Exercise Problems on Chapter 14. Multiple Regression Analysis This set of exercise problems has 22 problems, worth a total of 25 points. 1. Multiple regression analysis involves ____ independent variable(s) and _____ dependent variable(s). a. c. e. 2. autocorrelation or serial correlation homoscedasticity c. heteroscedasticity multicollinearity e. any of the above homoscedasticity b. multicollinearity d. only (c) and (d) of the above heteroscedasticity autocorrelation The R-squared in multiple regression is ______ the corresponding adjusted Rsquared. a. c. e. 5. one; more than one more than one; more than one If the assumption of equal variance of error terms is violated, we conclude that there is a _____ problem in regression estimation. a. c. e. 4. b. d. If the error (or residual) terms are correlated within themselves, we would find a _____ problem in regression estimation. a. b. d. 3. one; one more than one; one no; no greater than greater than or equal to the same as b. d. less than less than or equal to If independent variables are correlated among themselves in multiple regression, there is a ______ problem in estimation. a. c. e. homoscedasticity b. multicollinearity d. only (c) and (d) of the above heteroscedasticity autocorrelation 6. If there is a multicollinearity problem, you can alleviate this problem by _____. a. b. c. d. e. 7. If the intercept term is judged to be statistically equal to zero, it means that the intercept term is _____ in defining the regression equation. a. b. c. d. e. 8. important and significant unimportant but significant important but insignificant unimportant and insignificant useful If a slope coefficient in a regression equation is estimated to be 5 and its standard error, 3, you would _____ the null hypothesis of this coefficient being equal to zero if the appropriate critical (table)value is 2. a. b. 9. deleting one of the correlated independent variables. taking a ratio of the correlated independent variables. averaging the correlated variables. doing any of the above doing none of the above. reject accept Given 25 data points, if you found a calculated F value of 2.5 for a regression equation that has an intercept term and 4 independent variables, you would _____ the null hypothesis of all coefficients being zero at a 5% significance level because the appropriate critical (table) value is _____. a. d. accept; 2.87 reject; 5.80 b. e. reject; 2.87 accept; 3.51 c. accept; 5.80 Given the following description, answer Questions 10 - 22. Suppose that your boss asked you to analyze the impact of various factors on your company's quarterly profits. In order to write an analytical report to your boss, you collected the following quarterly data (in million dollars): Time period Profits Advertisement Sales Expenses Salaries Paid Y1 Q1 Y1 Q2 Y1 Q3 Y1 Q4 Y2 Q1 Y2 Q2 Y2 Q3 Y2 Q4 Y3 Q1 Y3 Q2 Y3 Q3 Y3 Q4 Y4 Q1 Y4 Q2 Y4 Q3 Y4 Q4 Y5 Q1 Y5 Q2 Y5 Q3 Y5 Q4 10. 200 190 200 240 210 200 250 260 250 200 240 300 280 270 280 340 330 320 350 370 50 45 50 60 55 53 60 65 65 60 60 63 62 60 61 65 65 67 70 70 Profits Sales all of the above b. d. Advertisement Expenses Salaries Paid If you are to conduct a regression analysis, the exogenous variable(s) in this case could be _____. a. c. e. 12. 10 5 10 12 6 7 8 10 10 5 8 4 10 11 15 13 13 12 14 17 If you are to conduct a regression analysis, the endogenous variable(s) in this case should be _____. a. c. e. 11. 