ECON S3412 - Introduction to Econometrics
Econometrics Lecture 6 - Nonlinear Regression I

(a) Finding X* (maximum):

First-order condition:

$\frac{\partial Y}{\partial X} = \beta_1 + 2\beta_2 X = 0$

$5 + 2(-0.5)X = 0$

$5 - X = 0$

$X^* = 5$

(b) Marginal effect at X = 3:

$\frac{\partial Y}{\partial X}\bigg|_{X=3} = 5 + 2(-0.5)(3) = 5 - 3 = 2$

해석: X = 3에서 X가 1 단위 증가하면 Y는 약 2 단위 증가

(c) Effect from X = 2 to X = 4 (After-Before):

$Y(2) = 10 + 5(2) + (-0.5)(4) = 10 + 10 - 2 = 18$

$Y(4) = 10 + 5(4) + (-0.5)(16) = 10 + 20 - 8 = 22$

$\Delta Y = 22 - 18 = 4$

Note: Average marginal effect = 4/2 = 2 (X=3일 때와 일치!)

(d) Shape:

$\beta_2 = -0.5 < 0$ → ∩-shape (inverted U)

Second derivative: $\frac{\partial^2 Y}{\partial X^2} = 2\beta_2 = -1 < 0$ → Concave

(e) Predicted value at X = 5:

$\hat{Y}(5) = 10 + 5(5) + (-0.5)(25) = 10 + 25 - 12.5 = 22.5$

Note: X = 5는 최대점이므로 이것이 Y의 최댓값입니다!

(a) Interpretation:

• Education: 1 year increase → 8% wage increase (log-linear model)
• Experience: 1 year increase → 5% wage increase

(b) Education from 12 to 16:

Approximate (작은 변화 근사):

$\Delta \ln(Wage) = 0.08 \times 4 = 0.32$ → 약 32% 증가

Exact (정확한 계산):

$\frac{Wage_{new}}{Wage_{old}} = e^{0.32} = 1.377$

정확한 증가율 = $(1.377 - 1) \times 100\% = 37.7\%$

큰 변화에서는 근사와 정확한 값의 차이가 큽니다!

(c) Combined effect:

$\Delta \ln(Wage) = 0.08(4) + 0.05(5) = 0.32 + 0.25 = 0.57$

Approximate: 57% increase

Exact: $e^{0.57} - 1 = 0.768$ → 76.8% increase

(d) SER = 0.15 meaning:

• Standard deviation of residuals in log scale

• Typical prediction error ≈ 15% of wage

• About 68% of observations within ±15% of predicted wage

(e) 95% CI for education coefficient:

CI = $0.08 \pm 1.96(0.01) = 0.08 \pm 0.0196 = [0.0604, 0.0996]$

해석: 95% 신뢰수준에서 교육 1년의 효과는 6.04%에서 9.96% 사이

(a) When PctEL = 0:

$\widehat{TestScore} = 686.3 - 1.12 \times STR$

(b) When PctEL = 20:

Substitute PctEL = 20:

$\widehat{TestScore} = 686.3 - 1.12 \times STR - 0.67(20) + 0.012 \times STR \times 20$

$= 686.3 - 13.4 + (-1.12 + 0.24) \times STR$

$= 672.9 - 0.88 \times STR$

(c) No effect point:

Set marginal effect = 0:

$\frac{\partial TestScore}{\partial STR} = -1.12 + 0.012 \times PctEL = 0$

$PctEL^* = \frac{1.12}{0.012} = 93.3$

의미: PctEL이 93.3%일 때 STR은 TestScore에 영향 없음 (비현실적!)

(d) Hypothesis test:

$H_0: \beta_3 = 0$ (STR 효과가 PctEL에 무관)

$H_1: \beta_3 \neq 0$

$t = \frac{0.012}{0.019} = 0.63$

$|t| = 0.63 < 1.96$ → Fail to reject $H_0$ at 5% level

(e) Economic interpretation:

• Positive interaction coefficient ($\beta_3 > 0$)

• STR의 negative effect가 PctEL이 높을수록 작아짐

• 하지만 통계적으로 유의하지 않음

(f) Effect when PctEL = 15:

Marginal effect = $-1.12 + 0.012(15) = -1.12 + 0.18 = -0.94$

STR을 2 감소시키면: $\Delta TestScore = -(-0.94)(2) = 1.88$ points increase

(a) Test A vs B (Linear vs Quadratic):

$H_0: \beta_2 = 0$ (Model A is adequate)

$H_1: \beta_2 \neq 0$ (Model B is better)

$t = \frac{-0.03}{0.01} = -3.0$

$|t| = 3.0 > 2.58$ (1% critical value) → Reject $H_0$

결론: Quadratic이 Linear보다 유의하게 좋음

(b) Testing B vs C:

불가능합니다! 이유:

• Non-nested models (서로 포함관계가 아님)
• Model B: polynomial in X
• Model C: logarithm of X
• t-test는 nested models에서만 가능

대안: AIC, BIC, cross-validation 등 사용

(c) Model choice:

Model C (Linear-log)를 선택합니다.

이유:

• 가장 높은 $R^2$ (0.53)
• 가장 낮은 RMSE (14.0)
• Diminishing returns 해석 가능
• Extrapolation에서 더 안정적
• 경제학적 해석이 명확

(d) Extrapolation concerns:

데이터 범위: X ∈ [10, 50]

• Model A (Linear):
  • X = 100에서도 같은 slope 가정
  • 매우 비현실적일 가능성
• Model B (Quadratic):
  • X > 50에서 급격히 감소할 수 있음
  • 특히 위험! (polynomial의 특성)
• Model C (Log):
  • 증가율이 점진적으로 감소
  • 가장 안전한 extrapolation

(a) Interpretation of black coefficient:

In a city with 0% black population, black players earn 19.8% less than white players.

This is the baseline racial gap when there's no same-race population support.

(b) Salary difference when percblack = 10:

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

$= -0.198 + 0.125 = -0.073$

Black players earn about 7.3% less than white players

(c) Equal salary point:

Set difference = 0:

$-0.198 + 0.0125 \times percblack^* = 0$

$percblack^* = \frac{0.198}{0.0125} = 15.84$

At 15.84% black population, salaries are equal

(d) Testing race effects:

Individual t-tests may not be significant due to multicollinearity

Must use joint F-test:

$H_0: \beta_{black} = \beta_{hispanic} = \beta_{black \times percblack} = \beta_{hispanic \times perchisp} = 0$

If F-stat is large → race matters for salary

(e) Adding performance variables:

Race coefficients would likely decrease (move toward zero)

Why? Omitted variable bias:

• Performance affects salary (positive)
• If minorities have lower average performance → positive OVB
• Adding performance controls reduces this bias
• Remaining coefficient = "pure" discrimination effect