斯托克计量经济学课后习题实证答案

P ART T WO Solutions to Empirical Exercises Chapter 3

Review of Statistics

Solutions to Empirical Exercises 1. (a)

Average Hourly Earnings, Nominal $’s

Mean SE(Mean) 95% Confidence Interval AHE199211.63 0.0 11.50 11.75

AHE200416.77 0.098 16.58 16.96

Difference SE(Difference) 95% Confidence Interval AHE2004 AHE1992 5.14 0.117 4.91 5.37

(b)

Average Hourly Earnings, Real $2004

Mean SE(Mean) 95% Confidence Interval AHE199215.66 0.086 15.49 15.82

AHE200416.77 0.098 16.58 16.96

Difference SE(Difference) 95% Confidence Interval AHE2004 AHE1992 1.11 0.130 0.85 1.37

(c) The results from part (b) adjust for changes in purchasing power. These results should be used.

(d)

Average Hourly Earnings in 2004

Mean SE(Mean) 95% Confidence Interval High School13.81 0.102 13.61 14.01

College20.31 0.158 20.00 20.62

Difference SE(Difference) 95% Confidence Interval College High School 6.50 0.188 6.13 6.87

Solutions to Empirical Exercises in Chapter 3 109 (e)

Average Hourly Earnings in 1992 (in $2004)

Mean SE(Mean) 95% Confidence Interval High School 13.48 0.091 13.30 13.65 College 19.07 0.148 18.78 19.36

Difference SE(Difference) 95% Confidence Interval College High School

5.59 0.173 5.25 5.93

(f) Average Hourly Earnings in 2004

Mean SE(Mean) 95% Confidence Interval AHE HS ,2004 AHE HS ,1992

0.33 0.137 0.06 0.60 AHE Col ,2004 AHE Col ,1992 1.24 0.217 0.82 1.66 Col–HS Gap (1992)

5.59 0.173 5.25 5.93 Col–HS Gap (2004) 6.50 0.188 6.13 6.87

Difference SE(Difference) 95% Confidence Interval Gap 2004 Gap 1992 0.91 0.256 0.41 1.41

Wages of high school graduates increased by an estimated 0.33 dollars per hour (with a 95%

confidence interval of 0.06 0.60); Wages of college graduates increased by an estimated 1.24

dollars per hour (with a 95% confidence interval of 0.82 1.66). The College High School gap increased by an estimated 0.91 dollars per hour.

(g) Gender Gap in Earnings for High School Graduates Year m Y s m n m w Y s w n w m Y w Y SE (m Y w Y )95% CI 1992 14.57 6.55 2770 11.86 5.21 1870 2.71 0.173 2.37 3.05 2004

14.88 7.16 2772 11.92 5.39 1574 2.96 0.192 2.59 3.34

There is a large and statistically significant gender gap in earnings for high school graduates.

In 2004 the estimated gap was $2.96 per hour; in 1992 the estimated gap was $2.71 per hour

(in $2004). The increase in the gender gap is somewhat smaller for high school graduates than

it is for college graduates. Chapter 4

Linear Regression with One Regressor Solutions to Empirical Exercises 1. (a) ·AHE 3.32 0.45 u Age

Earnings increase, on average, by 0.45 dollars per hour when workers age by 1 year.

(b) Bob’s predicted earnings 3.32 0.45 u 26 $11.70 Alexis’s predicted earnings 3.32 0.45 u 30 $13.70

(c) The R2 is 0.02.This mean that age explains a small fraction of the variability in earnings across

individuals. 2. (a)

There appears to be a weak positive relationship between course evaluation and the beauty index.

Course Eval 4.00 0.133 u Beauty. The variable Beauty has a mean that is equal to 0; the

(b) ·_

estimated intercept is the mean of the dependent variable (Course_Eval) minus the estimated

slope (0.133) times the mean of the regressor (Beauty). Thus, the estimated intercept is equal

to the mean of Course_Eval.

