Using the data in the file \(b r 5\), find least squares estimates of the following house-price relationships for houses sold in Baton Rouge during 2005.

a. Report the coefficient estimates and their standard errors.

b. Show how the estimates \(\left(\hat{\alpha}_{1}, \hat{\alpha}_{2}\right)\) can be found from the parameter estimates in the other two equations. How does the interpretation of \(\hat{\beta}_{2}\) differ from the interpretation of \(\hat{\alpha}_{2}\) ? What would you characterize as the omitted variable bias when estimating \(\alpha_{2}\) ? Is there evidence that BEDROOMS has a direct effect on \(\ln (P R I C E)\) ?

c. Estimate the equation \(\ln (P R I C E)=\theta_{1}+\theta_{2} S Q F T+e_{3}\). Compare the estimates \(\hat{\theta}_{2}\) and \(\hat{\beta}_{3}\). What was the effect of omitting BEDROOMS on the estimated coefficient for SQFT? What assumption about \(e_{3}\) is necessary for \(\theta_{2}\) to be given the causal interpretation: an increase in house size of 100 square feet leads to a \(\theta_{2}\) increase in \(\ln (P R I C E)\), when all other variables are held constant?

d. We will investigate whether this assumption might be violated. Estimate the following equation and report the results

e. A comparison of this equation with that in part (c) suggests \(e_{3}=\delta_{3} A G E+\delta_{4} A G E^{2}+e_{4}\). Assume \(E\left(e_{4} \mid S Q F T, A G E\right)=0\). We wish to investigate whether \(E\left(e_{3} \mid S Q F T\right)=0\). Show that \(E\left(e_{3} \mid S Q F T\right)=0\) if \(\delta_{3}=\delta_{4}=0\) or if \(E(A G E \mid S Q F T)=0\) and \(E\left(A G E^{2} \mid S Q F T\right)=0\).

f. Test the hypothesis \(H_{0}: \delta_{3}=\delta_{4}=0\) at a \(5 \%\) significance level.

g. Estimate the equations \(A G E=\lambda_{1}+\lambda_{2} S Q F T+u_{2}\) and \(A G E^{2}=\phi_{1}+\phi_{2} S Q F T+u_{3}\). Use a \(5 \%\) significance level to test the hypotheses \(H_{0}: \lambda_{2}=0\) and \(H_{0}: \phi_{2}=0\).

h. What do you conclude about the assumption \(E\left(e_{3} \mid S Q F T\right)=0\) ?

Transcribed Image Text:

In(PRICE)= + BEDROOMS + e In(PRICE) B+ BBEDROOMS + B3SQFT + e = SQFT=Y1+YBEDROOMS +u