1、PART TWO 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 0.64 11.5011.75 AHE200479816.5816.96 Difference SE(Difference) 95% Confidence Interval AHE2004 AHE19
2、92 5.14 0.117 4.915.37 (b) Average Hourly Earnings, Real $2004 Mean SE(Mean) 95% Confidence Interval AHE199215.6 0.86 15.4915.82 AHE20047 916.5816.96 Difference SE(Difference) 95% Confidence Interval AHE2004 AHE1992 1.11 0.130 0.851.37 (c) The results from part (b) adjust for changes in purchasing p
3、ower. These results should be used. (d) Average Hourly Earnings in 2004 Mean SE(Mean) 95% Confidence Interval High School 13.81 0.102 13.6114.01 College 20.31 0.158 20.0020.62 Difference SE(Difference) 95% Confidence Interval CollegeHigh School6.50 0.188 6.136.87 Solutions to Empirical Exercises in
4、Chapter 3 109 (e) Average Hourly Earnings in 1992 (in $2004) Mean SE(Mean) 95% Confidence IntervalHigh School 13.48 0.091 13.3013.65 College 19.07 0.148 18.7819.36 Difference SE(Difference) 95% Confidence Interval CollegeHigh School5.59 0.173 5.255.93 (f) Average Hourly Earnings in 2004 Mean SE(Mean
5、) 95% Confidence IntervalAHEHS,2004 AHEHS,19920.33 0.137 0.060.60 AHECol,2004 AHECol,19921.24 0.217 0.821.66 ColHS Gap (1992) 5.59 0.173 5.255.93 ColHS Gap (2004) 6.50 0.188 6.136.87 Difference SE(Difference) 95% Confidence Interval Gap2004 Gap1992 0.91 0.256 0.411.41 Wages of high school graduates
6、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) Ge
7、nder Gap in Earnings for High School Graduates Year mY sm nm wY sw nw mY wY SE(mY wY )95% CI 1992 14.57 6.55 2770 11.86 5.21 1870 2.71 0.173 2.373.05 2004 14.88 7.16 2772 11.92 5.39 1574 2.96 0.192 2.593.34 There is a large and statistically significant gender gap in earnings for high school graduat
8、es. 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.
9、 (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) Bobs predicted earnings 3.32 0.45 u 26 $11.70 Alexiss predicted earnings 3.32 0.45 u 30 $13.70 (c) The R2is 0.02.This mean that age explains a small fraction of the variability in earning
10、s across individuals. 2. (a) CourseEvaluationBeauty Index-2 -1 0 1 22345There appears to be a weak positive relationship between course evaluation and the beauty index. (b) _Course Eval 4.00 0.133 u Beauty. The variable Beauty has a mean that is equal to 0; the estimated intercept is the mean of the
11、 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 0.789. Thus Professor Watsons predicted course evaluations 4.00 0.133 u 0 u 0.789 4
12、.00 Professor Stocks predicted course evaluations 4.00 0.133 u 1 u 0.789 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 0.789. A one standard deviation increase in beauty is expected to increase
13、 course evaluation by 0.133 u 0.789 0.105, or 1/5 of a standard deviation of course evaluations. The effect is small. (e) The regression R2is 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 bui
14、lt 10 miles closer to where students go to high school, average years of college will increase by 0.073 years. (b) Bobs predicted years of completed education 13.96 0.073 u 2 13.81 Bobs predicted years of completed education if he was 10 miles from college 13.96 0.073 u 1 13.89 (c) The regression R2
15、is 0.0074, so that distance explains only a very small fraction of years of completed education. (d) SER 1.8074 years. 4. (a) Growth Trade Share0 .5 1 1.5 2 -5 0 5 10 Yes, there appears to be a weak positive relationship. (b) Malta is the “outlying” observation with a trade share of 2. (c) Growth 0.
16、64 2.31 u Tradeshare Predicted growth 0.64 2.31 u 1 2.95 (d) Growth 0.96 1.68 u Tradeshare Predicted growth 0.96 1.68 u 1 2.74 (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 Ma
17、lta (imports into Malta) and immediately transported to other countries (as exports from Malta). Thus, Maltas 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 a
18、nd 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
19、(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 (1.54) (0.05) The t-statistic is 0.69/0.05 13.06, which has a p-value of 0.
