Econ 141

56 CHAPTER 2 Review of Probability

2.5 Suppose that K , . . . ,Yn are i.i.d. random variables with a N(l. 4) distribu-tion. Sketch the probability density of Y when n 2. Repeat this for n = 10 and n = 100. In words, describe how the densities differ. What is the rela-tionship between your answer and the law of large numbers?

2.6 Suppose that Y\,...,Yn are i.i.d. random variables with the probability dis-tribution given in Figure 2.10a. You want to calculate Pr( Y < 0.1). Would it be reasonable to use the normal approximation if n -- 5? What about n = 25 or n - 100? Explain.

2.7 Y is a random variable with ixY = 0,aY 1, skewness - 0, and kurtosis = 100. Sketch a hypothetical probability distribution of Y. Explain why n ran-dom variables drawn from this distribution might have some large outliers.

Exercises

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58 CHAPTER 2 Review of Probability x Exercises 59

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T h a t i s , P r ( A = l , Y - 1 4 ) - 0.02, and so forth. a. Calculate the probability distribution, mean, and variance of Y. b. Calculate the probability distribution, mean, and variance of Y given

A"-8. c. Calculate the covanance and correlation between A'and Y.

205^ Compute the following probabilities: g ^ - l _ V v \ ' " p . If Yis distributed N(l,4),iind Pr(Y ^ 3).

b. If Yisdistributed;V(3,9),findPr(y > 0). \ W ^ l c. If Y is distributed /V(50.25),find Pr(40 < Y < 52).

d. If Y is distributed N(5.2). find Pr(6 < Y < 8). 2.11 Compute the following probabilities:

a. If Y is distributed * i find Pr( Y < 7.78). p ^ - ^ p - V ^ \ b. If y is distributed ^n . find P r ( K > 18.31).

~~ c. If y is distributed Fm^, find Pr( y > 1.83). d. Why are the answers to (b) and (c) the same? e. If Vis distributed*2!, find Pr(y < 1.0). (Hint: Use the definition of

the x\ distribution.) 2.12 Compute the following probabilities:

a. If y is distributed r15, find Pr( Y > 1.75). b. If y is distributed f90,find Pr(-1.99 < Y < 1.99).

c. If y is distributed N(0,1), find Pr(-1.99 < Y < 1.99) d. Why are the answers to (b) and (c) approximately the same? e. If Y is distributed F7.4, find Pr(Y > 4.12). f. If y is distributed F%m, find P r ( Y > 2.79).

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2.13 A is a Bernoulli random variable with P r ( A ^ 1) = 0.99, Y is distributed N(0,1), W is distributed N(0,100), and A, Y, and W are independent. Let

" 7 S - A T + ( l - A ) R ( T h a t i s , S - Y w h e n A = l , a n d = WwhenA = 0.) ; [ *51, ^ r ^ ^ b a. Show that (Y2) = 1 and (W2) = 100.

b. Show that ( y3) - 0 and ( IV3) - 0. (ffinc What is the skewness for a symmetric distribution?)

c Show that E( Y4) ~ 3 and E( W4) = 3 X 1002. {Hint: Use the fact that \\ \ the kurtosis is 3 for a normal distribution.)

C d* D e r i v e ( 5 ) - ( ^ 2 ) , (S3) and (54) . (Hint: Use the law of iterated "*** expectations conditioning on A ^ 0 and A = 1.) . * AA

e. Derive the skewness and kurtosis for S.^ ^^ ***? ^ C-N^ fi-o^^ N -^ -HKJ 2.1^ 98). c. In a random sample of size n = 64, find Pr( 101 ^ Y < 103).

2CXS Suppose Y{, i -= 1 ,2 , . . . , n, are i.i.d. random variables, each distributed N(10,4). a. Compute Pr(9.6 , < J s 10.4) when (i) n =-- 20, (ii) n = 100, and

b. Suppose c is a positive number. Show that Pr(10 - c < Y < 10 + c) . becomes close to 1.0 as n grows large. - -^* 2 ^ ^ . ^*Y * *"

c. Use your answer in (b) to argue that Y converges in probability to 10. 2.16 Y is distributed W(5,100) and you want to calculate Pr( Y < 3.6). Unfor-

tunately, you do not have your textbook and do not have access to a nor mal probability table like Appendix Table 1. However, you do have your computer and a computer program that can generate i.i.d. draws from the N(5,100) distribution. Explain how you can use your computer to compute an accurate approximation for Pr(Y < 3.6).

2.17 Yfri l, . . , , are i.i.d. Bernoulli random variables with p - 0.4. Let Y denote the sample mean. a. Use the central limit to compute approximations for

i. Pr( Y ^ 0.43) when n - 100. ii. Pr( Y

60 CHAPTER 2 Review of Probability Exercises 61

"V

b. How large would n need to be to ensure that Pr(0.39 s Y < 0.41) > 0.95? (Use the central limit theorem to compute an approximate answer.)

2.18 In any year, the weather can inflict storm damage to a home. From year to year, the damage is random. Let Y denote the dollar value of damage in any given year. Suppose that in 95% of the years Y ^ $0, but in 5% of the years Y - $20,000. a. What are the mean and standard deviation of the damage in any year? b. Consider an "insurance pool" of 100 people whose homes are suffi-

ciently dispersed so that, in any year, the damage to different homes can be viewed as independently distributed random variables. Let Y denote the average damage to these 100 homes in a year, (i) What is the expected value of the average damage Y? (ii) What is the proba-bility that Yexceeds $2000?

2.19 Consider two random variables X and Y. Suppose that Y takes on k values y\,>..,yk and that A takes on / values x\ ,X[. a. Show that Pr( Y - y}) = SLiPr(Y-y / |A '^j : I - ) Pr(A = x,). [Hint: Use

the definition of Pr(Y=-y}\X~*,-).] b. Use your answer to (a) to verify Equation (2.19). c. Suppose that X and Y are independent. Show that ...,im. The joint probability distribution of A, Y, Z is Pr(A - x, Y - y,Z z). and the conditional probability distribution of Y given A and Z is Pr( Y - y \ X = x, Z = z) - -^T^x^^z^z) U-a. Explain how the marginal probability that Y = y can be calculated

from the joint probability distribution. [Hint: This is a generalization of Equation (2.16).]

b. Show that E( Y) = E[E(Y'\X, Z)\ [Hint: This is a generalization of Equations (2.19) and (2.20).]

2.21 A is a random variable with moments E(X), E(X2), E(X3), and so forth. a. ShowE(X~ ti)^E(X3)~3[(X2)][E(X)] f 2[(A)]3. b. Show E(X - fi)4 = E(X4) - 4[E(A)1[E(A3)] -f 6[E(X)f[E(X2)]

3[E(X)f.

2.22 Suppose you have some money to invest for simplicity. $1 -and you are planning to put a fraction w into a stock market mutual fund and the rest, 1 - w, into a bond mutual fund. Suppose that $1 invested in a stock fund yields Rs after 1 year and that $1 invested in a bond fund yields Rb, sup-pose that Rs is random with mean 0.08 (8%) and standard deviation 0.07, and suppose that Rb is random with mean 0.05 (5%) and standard devia tion 0.04. The correlation between Rs and Rb is 0.25. If you place a fraction w of your money in the stock fund and the rest, 1 w, in the bond fund, then the return on your investment is R = wRs + (1 - w)Rb.

a. Suppose that w -= 0.5. Compute the mean and standard deviation of R. b. Suppose that w ~ 0.75. Compute the mean and standard deviation of R. c. What value of w makes the mean of R as large as possible? What is

the standard deviation of R for this value of w? d. (Harder) What is the value of w that minimizes the standard devia-

tion of Rl (Show using a graph, algebra, or calculus.)

2.23 This exercise provides an example of a pair of random variables A and Y for which the conditional mean of Ygiven Adependson AbutcorrfA, Y) = 0. Let X and Z be two independently distributed standard normal random variables, and let Y -= X2 + Z. a. ShowthatZf(Y|A)=A 2

b. Show that ixYLT 1. c Show thai E(AY) = 0. (Hint: Use the fact that the odd moments of a

standard normal random variable are all zero.) d. Show that cov(A, Y) ~- 0 and thus corr(A, Y) -- 0.

2.24 Suppose Yl is distributed i-i.d. N(0, a2) for i - 1, 2 , n.

a. Showtha t (Y? /o - 2 ) - l . b. Show that W - (l /o-2)^"-! Y2 is distributed Xl-c. Show that E( W) = n. [Hint: Use your answer to (a).]

d. Show that K - Y j n --1 is distributed t

2.25 (Review of summation notation.) Let x-, .., xn denote a sequence of num-bers ,^ , . . .,yn denote another sequence of numbers, and a, 6, and c denote three constants. Show that

62 CHAPTER 2 Review of Probability

âxiâ^xt HI i=]

HI (=i J=I

c. X a a HI n

d. y ( f l + ^+c>- I ) 2 =f l 2 + ft2^-l-c22)'/+2aft2^+2flc2>'/+ /=i /=] i=i i=i HI Ibc^XM

2.26 Suppose that Yu Y2,.. , Yn are random variables with a common mean p,y, a common variance oy, and the same correlation p (so that the correlation between Yt and Y} is equal to p for all pairs i and /, where / # ;). a. Show that cov( Y^ Yj) -* pa} for /" # ;. b. Suppose that n = 2. Show that (Y) - /Ay and var(Y) ^^YÎP^Y-c. Forrc > 2,showthat (Y) =p, y and var(Y) = cry/ J- [( - l)fn]p (W2). [//mf:Let h(Z) = g(Z)- E(AjZ) , so tha tK = [ A - E(X\Z)}-h(Z). Derive E(V2).]