100 90 95 120 110 100 120 130 120 100 110 130 120 110 110 150 130 120 140 160 Profits b. Advertisement Expenses Sales d. Salaries Paid any or all of the above except (a) Profits. If you estimate a regression analysis, using the above data for Profits (P) as a dependent variable and Advertisement Expenses (AD), Sales (S), and Salaries Paid (SP) as independent variables, you would find the following as the estimated regression equation. a. b. c. d. e. 13. The independent variable(s) that is/are significant at a 5% significance level is/are: a. c. e. 14. Sales all of the above Z Chi-square b. e. t either t or F c. F 20 16 b. e. 19 c. none of the above 17 If you are to conduct a significance testing of a regression coefficient via the critical-value approach, the (critical) table value at a 5% significance level will be _____. a. d. 17. b. d. If you are to find the appropriate degrees of freedom for significance testing of each of the 3 independent variables and an intercept with 20 data points (=observations), it would be _____. a. d. 16. Advertisement Expenses Salaries Paid only (b) and (c) of the above If you are to use the critical-value approach to a hypothesis testing of a regression coefficient, the appropriate test statistic to be used is the ____ statistic. a. d. 15. P = 15.988 + 20.236 AD + 0.7900 S + 0.4410 SP P = 15.988 - 0.1676 AD + 0.2184 S + 0.7672 SP P = 15.988 + 0.1676 AD - 0.2184 S - 0.7672 SP P = 15.988 - 0.1676 AD - 0.2184 S - 0.7672 SP P = 10.734 - 0.1676 AD + 0.2184 S + 0.7672 SP 2.0930 1.7459 b. e. 1.7291 1.96 c. 2.1199 The coefficient of determination of the regression model based on the above data set is _____. a. d. 0.9110 0.8155 b. e. 0.8300 8.1557 c. 0.7982 18. If you are test the joint significance of all regression coefficients, you would use a/an _____ test. a. d. 19. Z Chi-square b. e. t either t or F c. F If you are to test the joint significance of all coefficients in the regression equation, you would ______ the null hypothesis of all coefficients being zero at a 5% significance level. a. d. reject b. accept only (b) and (c) of the above c. e. fail to reject none of the above c. 2.5489 The following questions are worth 2 points each. 20. The Durbin-Watson (DW) statistic is _____. a. d. 21. b. e. 1.5973 3.9921 If you are to conduct a hypothesis testing of no autocorrelation for this estimated regression equation, you would find the critical (table) lower DW value at a 5% significance level to be _____ and the corresponding upper DW value to be _____. a. d. 22. 1.1235 3.2717 1.10; 1.54 0.77; 1.41 b. e. 1.00; 1.68 0.68; 1.57 c. 0.90; 1.83 If the calculated DW statistic is 2.88, and the critical lower and upper DW values are 0.5 and 1.54, respectively, then you would _____ the null hypothesis of no autocorrelation. a. d. reject b. be inconclusive about accept e. c. fail to reject only (b) and (c) of the above Answers to Homework Problems in Chapter 14. Multiple Regression Analysis This homework has 22 problems, worth a total of 25 points. 1. c.* more than one; one 2. a.* autocorrelation or serial correlation Autocorrelation is a special case of serial correlation. 3. b.* heteroscedasticity Homoscedasticity (=equal variance) is not a problem but heteroscedasticity (=unequal variance) is. 4. c.* greater than or equal to If an R-squared is 1, its adjusted R-squared is 1. Otherwise, the R-squared is always greater than the corresponding adjusted R-squared. 5. c.* multicollinearity 6. d.* doing any of the above 7. d.* unimportant and insignificant 8. b.* accept Since tc = (5-0)/3 = 1.667 < ttable = 2, you should accept the null hypothesis. 9. a.* accept; 2.87 Since Fc = 2.5 < Ftable = F204 , 0.05 = 2.87 , you should accept the null hypothesis. Note k=4 and n-k-1=25-4-1=20. 10. a* Profits An endogenous variable is a dependent variable. In this case, we are interested in knowing what variables determine Profits, not the other way around. 11. e* any or all of the above except (a) Profits Dr. Choi's GSB 420, Answers to HW Questions on Chapter 14 Multiple Regression Analysis, Page 1 Exogenous variables are independent variables. In this case, we are interested in knowing how other variables determine Profits. Since Profits is chosen as a dependent variable, other three variables can serve as independent variables. 