8.1 Given the Following Table (Table 8.1 from the Textbook)

Chapter 8 Exercises XXX

Chapter 8 Model Estimation

8.1 Given the following table (Table 8.1 from the textbook):

Table 8.1 — Hypothetical Demand for Oranges

Day(i) Quantity (Yi) Cents (Xi) (Yi – ) (Xi – ) (Xi – )(Yi – ) (Xi – )2 (Yi – )2

1 55 100 –45 30 –1,350 900 2025

2 70 90 –30 20 –600 400 900

3 90 80 –10 10 –100 100 100

4 100 70 0 0 0 0 0

5 90 70 –10 0 0 0 100

6 105 70 5 0 0 0 25

7 80 70 –20 0 0 0 400

8 110 65 10 –5 –50 25 100

9 125 60 25 –10 –250 100 625

10 115 60 15 –10 –150 100 225

11 130 55 30 –15 –450 225 900

12 130 50 30 –20 –600 400 900

Sums: 1,200 840 0 0 –3,550 2,250 6,300

Source: Jan Kmenta, Elements of Econometrics, second edition, Macmillan Publishing Company, 1986.

(a) Plot the scatter plot of X and Y on the following figure.

(b) Calculate the means of X and Y:

(d) The sign of the covariance indicates how Y varies with X. In this case, when X tends to be above its mean, Y tends to be below its mean, and when X tends to be below its mean, Y tends to be above its mean. Thus, these variables are (directly, inversely) related.

(e) Use the information in the table to calculate the variance of X and Y:

(f) The least-squares estimate of b is

(g) The least-squares estimate of a is

(h) Write the estimated least-squares equation:

(i) Verify that this equation passes through the point of the means by calculating at X = :

(j) Plot this equation on the same graph as the scatter plot. Interpret the estimated coefficients.

(k) Compute for each observation and complete the appropriate column in the following extension of Table 8.1:

Table 8.1 (continued)

Day(i) Quantity (Yi) Cents (Xi) (Yi – ) (Yi – )2 ( – ) ( – )2

1 55 100

2 70 90

3 90 80

4 100 70

5 90 70

6 105 70

7 80 70

8 110 65

9 125 60

10 115 60

11 130 55

12 130 50

Sums: 1,200 840

(l) Calculate i = Yi – i and i2 for each observation, and complete the appropriate columns in the table.

(m) Calculate and interpret the standard error of regression:

(n) Calculate and interpret the standard error of the estimated slope coefficient:

(o) Calculate and interpret R2 using

(p) Calculate the adjusted R2 using the following formula; is (smaller, larger) than R2:

8.2 The capital asset pricing model (CAPM) says that

mj = c + bj(mm – c)

where mj is the expected return to any asset j, c is a risk-free return, and mm is a measure of the return to the market as a whole. The term bj(mm – c) is called the risk-adjustment factor (or risk premium), the extra amount over the sure return c that is required to compensate for the risk of buying asset j. To implement the CAPM, we can estimate beta for any asset using historical data. The sure return c is subtracted from both sides of the equation, the empirically observed return rj for asset j is substituted for mj, and the empirically observed return rm is substituted for mm. The model to be estimated is

rjt – ct = aj + bj(rmt – ct) + et

where t is the periodic observation index, (rjt – ct) is the periodic rate of return for asset j above the risk-free rate ct in period t, aj is an intercept added to avoid forcing the estimated equation through the origin, and (rmt – ct) is the market return above the risk-free rate. As before, et is a random error term. Empirical experience with this model has been fairly successful with researchers obtaining estimates of betas that are statistically significant.