2 . 1 D e r i v a t i o n o f R e s u l t s in K e y C o n c e p t 2 . 3

This appendix derives the equations in Key Concept 2.3. Equation (2.29) follows from the definition of the expectation. To derive Equation (2.30), use the definition of the variance to write var(a + bY)

{[fl-^6Y-(fl + 6Y)]2} = {[6(Y-/*r)l2 = 6 2 f i [ (Y-^) 2 ] = 6 2 ^ .

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Derivation of Results in Key Concept 2.3 63

To derive Equation (2.31), use the definition of the variance to write

var(oA + bY) = E{[(aX+bY) - (anx-*-bfir)f} = E{[a(X - ^ ) + 6(Y-/*y)]2} = E[a2(X - fxx)2} + 2E\ab(X- fix)(Y - ixY)]

+ E[b2(Y ixY)2) = a2var(A) ^2a6cov(A, F) + 62 var(K) = a2ax + 2abaxY + b2a^ (2.49)

where the second equality follows by collecting terms, the third equality follows by expand-ing the quadratic, and the fourth equality follows by the definition of the variance and covariance.

To derive Equation (2.32), write E(Y2) = E{[(Y~fj.Y) + Ay]2} = E[{Y~py)2] + 2 iYE{ Y-py) + fiY=o-Y + fiy because E(Y- tY) = 0.

To derive Equation (2.33J, use the definition of the covariance to write

covffl + bX + cV, Y) = E{[a + bX + cV- E{a + bX + cV)][Y- fj.Y]} = E{\b(X-fix)+c(V fxv)}[Y-nY}\ = E {[b(X ^ ) ] [Y- M y]} + E {[c(V- fLV)\[Y-pY]} - bam + ca-yy, (2.50)

which is Equation (2.33). To derive Equation (2.34), write E(XY) = E{[(X- p.x) + fj,x][{Y fiy) + fiY]} =

E[(X - ix)(Y- tiy)} + /xxE(Y - fiY) + fj.YE(X- fxx) 4- /xxfXy = rrxy + (JLX IY. We now prove the correlation inequality in Equation (2.35); that is, j corr (A, X)' s 1.

Let a = -

0 *1 t *

96 CHAPTER 3 Review of Statistics

causal effect (84) treatment effect (84) scatterplot(91) sample covariance (91) sample correlation coefficient

(sample correlation) (92)

power of a test (77) one-sided alternative hypothesis (79) confidence set (79) confidence level (79) confidence interval (79) coverage probability (81) test for the difference between two

means (81)

Review the Concepts

3.1 Explain the difference between the sample average Y and the population mean.

3.2 Explain the difference between an estimator and an estimate. Provide an example of each.

3.3 A population distribution has a mean of 10 and a variance of 16. Determine the mean and variance of Y from an i.i.d. sample from this population for (a) n = 10; (b) n 100; and (c) n = 1000. Relate your answers to the law of large numbers.

3.4 What role does the central limit theorem play in statistical hypothesis test-ing? In the construction of confidence intervals?

3.5 What is the difference between a null and alternative hypothesis? Among size, significance level, and power? Between a one-sided alternative hypoth-esis and a two-sided alternative hypothesis?

3.6 Why does a confidence interval contain more information than the result of a single hypothesis test?

3.7 Explain why the differences-of-means estimator, applied to data from a randomized controlled experiment, is an estimator of the treatment effect.

3.8 Sketch a hypothetical scatterplot for a sample of size 10 for two random vari-ables with a population correlation of (a) 1.0; (b) -1.0; (c) 0.9; (d) 0.5; (e) 0.0.

Exercises

(?.i In a population, fiy 100 and

98 CHAPTER 3 Review of Statistics

/

A- * ^ Q t Vol

a. You are interested in the competing hypotheses H0: p = 0.5 vs. //i: p ^ 0.5. Suppose that you decide to reject H0 if \p 0.5 [ > 0.02. i. What is the size of this test?

ii. Compute the power of this test if p = 0.53. b. In the survey, p = 0.54.

i. Test H0: p 0.5 vs. Hj. p = 0.5 using a 5% significance level, ii. Test H0: p - 0.5 vs. H\: p > 0.5 using a 5% significance level,

iii. Construct a 95% confidence interval for p. iv. Construct a 99% confidence interval for p. v. Construct a 50% confidence interval for p.

c Suppose that the survey is carried out 20 times, using independently selected voters in each survey. For each of these 20 surveys, a 95 % confidence interval for p is constructed, i. What is the probability that the true value of p is contained in all

20 of these confidence intervals? ii. How many of these confidence intervals do you expect to contain

the true value of p? d. In survey jargon, the "margin of error" is 1.96 X SE(p); that is, it is

half the length of 95% confidence interval. Suppose you wanted to design a survey that had a margin of error of at most 1 %. That is, you wanted Pr( |p - p 1 > 0.01) < 0.05. How large should n be if the sur-vey uses simple random sampling?

Let Yj,.. , Yn be i.i.d.draws from a distribution with mean p. A test of HQ. p. = S^versus / A ^ ^ j V s i n g the usual /-statistic yields a\p-value of 0.03. a. Does the 95% confidence interval contain p. = 5? Explain. b. Can you determine ii p. 6is contained in the 95% confidence inter-you

val? Explain. .X

n a given population, 11 % of the likely voters are African American. A sur-ey using a simple random sample of 600 landline telephone numbers finds

8% African Americans. Is there evidence that the survey is biased? Explain. 3 $ A new version of the SAT test is given to^lp00o;andomly selected high

school seniors. The sample mean test score is 1110, and the sample standard deviation is 123. Construct a 95% confidence interval for the population mean test score for high school seniors.

Exerdses 99 \ 3.9

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Suppose that a lightbulb manufacturing plant produces bulbs with a mean life of 2000 hours and a standard deviation of 200 hours. An inventor claims to have developed an improved process that produces bulbs with a longer mean life and the same standard deviation. The plant manager randomly select^ 100 bulbs\produced by the process. She says that she will believe the inventor's claim if the sample mean life of the bulbs is greater than 2100 hours; otherwise, she will conclude that the new process is no better than the old process. Let p denote the mean of the new process. Consider the null and alternative hypothesis H0: p. 2000 vs. Hy. p, > 2000.

a. What is the size of the plant manager's testing procedure? b. Suppose the new process is in fact better and has a mean bulb life of

2150 hours. What is the power of the plant manager's testing procedure? ^ c p

i

a. The authors plan to administer the test to all third-grade students in New Jersey. Construct a 95% confidence interval for the mean score of all New Jersey third graders.

. What testing procedure should the plant manager use if she wants the size of her test to be 5%? ***" J> **"V* *~* - ^ *

' 'XlO4 Suppose a new standardized test is given to 100 randomly selected third-grade students in New Jersey. The sample average score Y on the test is 58 points, and the sample standard deviation, sY, is 8 points.

/t i. \&& - 2 ^ ^ ? ? ^ J b- Suppose the same test is given tq200_|andomly selected third V ~Z ,* *l* JJ- graders from Iowa, producing a sample average o{62 feoints and

x o*. v\~ \,co iro .^ sample standard deviation of 11 points. Construct a 90% confidence interval for the difference in mean scores between lov^a and New Jersey. *$*>*

^ , ^fe-^pA ^ ^ ^ - ^ Qan v o u c o n c i u d e with a high degree of confidence that the popula-*-1LAC(V ' \

100 CHAPTER 3 Review of Statistics ^M- WX.^ vTCcA YL 3CC X...2W30 ^~ 2o> Sv4= ?ao

Exercises 101

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Men Women

Average Salary ( V) $3100 $2900

Standard Deviation (sY) $200 $320

n 100 64

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ig^i&3, a. What do these data suggest about wage differences in the firm? Do k>^v they represent statistically significant evidence that average wages of

men and women are different? (To answer this question, first state the null and alternative hypothesis; second, compute the relevant f-statistic; third, compute the p-value associated with the r-statistic; and finally, use the p-value to answer the question.)

b. Do these data suggest that the firm is guilty of gender discrimination in its compensation policies? Explain. -

3.13 Data on fifth-gr'ade test scores (reading and mathematics) for yVj-v ,i\l~ SChOOl a- "V-^^ districts in California yield Y = 646.2 and standard deviation sY - 1 9 . 5 ^ 4 ^ , ^ ^ , ^ a. Construct a 95% confidence interval for the mean test score in the

population. b. When the districts were divided into districts with small classes ( < 20

students per teacher) and large classes ( ^ 20 students per teacher), the following results were found:

Class Size Small L>arge

v.

Average Score {Y) 657.4 650.0

Standard Deviation (sY) 19.4 17.9

n 238 182

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Is there statistically significant evidence that the districts with smaller classes have higher average test scores? Explain.

3.14 Values of height in inches (Xs) and weight in pounds (Y) are recorded from a sample of 300 male college students. The resulting summary statistics are X = 70.5 in., Y - 158 lb, sx = 1.8 in., sY - 14.2 lb, Sxy = 21.73 in. X lb, and rXY - 0-85. Convert these statistics to the metric system (meters and kilo-grams) .

3.15 Let Ya and Yb denote Bernoulli random variables from two different pop-ulations, denoted a and b. Suppose that E(Ya) pa and E( Yb) = pb. A ran dom sample of size na is chosen from population a, with sample average denoted pa, and a random sample of size nb is chosen from population b,

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CHAPTER 3 Review of Statistics

+-l * -- s \\ better on their second attempt after taking the prep course?

