Question: 101.

Refer to Q.6.18.

Q. 6.18: Commercial properties. A commercial real estate company evaluates vacancy rates, square footage, rental rates, and operating expenses for commercial properties in a large metropolitan area in order to provide clients with quantitative information upon which to make rental decisions.

The data below are taken from 81 suburban commercial properties that are the newest, best located, most attractive, and expensive for five specific geographic areas. Shown here are the age (X₁), operating expenses and taxes (X₂), vacancy rates (X₃), total square footage (X₄), and rental rates (Y).

Suppose this real estate company came up with a “rental quality index” and added this variable to the regression equation. The data are attached below and sent by e-mail (the rental quality index is on the last column). Please note that the total square footage is divided by 100,000.

13.5 1 5.02 0.14 1.23 25.48

12 14 8.19 0.27 1.04 11.15

10.5 16 3 0 0.4 6.25

15 4 10.7 0.05 0.57 18.26

14 11 8.97 0.07 0.6 13.17

10.5 15 9.45 0.24 1.01 10.87

14 2 8 0.19 0.31 18.90

16.5 1 6.62 0.6 2.48 31.73

17.5 1 6.2 0 2.15 30.09

16.5 8 11.78 0.03 2.51 23.98

17 12 14.62 0.08 2.91 22.55

16.5 2 11.55 0.03 2.08 29.99

16 2 9.63 0 0.82 23.02

16.5 13 12.99 0.04 3.6 24.64

17.23 2 12.01 0.03 2.66 31.65

17 1 12.01 0 2.99 34.16

16 1 7.99 0.14 1.89 30.30

14.63 12 10.33 0.12 3.66 26.25

14.5 16 10.67 0 3.5 19.92

14.5 3 9.45 0.03 0.85 19.78

16.5 6 12.65 0.13 2.36 26.35

16.5 3 12.08 0 1.3 23.73

15 3 10.52 0.05 0.41 20.11

15 3 9.47 0 0.41 18.67

13 14 11.62 0 0.46 6.86

12.5 1 5 0.33 1.2 25.35

14 15 9.89 0.05 0.81 9.86

13.75 16 11.13 0.06 1.54 11.33

14 2 7.96 0.22 0.97 23.55

15 16 10.73 0.09 2.76 17.34

13.75 2 7.95 0 0.9 22.29

15.63 3 9.1 0 1.84 25.36

15.63 3 12.05 0.03 1.85 27.98

13 16 8.43 0.04 0.96 11.16

14 16 10.6 0.04 1.06 9.00

15.25 13 10.55 0.1 1.36 13.49

16.25 1 5.5 0.21 1.8 27.69

13 14 8.53 0.03 3.15 21.19

14.5 3 9.04 0.04 0.43 20.73

11.5 15 8.2 0 0.3 5.87

14.25 1 6.13 0 0.6 21.63

15.5 15 8.32 0 0.74 8.61

12 1 4 0 0.5 23.39

14.25 15 10.1 0 0.51 6.66

14 3 5.25 0.16 0.32 17.68

16.5 3 11.62 0 1.68 27.53

14.5 4 5.31 0 0.7 18.44

15.5 1 5.75 0 0.27 19.13

16.75 4 12.46 0.03 1.3 21.35

16.75 4 12.75 0 1.3 22.94

16.75 2 12.75 0 1.3 24.77

16.75 2 11.38 0 2.09 28.01

17 1 5.99 0.57 2.2 32.28

16 2 11.37 0.27 0.6 22.46

14.5 3 10.38 0 1.1 22.99

15 15 10.77 0.05 1.01 10.09

15 17 11.3 0 2.89 16.30

16 1 7.06 0.14 1.05 23.22

15.5 14 12.1 0.05 2.76 20.45

15.25 2 10.04 0.06 0.33 19.72

16.5 1 4.99 0.73 2.1 27.96

19.25 0 7.33 0.22 2.4 32.29

17.75 18 12.11 0 2.82 16.54

18.75 16 12.86 0 4.21 25.25

19.25 13 12.7 0.04 4.84 31.96

14 20 11.58 0 2.34 11.85

14 18 11.58 0.03 2.31 12.61

18 16 12.97 0.08 2.97 16.86

13.75 1 4.82 0 0.32 19.92

15 2 9.75 0.03 0.39 19.39

15.5 16 10.36 0.02 1.1 9.99

15.9 1 8.13 0.23 2.36 32.02

15.25 15 13.23 0.05 2.43 17.86

15.5 4 10.57 0.04 1.22 21.48

14.75 20 11.22 0 1.28 5.84

15 3 10.34 0 0.72 21.00

14.5 3 10.67 0 0.43 19.36

13.5 18 8.6 0.08 0.59 5.58

15 15 11.97 0.14 2.55 16.90

15.25 11 11.27 0.03 4.35 31.49

14.5 14 12.68 0.03 2.02 16.5

a. Why was the total square footage divided by 100,000? What would have happened if I didn’t do this.

Answer in a single sentence.

b. Fit the regression equation including all five predictors. Make sure to include the option, “VIF/Tolerance.” State the estimated function of the equation.

c. Present the correlation matrix. Is there any unusual feature in the matrix? Can you detect a potential problem here?

d. Briefly comment on unusual features of this estimate. Pay attention to the overall F*, individual t*, standardized coefficients, and VIF.

e. Run a regression analysis without the “rental quality index” and present the estimated equation.

Compare this to the equation in b. in terms of R2s, coefficient estimates, standard error, etc. Is there anything noteworthy in the comparison of these two equations?

f. Regress the rental quality index on the four other predictors and present the estimated equation.

Check R² for this equation. State how this value is related to any of the VIF’s you obtained in b.

g. What is “wrong” with this equation? Identify the problem with this equation and make a guess on how the “rental quality index” has been created.