Suppose that you are contracted to explore the determinants of rental rates in m

Question

Suppose that you are contracted to explore the determinants of rental rates in major cities across the U.S.

In addition to rental rate data, you decide to collect data for the year 2015 on what you believe are three key explanatory variables: the population of each city, the average income in each city, and the total student enrollment in each city (collegiate and above).

A. What is the null hypothesis of the impact of population on rental rates, and what is the alternative hypothesis that you are testing?

Consider two potential versions of a model:

Model (1): regressing rental rates on population and average income

Model (2): regressing rental rates on population, average income, and the percentage of the city’s total population that are students (here “students” means college and above).

B. Estimate Models (1) and (2) using an OLS regression and show your output.

C. Why might we want to run Model (2) using the percentage of the city’s total population that are students (as we did), rather than total student enrollment (the raw data we collected)? Does including both a city’s total population as well as the student’s share of the total population pose any problems for your estimation of Model (2)? Explain in each case.

D. Comment on the statistical significance of your slope coefficients in Models (1) and (2), referring to both the t-statistics and p-values from your output, and commenting on any differences in statistical significance between the models.

E. You decide that a log-log model might be more appropriate here. Re-run Model (2) as a log-log model, show your output, and comment on the statistical significance of your explanatory variables.

F. Conduct an F-test for the joint significance of all of your explanatory variables in the log-log version of Model (2). What can you say about the joint significance of the included explanatory variables?

128 Source SS df MS Number of obs F (2 125 224.60 7104 43.9 Prob F Model 1420887.8 0.0000 Residual 395393.076 125 3163.14461 R-squared 0.7823 Adj R-squared 0.7788 Total 1816280.88 127 14301.4242 Root MSE 56.242 rent Coef Std Err. [95* Conf. Interval] 0001181 .0000633 -1 87 0.064 0002433 7.07e-06 0153808 0007294 21.09 0.000 0139372 0168243 1. 52 0.1 32 -13.13599 98.95224 42.90812 28.31763

Explanation / Answer

Solution

For Model 1

Model (1): regressing rental rates, say y, on population, say p and average income, say e.

Then, we have y = 0 + 1p + 2e + , where is the error or residual component.

Let least square sample estimates of 0 , 1 and 2 be respectively, b0, b1 and b2.

To test the impact of population on rental rates, null hypothesis is: H0 : 1 = 0 vs alternative HA: 1 0.

Test statistics, t = |(b1 - 0)|/SE(b1), where SE(b1) = standard error of b1.

From the given software output, b1 = - 0.0001181, SE(b1) = 0.0000633, t = - 1.87.

The output also gives the p-value of t i.e., [P > |t|] = 0.064 which is more than 0.05.

=> H0 is accepted at 5% (0.05) level of significance implying that

population does not impact rental rates ANSWER 1

For Model 2

Model (2): regressing rental rates, say y, on population, say p, average income, say e and % of student population, say s.

Then, we have y = 0 + 1p + 2e + 3.s + , where is the error or residual component.

Let least square sample estimates of 0 , 1 2 and 3 be respectively, b0, b1. b2 and b3

To test the impact of population on rental rates, null hypothesis is: H0 : 1 = 0 vs alternative HA: 1 0.

Test statistics, t = |(b1 - 0)|/SE(b1), where SE(b1) = standard error of b1.

From the given software output, b1 = 3.5e-06, SE(b1) = 0.0000646, t = 0.05.

The output also gives the p-value of t i.e., [P > |t|] = 0.959 which is more than 0.05.

=> H0 is accepted at 5% (0.05) level of significance implying that

population does not impact rental rates ANSWER 2

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