CCVWT&V,- Vr.ve -} A^w-w. *~(a pcxotf jjj Students may have performed better in their second attempt w^ - ^ \o.W Vv \v * ^ ' < w v. because of the prep course or because they gained test-taking

experience in their first attempt. Describe an experiment that would quantify these two effects.

3.17 Read the box "The Gender Gap of Earnings of College Graduates in the , . 24 3L: 2 i ^ L ^ - , A

132 CHAPTER A Linear Regression with One Regressor

2. The population regression line can be estimated using sample observations (Yh Xt), i - 1, . . . , n by ordinary least squares (OLS).The OLS estimators of the regression intercept and slope are denoted fa and fa.

3. The R2 and standard error of the regression (SER) are measures of how close the values of Y; are to the estimated regression line. The R2 is between 0 and 1, with a larger value indicating that the Y/s are closer to the line. The standard error of the regression is an estimator of the standard deviation of the regression error.

4. There are three key assumptions for the linear regression model: (1) The regression errors. ut, have a mean of zero conditional on the regressors X;% (2) the sample observations are i.i.d. random draws from the population; and (3) large outliers are unlikely. If these assumptions hold, the OLS estimators fa and fa are (1) unbiased, (2) consistent, and (3) normally distributed when the sample is large.

Key Terms

linear regression model with a single regressor (110)

dependent variable (110) independent variable (110) regressor (110) population regression line (110) population regression function (110) population intercept (110) population slope (110) population coefficients (110) parameters (110) error term (110) ordinary least squares (OLS)

estimators (114)

Review the Concepts

4.J Explain the difference between fa and fa; between the residual ut and the regression error ,; and between the OLS predicted value If and E(Yt\Xi).

4.2 For each least squares assumption, proyide an example in which the assumption is valid, then provide an example in which the assumption fails.

OLS regression line (114) sample regression line (114) sample regression function (114) predicted value (114) residual (115) regression^?2 (119) explained sum of squares (ESS) (119) total sum of squares (TSS) (119) sum of squared residuals (SSR) (120) standard error of the regression (SER)

(120) least squares assumptions (122)

Exercises 133

a. TV

4.3 Sketch a hypothetical scatterplot of data for an estimated regression with R2 = 0.9. Sketch a hypothetical scatterplot of data for a regression with R2 = 0.5.

/

Exercises y 4.1 Suppose that a researcher, using data on class size (C5) and average test

scores from 100 third-grade classes, estimates the OLS regression Jk. f

TestScore - 520.4 - 5.82 X CS, R2 = 0.08, SER = 11.5.

a. A classroom has 22 students. What is thejegression's prediction for that classroom's average test score? -Vn^Suf = _3*-3 ^ - s V i * i

b. Last year a classroom had 19 students, and this year it has 23 students. What is the regression's predict! qn_for,the .change in the classroom average test score? A -*5 .*% 1 \~JW "^ *

c. The sample average class size across the 100 classrooms is 21.4. What 2.\ A is the sample average of the test scores across the 100 classrooms?

(Hint: Review the formulas for the OLS estimators.) S & * - J - - - - ^^56. St{>-"Td. What is the sample standard deviation of test scores across the LOO

classrooms? (Hint: Review the formulas for the R2 and SER.) ^

Suppose that a random sample of 200twenty-year-old men is selected from a population and that these men's height and weight are recorded. A regres-

t*y sion of weight on height yields | .,=

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^ Weight - -99.41 +- 3.94 x Height. Rl -- 0.81, SER = 10.2,

,t.^v ~\ AN * ^ ^ ' ---"- -where Weight is measured in pounds and Height is measured in inches. \ ^ ^ , ^ A -ft. yjtf \5 a- What is the regression's weight prediction for someone who is 70 in.

tall? 65 in. tall? 74 in. tall? i - ,A*iS- (,_ ^ m a n ^yg a i a t e growth spurt and grows 1.5 in. over the course of a year.

* ' % s-oxxw \*t. 'UJJ .^ ^ate What is the regression's prediction for the increase in this man's weight? _ M '** r *^ c. Suppose that instead of measuring weight and height in pounds and

inches these variables are measured in centimeters and kilograms." What are the regression estimates from this new ceptimeter-kilogram regression? (Give all results,

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mmius nuiii una new L,cjrtJ*uciE;i-iviiugiuui s, estimated coefficients, Ryand SER.) .

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- 1 134 . CHAPTER A Linear Regression with One Regressor

1% - G?tfo n ~ - ^ y * S * ^ &iNv*ftv**A o ^ o W - KstA < g k V*. ,AO* with a normal distribution?) ^ - ^ . ^ ^ ^ S O n v.-^\axi wJ^ cwi (\\>J^)p^-*aVirt 8* The average age in this sample is 41.6 years. What is the average value C^ei**A u s let Xt denote the amount of time that the student has to complete the exam 4 A , . L\\ t e ^ ^ ^ S {X; = 90 or 120), and consider the regression model Yt = fa + faX{ + uh

MV^ w ^ o ^ Explain what the term ,- represents. Why will different students have ' l ^ c ^ V & f c v t f ^ ^ ^ V ^ \ different values of u{l

r^fwfc ^ o r 6 S ote w & f c ^ G**^ c# ^ r e t n e other assumptions in Key Concept 4.3 satisfied? Explain.

score of students given 90 minutes to complete the exam. Repeat for 120 minutes and 150 minutes.

ii. Compute the estimated gain in score for a student who is given an , _ . additional 10 minutes on the exam.

t * '" ^ v Wl A i 4.6 Show that the first least squares assumption, E(ui Xt) - 0, implies that B ^ 11 fa ft^^Ztt^jji^ = f}Q + plXh

4.7 Show that fa is an unbiased estimator of fa. (Hint: Use the fact that : is unbiased, which is shown in Appendix 4.3.)

4.8 Suppose that all of the regression assumptions in Key Concept 4.3 are sat-isfied except that the first assumption is replaced with E(ut Xt) 2. Which parts of Key Concept 4.4 continue to hold? Which change? Why? (Is j normally distributed in large samples with mean and variance given in Key Concept 4.4? What about 0?)

4.9 a. A linear regression yields fi\ s 0. Show that R~ - 0. b. A linear regression yields R2 - 0. Does this imply that fa - 0?

H^ Suppose that Yj = fa + faXt t-,-, where (A), u{) are i.i.d., and X{ is a

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7 7 ^ , ^ (Bernoulli random variable with Vr(X = 1) = 0.20. When X ^ 1. w, is N(0,4);

a. Show that the regression assumptions in Key Concept 4.3 are satisfied. b. Derive an expression for the large-sample variance of j81. [Hint: Eval-

uate/the terms in Equation (4.21).] 4.11 Consider the regression model Yj = fa + p^X, -*- ut.

a. Suppose you know that fa 0. Derive a formula for the least squares estimator of j3>. ^ L* g_. z _^

> '

136 CHAPTER A Linear Regression with One Regressor Empirical Exercises 137

b. Suppose you know that fa = 4. Derive a formula for the least squares estimator of Bv

4.12 a. Show that the regression R2 in the regression of Y on X is the squared value of the sample correlation between X and Y. That is, show that R2-r2XY.

b. Show that the R2 from the regression of Y on X is the same as the R2 from the regression of X on Y

c. Show that fa = rXy(sY/sx), where rXY is the sample correlation between X and Y, and sY and sx are the sample standard deviations of X and Y.

4.13 Suppose that 1,' = fa + faX, + KUi} where K is a non-zero constant and (Yh X{) satisfy the three least squares assumptions. Show that the large sample

, -, var[(X. - /0". variance of /3i is given by crx K " ^ -.[Hint: This equation is the variance given in equation (4.21) multiplied by jr.]

4.14 Show that the sample regression line passes through the point (X,Y).

Empirical Exercises

E4.1 On the text Web site http://www.pearsonhighered.com/stock_watson/, you will find a data file CPS08 that contains an extended version of the data set used in Table 3.1 for 2008. It contains data for full-time, full-year workers, age 25-34, with a high school diploma or B.A./B.S. as their highest degree. A detailed description is given in CPS08_Description, also available on the Web site. (These are the same data as in CPS92_08 but are limited to the year 2008.) In this exercise, you will investigate the relationship between a worker's age and earnings. (Generally, older workers have more job expe-rience, leading to higher productivity and earnings.) a. Run a regression of average hourly earnings (AHE) on age (Age).

What is the estimated intercept? What is the estimated slope? Use the estimated regression to answer this question: How much do earnings increase as workers age by 1 year?

b. Bob is a 26-year-old worker Predict Bob's earnings using the esti-mated regression. Alexis is a 30-year-old worker. Predict Alexis's earnings using the estimated regression.

c. Does age account for a large fraction of the variance in earnings across individuals? Explain.

K4.2 On the text Web site http://www.pearsonhighered.com/stock_watson/, you will find a data file TeachingRatings that contains data on course evalua-tions, course characteristics, and professor characteristics for 463 courses at the University of Texas at Austin.1 A detailed description is given in TeachingRatings ^ Description, also available on the Web site. One of the characteristics is an index of the professor's "beauty" as rated by a panel of six judges. In this exercise, you will investigate how course evaluations are related to the professor's beauty. a. Construct a scatterplot of average course evaluations (Course_Eval)

on the professor's beauty (Beauty). Does there appear to be a rela-tionship between the variables?

b. Run a regression of average course evaluations (Course _Eval) on the professor's beauty (Beauty). What is the estimated intercept? What is the estimated slope? Explain why the estimated intercept is equal to the sample mean of Course_Eval. (Hint: What is the sample mean of Beauty!)

c. Professor Watson has an average value of Beauty, while Professor Stock's value of Beauty is one standard deviation above the average. Predict Professor Stock's and Professor Watson's course evaluations.

d. Comment on the size of the regression's slope. Is the estimated effect of Beauty on Course_Eval large or small? Explain what you mean by "large" and "small."

e. Does Beauty explain a large fraction of the variance in evaluations , across courses? Explain.

