Question

1. an economist in a large city wanted to develop a model to estimate the vacancy rate of downtown office space (the percent of available office space that was unoccupied) based on the monthly rental price per square foot. she took a random sample of various office buildings in the downtown area and created three models using monthly rental prices (x) to predict vacancy rate (y).

i. a model using (x,y) produced a regression equation of \\( \hat { y } = 1.176 x - 3.019 \\) and the residual plot showed strong curvature.
ii. a model using \\( ( x, \sqrt { y } ) \\) produced a regression equation of \\( \sqrt { \hat { y } } = 0.157 x + 1.3497 \\) and the residual plot indicated that the variance increased as the rental prices increased (a \fanning\ effect).
iii. a model using \\( ( x, \ln ( y ) ) \\) produced a regression equation of \\( \ln ( \hat { y } ) = 0.038 x + 0.538 \\) and the residu plot showed no apparent pattern.
based on the above information, which transformation(s) would be most appropriate to linearly model the
a) i only
b) ii only
c) iii only
d) i and ii
e) none of the above would be appropriate

Explanation:

Step1: Analyze Model I

A good regression model should have a residual plot with no pattern. Model I has a residual plot with strong curvature. This indicates that the linear model \( \hat{y}=1.176x - 3.019\) does not fit the data well, as curvature in the residual plot suggests a non - linear relationship that the simple linear model (no transformation of \(y\)) fails to capture.

Step2: Analyze Model II

For a regression model to be valid in the context of least - squares regression (which assumes constant variance), the "fanning" effect (heteroscedasticity, where the variance of the residuals increases with the value of the independent variable \(x\)) is a problem. Model II, which uses the transformation \(\sqrt{y}\), has a residual plot with increasing variance (fanning effect). So, this model does not meet the constant - variance assumption of linear regression.

Step3: Analyze Model III

One of the key assumptions for a good regression model is that the residual plot should show no apparent pattern. Model III uses the transformation \(\ln(y)\), and its residual plot shows no apparent pattern. This suggests that the relationship between \(x\) (monthly rental price) and \(\ln(y)\) (where \(y\) is the vacancy rate) is linear (or at least meets the basic assumptions of a linear regression model better than the other two models) after the transformation.

Answer:

C. III only