The file collegetown contains data on 500 single-family houses sold in Baton Rouge, Louisiana during 2009-2013. 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 measured as a categorical variable \((A G E)\), with 1 representing the newest and 11 the oldest. Let \(\mathbf{X}\) denote all observations on \(S Q F T\) and \(A G E\). Use all observations to estimate the following regression model:

a. Report the results. Are the estimated coefficients significantly different from zero?

b. Write down expressions for the marginal effects \(\partial E(P R I C E \mid \mathbf{X}) / \partial S Q F T\) and \(\partial E(P R I C E \mid \mathbf{X}) / \partial A G E\). Interpret each of the coefficients. Given the categorical nature of the variable \(A G E\), what assumptions are being made?

c. Find point and \(95 \%\) interval estimates for the marginal effect \(\partial E(P R I C E \mid \mathbf{X}) / \partial S Q F T\) for houses that are (i) 5 years old, (ii) 20 years old, and (iii) 40 years old. How do these estimates change as \(A G E\) increases? (Refer to the file collegetown.def for the definition of \(A G E\).)

d. As a house gets older and moves from one age category to the next, the expected price declines by \(\$ 6000\). Set up this statement as a null hypothesis for houses with (i) 1500 square feet, (ii) 3000 square feet, and (iii) 4500 square feet. Using a \(5 \%\) significance level, test each of the null hypotheses against an alternative that the price decline is not \(\$ 6000\). Discuss the outcomes.

e. Find a \(95 \%\) prediction interval for the price of a 60 -year old house with 2500 square feet. In the data set there are four 60-year old houses with floor space between 2450 and 2550 square feet. What prices did they sell for? How many of these prices fall within your prediction interval? Is the model a good one for forecasting price?

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PRICE = B+BSQFT+B3(SQFT AGE) + e