Stock Watson 3U Exercise Solutions Chapter 7 Instructors PDF

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      Introduction to Econometrics rd (3 Updated Edition, Global Edition) by

James H. Stock and Mark W. Watson

Solutions to End-of-Chapter Exercises: Chapter 7* (This version August 17, 2014)

*Limited distribution: For Instructors Only. Answers to all odd-numbered questions are provided to students on the textbook website. If you find errors in the solutions, please pass them along to us at [email protected].

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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 1 _____________________________________________________________________________________________________

 7.1 and 7.2 Regressor Graduated high school (X1)

(1) 0.352**

(2) 0.373**

(0.021)

(0.021)

(0.021)

Male (X2)

0.458**

0.457**

0.451**

(0.021) Age (X3)

(3) 0.371**

(0.020)

(0.020)

0.011**

0.011**

(0.001)

(0.001)

North (X4)

0.175**

South (X5)

0.103**

East (X6)

−0.102*

(0.037) (0.033) (0.043) Intercept

12.84** (0.018)

12.471** (0.049)

12.390** (0.057)

(a) The t-statistic is 0.352/0.021 = 17.10, which is above 1.96 and significant at the 5% level. The confidence interval is given by

0.352  1.96  0.021  [0.312, 0.393] .

(b) The t-statistic is 0.458/0.021 = 22.29, which is above 1.96 and significant at the

5% level. The confidence interval is given by 0.458  1.96  0.021  [0.418, 0.498] .

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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 2 _____________________________________________________________________________________________________

7.3. (a) Yes, age is an important determinant of earnings. Using a t-test, the t-statistic is 0.011/0.001 = 11, which is greater than 2.58 and, hence, significant at the 1% level. The 95% confidence interval is 0.011  1.96  0.001.

(b) Age  [0.011  1.96  0.001] = 10  [0.011  1.96  0.001] = 0.11  1.96  0.001 = [0.108, 0.112] or 10.8 − 11.2% increase in weekly earnings.

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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 3 _____________________________________________________________________________________________________

7.4. (a) The F-statistic testing the coefficients on the regional regressors are zero is 21.87. The 1% critical value (from the F3,  distribution) is 3.78. Since 21.87  3.78, the regional effects are significant at the 1% level. (b) (i) The expected difference between Juan and Ali is (X4,Juan  X4,Ali)  4  4. Thus, a 95% confidence interval is −0.175  1.96  0.037. (ii) The expected difference between Juan and Mayank is (X4,Juan  X4,Mayank)   4  (X6,Juan  X6,Mayank)   6  4   6. A 95% confidence interval could be constructed using the general methods discussed in Section 7.3. In this case, an easy way to do this is to omit East from the regression and replace it with X6 = West. In this new regression, the coefficient on North measures the difference in wages between the North and East, and a 95%

confidence interval can be computed directly.

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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 4 _____________________________________________________________________________________________________

7.5. The t-statistic for the difference in the college coefficients is

t

ˆhigh school, 2007  ˆ high school, 1993 SE (ˆhigh school, 2007  ˆ high school, 1993)

.

Because ˆhigh school, 2007 and ˆhigh school, 1993 are computed from

cov( ˆhigh school, 2007  ˆhigh school, 1993 )  0 . Thus, var( ˆ high school, 2007  ˆ high school, 1993)  var( ˆ high school, 2007)  var(ˆ high school, 1993). 1 This implies that SE(ˆhigh school, 2007 ˆ high school, 1993)  (0.0212  0.0192 )2  0.0283.

0.373  0.301  2.54. Since the calculated t-statistic is greater than 1.96, 0.0283 the difference is significant at the 5% level.

Thus, t 

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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 5 _____________________________________________________________________________________________________

7.6. The estimated coefficients suggest a strong correlation between schooling and earnings. However, this effect cannot be interpreted as causal unless we are sure that there is no bias from omitted variables. We may believe, for example, that students who complete high school are relatively more able than those who do not. In this case, ability affects both high school graduation rates as well as earnings in the labor market, leading to an upward bias on the Highschool coefficient. This provides strong statistical evidence of the high returns to schooling in the labor market.