Suppose that you collect data on ten years of monthly returns rjt for two companies, Mobil (j=1) and Motorola (j=2). The market rate rmt used is a volume-adjusted composite monthly return based on all stock transactions from the New York Stock Exchange and the American Exchange over the same ten year-time span. The risk-free return ct is the return on 30-day U.S. Treasury bills. Ordinary least squares regression yields (t-ratios in parentheses):

Mobil: = 0.0079 + 0.7815(rmt – ct) = 0.3810

(1.019) (5.974)

Motorola: = 0.0063 + 1.3381(rmt – ct) = 0.5536

(0.672) (8.482)

(a) What is the estimated beta of the asset for Mobil? ______. What is the estimated beta of the asset for Motorola? ______.

(b) Comment on the magnitude and precision of each of the estimated betas. Is this about what you would expect? ______.

(c) Comment on the magnitude and precision of each of the estimated intercept terms. Is this about what you would expect? ______.

(d) Calculate the standard error of the estimate for b1. ______. Calculate the standard error of the estimate for b2. ______.

(e) How would you test a null hypothesis that each company’s risk is the same as the average risk over the entire market? That is, test that b = 1 against the alternative that b ≠ 1. Using the rule of thumb, are either of these estimated betas different from unity?

(f) The estimated CAPM can be used to measure the proportion of the variation in the dependent variable explained by the variation in the independent variable. In the CAPM context, R2 is said to be a measure of the impact of market (systematic) risk, and 1 – R2 is a measure of the impact of asset-specific (unsystematic) risk. For Mobil, what is the proportion of the total variation attributed to market risk, and what is the proportion of the total variation attributed to asset-specific risk? ______. For Motorola, what is the proportion of the total variation attributed to market risk, and what is the proportion of the total variation attributed to asset-specific risk? ______.

(g) According to the CAPM model, the beta of the jth asset is

Using the empirical results, is the Mobil asset more or less risky than the market as a whole? How do you know? ______. Is the Motorola asset more or less risky than the market as a whole? How do you know? ______.

8.3 Michael Lowell (“Tests of the Rational Expectations Hypothesis,” American Economic Review, March 1986) estimated the following model of gasoline mileage of various models of cars (standard errors in parentheses):

= 22.008 – 0.002Wi – 2.76Ai + 3.28Di + 0.415Ei = 0.82

(0.001) (0.71) (1.41) (0.097)

where Gi = miles per gallon of the ith model based on road tests

Wi = weight (in pounds) of the ith model

Ai = an automatic transmission dummy variable

Di = a diesel engine dummy variable

Ei = estimate of miles per gallon published by the U.S. EPA

A dummy variable takes a value of unity or zero depending on whether some condition does or does not hold.

(a) Comment on the overall goodness of fit. What is the advantage in using the adjusted R2? ______.

(b) Are the signs of the estimated coefficients consistent with your expectations? Comment on each. ______.

(d) Using the rule of thumb, comment on the precision of the coefficient estimate for each variable. ______.

(e) The variable Ei plays a unique role in this specification. If the goal of the research was to predict gas mileage, that variable probably would not be included. Why, then, did Lowell include Ei as an explanatory variable?

8.4 In a classic study of returns to on-the-job training, Jacob Mincer (Schooling, Experience and Earnings, Columbia University Press, 1974) used 31,093 observations from 1959 data on white, non-farm, non-student males up to the age 65. Mincer began by running a simple least-squares regression of log earnings on schooling (t-ratios in parentheses):

= 7.58 + 0.070S R2 = 0.067

(43.8)

where E is annual earnings and S is years of schooling. This implies a 7% annual return to schooling because the slope ∂lnE/∂S = 0.07 is the (constant) rate of return of an extra year of schooling for this specification.

(a) How reliable is the estimate of the rate of return to schooling in Mincer’s beginning regression estimate? ______.

(b) This specification implies that the rate of return is constant. Is that plausible? Why or why not? ______.

(c) When Mincer added experience (X) and its square (X2) to the equation, where X equals age minus S minus 6, he obtained:

= 6.20 + 0.107S + 0.081X – 0.0012X2 R2 = 0.285

(72.3) (75.5) (–55.8)

What is the new estimate of the rate of return to schooling? ______. Are the signs of the coefficients attached to X and X2 plausible? Why or why not? ______. Are these coefficients very precise? Why or why not? ______. What was the increase in the proportion of the variance in the dependent variable explained when X and X2 were added to the regression? ______.

