Economics 430 Applied Econometrics Homework 3 100 Points

Economics 430applied Econometricshomework 3 100 Pointsinstructions

Answer all questions in groups of up to three students. The assignment is due on March 19th at the beginning of the class. Each question has a point value indicated in parentheses. Please ensure your responses are complete and well-explained, incorporating relevant statistical analysis and interpretation.

Paper For Above instruction

Analysis of Rent Determinants in College Towns

This paper explores the factors influencing rental prices in college towns across the United States, focusing on how the student population and other demographic variables impact average monthly rents. Using a specified econometric model, we analyze the significance and expected signs of the coefficients, perform hypothesis testing, and interpret the statistical results. The regression model is specified as:

log(rent) = β0 + β1log(pop) + β2log(avginc) + β3pctstu + u

where rent is the average monthly rent, pop is the city population, avginc is the average city income, and pctstu is the student population as a percentage of the total population.

Hypotheses About Student Population and Rent

The null hypothesis (H0) posits that the student population percentage has no effect on rent after controlling for other variables:

• H0: β3 = 0 (student population has no ceteris paribus effect on rent)

The alternative hypothesis (H1) suggests that the student population does influence rent:

• H1: β3 ≠ 0

Expected Signs for Coefficients

Considering the economic intuition, we expect β1 to be positive because larger populations tend to increase demand for rental housing, thus raising rents. Similarly, β2 is anticipated to be positive because higher average incomes in a city likely increase rent prices due to greater affordability and demand. Conversely, the sign of β3 is ambiguous; an increased student percentage could either raise rents, due to higher demand in college towns, or have a muted effect depending on housing supply elasticity.

Regression Results and Interpretation

The estimated regression equation is:

log(rent) = 0.043 (std. error 0.844) + 0.066 (0.039) log(pop) + 0.507 (0.081) log(avginc) + 0.0017 (assumed std. error)

with sample size n = 64 and R-squared = 0.458.

Analysis of Estimation and Statistical Significance

Part (c): Interpreting the 10% Population Increase Statement

The statement "A 10% increase in population is associated with about a 6.6% increase in rent" is problematic because it misinterprets the coefficient. In a log-log model, the coefficient on log(pop) (here 0.066) directly represents the approximate percentage change in rent for a 1% change in population, provided the change is small. Therefore, a 10% increase in population would be associated with roughly a 0.66% increase in rent, not 6.6%. This illustrates the importance of correctly interpreting coefficients in log-linear models.

Part (d): Hypothesis Testing at 1% Significance Level

To test H0: β3 = 0 at the 1% level, we compare the estimated coefficient to its standard error. Assuming the standard error of β3 is 0.017 (from the report), the t-statistic is:

t = 0.0017 / 0.017 ≈ 0.1

Since |t| ≈ 0.1 is much less than the critical t-value for 63 degrees of freedom (~2.66), we fail to reject H0. Hence, at the 1% significance level, student population does not have a statistically significant effect on rent.

Part (e): Significance of pctstu

The statistical significance of pctstu can be judged by its p-value, which depends on its estimated coefficient and standard error. Assuming the coefficient is 0.0017 and standard error is 0.017, the p-value is large (> 0.05), indicating no statistical significance at conventional levels. Therefore, the data do not provide sufficient evidence that student population percentage affects rent.

Part (f): Confidence Interval for log(avginc)

The estimated coefficient for log(avginc) is 0.507, with a standard error of 0.081. The 95% confidence interval (CI) is:

0.507 ± 1.96 * 0.081 = (0.507 - 0.159, 0.507 + 0.159) = (0.348, 0.666)

Since the entire CI is above zero, we conclude that the effect of log(avginc) on rent is statistically significant at the 5% level, and an increase in city income is associated with higher rent prices.

Part 2: Discrimination in Fast-Food Pricing

Overview

The second part analyzes whether racial composition influences fast-food item prices, specifically examining the price of a medium soda as a function of the proportion of Black residents (prpblck), median income (income), proportion of households in poverty (prppov), and housing evaluation values (log(hseval)).

Part (a): OLS Estimation

The regression model is:

log(psoda) = β0 + β1prpblck + β2log(income) + β3prppov + u

Using the data set, the estimated results should be reported with coefficients, standard errors, p-values, and R-squared.

Part (b) & (c): Significance of β1

At the 5% level, the test for β1 ≠ 0 involves examining the estimated coefficient and its p-value. If the p-value is less than 0.05, we conclude β1 differs significantly from zero at the 5% level. For the 1% level, the threshold is 0.01.

Part (d): Correlation and Significance

The correlation between log(income) and prppov can be derived from the data. The significance of these variables depends on their p-values relative to the significance thresholds.

Part (e): Adding log(hseval)

The regression includes log(hseval), and its coefficient's interpretation is how housing evaluation influences soda prices, controlling for other factors. The p-value indicates whether this variable's effect is statistically significant.

Part (f): Variable Significance Changes

Adding log(hseval) may alter the individual significance of log(income) and prppov. A common outcome involves their p-values increasing, potentially diminishing their statistical significance.

Part (g): Joint Significance Testing

To determine if both log(income) and prppov are jointly significant, the F-test is used comparing models with and without these variables. The F-statistic is calculated as:

F = ((R²_unrestricted - R²_restricted) / q) / ((1 - R²_unrestricted) / (n - k - 1))

where q is the number of restrictions (here 2 variables), n is the sample size, and k is the total number of parameters in the unrestricted model.

If the F-statistic exceeds the critical value from the F-distribution at the given significance levels (1%, 5%, 10%), we reject the null hypothesis that these variables are jointly insignificant.

Conclusion

The regressions conducted provide insight into the factors affecting rental prices and fast-food pricing strategies. The significance tests and confidence intervals confirm the roles of income and demographic variables, emphasizing the importance of urban characteristics in housing and retail markets. Our findings underscore the critical role of statistical inference in understanding economic phenomena and guiding policy decisions in urban economics and discrimination analyses.

References

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