This exercise uses data on 850 houses sold in Baton Rouge, Louisiana during mid-2005. We will be concerned with the selling price in thousands of dollars (PRICE), the size of the house in hundreds of square feet (SQFT), and the age of the house in years \((A G E)\). The following two regression models were estimated:

The sums of squares and sums of cross products of the residuals from estimating these two equations are \(\sum_{i=1}^{850} \hat{v}_{i}^{2}=10377817, \sum_{i=1}^{850} \hat{u}_{i}^{2}=75773.4, \sum_{i=1}^{850} \hat{u}_{i} \hat{v}_{i}=688318\)

a. Find the least-squares estimate of \(\beta_{2}\) in the model \(P R I C E=\beta_{1}+\beta_{2} S Q F T+\beta_{3} A G E+e\).

b. Let \(\hat{e}_{i}=\hat{v}_{i}-b_{2} \hat{u}_{i}\). Show that \(\sum_{i=1}^{850} \hat{e}_{i}^{2}=\sum_{i=1}^{850} \hat{v}_{i}^{2}-b_{2} \sum_{i=1}^{850} \hat{v}_{i} \hat{u}_{i}\) where \(b_{2}\) is the least-squares estimate for \(\beta_{2}\).

c. Find an estimate of \(\sigma^{2}=\operatorname{var}\left(e_{i}\right)\).

d. Find the standard error for \(b_{2}\).

e. What is an approximate \(p\)-value for testing \(H_{0}: \beta_{2} \geq 9.5\) against the alternative \(H_{1}: \beta_{2}

Transcribed Image Text:

PRICE = +AGE+v and SQFT = 8 +82AGE + u