E4.3 On tne text Web site http://www.pearsonhighered.com/stock_watson/, you will find a data file CollegeDistance that contains data from a random sam-ple of high school seniors interviewed in 1980 and re-interviewed in 1986. In this exercise, you will use these data to investigate the relationship between the number of completed years of education for young adults and the distance from each student's high school to the nearest four-year col-lege. (Pr6ximity to college lowers the cost of education, so that students who-live closer to a four-year college should, on average, complete

1 These data were provided by Professor Daniel Hamermesh of the University of Texas at Austin and were used in his paper with Amy Parker, "Beauty in the Classroom: Instructors' Pulchritude and Puta-tive Pedagogical Productivity," Economics of Education Review, August 2005.24(4): 369 -376.

~?r ^ -* ,""Ba^

138 CHAPTER 4 Linear Regression with One Regressor Derivation of the OLS Estimators 139

E4.4

more years of higher education.) A detailed description is given in College Distance ^ Description, also available on the Web site.2

a. Run a regression of years of completed education (ED) on distance to the nearest college (Dist), where Dist is measured in tens of miles. (For example, Dist = 2 means that the distance is 20 miles.) What is the estimated intercept? What is the estimated slope? Use the esti-mated regression to answer this question: How does the average value of years of completed schooling change when colleges are built close to where students go to high school?

b. Bob's high school was 20 miles from the nearest college. Predict Bob's years of completed education using the estimated regression. How would the prediction change if Bob lived 10 miles from the nearest college?

c. Does distance to college explain a large fraction of the variance in educational attainment across individuals? Explain.

d. What is the value of the standard error of the regression? What are the units for the standard error (meters, grams, years, dollars, cents, or something else)?

On the text Web site http://www.pearsonhighered.com/stock watson/, you will find a data file Growth that contains data on average growth rates from 1960 through 1995 for 65 countries along with variables that are potentially related to growth. A detailed description is given in Growth,Description, also available on the Web site. In this exercise, you will investigate the rela-tionship between growth and trade.3

a. Construct a scatterplot of average annual growth rate (Growth) on the average trade share (TradeShare). Does there appear to be a rela-tionship between the variables?

b. One country, Malta, has a trade share much larger than the other coun-tries. Find Malta on the scatterplot. Does Malta look like an outlier?

c. Using all observations, run a regression of Growth on TradeShare. What is the estimated slope? What is the estimated intercept? Use the

: These data were piovided by Professor Cecilia Rouse of Princeton University and were used in her paper "Democratization or Diversion? The Effect of Community Colleges on Educational Attain-ment," Journal of Business and Economic Statistics, April 1995,12(2): 217-224. 3 rhese data were provided by Professor Ross Levine of Brown University and were used in his paper with Thorsten Beck and Norman Ix>ayza, "Finance and the Sources of Growth." Journal of Financial Economics, 2000,58:26! -300.

regression to predict the growth rate for a country with a trade share of 0.5 and with a trade share equal to 1.0.

d. Estimate the same regression excluding the data from Malta. Answer the same questions in c..

e. Where is Malta? Why is the Malta trade share so large? Should Malta be included or excluded from the analysis?

4 . 1 T h e C a l i f o r n i a T e s t S c o r e D a t a S e t

The California Standardized Testing and Reporting data set contains data on test perfor-mance, school characteristics, and student demographic backgrounds. The data used here are from all 420 K-6 and K-8 districts in California with data available for 1999. Test scores are the average of the reading and math scores on the Stanford 9 Achievement Test, a stan-dardized test administered to fifth-grade students. School characteristics (averaged across the district) include enrollment, number of teachers (measured as "full-time equivalents"), number of computers per classroom, and expenditures per student. The student-teacher ratio used here is the number of students in the district divided by the number of full-time equivalent teachers. Demographic variables for the students also are averaged across the district. The demographic variables include the percentage of students who are in the pub-lic assistance program CalWorks (formerly AFDC), the percentage of students who qual-ify for a reduced price lunch, and the percentage of students who are English learners (that is, students for whom English is a second language). All of these data were obtained from the California Department of Education (www.cde.ca.gov).

4 . 2 D e r i v a t i o n , ^ t h e O L S E s t i m a t o r s

This appendixWs calculus to derive the formulas for the OLS estimators given in Key Concept 4.2. To minimize the sum of squared prediction mistakes X"=i(X^^u - b\KY [Equation (4.6)]. first take the partial derivatives with respect to b0 and b\.

'~^J(Yi-^-biXi)2--2%(Yr fco-MQand ob0 1 ,=i -r'kiY^bu 'b.Xi)222(X-bu-blXi)X,

168 CHAPTER 5 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals

{

M -1&

^h?

coefficient multiplying Dt (154) coefficient on D-t (154) heteroskedasticity and

homoskedasticity (156) homoskedasticity-only standard

errors (158) heteroskedasticity-robust standard

error(159)

Review the Concepts

Gauss-Markov theorem (162) best linear unbiased estimator

(BLUE) (163) weighted least squares (163) homoskedastic normal regression

assumptions (164) Gauss--Markov conditions (176)

5.1 Outline the procedures for computing the p-value of a two-sided test of H0: p,y = 0 using an i.i.d. set of observations Yh i = 1 , . . . , n. Outline the pro-cedures for computing the p-value of a two-sided test of HQ: fa = 0 in a regression model using an i.i.d. set of observations (Yh X;), i\~ 1 , . . . , n.

$ 5.2 Explain how you could use a regression model to estimate the wage gen-der gap using the data on earnings of men and women. What are the depen-dent and independent variables?

5.3 Define homoskedasticity and heteroskedasticity. Provide a hypothetical empirical example in which you think the errors would be heteroskedastic and explain your reasoning.

Exercises

5.1 Suppose that a researcher, using data on class size (CS) and average test scores from 100 third-grade classes, estimates the OLS regression

- 7

170 CHAPTER 5 Regression with a Single Regressor Hypothesis Tests and Confidence intervals Exercises 171

v j

5t

5.4 Read the box "The Economic Value of a Year of Education: Homoskedas-ticity or Heteroskedasticity?" in Section 5.4. Use the regression reported in Equation (5.23) to answer the following. a. A randomly selected 30-year-old worker reports an education level of

16 years. What is the worker's expected average hourly earnings? b. A high school graduate (12 years of education) is contemplating

going to a community college for a 2-year degree. How much is this worker's average hourly earnings expected to increase?

c. A high school counselor tells a student that, on average, college grad uates earn $10 per hour more than high school graduates. Is this state-ment consistent with the regression evidence? What range of values is consistent with the regression evidence?

5.5Xi In the 1980s, Tennessee conducted an experiment in which kindergarten stu-dents were randomly assigned to "regular" and "small" classes, and given standardized tests at the end of the year. (Regular classes contained approx-imately 24 students, and small classes contained approximately 15 students.) Suppose that, in the population, the standardized tests have a mean score of 925 points and a standard deviation of 75 points. Let SmallClass denote a binary variable equal to 1 if the student is assigned to a small class and equal to 0 otherwise. A regression of TestScore on SmallClass yields

TestScore - 918.0 + 13.9 X SmallClass, R- - 0 0 1 , SER (1.6) (2.5)

74.6.

^ t \ r - %s*

\ 5 ^ f 2 ^ ( :

y

a. Do small classes improve test scores? By how much? Is the effect large? Explain. ^ - 2 ^ ^ ^ w >l Su*-2> KW>'

b. Is the estimated effect of class size on test scores statistically signifi-cant? Carry out a test at the 5% level. 5 ^ *&. r t u \ ^

c. Construct a 99% confidence interval for the effect of SmallClass on test score.

5.6" \ Refer to the regression described in Exercise 5.5. a. Do you think that the regression errors plausibly are homoskedastic?

Explain. ^ ^ ^ ^ p r ^ v VcjVW ^ ^ - A ^ W ^ 0 ^ $ S ; , o ^ b. SE(fa) was computed using Equation (5.3). Suppose that the regres-

sion errors were homoskedastic: Would this affect the Validity of the confidence interval constructed in Exercise 5.5(c)? Explain.

\ 1UA i4 l

r 5.7 ) Suppose that (Y Xf) satisfy the assumptions in Key Concept 4.3. A random sample of size n = 250 is drawn and yields ^J

Y - 5.4 f 3.2X, R2 = 0.26, SER -= 6.2. (3.1) (1.5)

a. Test H0: fa = 0 vs.H< fa^Qat the 5% level. b. Construct a 95% confidence interval for fa. c Suppose you learned that Yj and X{ were independent. Would you be

surprised? Explain. d. Suppose that Yi and Xi are independent and many samples of size

n = 250 are drawn, regressions estimated, and (a) and (b) answered. In what fraction of the samples would H0 from .(a) be rejected? In what fraction of samples would the value B: 0 be included in the confidence interval from (b)?

