Chapter 7Hypothesis Tests and Confidence Intervals in Multiple Regression Solutions to Exercises 1. Regressor College (X1) Female (X2) Age (X3) Ntheast (X4) Midwest (X5) South (X6) Intercept 12.69** (0.14) 4.40** (1.05) (1) 5.46** (0.21) −2.64** (0.20) (2) 5.48** (0.21) −2.62** (0.20) 0.29** (0.04) (3) 5.44** (0.21) −2.62** (0.20) 0.29** (0.04) 0.69* (0.30) 0.60* (0.28) −0.27 (0.26) 3.75** (1.06) (a) The t-statistic is 5.46/0.21 = 26.0 > 1.96, so the coefficient is statistically significant at the 5% level. The 95% confidence interval is 5.46 ± 1.96 × 0.21. (b) t-statistic is −2.64/0.20 = −13.2, and 13.2 > 1.96, so the coefficient is statistically significant at the 5% level. The 95% confidence interval is −2.64 ± 1.96 × 0.20. 3. 0.29 (a) Yes, age is an important determinant of earnings. Using a t-test, the t-statistic is 0.04 = 7.25, with −13 a p-value of 4.2 × 10 , implying that the coefficient on age is statistically significant at the 1% level. The 95% confidence interval is 0.29 ± 1.96 × 0.04. (b) ∆Age × [0.29 ± 1.96 × 0.04] = 5 × [0.29 ± 1.96 × 0.04] = 1.45 ± 1.96 × 0.20 = $1.06 to $1.84 4. (a) The F-statistic testing the coefficients on the regional regressors are zero is 6.10. The 1% critical value (from the F3, ∞ distribution) is 3.78. Because 6.10 > 3.78, the regional effects are significant at the 1% level. (b) The expected difference between Juanita and Molly is (X6,Juanita − X6,Molly) × β6 = β6. Thus a 95% confidence interval is −0.27 ± 1.96 × 0.26. 1992 ) = (0. Therefore. Gender discrimination means that two workers.002 ± 2.1998 .Juanita − X6.30. 7.08 < 2.485 2.96. 5. The t-statistic for the difference in the college coefficients is ˆ ˆ ˆ ˆ ˆ ˆ t = (β college.3. Because 0. Thus. 6. but that is a question about something other than gender discrimination.Juanita − X5. (c) The 99% confidence interval for effect of lot size on price is 2000 × [.29 (0. identical in every way but gender.21 + 0.1998 college. t 5.Jennifer) × β6 = −β5 + β6.1992 college.1992 are computed ˆ ˆ from independent samples.6552.002. (If this were true.1998 − β college. an easy way to do this is to omit Midwest from the regression and replace it with X5 = West. the results in (a) says little about the conventional wisdom. are paid different wages. . This implies that college. If the lot size were measured in thousands of square feet. the 5% critical value. In isolation.58 × . (d) Choosing the scale of the variables should be done to make the regression results easy to read and to interpret. then they may be responsible for the difference in mean wages. = 0.1998 1 2 2 2 act ˆ ˆ SE ( β = college.202 ) 2 1 = 0.) These are potentially important omitted variables in the regression that will lead to bias in the OLS coefficient estimator for Female. the coefficients are not jointly significant at the 10% level. etc.1992 ).1998 − β college. In this new regression the coefficient on South measures the difference in wages between the South and the Midwest.) If these characteristics are systematically different between men and women. There is no significant change since the calculated t-statistic is less than 1. years of experience.Introduction to Econometrics .