Question

Are rent rates influenced by the student population in a college town? Let rent be the average monthly rent paid on rental units in a college town in the United States. Let pop denote the total city population, avginc the average city income, and pctstu the student population as a percentage of the total population. One model to test for a relationship is $$\log (\text {rent})=\beta_{0}+\beta_{1} \log (p o p)+\beta_{2} \log (\text {avginc})+\beta_{3} p c t s t u+u$$ i. State the null hypothesis that size of the student body relative to the population has no ceteris paribus effect on monthly rents. State the alternative that there is an effect. ii. What signs do you expect for \(\beta_{1}\) and \(\beta_{2} ?\) iii. The equation estimated using 1990 data from RENTAL for 64 college towns is $$\begin{aligned} &\widehat{\log (\text {rent})}=.043+.066 \log (\text {pop})+.507 \log (\text {avginc})+.0056 \text { pctstu}\\\ &\begin{aligned} (.844)(.039) &(.081) \\ n=64, R^{2}=.458 \end{aligned} \end{aligned}$$ What is wrong with the statement: "A 10\% increase in population is associated with about a \(6.6 \%\) increase in rent"? iv. Test the hypothesis stated in part (i) at the \(1 \%\) level.

Short answer

1. \(H_0: \beta_3 = 0\); \(H_1: \beta_3 \neq 0\). 2. \(\beta_1 > 0\), \(\beta_2 > 0\). 3. Correct interpretation: 10% increase in pop ≈ 0.66% rent increase. 4. Do not reject \(H_0\) at 1% level.

Step-by-step solution

State the Hypotheses

In part (i), we are asked to state the null and alternative hypotheses. The null hypothesis (\(H_0\)) is that the student population percentage, \(\beta_3\), has no effect on rent: \(H_0: \beta_3 = 0\). The alternative hypothesis (\(H_1\)) is that the student population percentage does have an effect on rent: \(H_1: \beta_3 eq 0\).

Predict Signs for β1 and β2

In part (ii), we need to predict the signs of \(\beta_1\) and \(\beta_2\). - \(\beta_1\), the coefficient of \(\log(\text{pop})\), is expected to be positive, as an increase in population usually causes demand for housing to rise, which can increase rent.- \(\beta_2\), the coefficient of \(\log(\text{avginc})\), is also expected to be positive because higher average income often allows people to afford higher rents.

Analyze the Interpretation of β1

In part (iii), we evaluate the interpretation of \(\beta_1 = 0.066\). The error in the statement "A 10\% increase in population is associated with about a 6.6\% increase in rent" is assuming a linear percentage increase. The log-log model implies elasticity, so a 10\% increase in population leads to a 0.066 * 10\% = 0.66\% increase in rent.

Hypothesis Testing at 1% Level

In part (iv), we test the null hypothesis \(H_0: \beta_3 = 0\). - Given \(\beta_3 = 0.0056\) with a standard error of 0.081, the t-statistic is \(t = \frac{0.0056}{0.081} = 0.069\).- With \(n = 64\), the degrees of freedom is \(n - 4 = 60\).- At a 1% significance level, the critical t-value for a two-tailed test is approximately 2.660 (from t-distribution tables).- Because \(0.069 < 2.660\), we do not reject \(H_0\). Thus, we conclude that there is no statistically significant effect of the student population percentage on rent at the 1% level.

Key concepts

Student Population

The concept of 'Student Population' is crucial in understanding how the demographic composition of a college town might influence various economic factors, such as rent rates. The student population, often composed of young adults seeking higher education, can have significant effects on local housing markets. In our context, 'pctstu,' which stands for the percentage of the student population relative to the total city population, is the key variable. An increase in the student population could lead to higher demand for rental accommodations, potentially driving up rent prices as a result.

When analyzing economic models like the one in our exercise, understanding the student population's impact allows us to assess whether local policies or economic conditions might need adjustments to accommodate changes in demand.

Rent Rates

Rent rates are a vital component of the housing market, particularly in college towns where students form a substantial market segment. In econometrics, analyzing rent rates involves understanding various influencing factors such as population size, average income, and the student population percentage.

Economic theory suggests that factors affecting rent rates include:

• Supply and demand dynamics: If there is a larger population, demand for housing usually rises, leading to higher rent prices.
• Income levels: Higher average city incomes often correlate with the capacity to pay more for housing, thereby increasing rent rates.
• Specific local demographics: A high student population can create a unique housing demand, as students usually rent rather than own homes.

In the exercise, the model predicts how these factors might influence rents using a regression analysis framework. Understanding these economic relationships thoroughly helps policymakers and stakeholders make informed decisions.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on sample data. This method involves stating a null hypothesis, which represents a statement of no effect or no difference, and an alternative hypothesis, which is what the researcher aims to support.

In our exercise, the hypothesis testing focuses on the student population percentage's (\(\beta_3\)) effect on rent rates. We start with the null hypothesis (\(H_0\)), stating that the student population has no effect on rent, \(\beta_3 = 0\), and the alternative hypothesis (\(H_1\)), suggesting an effect exists, \(\beta_3 eq 0\).

To test these hypotheses, we calculate a t-statistic. A calculated t-statistic lower than the critical value from the t-distribution implies that we fail to reject the null hypothesis. In our analysis, this means concluding that there's no significant measurable effect of the student population percentage on rent at the 1% level of significance. Hypothesis testing thus provides a structured approach to determine statistical significance.

Regression Analysis

Regression analysis is a statistical tool used to explore relationships between variables. It helps in understanding how the typical value of a dependent variable (e.g., rent) changes when any one of the independent variables (e.g., population, income, student percentage) is varied while the others are held fixed.

The given model employs a particular kind of regression: a log-linear model. This involves taking the logarithm of the variables, which can make relationships easier to interpret as elasticities. In this case, elasticity refers to the percentage change in rent resulting from a 1% change in the independent variables.

• \(\beta_1\) and \(\beta_2\) are coefficients interpreted as elasticities for population and income, suggesting how these changes proportionally influence rent.
• \(\beta_3\) represents the linear effect of the student percentage on rent.

Understanding regression analysis allows us to quantify these influences, helping in forecasting and planning. Proper interpretation of the regression coefficients helps policymakers and economists devise better strategies to manage housing markets efficiently.