Suppose that (Yh %) satisfy the assumptions in Key Concept 4.3 and, in addition, u( is N(0, a2) and is independent of X,. A sample of size n - 30 yields ^

Y = 43.2 + 61.5JT, R2 = 0.54, SER = 1.52, (10.2) (7.4)

^wKeYelhe numbers in parentheses are the homoskedastic-onlv standard errors fof the regression coefficients. a. Construct a 95% confidence interval for fa. b. Test H0: fa = 55 vs. H^. fa * 55 at the 5% l e v e l . ^ ^ c. Test //: fa - 55 vs. Hx: fa > 55 at the 5% level. ^ " ^

5.9 Consider the regression model

where u{ and X{ satisfy the assumptions in Key Concept 4.3. Let ~$ denote an estimator of /? that is constructed as 0 - YjA\ where Y and X are the sample means of Yt and X respectively. a. Show that 0 is a linear function of Yh Y2)..., Y. b. Show that 0 is conditionally unbiased.

y

V "

PS k-

?* ^ & & A 9~+

172 CHAPTER 5 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals

5.10 Let X-t denote a binary variable and consider the regression Yt~ fa + faXi +- ,. Let if) denote the sample mean for observations with X = 0 and yj denote the sample mean for observations with X~\. Show that fa = Y0, fa + fa-- % and fa = YX- %

5.11 A rapdoin^amrjle of workers contains nm = 120 men and nw ~ 131 women. The fcample averagejof men's weekly earnings [Ym^ (l/nm)2w=i^,,-] is $ 5 2 3 . 1 0 , a ^ t h e ^ ^ g s f j n ^ a i r l j h ^ t j ^ i [sm = V^^I^(YmJ - Ym)2] is $68.1. The corresponding values for women are Yw ~ $485.10 and sw - $51.10. Let Women denote an indicator variable that is equal to 1 for women and 0 for men and suppose that alf 251_^bservations are used in the regression Yt fa 4 faWomen, f ut. Find the OLS estimates of 8$ and Bx and their corresponding standard errors.

5.12 Starting from Equation (4.22), derive the variance of fa under homoskedasticity given in Equation (5.28) in Appendix 5.1.

5.13 Suppose that (Yh Xt) satisfy the assumptions in Key Concept 4.3 and, in addition, ut is N(Q, cr2u) and is independent of Xj. a. Is B i conditionally unbiased? b. Is fa the best linear conditionally unbiased estimator of B{1 c. How would your answers to (a) and (b) change if you assumed only

that (Yh Xi) satisfied the assumptions in Key Concept 4.3 and var(,j^- = x) is constant?

d. How would your answers to (a) and (b) change if you assumed only that (Yn X^ satisfied the assumptions in Key Concept 4.3?

5.14 Suppose that Yj = BXt -f ,-, where (,-, Xj) satisfy the Gauss-Markov con-ditions given in Equation (5.31). a. Derive the least squares estimator of (5 and show that it is a linear

function of Y,..., Yn. b. Show that the estimator is conditionally unbiased. c. Derive the conditional variance of the estimator. d. Prove that the estimator is BLUE. ^ _

5.15 A researcher has two independent samples of observations on (Yb X^). To be specific, suppose that Y} denotes earnings, Xt denotes years of schooling, and the independent samples are for men and women. Write the regression for men as Ymi - Bm0 + BmlXmi + umi and the regression for women as Kj ~ Pw.o + BwlXwi + uwj. Let jSMil denote the OLS estimator constructed

Empirical Exercises 173

using the sample of men, Bti denote the OLS estimator constructed from the sample of women, and SE(j3ml) and SE(fatl) denote the correspond-ing standard errors. Showthatjhe standard error of (3mi - fatl is given by S E ( / W ~k,i) = V[SE(pmtl)V + [SE(fa7l)i2.

Empirical Exercises

E5.1

E5.2

E5.2

Using the data set CPS08 described in Empirical Exercise E4.1, run a regression of average hourly earnings (A HE) on Age and carry out the fol-lowing exercises. a. Is the estimated regression slope coefficient statistically significant?

That is, can you reject the null hypothesis H0: B[ 0 versus a two-sided alternative at the 10%, 5%, or 1% significance level? What is the p-value associated with coefficient's /-statistic?

b. Construct a 95% confidence interval for the slope coefficient. , c. Repeat (a) using only the data for high school graduates.

d. Repeat (a) using only the data for college graduates. e. Is the effect of age on earnings different for high school graduates

than for college graduates? Explain. (Hint: See Exercise 5.15.) Using the data set TeachingRatings described in Empirical Exercise E4.2, run a regression of Course_Eval on Beauty. Is the estimated regression slope coefficient statistically significant? That is, can you reject the null hypothe-sis H0: /3I = 0 versus a two-sided alternative at the 10%, 5%, or 1% signifi-cance level? What is the p-value associated with coefficient's r-statistic? Using the data set CoIlegeDistance described in Empirical Exercise E4.3, run a regression of years of completed education (ED) on distance to the nearest college (Dist) and carry out the following exercises.

a. Is the estimated regression slope coefficient statistically significant? That is, can you reject the null hypothesis H0: B-^-Q versus a two-sided alternative at the 10%, 5%, or 1% significance level? What is the p-value associated with coefficient's r-statistic?

b. Construct a 95% confidence interval for the slope coefficient. c. Run the regression using data only on females and repeat (b). d. Run the regression using data only on males and repeat (b). e. Is the effect of distance on completed years of education different for

men than for women? (Hint: See Exercise 5.15.)

y

204 CHAPTER 6 Linear Regression with Multiple Regressors

associated with a 1-unit change in X*, holding the other regressors constant. The other regression coefficients have an analogous interpretation. The coefficients in multiple regression can be estimated by OLS. When the four least squares assumptions in Key Concept 6.4 are satisfied, the OLS esti-mators are unbiased, consistent, and normally distributed in large samples. Perfect multicollinearity, which occurs when one regressor is an exact linear function of the other regressors, usually arises from a mistake in choosing which regressors to include in a multiple regression. Solving perfect multi-collinearity requires changing the set of regressors.

5. The standard error of the regression, the R2, and the R2 are measures of fit for the multiple regression model.

3.

4.

Key Terms

omitted variable bias (180) multiple regression model (186) population regression line (186) population regression function (186) intercept (186) slope coefficient of Xu (186) coefficient on Xu (186) slope coefficient of X2j (186) coefficient on X2i (186) holding X2 constant (187) controlling for X7 (187) partial effect (187) population multiple regression model

(188)

Review the Concepts

constant regressor (188) constant term (188) homoskedastic (188) heteroskedastic (188) ordinary least squares (OLS)

estimators of fa, Bh...,Bk (190) OLS regression line (190) predicted value (190) OLS residual (190) R2 (193) adjusted/?2 (tf2) (194) perfect multicollinearity (197) dummy variable trap (201) imperfect multicollinearity (202)

6.1 A researcher is interested in the effect on test scores of computer usage. Using school district data like that used in this'6hapter. she regresses dis-trict average test scores on the number of computers per student. Will fa be an unbiased estimator of the effect on test scores of increasing the num-ber of computers per student? Why or why not? If you think B-\ is biased, is it biased up or down? Why?

\

Exercises 205

6.2 A multiple regression includes two regressors: Y{ = fa -f faX\i + faX2i + ;. What is the expected change in Y if X\ increases by 3 units and X2 is unchanged? What is the expected change in Yif X2 decreases by 5 units and Xj is unchanged? What is the expected change in Y if Xy increases by 3 units and X2 decreases by 5 units?

6.3 Explain why two perfectly multicollinear regressors cannot be included in a linear multiple regression. Give two examples of a pair of perfectly mul-ticollinear regressors.

6.4 Explain why it is difficult to estimate precisely the partial effect of Xx.. hold-ing X2 constant, if Xx and X2 are highly correlated.

Exercises

The first four exercises refer to the table of estimated regressions on page 206. computed using data for 1998 from the CPS.The data set consists of infor-

0 mation on^OOOfilfTtime full-year workers.The highest educational achieve-ment for each worker was either a high school diploma or a bachelor's degree. The worker's ages ranged from 25 to 34 years. The data set also con-tained information on the region of the country where the person lived, mar ital status, and number of children. For the purposes of these exercises, let AHE = average hourly earnings (in 1998 dollars) College - binary variable (1 if college, 0 if high school) Female ~ binary variable (1 if female, 0 if male) Age = age (in years) Ntheast - binary variable (1 if Region = Northeast, 0 otherwise) Midwest binary variable (1 if Region = Midwest, 0 otherwise) South - binary variable (1 if Region ~ South, 0 otherwise) West ~ binary variable (1 if Region - West, 0 otherwise) Compute R2 for each of the regressions. Using the regression results in column (1);

a. Do workers with college degrees earn more, on average, than workers with only high school degrees? How much more? ^ ctvD* ^ >w-r^

6.1 v6.2

6.3 b. Do men earn more than women on average? How much more? Using the regression results in column (2): a. Is age an important determinant of earnings? Explain.

n v ^


A -to * 5 A 1 1, Sajjy 1S a 29-year-old female college graduate. Betsy is a 34-year-old female college graduate. Predict Sally's and Betsy's earnings.

sing the regression results in column (3): a.yiDo there appear to be important regional differences? . Why is the regressor West omitted from the regression? What would

happen if it was included? . v \ ytffctfk Vll^ et fo-^-O ^ -VW A: (PQuyAll fft^P^ofS^

A y ^ -Nfc*

_ CV 7-i- %t

Dependent variable: average hourly earnings (AHE).

Regressor (1) (2) (3) College (#;) 5 46 Female [Xz) -2.t Age (A3) Northeast (X,)

5.48 -2.62

0.29

5.44 2.62 0.29 0.69

/

Midwest (X5) South (X6) Intercept Summary Statistics SER Rz

12.69

6.27 0.176

4.40

6.22 0.190

0.60 -0.27

3.75

6.21 0.194

Exercises 207

R2

n 4000 4000 4000

^ c. Juanita is a 28-year-old female college graduate from the South. Jennifer is a 28-year-old female college graduate from the Midwest. Calculate the expected difference in earaifigs between Juanita and Jennifer.

