Essentials of EconometricsTutorial Exercises SW 8
8.1 Sales in a company are $196 million in 2009 and increase to $198 million in 2010.
−Sales2009
(a) Compute the percentage increase in sales using the usual formula 100 × Sales2010
.
Sales2009
Compare this value to the approximation 100 × [ln(Sales2010 ) − ln(Sales2009 )]
(b) Repeat (a) assuming Sales2010 = 205; Sales2010 = 250; Sales2010 = 500.
(c) How good is the approximation when the change is small? Does the quality of the
approximation deteriorate as the percentage change increases?
8.2 Suppose that a researcher collects data on houses that have sold in a particular neighborhood
over the past year and obtains the regression results in the table shown below.
(a) Using the results in column (1), what is the expected change in price of building a 500square-foot addition to a house? Construct a 95% confidence interval for the percentage
change in price.
(b) Comparing columns (1) and (2), is it better to use Size or ln(Size) to explain house
prices?
(c) Using column (2), what is the estimated effect of pool on price? (Make sure you get the
units right.) Construct a 95% confidence interval for this effect.
(d) The regression in column (3) adds the number of bedrooms to the regression. How large
is the estimated effect of an additional bedroom? Is the effect statistically significant?
Why do you think the estimated effect is so small? (Hint:Which other variables are being
held constant?)
(e) Is the quadratic term ln(Size)2 important?
(f ) Use the regression in column (5) to compute the expected change in price when a pool is
added to a house without a view. Repeat the exercise for a house with a view. Is there
a large difference? Is the difference statistically significant?
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8.3 After reading this chapter’s (Stock and Watson Chapter 8) analysis of test scores and
class size, an educator comments, “In my experience, student performance depends on
class size, but not in the way your regressions say. Rather, students do well when class
size is less than 20 students and do very poorly when class size is greater than 25. There
are no gains from reducing class size below 20 students, the relationship is constant in
the intermediate region between 20 and 25 students, and there is no loss to increasing
class size when it is already greater than 25.” The educator is describing a “threshold
effect” in which performance is constant for class sizes less than 20, then jumps and is
constant for class sizes between 20 and 25, and then jumps again for class sizes greater
than 25. To model these threshold effects, define the binary variables
ST Rsmall = 1 if ST R < 20, and ST Rsmall = 0 otherwise;
ST Rmoderate = 1 if 20 ≤ ST R ≤ 25, and ST Rmoderate = 0 otherwise; and
ST Rlarge = 1 if ST R > 25, and ST Rl arg e = 0 otherwise.
(a) Consider the regression T estScorei = β0 + β1 ST Rsmalli + β2 ST Rlargei + ui . Sketch the
regression function relating T estScore to ST R for hypothetical values of the regression
coefficients that are consistent with the educator’s statement.
(b) A researcher tries to estimate the regression T estScorei = β0 +β1 ST Rsmalli +β2 ST Rmoderatei +
β3 ST Rlargei + ui and finds that her computer crashes. Why?
8.4 Read the box “The Returns to Education and the Gender Gap” in Section 8.3 of the
Stock and Watson textbook.
(a) Consider a man with 16 years of education, and 2 years of experience, who is from a
western state. Use the results from column (4) of Table 8.1 and the method in Key
Concept 8.1 to estimate the expected change in the logarithm of average hourly earnings
(AHE) associated with an additional year of experience.
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(b) Repeat (a) assuming 10 years of experience.
(c) Explain why the answers to (a) and (b) are different.
(d) Is the difference in the answers to (a) and (b) statistically significant at the 5% level?
Explain.
(e) Would your answers to (a)–(d) change if the person was a woman? From the South?
Explain.
(f ) How would you change the regression if you suspected that the effect of experience on
earnings was different for men than for women?
8.5 Read the box “The Demand for Economics Journals” in Section 8.3 of the Stock and
Watson textbook.
(a) The box reaches three conclusions. Looking at the results in the table, what is the
basis for each of these conclusions?
(b) Using the results in regression (4), the box reports that the elasticity of demand for
an 80-year-old journal is -.28.
(i) How was this value determined from the estimated regression?
