Use the data in RENTAL for this exercise. The data for the...

Use the data in RENTAL for this exercise. The data for the years 1980 and 1990 include rental prices and other variables for college towns. The idea is to see whether a stronger presence of students affects rental rates. The unobserved effects model is $\log \left(r e n t_{i t}\right)=\beta_{0}+\delta_{0} y 90_{t}+\beta_{1} \log \left(p o p_{i t}\right)+\beta_{2} \log \left(\operatorname{avginc}_{i i}\right)+\beta_{3} p c t s t u_{i t}+a_{i}+u_{i l}$ where pop is city population, avginc is average income, and pctstu is student population as a percentage of city population (during the school year). (i) Estimate the equation by pooled OLS and report the results in standard form. What do you make of the estimate on the 1990 dummy variable? What do you get for $\hat{\beta}_{\text { perstit }} ?$ (ii) Are the standard errors you report in part (i) valid? Explain. (iii) Now, difference the equation and estimate by OLS. Compare your estimate of $\beta_{p c i s r u}$ with that from part (ii). Does the relative size of the student population appear to affect rental prices? (iv) Obtain the heteroskedasticity-robust standard errors for the first-differenced equation in part (iii). Does this change your conclusions?

Use the data in RENTAL for this exercise. The data for the years 1980 and 1990 include rental prices
and other variables for college towns. The idea is to see whether a stronger presence of students affects
rental rates. The unobserved effects model is
$\log \left(r e n t_{i t}\right)=\beta_{0}+\delta_{0} y 90_{t}+\beta_{1} \log \left(p o p_{i t}\right)+\beta_{2} \log \left(\operatorname{avginc}_{i i}\right)+\beta_{3} p c t s t u_{i t}+a_{i}+u_{i l}$
where pop is city population, avginc is average income, and pctstu is student population as a percentage
of city population (during the school year).
(i) Estimate the equation by pooled OLS and report the results in standard form. What do you make of the estimate on the 1990 dummy variable? What do you get for $\hat{\beta}_{\text { perstit }} ?$
(ii) Are the standard errors you report in part (i) valid? Explain.
(iii) Now, difference the equation and estimate by OLS. Compare your estimate of $\beta_{p c i s r u}$ with that from part (ii). Does the relative size of the student population appear to affect rental prices?
(iv) Obtain the heteroskedasticity-robust standard errors for the first-differenced equation in
part (iii). Does this change your conclusions?

Key Concepts

First-Difference Estimation

First-difference estimation is a method used in panel data econometrics to remove unobserved time-invariant individual effects by subtracting the previous time period’s observation from the current period. This transformation helps eliminate fixed effects and reduces bias from omitted variables that do not change over time. However, it requires careful handling of potential serial correlation and other issues in the error structure.

Panel Data Analysis

Panel data analysis involves the study of multi-dimensional data involving measurements over time. In econometrics, this type of data allows researchers to control for individual heterogeneity by observing the same entities (such as cities in this example) across different time periods, which makes it possible to more accurately estimate relationships among variables while accounting for unobserved factors that are constant over time for each cross-sectional unit.

Unobserved Effects and Fixed Effects Models

Unobserved effects refer to characteristics that vary across individuals or entities but remain constant over time and can influence the dependent variable. In fixed effects models, these unobserved characteristics are accounted for by allowing each entity to have its own intercept. This framework is important when using panel data, as it helps to control for omitted variable bias by differencing out or otherwise controlling for time-invariant unobservable factors.

Pooled Ordinary Least Squares (OLS)

Pooled OLS is an estimation method where data from all time periods and entities are combined into a single cross-sectional dataset, and a common regression is estimated. While pooled OLS is straightforward to implement, it assumes that the unobserved individual effects are uncorrelated with the regressors. If this assumption is violated, as often happens in panel settings, the pooled OLS estimates can be biased and inconsistent.

Dummy Variables in Regression Analysis

Dummy variables are used in regression analysis to represent categorical variables, such as time period indicators. They allow researchers to capture shifts in the dependent variable that are associated with different groups or time periods. In the context of the example, the 1990 dummy variable helps to isolate the effect of factors specific to the year 1990, controlling for other variables in the model.

Heteroskedasticity and Robust Standard Errors

Heteroskedasticity occurs when the variance of the error terms is not constant across observations, potentially leading to invalid standard errors in OLS estimates. Robust standard errors, also known as heteroskedasticity-robust standard errors, correct for this problem by providing more reliable estimates of the variability of the coefficients, which in turn leads to more trustworthy hypothesis tests and confidence intervals.

Interpretation of Coefficients in Log-Linear Models

In log-linear models, the coefficients represent percentage changes rather than absolute changes. For example, a coefficient on a logarithmically transformed independent variable indicates the elasticity of the dependent variable with respect to that independent variable. This interpretation is crucial when analyzing the impact of variables such as city population, income, or the share of the student population on rental prices.

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