6.5 Data were collected from a random sample of 220 home sales from a com-munity in 2003. Let Price denote the selling price (in $1000), BDR denote the number of bedrooms, Bath denote the number of bathrooms, Hsize denote the size of the house (in square feet), Lsize denote the lot size

(in square feet), Age denote the age of the house (in years), and Poor denote a binary variable that is equal to 1 if the condition of the house is reported as "poor." An estimated regression yields

Prices 119.2 + 0AS5BDR + 73ABath + 0.\56Hske + 0.002Lsize t- 0.090A#e 48.c\Poor, R2 = 0.72, SER - 41.5.

a. Suppose that a homeowner converts part of an existing family room in her house into a new bathroom. What is the expected increase in the value of the house?

b. Suppose that a homeowner adds a new bathroom to her house, which increases the size of the house by 100 square feet. What is the expected increase in the value of the house?

c. What is the loss in value if a homeowner lets his house run down so that its condition becomes "poor"?

-$. Compute the R2 for the regression. 6.6 A researcher plans to study the causal effect of police on crime using data

from a random sample of U.S. counties. He plans to regress the county's crime rate on the (per capita) size of the county's police force. a. Explain why this regression is likely to suffer from omitted variable

bias. Which variables would you add to the regression to control for important omitted variables?

b. Use your answer to (a) and the expression for omitted variable bias given in Equation (6.1) to determine whether the regression will likely over- or underestimate the effect of police on the crime rate. (That is, do you think that fa > fa or fa < fal)

6.7' Critique each of the following proposed research plans. Your critique should explain any problems with the proposed research and describe how the research plan might be improved. Include a discussion of any additional data that need to be collected and the appropriate statistical techniques for analyzing the data. a. A researcher is interested in determining whether a large aerospace

firm is guilty of gender bias in setting wages. To determine potential bias, the researcher collects salary and gender information for all of the firm's engineers. The researcher then plans to conduct a "differ-ence in means" test to determine whether the average salary for women is significantly less than the average salary for men.


b. A researcher is interested in determining whether time spent in prison has a permanent effect on a person's wage rate. He collects data on a random sample of people who have been out of prison for at least 15 years. He collects similar data on a random sample of peo-ple who have never served time in prison. The data set includes infor-mation on each person's current wage, education, age, ethnicity, gender, tenure (time in current job), occupation, and union status, as well as whether the person was ever incarcerated. The researcher plans to estimate the effect of incarceration on wages by regressing wages on an indicator variable for incarceration, including in the regression the other potential determinants of wages (education, tenure, union status, and so on).

6.8 A recent study found that the death rate for people who sleep 6 to 7 hours per night is lower than the death rate for people who sleep 8 or more hours. The 1.1 million observations used for this study came from a random sur-vey of Americans aged 30 to 102. Each survey respondent was tracked for 4 years. The death rate for people sleeping 7 hours was calculated as the ratio of the number of deaths over the span of the study among people sleeping 7 hours to the total number of survey respondents who slept 7 hours. This calculation was then repeated for people sleeping 6 hours, and so on. Based on this summary, would you recommend that Americans who sleep 9 hours per night consider reducing their sleep to 6 or 7 hours if they want to prolong their lives? Why or why not? Explain.

6.91 (Yh X\i, X2i) satisfy the assumptions in Key Concept 6.4. You are interested in fa, the causal effect of X\ on Y. Suppose that X\ and X2 are uncorre-cted. You estimate fa by regressing Y onto Xv (so that X2 is not included in the regression). Does this estimator suffer from omitted variable bias? Explain.

6.10 (Yj, Xu, X2i) satisfy the assumptions in Key Concept 6.4; in addition, var(u, | Xu, X2i) = 4 and vai(Xu) - 6. A random sample of size n = 400 is drawn from the population. a. Assume that Xx and X2 are uncorrected. Compute the variance of Br.

[Hint: Look at Equation (6.17J in the Appendix 6.2. b. Assume that cor(Xi, X2) - 0.5. Compute the variance of fa. c. Comment on the following statements: "When X\ and X2 are corre-

lated. the variance of 8 L is larger than it would be if X\ and X2 were


uncorrected. Thus, if you are interested in , it is best to leave out of the regression if it is correlated with Xxr

6.11 (Requires calculus) Consider the regression model

Y^faXu^faX2iUl

for / 1 , . . . , n. (Notice that there is no constant term in the regression.) Following analysis like that used in Appendix 4.2; a. Specify the least squares function that is minimized by OLS. b. Compute the partial derivatives of the objective function with respect

to hi and bi-c. Suppose XUX-jXu = 0. Show that fa = S ^ A - t y ^ - i A * . d. Suppose S/Li-Jq/Aa ^ 0- Derive an expression for fa as a function of

thedata(Yi,Xu,X2i),i = l,...,n. e. Suppose that the model includes an intercept:

Yj = fa + faXu 4- faX2i + ,-. Show that the least squares estimators satisfy fa = Y - faX\ - faX2.

f. As in (e), suppose that the model contains an intercept. Also suppose that 2r=i(Ai, - X\)(X2i - X2) = 0. Show that jSi-2?-i(Ai,- -XiXYi-YWZUiXu- X,)2. How does this compare to the OLS estimator of B} from the regression that omits AV?

Empirical Exercises

E6.1 Using the data set Teaching Ratings described in Empirical Exercises 4.2, carry out the following exercises. a. Run a regression of Course Eval on Beauty. What is the estimated

slope? b. Run a regression of Course_Eval on Beauty, including some addi-

tional variables to control for the type of course and professor charac-teristics. In particular, include as additional regressors Intro, OneCredit, Female, Minority, and AWEnglish. What is the estimated effect of Beauty on Course_EvaH Does the regression in (a) suffer from important omitted variable bias?


c. Estimate the coefficient on Beauty for the multiple regression model in (b) using the three-step process in Appendix 6.3 (the Frisch-Waugh theorem). Verify that the three-step process yields the same estimated coefficient for Beauty as that obtained in (b).

d. Professor Smith is a black male with average beauty and is a native English speaker. He teaches a three-credit upper-division course. Pre-dict Professor Smith's course evaluation.

E6.2 Using the data set CollegeDistance described in Empirical Exercise 4.3. carry out the following exercises. a. Run a regression of years of completed education (ED) on distance

to the nearest college (Dist). What is the estimated slope? b. Run a regression of ED on Dist, but include some additional regres-

sors to control for characteristics of the student, the student's family, and the local labor market. In particular, include as additional regressors Bytest, Female. Black, Hispanic, Incomehi, Ownhome, DadColl, CueSO, and Stwmfg80. What is the estimated effect of Dist on ED'l

c. Is the estimated effect of Dist on ED in the regression in (b) substan-tively different from the regression in (a)? Based on this, does the regression in (a) seem to suffer from important omitted variable bias?

d. Compare the fit of the regression in (a) and (b) using the regression standard errors, R2 and R2. Why are the R2 and R2 so similar in regression (b)? t

e. The value of the coefficient on DadColl is positive. What does this coefficient measure?

f. Explain why CueSO and Swmfg80 appear in the regression. Are the signs of their estimated coefficients (+ or - ) what you would have believed9 Interpret the magnitudes of these coefficients.

g. Bob is a black male. His high school was 20 miles from the nearest college. His base-year composite test score (Bytest) was 58. His family income in 1980 was $26,000, and his family owned a home. His mother attended college, but his father did not. The unemployment rate in his county was 7.5%. and the state average manufacturing hourly wage was $9.75. Predict Bob's years of completed schooling using the regression in (b).

Derivation of Equation (6.1) 211

h. Jim has the same characteristics as Bob except that his high school was 40 miles from the nearest college. Predict Jim's years of com-pleted schooling using the regression in (b).

E6.3 Using the data set Growth described in Empirical Exercise 4.4, but exclud-ing the data for Malta, carry out the following exercises. a. Construct a table that shows the sample mean, standard deviation,

and minimum and maximum values for the series Growth, TradeShare, YearsSchool, Oil, Rev_Coups, Assassinations, RGDP60. Include the appropriate units for all entries.

b. Run a regression of Growth on TradeShare, YearsSchool, Rev _Coups, Assassinations and RGDP60. What is the value of the coefficient on RevJZoupsI Interpret the value of this coefficient. Is it large or small in a real-world sense?

c. Use the regression to predict the average annual growth rate for a country that has average values for all regressors.

d. Repeat (c) but now assume that the country's value for TradeShare is one standard deviation above the mean.

e. Why is Oil omitted from the regression? What would happen if it were included?

6.1 D e r i v a t i o n o f E q u a t i o n ( 6 . 1 )

This appendix presents a derivation of the formula for omitted variable bias in Equa-tion (6.1). Equation (430) in Appendix 4.3 states that

1 i = / M

v\2 P

(6.16)

crx and Under the last two assumptions in Key Concept 4.3, (l/)2,-=i(^- - X (Vn)l!l=i(Xi-X)ut ^- COV(M,-,A;) = pXuo-uax. Substitution of these limits into Equation (6.16) yields Equation (6.1).

I

242 CHAPTER 7 Hypothesis Tests and Confidence Intervals in Multiple Regression Exercises 243

hypothesis that fa = 0 and fa ~ 0. Why isn't the result of the joint test implied by the results of the first two tests?