20、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 E1coefficients is 1, 1,College HighScoolEE 0.69 0.26 0.43. The standard error of for the estimated difference is SE1, 1,()College HighScoolEE (0.032 0.052)1/2 0.
21、06, so that a 95% confidence interval for the difference is 0.43 r 1.96 u 0.06 0.32 to 0.54 (dollars per hour). 2. _4.00.13Course 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
22、 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.
23、047. (c) Ed 13.94 0.064 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 E1coefficients is 1, 1,Female MaleEE 0.064 (0.084) 0.020. The standard error of for the estimated difference is SE1, 1,()Female
24、 MaleEE (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 0.064. The difference is not statistically different. Chapter 6 Linear Regression with Multiple Regressors Solutions to Empirical Exercises 1. Regressions used in (a) and (b) Mo
25、del 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 0.545 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
26、appear to be large omitted variable bias. (c) Professor Smiths predicted course evaluation (0.166 u 0) 0.011 u 0) (0.634 u 0) (0.173 u0) (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
27、.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 2R 0.007 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 s
28、uffer from omitted variable bias. (d) The regression in (b) fits the data much better as indicated by the R2, 2,R and SER. The R2and 2R are similar because the number of observations is large (n 3796). (e) Students with a “dadcoll 1” (so that the students father went to college) complete 0.696 more
29、years of education, on average, than students with “dadcoll 0” (so that the students 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 t
30、he 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) Bobs predicted years of education 0.0315 u 2
31、 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) Jims expected years of education is 2 u 0.0315 0.0630 less than Bobs. Thus, Jims expected years of education is 14.75 0.063 14.69. 3. Variable Mean Standard Deviation Units growth 1.86 1.
32、82 Percentage Points rgdp60 3131 2523 $1960 tradeshare 0.542 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 01 indicator variable (b) Estimated Regression (in table format): Regressor Coefficient tradeshare 1.
33、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 2R 0.23 116 Stock/Watson - Introduction to Econometrics - Second Edition The coefficient on Rev_Coups is 2.15. An additional coup in a five year perio
34、d, 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 .4325 = 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 s
35、ignificant 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.
36、18) Bachelor 6.87 (0.19) Intercept 3.32 (0.97) SER 8.66 7.88 R20.023 0.19 2R 0.022 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. Evi
37、dently the regression in (a) does not suffer from important omitted variable bias. (d) Bobs predicted average hourly earnings 0.44 u 26 3.17 u 0 6.87 u 0 3.32 $11.44 Alexiss 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.
38、 Gender and education are important predictors of earnings. The R2and 2R 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 coef
39、ficients 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, Fe
40、male) 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) 0.64* (0.10) Female 0.17* (0.05) 0.17* (0.05) Minority 0.17* (0.07) 0.16* (0.07) NN
41、English 0.24* (0.09) 0.25* (0.09) Intercept 4.00* (0.03) 4.07* (0.04) 4.07* (0.04) SER 0.545 0.513 0.513 R2 0.036 0.155 0.155 2R 0.034 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
42、% 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 0.354* (0.067) 0.338* (0.069) h
43、ispanic 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
44、 (0.099) intercept 13.956*(0.038) 8.861* (0.241) 8.893* (0.243) F-statitistic for urban and tuition SER 1.81 1.54 1.54 R2 0.007 0.282 0.284 2R 0.007 0.281 0.281 (a) The groups claim is that the coefficient on Dist is 0.075 ( 0.15/2). The 95% confidence for EDistfrom column (a) is 0.073 r 1.96 u 0.01
45、3 or 0.099 to 0.046. The groups 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 coef
46、ficient 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 EDist is 0.031 r 1.96 u 0.012
47、or 0.055 to 0.008. Similar results are obtained from the regression in (c). (c) Yes, the estimated coefficients EBlack and EHispanicare 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) 0.16