(d) Mincer also reports estimates of a model allowing for interactions between schooling and experience (SX), and he allowed for diminishing returns to schooling:

= 4.87 + 0.255S – 0.0029S2 – 0.0043SX + 0.148X – 0.0018X2 R2 = 0.309

(2.34) (–7.1) (–31.8) (63.7) (–66.2)

This implies that the effect of schooling on log-earnings is given by this partial derivative:

∂lnE/∂S = 0.255 – 0.0058S – 0.0043X

If X = 8, find the marginal rates of returns to schooling at 8 years, 12 years, and 16 years: ______.

(e) From the estimated earnings equation in (d), the effect of experience on earnings at various levels of schooling is given by

∂lnE/∂X = 0.148 – 0.0043S – 0.0036X

Set this equation equal to zero to find the X* that maximizes ln E; then evaluate your expression at S = 8, 12, and 16 to find the different X* values. Finally, to find the age at which lnE is maximized, add (S + 6) to each of these X* values. Is there much of a difference? ______.

(f) Using the information from (d) and (e), what shape would you attribute to the age-earnings profile of more highly educated men compared to those with less education? Explain. ______.

8.5 Price indexes play an important role in today’s economies, and so it is important that they be constructed with accuracy. When looking at the price of any product over time, however, the quality of the product is likely to be changing. Multivariate regression analysis can be used to measure the extent to which prices have changed over time, controlling for changes in quality, using what is called a “hedonic” price equation.

In one such application, Gregory Chow (“Technological Change and the Demand for Computers,” American Economic Review, December 1967) used data on computer rental rates over the 1955-1965 time period. During this era, most computers were bundled with software. Chow chose three variables to control for quality: Multiply — the time in microseconds required to do a multiplication instruction; Memory — measured as the product of the number of words in main memory (in thousands) and the number of binary digits per word; and Access — the average time needed to access information from memory. Chow argued that other potential explanatory variables are likely to be highly correlated with the three included variables. Because pre-1960 data were scarce, and to avoid mixing data on first-generation computers with more advanced models, Chow pooled his data on different computers for the years 1960 to 1965 and used dummy variable intercept terms for each year 1961 to 1965. His estimated equation is:

Price = – 0.1045 – 0.0654Multiply + 0.5793Memory – 0.1406Access

(0.0284) (0.0354) (0.0293)

– 0.1398D61 –0.4891D62 –0.5938D63 –0.9248D64 –1.1630D65

(0.1665) (0.1738) (0.1661) (0.1663) (0.1660)

where D61 through D65 are yearly dummy variables. All other variables are in natural logarithms. From 82 observations, Chow reported that R2 = 0.908. Standard errors are in parentheses.

(a) What is the proportion of the variation in the dependent variable explained by this regression? ______.

(b) What problem may have been encountered if other quality variables had been included in the regression? ______.

(d) Judge the precision of the estimated coefficients for each of the dummy variables. ______.

(e) Examine the signs and magnitudes of each of the dummy variable coefficients. What implications does this have about the price of computers over this period? ______.

(f) If you were to conduct a similar study today using more recent data, can you think of any other quality variables that might be included? ______.

8.6 For the preceding problem, use a calculator to compute the antilogarithms of the dummy variable coefficients, then divide these by the antilogarithm of the intercept. These ratios can be interpreted as quality adjusted price indexes for the years 1961 to 1965 with 1960 as the base year. Complete the following table:

Year Coefficient Antilog Price Index

1960 –0.1045 1.0000

1961 –0.1398

1962 –0.4891

1963 –0.5938

1964 –0.9248

1965 –1.1630

Based on these estimated price indexes, calculate the average annual growth rate in the quality-adjusted price of computers from 1960 to 1965. ______.