Are rent rates influenced by the student population in a...

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.

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.

Key Concepts

Hypothesis Testing in Regression

This concept involves setting up a null hypothesis and an alternative hypothesis to statistically test whether a particular independent variable has a significant effect on the dependent variable in a regression model. In the broader context, the null hypothesis generally posits that there is no relationship (or effect) between the variable of interest and the outcome, while the alternative hypothesis posits that such a relationship exists. This framework allows researchers to use data to determine if observed coefficients are statistically different from zero, thereby providing evidence of an effect or relationship.

Ceteris Paribus (All Else Equal) Analysis

In multiple regression analysis, the ceteris paribus assumption means that the estimated effect of one variable on the dependent variable is isolated while holding all other variables constant. This principle is crucial because it allows for a clearer interpretation of the individual impact of each explanatory variable, ensuring that the effect is not confounded by the influence of other variables included in the model.

Log?Linear Models and Elasticity Interpretation

Log?linear models involve taking the natural logarithm of one or more variables in a regression equation. When both the dependent and independent variables are log-transformed, the coefficients can be interpreted as elasticities—the percentage change in the dependent variable associated with a one percent change in an independent variable, assuming all other variables remain constant. This transformation is widely used because it stabilizes variance and facilitates the interpretation of proportional changes between variables.

Coefficient Interpretation and Percentage Change Misinterpretation

Interpreting coefficients in log-transformed models requires careful consideration. A common pitfall is directly equating the estimated coefficient in a log-linear model to a percentage change without accounting for the transformation's properties. In a log-log model, the coefficient represents an elasticity, meaning it directly indicates the percentage change in the dependent variable for a one percent change in the independent variable. However, when such a statement is made without considering the fact that the relationship is multiplicative rather than additive, it can lead to misinterpretation of the magnitude of the effect.

Transcript

00:01 This question says, are rent rates influenced by the student population in a college town? let's rent with the average monthly rent paid on the rental unit in the college town in the us.

00:15 Let's pop notes denote the total city population in the average city income.

00:22 I let pct stu, the student population, as a percentage of the total population.

00:28 And one model to test for a relationship is given in the question.

00:33 And the first question wants us to state the null hypothesis that size of the student body relative to the population has no certain risk parables effect on monthly rent and states the alternative that there is an effect.

00:48 So the first question.

00:54 So this is the coefficient of independent variables that represents the student population as a percentage of the total population.

01:01 So coefficient of independent variables, student population, as a percentage of total population.

01:49 So the null hypothesis that size of student body relative to the population has no set risk, but effect on the monthly rental is given by the null hypothesis is given by h .o.

02:10 Equals to 0 and the alternative hypothesis is given by h1 is not equal to 0.

02:31 So that's the answer.

02:33 Now the next question.

02:37 What signs do you expect for b1 and b2? so b1 indicates that the impact of log of population or log of monthly rent so it indicates the impact of the log of population and monthly rent.

03:01 So a large population will mean that will upsurge the demand for housing and will lead to an increase in leases right.

03:09 So as the b1 will be positive.

03:18 Now, the b2, on the other hand, the demand for overall housing will be higher if the average city income will be higher.

03:28 Therefore, the cost of housing will be pushed up, including the rental rates and the sign of b22.

03:34 Will be positive, all other things remain constant.

03:42 Now the next question, the equation estimated using the 1990 data from rental rowe for 64 college towns is given in the question and what is wrong with the statement a 10 % increase in population is associated with about 6 .6 % increase in rent...

Please give Ace some feedback