∞ distribution is 2. In this case.30.Second Edition (c) The expected difference between Juanita and Jennifer is (X5. which means that cov( β college. var( β ) = var( β ) + var( β ).212 + 0.20 ) . (b) The coefficient on BDR measures the partial effect of the number of bedrooms holding house size (Hsize) constant. it is premature to reach a conclusion about gender discrimination. Thus. they are independent.Jennifer) × β5 + (X6.1998 − β college.48−5.61 significantly different from zero. Thus.96.48 (in thousands of dollars).38 Stock/Watson .1992 )/SE ( β college. Because β college. it would raise an interesting and important question of why women tend to have less education or less experience than men. and a 95% confidence interval can be computed directly. these results do imply gender discrimination.1998 and β college.52 to 6. Yet. (e) The 10% critical value from the F2. Since these characteristics were not controlled for in the statistical analysis. the coefficient on BDR is not statistically (a) The t-statistic is 0. it is also important to control for characteristics of the workers that may affect their productivity (education. the typical 5-bedroom house is much larger than the typical 2-bedroom house. the estimate coefficient would be 2 instead of 0. β college.186 < 1.1998 college.00048] or 1.1992 ) = 0 ˆ ˆ ˆ ˆ −β Thus. A 95% confidence interval could be contructed using the general methods discussed in Section 7. 629 420 − 1 420 − 4 − 1 (1 − 0.051 420 − 1 420 − 2 − 1 (1 − 0.775 − 0. β 4 ≠ 0 2 = 0. (b) Estimate Yi = β 0 + γ X1i + β 2 ( X2i − aX1i ) + ui and test whether γ = 0.775 420 − 1 420 − 3 − 1 (1 − 0.427 420 − 1 420 − 3 − 1 (1 − 0. Rrestricted = 0.00 so Ho is rejected at the 5% level.00 = 4. (d) −1. n − k −1 n −1 420 − 1 − 1 (1 − 0. algebra shows that R2 = 1 − n −1 n − k −1 (1 − R 2 ).27 9.921 and t4 = 0. Thus the null hypothesis is rejected at the 1% level.22 (1 − 0. q = 2.174 = = = 322. (a) Estimate Yi = β 0 + γ X1i + β 2 ( X1i + X2i ) + ui and test whether γ = 0.775 420 − 1 Column 1: R 2 = 1 − Column 2: R2 = 1 − Column 3: R2 = 1 − Column 4: R2 = 1 − Column 5: R2 = 1 − (b) H0 : β 3 = β 4 = 0 H1 : β 3 ≠.01 ± 2. q = 2 2 (1 − Runrestricted )/(n − kunrestricted − 1) FHomoskedasticityOnly = = (0.00054 5% Critical value form F2.775)/(420 − 4 − 1) (0. Runrestricted 2 Restricted regression (Column 2): Y = β 0 + β1 X1 + β 2 X2 . |t3| > c (Where c = 2.61.225)/415 0. n = 420. so R2 = 1 − (1 − R 2 ).Solutions to Exercises in Chapter 7 39 8. FHomoskedasticityOnly > F2. the 1% Benferroni critical value from Table 7.814.626) = 0. (a) Using the expressions for R2 and R 2. (c) t3 = −13.807. .773) = 0.424) = 0.427 2 2 ( Runrestricted − Rrestricted )/q .58 × 0.3).775 Unrestricted regression (Column 5): Y = β 0 + β1 X1 + β 2 X2 + β 3 X3 + β 4 X4 .427)/2 0.049) = 0.348/2 0.773) = 0. kunrestricted = 4. 40 Stock/Watson . Because R 2 = 1 − SSR . Thus F= = = 2 2 ( Runrestricted − Rrestricted )/q 2 (1 − Runrestricted )/(n − kunrestricted − 1) SSRrestricted − SSRunrestricted TSS SSRunrestricted TSS /q /(n − kunrestricted − 1) (SSRrestricted − SSRunrestricted )/q SSRunrestricted /(n − kunrestricted − 1) . 2 2 10. Runrestricted − Rrestricted = TSS SSRrestricted − SSRunrestricted TSS 2 and 1 − Runrestricted = SSRunrestricted TSS .Introduction to Econometrics .Second Edition (c) Estimate Yi − X1i = β 0 + γ X1i + β 2 ( X2 i − X1i ) + ui and test whether γ = 0.