinearity.

(c) Interpretation of negative interaction ($\beta_4 < 0$):

If $\beta_4 < 0$, it means:

• The negative effect of skipping class is worse for students with boyfriends/girlfriends
• Or equivalently: Having a boyfriend/girlfriend is more harmful for students who skip classes frequently
• This suggests that relationship distractions compound with poor attendance habits