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

**The following question uses the “rental” Wooldridge dataset.**

**The following question uses the “rental” Wooldridge dataset.**

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?

B. 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).

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

*Hint: You will need to create a new variable that is the percentage of the city’s total population that are students.*

B. 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.

C. 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.

*Hint* *1:* *You* *don’t* *need* *to* *convert* *your* *newly-created* *“student* *share”* *variable Hint 2: use the “gen” command to create new variable and “log” tocalculate*

*the* *natural* *log* *of* *a* *variable,* *Stata* *calculates* *a* *natural* *logarithm* *by* *default.*

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?