# Question: This data table gives annual costs of 223 commercial leases

This data table gives annual costs of 223 commercial leases. All of these leases provide office space in a Midwestern city in the United States. The cost of the lease is measured in dollars per square foot, per year. The number of square feet is as labeled, and Parking counts the number of parking spots in an adjacent garage that the realtor will build into the cost of the lease. Fit the multiple regression of Cost per Sq Ft on 1 / Sq Ft and Parking / Sq Ft. (Recall that the slope of 1 / Sq Ft captures the fixed costs of the lease, those present regardless of the number of square feet.)

(a) Thinking marginally for a moment, should there be a correlation between the number of parking spots and the fixed cost of a lease?

(b) Interpret the coefficient of Parking / Sq Ft. Once you figure out the units of the slope, you should be able to get the interpretation.

(c) One of the two explanatory variables explains slightly more than statistically significant variation in the price. Had the two explanatory variables been uncorrelated (and produced these estimates), would the variation have been more clearly statistically significant? Use the VIF to see.

(d) We can see the effects of collinearity by constructing a plot that shows the slope of the multiple regression. To do this, we have to remove the effect of one of the explanatory variables from the other variables. Here’s how to make a so-called partial regression leverage plot for these data. First, regress Cost / Sq Fit on Parking / Sq Fit and save the residuals. Second, regress 1/ Sq Fit on Parking / Sq Fit and save these residuals. Now, make a scatterplot of the residuals from the regression of Cost / Sq Ft on Parking / Sq Fit on the residuals from the regression of 1 / Sq Fit on Parking>Sq Ft. Fit the simple regression for this scatterplot, and compare the slope in this fit to the partial slope for 1 / Sq Ft in the multiple regression. Are they different?

(e) Compare the scatterplot of Cost / Sq Ft on 1 / Sq Ft to the partial regression plot constructed in part (d). What has changed?

(a) Thinking marginally for a moment, should there be a correlation between the number of parking spots and the fixed cost of a lease?

(b) Interpret the coefficient of Parking / Sq Ft. Once you figure out the units of the slope, you should be able to get the interpretation.

(c) One of the two explanatory variables explains slightly more than statistically significant variation in the price. Had the two explanatory variables been uncorrelated (and produced these estimates), would the variation have been more clearly statistically significant? Use the VIF to see.

(d) We can see the effects of collinearity by constructing a plot that shows the slope of the multiple regression. To do this, we have to remove the effect of one of the explanatory variables from the other variables. Here’s how to make a so-called partial regression leverage plot for these data. First, regress Cost / Sq Fit on Parking / Sq Fit and save the residuals. Second, regress 1/ Sq Fit on Parking / Sq Fit and save these residuals. Now, make a scatterplot of the residuals from the regression of Cost / Sq Ft on Parking / Sq Fit on the residuals from the regression of 1 / Sq Fit on Parking>Sq Ft. Fit the simple regression for this scatterplot, and compare the slope in this fit to the partial slope for 1 / Sq Ft in the multiple regression. Are they different?

(e) Compare the scatterplot of Cost / Sq Ft on 1 / Sq Ft to the partial regression plot constructed in part (d). What has changed?

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