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 residual plot showed no apparent pattern.

Explanation:

Step1: Analyze Model I

The residual plot shows strong curvature. In regression, a non - random pattern (like curvature) in the residual plot indicates that the linear model \( \hat{y}=1.176x - 3.019\) is not a good fit. A good linear model should have residual plots with no discernible pattern.

Step2: Analyze Model II

The residual plot shows that the variance increases as \(x\) (rental price) increases (fanning effect). One of the assumptions of linear regression (even after transformation) is constant variance (homoscedasticity). Since this assumption is violated for the model \( \sqrt{\hat{y}}=0.157x + 1.3497\), it is not a good fit.

Step3: Analyze Model III

The residual plot for the model \( \ln(\hat{y})=0.038x + 0.538\) shows no apparent pattern. A residual plot with no pattern (random scatter of points) is a sign that the linear regression model (after the \(y\) - transformation \(\ln(y)\)) satisfies the basic assumptions of linear regression (linear relationship between the transformed variables, constant variance of residuals, etc.)

Answer:

Model III is the most appropriate to linearly model the data.