(c) The standard deviation of Beauty is 0.7. Thus

Professor Watson’s predicted course evaluations 4.00 0.133 u 0 u 0.7 4.00

Professor Stock’s predicted course evaluations 4.00 0.133 u 1 u 0.7 4.105

Solutions to Empirical Exercises in Chapter 4 111

(d) The standard deviation of course evaluations is 0.55 and the standard deviation of beauty is

0.7. A one standard deviation increase in beauty is expected to increase course evaluation by

0.133 u 0.7 0.105, or 1/5 of a standard deviation of course evaluations. The effect is small.

(e) The regression R2 is 0.036, so that Beauty explains only 3.6% of the variance in course

evaluations.

3. (a) ?Ed 13.96 0.073 u Dist. The regression predicts that if colleges are built 10 miles closer

to where students go to high school, average years of college will increase by 0.073 years.

(b) Bob’s predicted years of completed education 13.96

0.073 u 2 13.81

Bob’s predicted years of completed education if he was 10 miles from college 13.96 0.073 u

1 13.

(c) The regression R2 is 0.0074, so that distance explains only a very small fraction of years of

completed education. (d) SER 1.8074 years. 4. (a)

Yes, there appears to be a weak positive relationship. of 2.

(c) ·Growth 0. 2.31 u Tradeshare Predicted growth 0. 2.31 u 1 2.95 (d) ·Growth 0.96 1.68 u Tradeshare Predicted growth 0.96 1.68 u 1 2.74

(b) Malta is the “outlying” observation with a trade share

(e) Malta is an island nation in the Mediterranean Sea, south of Sicily. Malta is a freight transport

site, which explains its large “trade share”. Many goods coming into Malta (imports into Malta)

and immediately transported to other countries (as exports from Malta). Thus, Malta’s imports

and exports and unlike the imports and exports of most other countries. Malta should not be

included in the analysis. Chapter 5

Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals Solutions to Empirical Exercises 1. (a) ·AHE 3.32 0.45 u Age (0.97) (0.03)

The t -statistic is 0.45/0.03 13.71, which has a p -value of 0.000, so the null hypothesis can be

rejected at the 1% level (and thus, also at the 10% and 5% levels).

(b) 0.45 r 1.96 u 0.03 0.387 to 0.517 (c) ·AHE 6.20 0.26 u Age (1.02) (0.03)

The t -statistic is 0.26/0.03 7.43, which has a p -value of 0.000, so the null hypothesis can be

rejected at the 1% level (and thus, also at the 10% and 5% levels).

(d) ·AHE 0.23 0.69 u Age (1.) (0.05)

The t -statistic is 0.69/0.05 13.06, which has a p -value of 0.000, so the null hypothesis can be

rejected at the 1% level (and thus, also at the 10% and 5% levels).

(e) The difference in the estimated E 1 coefficients is 1,1,??College HighScool E E 0.69 0.26 0.43. The

standard error of for the estimated difference is SE 1,1,??()College HighScool

E E (0.032 0.052)1/2 0.06, so that a 95% confidence interval for the difference is 0.43 r 1.96 u 0.06 0.32 to 0.

(dollars per hour). 2. ·_ 4.000.13Course Eval Beauty u (0.03) (0.03)

The t -statistic is 0.13/0.03 4.12, which has a p -value of 0.000, so the null hypothesis can be rejected

at the 1% level (and thus, also at the 10% and 5% levels). 3. (a) ?Ed

13.96 0.073 u Dist (0.04) (0.013)

The t -statistic is 0.073/0.013 5.46, which has a p -value of 0.000, so the null hypothesis can be rejected at the 1% level (and thus, also at the 10% and 5% levels).

(b) The 95% confidence interval is 0.073 r 1.96 u 0.013 or 0.100 to 0.047.

(c) ?Ed

13.94 0.0 u Dist (0.05) (0.018)

Solutions to Empirical Exercises in Chapter 5 113 (d) ?Ed

13.98 0.084 u Dist (0.06) (0.013)

(e) The difference in the estimated E 1 coefficients is 1,1,??Female Male E E 0.0 ( 0.084) 0.020.

