In this exercise we reconsider the premium wine data in the file wine 1. Please see Exercise 6.32 and wine1.def for details.

a. Estimate the following equation using (i) only cabernet wines, (ii) only pinot wines, and (iii) all other varieties:

Using casual inspection, do you think separate equations are needed for the different varieties?

b. We can develop an \(F\)-test to test whether there is statistical evidence to suggest the coefficients in the three equations are different. The unrestricted sum of squared errors for such a test is

Compute \(S S E_{U}\).

c. What is the total number of parameters from the three equations? How many parameters are there when we estimate one equation for all varieties? How many parameter restrictions are there if we restrict corresponding coefficients for all varieties to be equal?

d. Estimate one equation for all varieties. This is the restricted model where corresponding coefficients for the different varieties are assumed to be equal.

e. Using a 5\% significance level, test whether there is evidence to suggest there should be different equations for different varieties. What is the null hypothesis for this test? Develop some notation that enables you to state the null hypothesis clearly and precisely.

Data From Exercise 6.32:-

In their study of the prices of Californian and Washington red wines, Costanigro, Mittelhammer and McCluskey \(^{21}\) categorize the wines into commercial, semipremium, premium, and ultrapremium. Their data for premium wines are stored in the file wine 1 ; those for ultrapremium wines are in the file wine 2. We will be concerned with the variables PRICE (bottle price, CPI adjusted), SCORE (score out of 100 given by the Wine Spectator Magazine), \(A G E\) (years of aging), and CASES (number of cases produced in thousands).

a. What signs would you expect on the coefficients \(\left(\beta_{2}, \beta_{3}, \beta_{4}\right)\) in the following model? Why?

b. Estimate separate equations for premium and ultrapremium wine, and discuss the results. Do the coefficients have the expected signs? If not is there an alternative explanation? Is SCORE more important for premium wines or ultrapremium wines? Is \(A G E\) more important for premium wines or ultrapremium wines?

c. Find point and \(95 \%\) interval estimates for i. \(E[\ln (P R I C E) \mid S C O R E=90, A G E=2, C A S E S=2]\) for premium wines, and ii. \(E[\ln (P R I C E) \mid S C O R E=93, A G E=3, C A S E S=1]\) for ultrapremium wines. Do the intervals overlap, or is there a clear price distinction between the two classes?

d. Using the "corrected predictor"-see Section 4.5.3-predict the prices for premium and ultrapremium wines for the settings in parts \(\mathrm{c}(\mathrm{i})\) and c(ii), respectively.

e. Suppose that you are a wine producer choosing between producing 1000 cases of ultrapremium wine that has to be aged three years and is likely to get a score of 93, and 2000 cases of premium wine that is aged two years and is likely to get a score of 90 . Which choice gives the higher expected bottle price? Which choice gives the higher expected revenue? (There are 12 bottles in a case of wine.)

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In(PRICE)=, + BSCORE + B3AGE + B4 CASES + e