(ii) The box reports that the standard error for the estimated elasticity is 0.06. How
would you calculate this standard error? (Hint: See the discussion “Standard
errors of estimated effects” on page 302 of the Stock and Watson textbook.)
(iii) Suppose that the variable Characters had been divided by 1,000 instead of
1,000,000. How would the results in column (4) change?
8.6 Refer to Table 8.3.
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(a) A researcher suspects that the effect of %Eligible f or subsidized lunch has a nonlinear
effect on test scores. In particular, he conjectures that increases in this variable from
10% to 20% have little effect on test scores, but that changes from 50% to 60% have a
much larger effect.
(i) Describe a nonlinear specification that can be used to model this form of nonlinearity.
(ii) How would you test whether the researcher’s conjecture was better than the linear
specification in column (7) of Table 8.3?
(b) A researcher suspects that the effect of income on test scores is different in districts with
small classes than in districts with large classes.
(i) Describe a nonlinear specification that can be used to model this form of nonlinearity.
(ii) How would you test whether the researcher’s conjecture was better than the linear
specification in column (7) of Table 8.3?
8.7 This problem is inspired by a study of the “gender gap” in earnings in top corporate jobs
[Bertrand and Hallock (2001)]. The study compares total compensation among top executives
in a large set of U.S. public corporations in the 1990s. (Each year these publicly traded
corporations must report total compensation levels for their top five executives.)
(a) Let F emale be an indicator variable that is equal to 1 for females and 0 for males. A
regression of the logarithm of earnings onto F emale yields
\
ln(Earnings)
= 6.48 − 0.44 F emale, SER = 2.65.
(0.01)
(0.05)
(i) The estimated coefficient on F emale is −0.44. Explain what this value means.
(ii) The SER is 2.65. Explain what this value means.
(iii) Does this regression suggest that female top executives earn less than top male
executives? Explain.
(iv) Does this regression suggest that there is gender discrimination? Explain.
(b) Two new variables, the market value of the firm (a measure of firm size, in millions of
dollars) and stock return (a measure of firm performance, in percentage points), are
added to the regression:
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\
ln(Earnings)
= 3.86 − 0.28 F emale + 0.37 ln(M arketV alue) + 0.004Return, n =
(0.03)
(0.04)
(0.004)
2
(0.003)
46, 670, R = 0.345.
(i) The coefficient on ln(M arketV alue) is 0.37. Explain what this value means.
(ii) The coefficient on F emale is now −0.28. Explain why it has changed from the
regression in (a).
(c) Are large firms more likely to have female top executives than small firms? Explain.
8.8 X is a continuous variable that takes on values between 5 and 100. Z is a binary variable.
Sketch the following regression functions (with values of X between 5 and 100 on the horizontal
axis and values of Yb on the vertical axis):
(a) Yb = 2.0 + 3.0 × ln(X)
(b) Yb = 2.0 − 3.0 × ln(X)
(c)
(d)
i. Yb = 2.0 + 3.0 × ln(X) + 4.0Z, with Z = 1
ii. Same as (i), but with Z = 0.
i. Yb = 2.0 + 3.0 × ln(X) + 4.0Z − 1.0 × Z × ln(X), with Z = 1
ii. Same as (i), but with Z = 0.
(e) Yb = 1.0 + 125.0X − 0.01X 2
8.9 Explain how you would use “Approach #2” of Section 7.3 to calculate the confidence interval
discussed below Equation (8.8). [Hint : This requires estimating a new regression using a
different definition of the regressors and the dependent variable. See Exercise (7.9).]
8.10 Consider the regression model Yi = β0 X1i + β2 X2i + β3 (X1i × X2i ) + ui . Use Key Concept 8.1
in the Stock and Watson textbook to show:
∆Y
= β1 + β3 X2 (effect of change in X1 holding X2 constant).
(a) ∆X
1
∆Y
= β2 + β3 X1 (effect of change in X2 holding X1 constant).
(b) ∆X
2
(c) If X1 changes by ∆X1 and X2 changes by ∆X2 , then ∆Y = (β1 + β3 X2 )∆X1 + (β2 +
β3 X1 )∆X2 + β3 ∆X1 ∆X2 .
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