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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 6 _____________________________________________________________________________________________________

7.7. (a) The t-statistic is 0.567/1.23 = 0.461 < 1.96. Therefore, the coefficient on BDR is not statistically significantly different from 0. (b) The coefficient on BDR measures the partial effect of the number of bedrooms holding house size (Hsize) constant. Yet the typical 4-bedroom house is larger than the typical 3-bedroom house. Thus, the results in (a) say little about the conventional wisdom. (c) The 95% confidence interval for effect of lot size on price is 2500  [0.005  1.96  0.00072] or 8.972 to 16.028 (in thousands of dollars). (d) Choosing the scale of the variables should be done to make the regression results easy to read and interpret. If the lot sizes were measured in thousands of square feet, the estimate coefficient would be 5 instead of 0.005. (e) The 10% critical value from the F2,  distribution is 2.30. Because 2.38 > 2.30, the coefficients are jointly significant at the 10% level.

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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 7 _____________________________________________________________________________________________________

7.8. (a) Using the expressions for R2 and R2, algebra shows that 2

R  1

n 1 n  k 1 2 2 2 (1  R ), so R  1  (1 R ).    n k 1 n 1

Column 1: R  1 

10973  2 1 (1  0.0710)  0.0712 10973  1

Column 2: R 2  1 

10973  3  1 (1 0.0761)  0.0764 10973  1

Column 3: R 2  1 

10973  6 1 (1  0.0814)  0.0819 10973  1

2

H0 : 4  5   6  0

(b)

H1 : i  0 for i  4,5,6. Unrestricted regression (Column 3): 2 Y   0   1 X 1   2 X 2   3 X 3   4 X 4   5 X 5   6 X 6 , R unrestricted  0.0819.

Restricted regression (Column 2): 2

Y   0   1 X 1   2 X 2   3 X 3 , R restricted  0.0764. 2

2

( Runrestricted  Rrestricted )/ q , n  10973, kunrestricted  6, q  3 2 (1 Runrestricted )/(n  kunrestricted  1) (0.0819  0.0764)/3   21.898. (1 0.0819)/(10973 6 1)

FHomoskedasticity-only 

1% Critical value for F3,  3.78; so Ho is rejected at the 1% level. (continued on the next page)

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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 8 _____________________________________________________________________________________________________

7.8 (continued)

(c) t4 t5 and t6 –q  3; ti  c (where c = 2.935, the 1% Benferroni critical value from Table 7.2). Thus, the null hypothesis is rejected at the 1% level.

(d) .371  2.58  0.021.

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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 9 _____________________________________________________________________________________________________

7.9. (a) Estimate

Yi   0   X1i  2 ( X1i  X 2i )  ui and test whether   0.

(b) Estimate

Yi  0   X 1i   2 ( X 2i  aX 1i )  ui and test whether   0.

(c) Estimate

Yi  X 1i   0   X 1i  2 ( X 2i  X 1i )  ui and test whether   0.

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Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 10 _____________________________________________________________________________________________________

2 2 7.10. Because R2  1  SSR TSS , R unrestricted  R restricted 

1  R 2unrestricted 

SSR unrestricted TSS

SSRrestricted  SSRunrestri cted TSS

. Thus

F

 

2 2  Rrestricted ( Runrestricted )/q 2 (1  Runrestricted )/(n  k unrestricted  1)

SSRrestricted  SSRunrestricted TSS SSR unrestricted TSS

/q

/(n  kunrestricted  1)

(SSRrestricted  SSRunrestricted )/q SSRunrestricted / (n  k unrestricted 1)

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and

Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 7 11 _____________________________________________________________________________________________________

7.11. (a) Treatment (assignment to small classes) was not randomly assigned in the population (the continuing and newly-enrolled students) because of the difference in the proportion of treated continuing and newly-enrolled students. Thus, the treatment indicator X1 is correlated with X2. If newly-enrolled students perform systematicallydifferently on standardized tests than continuing students (perhaps because of adjustment to a new school), then this becomes part of the error term u in (a). This leads to correlation between X1 and u, so that E(u|Xl) ≠ 0. Because E(u|Xl) ≠ 0, the ˆ1 is biased and inconsistent. (b) Because treatment was randomly assigned conditional on enrollment status (continuing or newly-enrolled), E(u|X1, X2) will not depend on X1. This means that the assumption of conditional mean independence is satisfied, and 1ˆ is unbiased and consistent. However, because X2 was not randomly assigned (newly-enrolled students may, on average, have attributes other than being newly enrolled that affect test scores), E(u|X1, X2) may depend of X2, so that 2ˆ may be biased and inconsistent. 

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