7.2 Provide an example of a regression that arguably would have a high value of R2 but would produce biased and inconsistent estimators of the regres-sion coefficients). Explain why the R2 is likely to be high. Explain why the OLS estimators would be biased and inconsistent.

b. Sally is a 29-year-old female college graduate. Betsy is a 34-year-old female college graduate. Construct a 95 % confidence interval for the expected difference between their earnings.

7.4 Using the regression results in column (3): a. Do there appear to be important regional differences? Use an appro-

priate hypothesis test to explain your answer.

Exercises

The first six exercises refer to the table of estimated regressions on page 243, computed using data for 1998 from the CPS. The data set consists of infor-mation on 4000 full-time full-year workers. The highest educational achieve-ment for each worker was either a high school diploma or a bachelor's degree.The worker's ages ranged from 25 to 34 years.The data set also con-tained information on the region of the country wlj^fe the person lived, mar-ital status, and number of children. For the purposes of these exercises, let AHE - average hourly earnings (in 1998 dollars) College = binary variable (1 if college, 0 if high school) Female = binary variable (1 if female, 0 if male) Age = age (in years) Ntheast binary variable (1 if Region - Northeast, 0 otherwise) Midwest = binary variable (1 if Region = Midwest, 0 otherwise) South - binary variable (1 if Region - South, 0 otherwise) West - binary variable (1 if Region - West, 0 otherwise)

7.1 Add "*" (5%) and "**" (1%) to the table to indicate the statistical signifi-cance of the coefficients.

7.2 Using the regression results in column (1): a. Is the college-high school earnings difference estimated from this

regression statistically significant at the 5% level? Construct a 95% confidence interval of the difference.

b. Is the male-female earnings difference estimated from this regression statistically significant at the 5% level? Construct a 95% confidence interval for the difference.

7.3 Using the regression results in column (2): a. Is age an important determinant of earnings? Use an appropriate sta-

tistical test and/or confidence interval to explain your answer.

f

Dependent Variable: average hourly earnings (AHE).

Regressor (1) i)Uege(Ai)

Feinale (X2)

(2) 5.46

m\) 2.64

(0.20)

5.48 (0.21)

-2.62 (0.20)

(3) 5.44

(0.21) -2.62 (0.20)

Aget-Y,)

Northeast (JtT4)

Midwest (X5)

Soulh (Xb)

Intercept

Summary Statistics and Joint Tests ^-statistic for regional effects 0 SER R1 n

12.69 (0.14)

6.27 0.176

4000

0.29 (0.04)

4.40 (1.05)

6.22 0.190

4000

0.29 (0.04) 0.69

(0-30) 0.60

(0.28) -0.27 (0.26) 3.75

(1.06)

6.10 6.21 0.194

4000

b. Juanita is a 28-year-old female college graduate from the South. Molly is a 28-ycar-old female college graduate from the West. Jennifer is a 28-year-old female college graduate from the Midwest. i. Construct a 95 % confidence interval for the difference in

expected earnings between Juanita and Molly.

244 CHAPTER 7 Hypothesis Tests and Confidence Intervals in Multiple Regression

ii. Explain how you would construct a 95% confidence interval for the difference in expected earnings between Juanita and Jennifer. (Hint: What would happen if you included West and excluded Midwest from the regression?)

7.5 The regression shown in column (2) was estimated again, this time using data from 1992 (4000 observations selected at random from the March 1993 CPS, converted into 1998 dollars using the consumer price index). The results are

AHE ^ 0.77 + 5.2QCollege - 2.59Female + 0 .404^, SER = 5.85. R2 = 0^ (0.98) (0.20) (0.18) (0.03)

Comparing this regression to the regression for 1998 shown in column (2), was there a statistically significant change in thf coefficient on College?

7.6 Evaluate the following statement: "In all of tne regressions, the coefficient on Female is negative, large, and statistically significant. This provides strong statistical evidence of gender discrimination in the U.S. labor market."

7.7 Question 6.5 reported the following regression (where standard errors have been added):

Price = 119.2 +- U.485BDR J- 23ABath + 0. l56Hsize + Q.002Lsize ^ (23.9) (2.61) (8.94) (0.011) (0.00048) f 0.090Age - 48.8Poor, R2 ~- 0.72. SER = 41.5 (0.311) (10.5)

a. Is the coefficient on BDR statistically significantly different from zero? b. Typically five-bedroom houses sell for much more than two-bedroom

houses. Is this consistent with your answer to (a) and with the regres-sion more generally?

c. A homeowner purchases 2000 square feet from an adjacent lot. Con-struct a 99% confident interval for the change in the value of her house.

d. Lot size is measured in square feet. Do you think that another scale might be more appropriate? Why or why not?

e. The F-statistic for omitting BDR and Age from the regression is F 0.08. Are the coefficients on BDR and Age statistically different from zero at the 10% level?

Exercises 245 I !

7.8 Referring to Table 7.1 in the text: a. Construct the R2 for each of the regressions. b. Construct the homoskedasticity-only ^-statistic for testing

03 ~ 4 = 0 in the regression shown in column (5). Is the statistic sig nificant at the 5% level?

c. Test fa 84 = 0 in the regression shown in column (5) using the Bonferroni test discussed in Appendix 7.1.

d. Construct a 99% confidence interval for fa for the regression in column (5). '

7.9 Consider the regression model Yj = fa ^-/B1 Xh + faX2i + ,-. Use Approach #2 from Section 7.3 to transform the regression so that you can use a /-statistic to test a. fa^fa; b. Bx + a fa ~ 0, where a is a constant; c. JSJ -t- fa =- 1. (Hint: You must redefine the dependent variable in the

regression.) 7.10 Equations (7.13) and (7.14) show two formulas for the homoskedasticity-

' only F-statistic. Show that the two formulas are equivalent. 7.11 A school district undertakes an experiment to estimate the effect of class

size on test scores in second grade classes. The district assigns 50% of its previous year's first graders to small second-grade classes (18 students per classroom) and 50% to regular-size classes (21 students per class-room). Students new to the district are handled differently: 20% are ran-domly assigned to small classes and 80% to regular-size classes. At the end of the second-grade school year, each student is given a standardized exam. Let ^denote the exam score for the /'th student, ,Y1( denote a binary variable that equals 1 if the student is assigned to a small class, and Xy denote a binary variable that equals 1 if the student is newly enrolled. Let Bi denote the causal effect on test scores of reducing class size from reg-ular to small. a. Consider the regression Yj - fa + 8{XU t- ur Do you think that

E(ut\X\j) ~ 0 ? Is the OLS estimator of B: unbiased and consistent? Explain.

b. Consider the regression Yf = fa + B^Xy, + faX2l + ,-. Do you think that E(Uj\X\j, X2i) depends on X{> Is the OLS estimator of 8: unbiased

246 CHAPTER 7 Hypothesis Tests and Confidence Intervals in Multiple Regression

and consistent? Explain. Do you think that E(ui\Xu. X2j) depends on X2? Will the OLS estimator of fa provide an unbiased and consistent estimate of the causal effect of transferring to a new school (that is, being a newly-enrolled student)? Explain.

Empirical Exercises

E7.1 Use the data set CPS08 described in Empirical Exercise 4.1 to answer the following questions. a. Run a regression of average hourly earnings (AHE) on age (Age).

What is the estimated intercept? What is the estimated slope? b. Run a regression oiAHE on Age, gender LEemale), and education

(Bachelor). What is the estimated effect of Age on earnings? Construct a 95% confidence interval for the coefficient on Age in the regression.

c. Are the results from the regression in (b) substantively different from the results in (a) regarding the effects of Age and AHFf. Does the regression in (a) seem to suffer from omitted variable bias?

d. Bob is a 26-year-old male worker with a high school diploma. Predict Bob's earnings using the estimated regression in (b). Alexis is a 30-year-old female worker with a college degree. Predict Alexis's earnings using the regression.

e. Compare the fit of the regression in (a) and (b) using the regression standard errors, R2 and R2 Why are the R2 and R2 so similar in regression (b)?

f. Are gender and education determinants of earnings? Test the null hypothesis that Female can be deleted from the regression. Test the null hypothesis that Bachelor can be deleted from the regression. Test the null hypothesis that both Female and Bachelor can be deleted from the regression.

g. A regression will suffer from omitted variable bias when two condi-tions hold. What are these two conditions? Do these conditions seem to hold here?

E7.2 Using the data set TeachingRatings described in Empirical Exercise 4.2, carry out the following exercises.


a. Run a regression of Course_Eval on Beauty. Construct a 95% confi-dence interval for the effect of Beauty on Course_Eval.

b. Consider the various control variables in the data set. Which do you think should be included in the regression? Using a table like Table 7.1, examine the robustness of the confidence interval that you constructed in (a). What is a reasonable 95% confidence interval for the effect of Beauty on Course_EvaP.

E7.3 Use the data set CollegeDistance described in Empirical Exercise 4.3 to answer the following questions. a. An education advocacy group argues that, on average, a person's edu-

cational attainment would increase by approximately 0.15 year if dis-tance to the nearest college is decreased by 20 miles. Run a regression of years of completed education (ED) on distance to the nearest col-lege (Dist). Is the advocacy groups' claim consistent with the esti-mated regression? Explain.

b. Other factors also affect how much college a person completes. Does controlling for these other factors change the estimated effect of dis-

s tance on college years completed? To answer this question, construct a table like Table 7.1. Include a simple specification [constructed in (a)], a base specification (that includes a set of important control variables), and several modifications of the base specification. Dis-cuss how the estimated effect of Dist on ED changes across the specifications.

c. It has been argued that, controlling for other factors, blacks and His-panics complete more college than whites. Is this result consistent with the regressions that you constructed in part (b)?

