Exercises

(More difficult exercises are marked with a *)

1. Comment briefly on the meaning of each of the following.

(a) Estimated coefficient.

(b) Standard error.

(c) r-statistic.

(d) i?-squared.

(e) Sum of squared residuals.

(f) Standard error of the regression.

(g) Best linear unbiased estimator.

2. A store manager selling TV sets observes the following sales on 10 different days. Calculate the regression of on : where

= number of TV sets sold

: = number of sales representatives

Present all the items mentioned in Exercise 1. 3. The following data present experience and salary structure of University of Michigan economists in 1983-1984. The variables are

= salary (thousands of dollars)

: = years of experience (defined as years since receiving Ph.D.)

Source: R. . Frank, "Are Workers Paid Their Marginal Products?" Economic Review, September 1984. Table 1, p. 560.

The American

Calculate the regression of on x. Present all the items mentioned in Exercise 1. Give reasons why the regression does or does not make sense. Calculate the residuals to see whether there are any outliers. Would you discard these observations or look for other explanations? 4. Show that the simple regression line of against : coincides with the simple regression line of : against if and only if = 1 (where r is the sample correlation coefficient between jc and y).

(a) Calculate a regression of on x.

= a + Px + M

(b) Construct a 95% confidence interval forip.

(c) Test the hypothesis : P = 0 against the alternative p 5 0 at the 5% significance level.

(d) Construct a 90% confidence interval for tr = ( ).

(e) What is likely to be wrong with the assumptions of the classical normal linear model in this case? Discuss.

7. Let u, be the residuals in the least squares fit of y, against x, (/ = 1, 2, . . . , ). Derive the following results:

X «. = 0 and 2 x,M, = 0 „1 ,=1

8. Given data on and x explain what functional form you will use and how you will estimate the parameters if

(a) is a proportion and lies between 0 and 1.

(b) X > 0 and X assumes very large values relative to y.

(c) You are interested in estimating a constant elasticity of demand function.

9. At a large state university seven undergraduate students who are majoring in economics were randomly selected from the population and surveyed. Two of the survey questions asked were: (1) What was your grade-point average (GPA) in the preceding term? (2) What was the average number of hours spent per week last term in the Orange and Brew? The Orange and Brew is a favorite and only watering hole on campus. Using the data below, estimate with ordinary least squares the equation

G = a +

5. In the regression model y, = a + p,x, + ii, if the sample mean of x of x is zero, show the cov(a, p) = 0, where a and p are the least squares estimators of a and p.

6. The following are data on

= quit rate per 100 employees in manufacturing X = unemployment rate

The data are for the United States and cover the period 1960-1972.

Hours per Week in

10. Two variables and x are believed to be related by the following stochastic equation:

= a + ( + M

where is the usual random disturbance with zero mean and constant variance . To check this relationship one researcher takes a sample size of 8 and estimates (3 with OLS. A second researcher takes a different sample size of 8 and also estimates p with OLS. The data they used and the results they obtained are as follows:

Researcher 1 Researcher 2

Can you explain why the standard error of p for the first researcher is larger than the standard error of p for the second researcher? 11. Since the variance of the regression coefficient p varies inversely with the variance of x, it is often suggested that we should drop all the observations in the middle range of x and use only the extreme observations on x in the calculation of p. Is this a desirable procedure?

where G is GPA and H is hours per week in Orange and Brew. What is the expected sign for (3? Do the data support your expectations?