The standard error of for the estimated difference is SE 1,1,??()Female Male E E (0.0182 0.0132)1/2

0.022, so that a 95% confidence interval for the difference is 0.020 r 1.96 u 0.022 or 0.022 to

0.0. The difference is not statistically different. Chapter 6

Linear Regression with Multiple Regressors Solutions to Empirical Exercises 1. Regressions used in (a) and (b) Regressor a b Beauty 0.133 0.166 Intro 0.011 OneCredit 0.634 Female 0.173 Minority 0.167 NNEnglish 0.244 Intercept 4.00 4.07 SER 0.5 0.513 R2 0.036 0.155

(a) The estimated slope is 0.133

(b) The estimated slope is 0.166. The coefficient does not change by an large amount. Thus, there

does not appear to be large omitted variable bias.

(c) Professor Smith’s predicted course evaluation (0.166 u 0) 0.011 u 0) (0.634 u 0) (0.173 u

0) (0.167 u 1) (0.244 u 0) 4.068 3.901 2. Estimated regressions used in question Model Regressor a b dist 0.073 0.032 bytest 0.093 female 0.145 black 0.367 hispanic 0.398

incomehi 0.395 ownhome 0.152 dadcoll 0.696 cue80 0.023 stwmfg80 0.051 intercept 13.956 8.827 SER 1.81 1.84 R2 0.007 0.279 R0.007 0.277

Solutions to Empirical Exercises in Chapter 6 115 (a) 0.073 (b) 0.032

(c) The coefficient has fallen by more than 50%. Thus, it seems that result in (a) did suffer from

omitted variable bias.

(d) The regression in (b) fits the data much better as indicated by the R2, 2,

R and SER. The R2 and R are similar because the number of observations is large (n 3796).

(e) Students with a “dadcoll 1” (so that the student’s father went to college) complete 0.696 more

years of education, on average, than students with “dadcoll 0” (so that the student’s father did

not go to college).

(f) These terms capture the opportunity cost of attending college. As STWMFG increases, forgone

wages increase, so that, on average, college attendance declines. The negative sign on the

coefficient is consistent with this. As CUE80 increases, it is more difficult to find a job, which

lowers the opportunity cost of attending college, so that college attendance increases. The

positive sign on the coefficient is consistent with this. (g) Bob’s predicted years of education 0.0315 u 2 0.093 u 58 0.145 u 0 0.367 u 1 0.398 u

0 0.395 u 1 0.152 u 1 0.696 u 0 0.023 u 7.5 0.051 u 9.75 8.827 14.75

(h) Jim’s expected years of education is 2 u 0.0315 0.0630 less than Bob’s. Thus, Jim’s expected

years of education is 14.75 0.063 14.69. 3.

Variable Mean Standard Deviation Units

growth 1.86 1.82 Percentage Points rgdp60 3131 2523 $1960 tradeshare 0.2 0.229 unit free yearsschool 3.95 2.55 years

rev_coups 0.170 0.225 coups per year

assasinations 0.281 0.494 assasinations per year

oil 0 0 0–1 indicator variable (b) Estimated Regression (in table format):

Regressor Coefficient tradeshare 1.34 (0.88)

yearsschool 0.56** (0.13)

rev_coups 2.15* (0.87)

assasinations 0.32 (0.38)

rgdp60 0.00046** (0.00012) intercept 0.626 (0.869) SER 1.59 R2 0.29 R0.23

116 Stock/Watson - Introduction to Econometrics - Second Edition

The coefficient on Rev_Coups is í2.15. An additional coup in a five year period, reduces the

average year growth rate by (2.15/5) = 0.43% over this 25 year period. This means the GPD in 1995 is expected to be approximately .43×25 = 10.75% lower. This is a large effect.

(c) The 95% confidence interval is 1.34 r 1.96 u 0.88 or 0.42 to 3.10. The coefficient is not

statistically significant at the 5% level.

(d) The F-statistic is 8.18 which is larger than 1% critical value of 3.32.

Chapter 7

Hypothesis Tests and Confidence Intervals in Multiple Regression

Solutions to Empirical Exercises 1. Estimated Regressions Model Regressor a b Age 0.45 (0.03) 0.44 (0.03) Female 3.17

(0.18) Bachelor 6.87 (0.19) Intercept 3.32 (0.97) SER 8.66 7.88 R20.023 0.190 2

R0.022 0.190

(a) The estimated slope is 0.45

(b) The estimated marginal effect of Age on AHE is 0.44 dollars per year. The 95% confidence

interval is 0.44 r 1.96 u 0.03 or 0.38 to 0.50.