12. b.* P = 15.988 - 0.1676 AD + 0.2184 S + 0.7672 SP This and other answers given hereafter are based on the following regression result: Intercept AD Sales Salaries 13. b.* Coefficients 15.98811 -0.16762 0.218468 0.767207 Standard Error 20.23665 0.741557 0.075278 0.550132 t Stat 0.790057 -0.22603 2.902137 1.394586 P-value 0.441047 0.824037 0.010395 0.182203 Sales Since the prob-value or p-value for Sales is 0.01039 which is less than the chosen significance level of 0.05, we reject the null hypothesis that the Sales coefficient is (statistically) equal to zero that is, (we accept the alternative hypothesis that) the Sales variable is significant(ly different from zero) Sales is an important/significant variable explaining/determining the independent variable of Profits. 14. b.* t Significance testing of individual coefficients is done via a t-test. Significance testing most often means that the null hypothesis is stated as: H 0 : i = 0 , unless specifically stated as otherwise. That is, it almost always means a two-tail test. 15. d.* 16 d.f. = n - k - 1= 20 - 3 - 1 = 16 16. c.* 2.1199 The table value of t-statistic at 0.05/2 and (n-k-1) degrees of freedom is 2.1199. 17. b.* 0.8300 The Excel output calls the coefficient of determination the R-square. When you square Multiple R, you get the R-square. Regression Statistics Dr. Choi's GSB 420, Answers to HW Questions on Chapter 14 Multiple Regression Analysis, Page 2 Multiple R R Square Adjusted R Square Standard Error Observations 18. c.* 0.911095 0.830094 0.798236 8.155717 20 F F-tests are used for testing joint significance of all coefficients whereas t-tests for individual significance of each coefficient. 19. a.* reject Because the p-value for F is 2.14E-06, which is almost equal to zero and less than the chosen significance level of 0.05, we reject the null hypothesis That is, not all coefficients are (statistically) equal to zero. ANOVA df Regression Residual Total 20. b.* SS 5199.498 1064.252 6263.75 3 16 19 MS 1733.166 66.51572 F 26.05649 Significance F 2.14E-06 1.5973 n Because DW = (e t =2 t et 1 ) 2 n e t =1 2 t = 1699.941 = 1.5973 1064.252 RESIDUAL OUTPUT Residual Observation 1 2 3 4 5 6 7 8 9 10 Predicted Profits 96.36583 91.18321 96.36583 112.4414 103.057 99.1703 115.2965 120.982 118.7973 104.876 Residuals 3.634169 -1.18321 -1.36583 7.558624 6.942986 0.829695 4.703476 9.017999 1.202677 -4.87599 Squared 13.20718 1.399979 1.865495 57.1328 48.20506 0.688394 22.12268 81.3243 1.446431 23.77526 Difference Lagged Residual NA 3.634169 -1.18321 -1.36583 7.558624 6.942986 0.829695 4.703476 9.017999 1.202677 e(t-1)e(t) NA 4.817376 0.182624 -8.92446 0.615638 6.113291 -3.87378 -4.31452 7.815322 6.078665 Squared NA 23.20711 0.033352 79.6459 0.37901 37.37232 15.00617 18.61511 61.07926 36.95016 Dr. Choi's GSB 420, Answers to HW Questions on Chapter 14 Multiple Regression Analysis, Page 3 11 12 13 14 15 16 17 18 19 20 113.1118 129.192 123.0497 119.163 121.4444 137.9566 135.7719 135.2892 143.8097 147.6762 -3.11185 0.807996 -3.04974 -9.16303 -11.4444 12.04343 -5.77189 -15.2892 -3.80967 12.32383 Sum = DW = 21. b.* 9.683588 0.652857 9.300891 83.96106 130.9752 145.0442 33.31475 233.7611 14.51355 151.8768 -4.87599 -3.11185 0.807996 -3.04974 -9.16303 -11.4444 12.04343 -5.77189 -15.2892 -3.80967 12.32383 1064.252 -1.76414 -3.91984 3.857732 6.113291 2.281414 -23.4879 17.81532 9.517353 -11.4796 -16.1335 3.112195 15.36516 14.88209 37.37232 5.204851 551.6801 317.3857 90.58002 131.7808 260.2897 1699.941 1.597312 1.00; 1.68 Given k=3 and n=20, the critical lower and upper DW values are 1.00 and 1.68, respectively. 22. d.* be inconclusive about Dr. Choi's GSB 420, Answers to HW Questions on Chapter 14 Multiple Regression Analysis, Page 4