Assuming that the demand for oranges is given by the linear equation

= a + fx +

estimate the parameters of this equation. Calculate a 90% confidence interval for the quantity of oranges sold in week seven if the price is 25 cents per orange during that week.

14. In Exercise 9 we considered the relationship between grade-point average (GPA) and weekly hours spent in the campus pub. Suppose that a freshman economics student has been spending 15 hours a week in the Orange and Brew during the first two weeks of class. Calculate a 90% confidence interval for his GPA in his first quarter of college if he continues to spend 15 hours per week in the Orange and Brew. Suppose that this freshman remains in school for four years and completes the required 12 quarters for graduation. Calculate a 90% confidence interval for his final cumulative GPA. Suppose that the minimum requirements for acceptance into most graduate schools for economics is 3.25. What are the chances that this student will be able to go to graduate school after completing his undergrate studies?

15. A local night entertainment establishment in a small college town is trying to decide whether they should increase their weekly advertising expenditures on the campus radio station. The last six weeks of data on monthly revenue (y) and radio advertising expenditures (x) are given in the accompanying table. What would their predicted revenues be next week if they spent $500 on radio commercials. Give a 90% confidence interval for next weeks revenues. Suppose that they spend $500 per week for the next 10 weeks. Give a 90% confidence interval for the average revenue over the next 10 weeks.

12. Suppose you are attempting to build a model that explains aggregate savings behavior as a function of the level of interest rates. Would you rather sample during a period of fluctuating interest rates or a period of stable interest rates? Explain.

13. A small grocery store notices that the price it charges for oranges varies greatly throughout the year. In the off-season the price was as high as 60 cents per orange, and during the peak season they had special sales where the price was as low as 10 cents, 20 cents, and 30 cents per orange. Below are six weeks of data on the quantity of oranges sold (y) and price (x):

6. In the model y, = a + px, + „ / = 1, . . . , iV, the following sample moments have been calculated from 10 observations:

Sy, = 8 Ix, = 40 = 26 2x] = 200 x.y, = 20

(a) Calculate the predictor of for x = 10 and obtain a 95% confidence interval for it.

(b) Calculate the value of x that could have given rise to a value of = I and explain how you would find a 95% confidence interval for it.

7. (Instrumental variable method) Consider the linear regression model

y, = a + px, + u,

One of the assumptions we have made is that x, are uncorrelated with the errors u,. If X, are correlated with u„ we have to look for a variable that is uncorrelated with u, (but correlated with x,). Let us call this variable z,. z, is called an instrumental variable. Note that as explained in Section 3.3, the assumptions E(u) = 0 and cov(x, ) = 0 are replaced by

lu, = 0 and Ix.u, = 0

However, since x and are correlated we cannot use the second condition. But since z and are uncorrelated, we use the condition co\{z, u) = 0. This leads to the normal equations

= 0 and IzA = 0

The estimates of a and p obtained using these two equations are called the instrumental variable estimates. From a sample of 100 observations, the following data are obtained:

Sy? = 350 2x,y, = 150 IxJ = 400 Sz,y, = 100 lz,x, = 200 Izj = 400 ly, = 100 Ix, = 100 Iz, = 50 Calculate the instrumental variables estimates of a and p.

Let the instrumental variable estimator of p be denoted by p* and the least squares estimator p be p. Show that p* = Sy/S with Sy and S„ defined in a similar manner to 5„, 5„ and 5,. Show that

Hence, var(p*) > var(p). To test whether x and are correlated or not, the following test has been suggested (Hausmans test discussed in Chapter 12).

t= -N(0, 1)

Apply this test to test whether cov(x, ) = 0 with the data provided. *18. (Stochastic regressors) In the linear regression model

y, = a + px, + u,

Suppose that x, ~ IN(0, k). What is the (asymptotic) distribution of the least squares estimator p? Just state the result. Proof not required. *19. Consider the regression model

y, = a + px, + M,

u, ~ IN(0, 1) / - 1, 2, . . . ,

Suppose that the model refers to semiannual data, but the data available are either:

1. Annual data, that is

= + = + 4 = + etc. with , 2, , . . . , defined analogously (assuming that is even), or

2. Moving average data, that is,

=

= -2-

with xj, 2, , . . . defined analogously.

(a) What are the properties of the error term in the regression model with each sets of data: (y„ x,), (y„ x,), and (y*, x*)?

(b) How would you estimate p in the case of annual data?

(c) How would you estimate p in the case of moving average data?

(d) Would you rather have the annual data or moving average data? Why?

*20. Given data on and x explain how you will estimate the parameters in the following equations by using the ordinary least squares method. Specify the assumptions you make about the errors.

(a) = ctxP

(b) = ae

5-, / 1 \