(c) The results are quite similar. Evidently the regression in (a) does not suffer from important

omitted variable bias.

(d) Bob’s predicted average hourly earnings 0.44 u 26 3.17 u 0 6.87 u 0 3.32 $11.44

Alexis’s predicted average hourly earnings 0.44 u 30 3.17 u 1 6.87 u 1 3.32 $20.22 (e) The regression in (b) fits the data much better. Gender and education are important predictors of

earnings. The R2 and R are similar because the sample size is large (n 7986).

(f) Gender and education are important. The F-statistic is 752, which is (much) larger than the 1%

critical value of 4.61.

(g) The omitted variables must have non-zero coefficients and must correlated with the included

regressor. From (f) Female and Bachelor have non-zero coefficients; yet there does not seem to be important omitted

variable bias, suggesting that the correlation of Age and Female and Age and Bachelor is small. (The sample correlations are ·Cor(Age, Female) 0.03 and

·Cor(Age,Bachelor) 0.00).

118 Stock/Watson - Introduction to Econometrics - Second Edition

2. Model

Regressor a b c Beauty 0.13** (0.03) 0.17** (0.03) 0.17 (0.03) Intro 0.01 (0.06)

OneCredit 0.63** (0.11) 0.** (0.10) Female 0.17** (0.05) 0.17** (0.05) Minority 0.17** (0.07) 0.16** (0.07) NNEnglish 0.24** (0.09) 0.25** (0.09) Intercept 4.00** (0.03) 4.07** (0.04) 4.07** (0.04)

SER 0.5 0.513 0.513

R2 0.036 0.155 0.155 2

R0.034 0.144 0.145

(a) 0.13 r 0.03 u 1.96 or 0.07 to 0.20

(b) See the table above. Intro is not significant in (b), but the other variables are significant.

A reasonable 95% confidence interval is 0.17 r 1.96 u 0.03 or 0.11 to 0.23.

Solutions to Empirical Exercises in Chapter 7 119 3. Model

Regressor (a) (b) (c) dist 0.073** (0.013) 0.031** (0.012) 0.033** (0.013) bytest 0.092** (0.003) 0.093** (.003) female 0.143** (0.050) 0.144** (0.050) black 0.3**

(0.067) 0.338** (0.069) hispanic 0.402** (0.074) 0.349** (0.077) incomehi 0.367** (0.062) 0.374** (0.062) ownhome 0.146* (0.065) 0.143* (0.065) dadcoll 0.570** (0.076) 0.574** (0.076)

momcoll 0.379** (0.084) 0.379** (0.084) cue80 0.024** (0.009) 0.028** (0.010) stwmfg80 0.050* (0.020) 0.043* (0.020) urban 0.0652 (0.063) tuition 0.184 (0.099)

intercept 13.956** (0.038) 8.861** (0.241) 8.3** (0.243) F-statitistic

for urban and tuition SER 1.81 1. 1. R2 0.007 0.282 0.284 R0.007 0.281 0.281

(a) The group’s claim is that the coefficient on Dist is 0.075 ( 0.15/2). The 95% confidence for

E Dist from column (a) is 0.073 r 1.96 u 0.013 or 0.099 to 0.046. The group’s claim is included

in the 95% confidence interval so that it is consistent with the estimated regression.

120 Stock/Watson - Introduction to Econometrics - Second Edition

(b) Column (b) shows the base specification controlling for other important factors. Here the

coefficient on Dist is 0.031, much different than the results

from the simple regression in (a);

when additional variables are added (column (c)), the coefficient on Dist changes little from the result in (b). From the base specification (b), the 95% confidence interval for E Dist is 0.031 r

1.96 u 0.012 or 0.055 to 0.008. Similar results are obtained from the regression in (c).

(c) Yes, the estimated coefficients E Black and E Hispanic are positive, large, and statistically significant.

