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ECMT6007/6702: Econometric Applications Problem Set 8
发布时间:2022-10-28
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ECMT6007/6702: Econometric Applications
Problem Set 8 Solutions
Semester 2 2022
Question 1. Logit Model Calculations
(i) 10 hours per week:
P (grad = 1 | hsAT—AR= 0.75, study = 10) = Λ (−1.17 + 1.9 × 0.75 + 0.073 × 10)
= Λ (0.985)
exp (0.985)
=
1 + exp (0.985)
= 0.7281
5 hours per week:
P (grad = 1 | hsAT—AR= 0.75, study = 5) = Λ (−1.17 + 1.9 × 0.75 + 0.073 × 5)
= Λ (0.620)
exp (0.620)
=
1 + exp (0.620)
= 0.6502
Difference in graduation probabilities: 0.7281 - 0.6502 = 0.0779
Notice how wrong the calculation would be if we simply took the coefficient on study, (i.e. 0.073), and multiplied it by (10 − 5) = 5 as we would for the linear regression model.
(ii) 5 hours per week:
P (grad = 1 | hsAT—AR= 0.75, study = 5) = 0.6502
0 hours per week:
P (grad = 1 | hsT—ER= 0.75, study = 0) = Λ (−1.17 + 1.9 × 0.75 + 0.073 × 0)
= Λ (0.255)
exp (0.255)
=
1 + exp (0.255)
= 0.5634
Difference in graduation probabilities: 0.6502 - 0.5634 = 0.0868
Question 2. Computer Exercise: Women in the labour force
(i) Dependent variable: P (infl = 1)
Table 1: Table of Estimation Results
Variable |
LPM |
Probit (UR) |
Probit (R) βˆ Std. Error |
|||||
βˆ |
Std. Error |
βˆ Std. Error Margin Std. Error |
||||||
nwifeinc |
−0.0034 |
(0.0014) |
−0.0120 |
(0.0048) |
−0.0036 |
(0.0014) |
−0.0113 |
(0.0046) |
educ |
0.0380 |
(0.0074) |
0.1309 |
(0.0253) |
0.0394 |
(0.0072) |
0.1051 |
(0.0240) |
exper |
0.0395 |
(0.0057) |
0.1233 |
(0.0187) |
0.0371 |
(0.0052) |
0.1253 |
(0.0182) |
expersq |
−0.0006 |
(0.0002) |
−0.0019 |
(0.0006) |
−0.0006 |
(0.0002) |
−0.0020 |
(0.0005) |
age |
−0.0161 |
(0.0025) |
−0.0529 |
(0.0085) |
−0.0159 |
(0.0024) |
−0.0287 |
(0.0068) |
kidslt6 |
−0.2618 |
(0.0335) |
−0.8683 |
(0.1185) |
−0.2612 |
(0.0319) |
– |
– |
kidsge6 |
0.0130 |
(0.0132) |
0.0360 |
(0.0435) |
0.0108 |
(0.0131) |
– |
– |
cons |
0.5856 |
(0.1542) |
0.2701 |
(0.5086) |
– |
– |
−0.6190 |
(0.4165) |
R2 / |
0.2642 |
|
0.2206 |
|
|
|
0.1594 |
|
llf |
|
|
−401.302 |
|
|
|
−432.809 |
|
n |
753 |
|
753 |
|
|
|
753 |
|
(ii) β2 is the change in the expected probability a woman will participate in the labour market due to one extra year of education, holding other factor constant. (This is a marginal effect which is constant for all values of educ).
A t-test demonstrates that this coefficient is statistically significant (two-tailed test p-value is < 0.001).
Is this effect practically large? This depends on the magnitude of βˆ2 which is 0.038. An extra year of education, cet. par., increased expected probability of labour force participation by almost 4 percentage-points. Sample mean infl is 56.9 percentage-points, a 4-year degree implies a difference of 16 percentage-points. Education does have a substantial effect on the likelihood a woman will work.
(iii) See Table 1
(iv) In the probit model, β2 is the change in the ‘net benefit’ or ‘utility’ of being in the labour force due to an extra year of education, other things equal. The scale of the underlying (latent) index is not very meaningful, and the magnitude of the coefficient is not comparable to the
corresponding estimate for the LPM. Note however we could use Amemiya’s rule of thumb, that βˆLPM ≈ 0.4 × βˆProbit and take 0.4 × 0.1309 = 0.0524 which is a rough approximation to the (maximum) marginal effect.
(v) Average value:
工P (inl
1 | xi) = 0.5701
Note: the sample average value of inlfi, is given by inlfi = 0.5684.
(vi) See Table 1
(vii) The marginal effect of education is a calculation of the change in the predicted probability a woman will be in the labour due to one more year of education, other factors held constant. That is, the marginal effect measures:
∂P (in—lfi= 1)
∂educ
Note: there are a number of ways to do this. STATA does this by calculating this marginal effect for each observation in the sample, and then taking the average. In the textbook this is called the Average Partial Effect (APE). Another way to do it is to take the average value of all the x variables and do the calculation. In the textbook this is called the Partial Average
Effect (PAE). The way the current version of STATA does the marginal effect calculation is superior (i.e. using the APE).
From the output, Eeduc = 0.03937. This marginal effect is comparable to the coefficient on education in the Linear Probability Model in part (i).
Test:
H0 : MEeduc = 0
H1 : MEeduc 0
Test Statistic:
图
E 图
图se (
E) 图
图0.03937 图
= 图0.00722 图
= 5.45
Rejection Rule: Reject H0 in favour of H1 if t > c, where c is the critical value from the
standard normal distribution. Now t = 5.45 and c = 2.576.
Decision: Since t > c we reject the null at the 1% significance level.
Conclusion: The marginal effect of education on the likelihood a woman will be in the labour force is statistically significant.
(viii) Likelihood Ratio Test Test:
H0 : β6 = 0, β7 = 0
H1 : H0 is false
Test Statistic:
LR = 2 (Lur − Lr) ∼ χq(2)
= 2 (−401.30219 − −432.80875)
= 2 (31.5066)
= 63.0131
Rejection Rule: Reject H0 in favour of H1 if LR > c, where c is the critical value for the χ2 distribution with q = 2 degrees of freedom and a 1% significance level. Now LR = 63.0131 and c = 9.21.
Decision: Since LR > c we reject the null at the 1% significance level.
Conclusion: The number of infant and older dependent children have a statistically signifi- cant impact on the likelihood of a woman’s labour force participation, holding other income, education, experience and age constant.