Chapter 8

Nonlinear Regression Functions Solutions to Empirical Exercises

1. This table contains the results from seven regressions that are referenced in these answers.

Data from 2004

(1) (2) (3) (4) (5) (6) (7) (8) Dependent Variable

AHE ln(AHE) ln(AHE) ln(AHE) ln(AHE) ln(AHE) ln(AHE) ln(AHE) Age 0.439** (0.030) 0.024** (0.002) 0.147** (0.042) 0.146** (0.042) 0.190** (0.056) 0.117* (0.056) 0.160

(0.0) Age2 0.0021** (0.0007) 0.0021** (0.0007) 0.0027** (0.0009) 0.0017 (0.0009) 0.0023 (0.0011) ln(Age) 0.725** (0.052)

Female u Age 0.097 (0.084) 0.123 (0.084) Female u Age2 0.0015 (0.0014) 0.0019 (0.0014) Bachelor u Age 0.0 (0.083) 0.091 (0.084)

Bachelor u Age2 0.0009 (0.0014) 0.0013 (0.0014) Female 3.158* *

(0.176) 0.180** (0.010) 0.180** (0.010) 0.180** (0.010)

0.210** (0.014) 1.358* (1.230) 0.210** (0.014) 1.7 (1.239)

Bachelor 6.865** (0.185) 0.405** (0.010) 0.405** (0.010) 0.405** (0.010) 0.378** (0.014) 0.378** (0.014) 0.769 (1.228) 1.186 (1.239)

Female u Bachelor 0.0** (0.021) 0.063** (0.021) 0.066** (0.021) 0.066** (0.021)

Intercept 1.884 (0.7) 1.856** (0.053) 0.128 (0.177) 0.059 (0.613) 0.078 (0.612) 0.633 (0.819) 0.604 (0.819) 0.095 (0.945)

F-statistic and p-values on joint hypotheses (a) F-statistic on terms involving Age 98. (0.00) 100.30 (0.00) 51.42 (0.00) 53.04 (0.00) 36.72 (0.00)

(b) Interaction terms with Age2

4.12 (0.02) 7.15 (0.00) 6.43 (0.00)

SER 7.884 0.457 0.457 0.457 0.457 0.456 0.456 0.456 R0.17 0.1921 0.1924 0.1929 0.1937 0.1943 0.1950 0.1959 Significant at the *5% and **1% significance level.

122 Stock/Watson - Introduction to Econometrics - Second Edition

(a) The regression results for this question are shown in column (1) of the table. If Age increases

from 25 to 26, earnings are predicted to increase by $0.439 per hour. If Age increases from

33 to 34, earnings are predicted to increase by $0.439 per hour. These values are the same

because the regression is a linear function relating AHE and Age .

(b) The regression results for this question are shown in column (2) of the table. If Age increases

from 25 to 26, ln(AHE ) is predicted to increase by 0.024. This means that earnings are predicted to increase by 2.4%. If Age increases from 34 to 35, ln(AHE ) is predicted to increase by 0.024.

This means that earnings are predicted to increase by 2.4%. These values, in percentage terms,

are the same because the regression is a linear function relating ln(AHE ) and Age .

(c) The regression results for this question are shown in column (3) of the table. If Age increases

from 25 to 26, then ln(Age ) has increased by ln(26) ln(25) 0.0392 (or 3.92%). The predicted

increase in ln(AHE ) is 0.725 u (.0392) 0.0284. This means that earnings are predicted to

increase by 2.8%. If Age increases from 34 to 35, then ln(Age ) has increased by ln(35) ln(34) .0290 (or 2.90%). The predicted increase in ln(AHE ) is 0.725 u (0.0290) 0.0210. This means

that earnings are predicted to increase by 2.10%.

(d) When Age increases from 25 to 26, the predicted change in ln(AHE ) is

(0.147 u 26 0.0021 u 262) (0.147 u 25 0.0021 u 252) 0.0399. This means that earnings are predicted to increase by 3.99%. When Age increases from 34 to 35, the predicted change in ln(AHE ) is

(0. 147 u 35 0.0021 u 352) (0. 147 u 34 0.0021 u 342) 0.0063. This means that earnings are predicted to increase by 0.63%. (e) The regressions differ in their choice of one of the regressors. They can be compared on the basis of the .R The regression in (3) has a (marginally) higher 2,R so it is preferred.

(f) The regression in (4) adds the variable Age 2 to regression (2). The coefficient on Age 2 is

statistically significant ( t 2.91), and this suggests that the addition of Age 2 is important. Thus,

(4) is preferred to (2).

(g) The regressions differ in their choice of one of the regressors. They can be compared on the basis of the .R The regression in (4) has a (marginally) higher 2,R so it is preferred.

(h)

Solutions to Empirical Exercises in Chapter 8 123 The regression functions using Age (2) and ln(Age) (3) are similar. The quadratic regression (4) is different. It shows a decreasing effect of Age on ln(AHE) as workers age.

The regression functions for a female with a high school diploma will look just like these, but they will be shifted by the amount of the coefficient on the binary regressor Female. The

regression functions for workers with a bachelor’s degree will also look just like these, but they would be shifted by the amount of the coefficient on the binary variable Bachelor.

(i) This regression is shown in column (5). The coefficient on the interaction term Female u

Bachelor shows the “extra effect” of Bachelor on ln(AHE) for women relative the effect for men.

Predicted values of ln(AHE):

Alexis: 0.146 u 30 0.0021 u 302 0.180 u 1 0.405 u 1 0.0 u 1 0.078 4.504

Jane: 0.146 u 30 0.0021 u 302 0.180 u 1 0.405 u 0 0.0 u 0 0.078 4.063

Bob: 0.146 u 30 0.0021 u 302 0.180 u 0 0.405 u 1 0.0 u 0 0.078 4.651

Jim: 0.146 u 30 0.0021 u 302 0.180 u 0 0.405 u 0 0.0 u 0 0.078 4.273

Difference in ln(AHE): Alexis Jane 4.504 4.063 0.441 Difference in ln(AHE): Bob Jim 4.651 4.273 0.378

Notice that the difference in the difference predicted effects is 0.441 0.378 0.063, which is the value of the coefficient on the interaction term.

(j) This regression is shown in (6), which includes two additional regressors: the interactions of Female and the age variables, Age and Age2. The F-statistic testing the restriction that the coefficients on these interaction terms is equal to zero is F 4.12 with a p-value of 0.02. This implies that there is statistically significant evidence (at the 5% level) that there is a different effect of Age on ln(AHE) for men and women.

(k) This regression is shown in (7), which includes two additional regressors that are interactions of Bachelor and the age variables, Age and Age2. The F-statistic testing the restriction that the coefficients on these interaction terms is zero is 7.15 with a p-value of 0.00. This implies that there is statistically significant evidence (at the 1% level) that there is a different effect of Age on ln(AHE) for high school and college graduates.

(l) Regression (8) includes Age and Age2 and interactions terms involving Female and Bachelor.

The figure below shows the regressions predicted value of ln(AHE) for male and females with high school and college degrees.

124 Stock/Watson - Introduction to Econometrics - Second Edition

The estimated regressions suggest that earnings increase as workers age from 25–35, the range

of age studied in this sample. There is evidence that the quadratic term Age2 belongs in the

regression. Curvature in the regression functions in particularly important for men.

Gender and education are significant predictors of earnings, and there are statistically significant interaction effects between age and gender and age and education. The table below summarizes the regressions predictions for increases in earnings as a person ages from 25 to 32 and 32 to 35

Gender, Education Predicted ln(AHE) at Age(Percent per year) 25 32 35 25 to 32 32 to 35

Males, High School 2.46 2.65 2.67 2.8% 0.5% Females, BA 2.68 2. 2.93 3.0% 1.3% Males, BA 2.74 3.06 3.09 4.6% 1.0%

Earnings for those with a college education are higher than those with a high school degree, and

earnings of the college educated increase more rapidly early in their careers (age 25–32). Earnings for men are higher than those of women, and earnings of men increase more rapidly early in their

careers (age 25–32). For all categories of workers (men/women, high school/college) earnings

increase more rapidly from age 25–32 than from 32–35.