E7.4 Using the data set Growth described in Empirical Exercise 4,4, but exclud-ing the data for Malta, carry out the following exercises. a. Run a regression of Growth on TradeShare, YearsSchool, Rev_Coups,

Assassinations, and RGDP60. Construct a 95% confidence interval for the coefficient on TradeShare. Is the coefficient statistically signifi-cant at the 5% level?

b. Test whether, taken as a group, YearsSchool, Rev_Coups, Assassinations, and RGDP60 can be omitted from the regression. What is the p-value of the /^statistic?

/

296 CHAPTER 8 Nonlinear Regression Functions

R e v i e w t h e C o n c e p t s

, 8.1 Sketch a regression function that is increasing (has a positive slope) and is steep for small values of X but less steep for large values of X. Explain how you would specify a nonlinear regression to model this shape. Can you think of an economic relationship with a shape like this?

8.2 A "Cobb-Douglas" production function relates production (Q) to factors of production, capital (K), labor (/7h and raw materials (M), and an error term u using the equation Q - XKplL^2M^eu, where A, fat fa, and fa are production parameters. Suppose that you have data on production and the factors of production from a random sample of firms with thV-sanie Cobb--Douglas production function. How would you use regression analy-sis to estimate the production parameters?

8.3 A standard "money demand" function used by macroeconomists has ffifrfprm ln(m) - fa -I- B^^GDP) f B2R, where m is the quantity of (real) money, GDP is the value of (real) gross domestic product, and R is the value of the nominal interest rate measured in percent per year. Suppose that /3, = 1.0 and fa - ~ 0.02. What will happen to the value of m if GDP increases by 2%? What will happen to m if the interest rate increases from 4% to 5%?

8.4 You have estimated a linear regression model relating Y to X. Your pro-fessor says, "I think that the relationship between Y and X is nonlinear." Explain how you would test the adequacy of your linear regression.

8.5 Suppose that in Exercise 8.2 you thought that the value of B2 was not con-stant, but rather increased when K increased. How could you use an inter-action term to capture this effect?

ieo --' 7 , Fh

Exerc ises

' 8.1N Sales in a company are $196 million in 2009 and increase to $198 million in 2010.

iCC fh> / tit, ' ' ^ n> a. Compute the percentage increase in sales using the usual formula . Compare this value to the approximation (5flte,mii - Sales ) 100 X m" M Sfes?oti9

100 X [ln(Sales2Qio) - ln(Sa/es2oo9)]-b. Repeat (a) assuming Sales2QW = 205, Sales20m 250, and Sales2Q\(\ = 500. c. How good is the approximation when the change is small? Does the

quality of the approximation deteriorate as the percentage change increases? QJ ^ ^ o ^ ^ e ^ C , O ^ T O K O ^ O A

Exercises 297

8.2/ Suppose that a researcher collects data on houses that have sold in a par-ticular neighborhood over the past year and obtains the regression results in the table shown below.

f a. Using the results in column (1), what is the expected change in price of building a 500-square-foot addition to a house? Construct a 95% confidence interval for the percentage change in pricey

/-. . oQpAr2 (j&S) ^ ^C\_ ^ v i & y ^ t f W fo^to^S juEfy'*-S E r _^sl-^?5)-5-- 1

Dependent variable: InWrice)

Regressor (1) (2) (3) (4) Size

\n(Size)

0.00042 (0.000038)

0.69 (0.054)

0.68 (0.087)

0.57 (2.03)

ln(Size)2 0.0078 (0.14)

Bedrooms 0.0036 (0.037)

(5)

0.69 (0.055)

Pool

View

Pool X View

Condition

Intercept

Summary Statistics SER R1

0.082 (0.032) 0.037

(0.029)

0.13 (0.045) 10.97 (0 069)

0.102 0.72

0.071 (0.034) 0.027

(0.028)

0.12 (0.035) 6.60

(0.39)

0.098 0.74

0.071 (0.034) 0.026

(0.026)

0.12 (0.035) 6.63

(0.53)

0.099 0.73 . _ *..

0.071 (0.036) 0.027

(0.029)

0.12 (0.036) 7.02

(7.50)

0.099 0.73

0.071 (0.035) 0.027

(0.030) 0.0022

(0.10) 0.12

(0.035) 6.60

(0.40)

0.099 0.73

Variable definitions: Price = sale price ($); Size = house size (in square feet); Bedrooms = number of bedrooms; Pool = binary variable (1 if house has a swimming pool, 0 otherwise): View - binary variable (1 if house has a nice view. 0 otherwise); Condition - binary variable (1 if real estate agent reports house is in excellent condition. 0 otherwise).

^ .. . ....... J


UM^ -\o cO(wjta(^ Q^ C^VvW b* Comparing columns (I) and (2), is it better to use Size or ln(SZze) to || vJ \fi{$\ 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. ^

2iX ^ A\V /\>?


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 bet-ter than the linear specification in column (7) of Table 8.3?

b. A researcher suspects that the effect oT income on test scores is differ-ent 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 bet-

ter than the linear specification in column (7) of Table 8T3?

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 corpora-tions in the 1990s. (Each year these publicly traded corporations must report total compensation levels for their top five executives.) a. Let Female be an indicator variable that is equal to 1 for females and 0

for males. A regression of the logarithm of earnings onto Female yields

]r7(Ea7nings) - 6.48 -- 0A4Female, SER = 2.65. (0.01) (0.05)

i. The estimated coefficient on Female is -0.44, Explain what this value means,

ii. The SER is 2.65. Explain what this value means. hi. 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. "I"wo new variables, the market value of the firm (a measure of firm size, in millions of dollars) and stock return (a measure of firm perfor-mance, in percentage points), are added to the regression:

1

bi(Earnings) - 3.86 0.28Female + 0.371n(MarketValue) + O.QOAReturn, (0.03) (0.04) (0.004) (0.003)

n - 46,670. R2 = 0.345.

Exercises 301

i. The coefficient on In(MarketValue) is 0.37. Explain what this value means.

ii. The coefficient on Female 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 Xis 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 Y on the vertical axis): a. F = 2.0 + 3 . 0 x l n ( ^ ) . b. r = 2.0-3.0xin(AT). c. i. r - 2 . 0 + 3.0xln(A') f-4.0Z, with Z = 1.

ii. Same as (1), but with Z 0.

d. i. f - 2 . 0 + 3.0xln(^T) f 4 .0Z-1 .0 X Z X In(,Y), wi thZ - 1. ii. Same as (i), but with Z = 0.

e. f = 1 .0 -^125 .0^ -0 .01^ . 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 Yt^fa + faXu + faX2i -f fa(Xu X X2i) +,. Use Key Concept 8.1 to show: a. ^Y|^XX = fa + B3X2 (effect of change in XY holding X2 constant). b. A y/AX> = fa + faX: (effect of change in X2 holding X constant). c. If Xt changes by AA^ and X2 changes by AX2, then A Y ~

(fa - faX2)AX, -f (fa f faX,)AX2 + faAX^X2. 8.11 Derive the expressions for the elasticities given in Appendix 8.2 for the lin-

ear and log-log models. (Hint: For the log-log model assume that u and X are independent as is done in Appendix 8.2 for the log-linear model.)

8.12 The discussion following Equation (8.28) interprets the coefficient on inter-acted binary variables using the conditional mean zero assumption. This

302 CHAPTER 8 Nonlinear Regres^ ion-FwflGtions Empirical Exercises 303

exercise shows that interpretation also applies under conditional mean independence. Consider the hypothetical experiment in Exercise 7.11. a. Suppose that you estimate the regression Yt yQ +- y\XXi + ,- using

only the data on returning students. Show that y1 is the class size effect for returning students, that is, that y, = E(Yi\Xli^ 1, X^-O) -E{Yi\Xu-'-{),X2l-)- Explain why yy is an unbiased estimator of y^.

b. Suppose that you estimate the regression Yj 80 f S^Xn + ,- using only the data on new students. Show that 81 is the class size effect for new students, that is, that 8t ~- E(Yj\X^= 1; X2i = 1) - E(Y;\XU = 0, X2i = 1). Explain why 6j is an unbiased estimator of 8y

c. Consider the regression for both retijrning and new students, Yj = fa + faXu +- faX2j + fa(Xu X Xtf + Uj. Use the conditional mean independence assumption E(UJ\X\J, X2j) = E(Uj\X2j) to show that fa = yh By + B3 = 8U and fa - Si - yx (the difference in the class size effects).

d. Suppose that you estimate the interaction regression in (c) using the combined data and that E(ut\X^X2iy-**E(uj\X2i). Show that Bj and fa are unbiased but that fa '1S m general biased.

Empirical Exercises

E8.1 Use the data set CPS08 described in Empirical Exercise 4.1 to answer the following questions. a. Run a regression of average hourly earnings (AHE) on age (Age).

gender (Female), and education (Bachelor). If Age increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?

b. Run a regression of the logarithm of average hourly earnings, ln(,4//), on Age, Female, and Bachelor. If Age increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34. how are earnings expected to change?

c. Run a regression of the logarithm of average hourly earnings. \n(AHE), on \n(Age), Female, and Bachelor. UAge increases from 25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?

d. Run a regression of the logarithm of average hourly earnings, \n(AHE), on Age. Age2. Female, and Bachelor. UAge increases from

E8.2

&

25 to 26, how are earnings expected to change? If Age increases from 33 to 34, how are earnings expected to change?

e. Do you prefer the regression in (c) to the regression in (b)? Explain. f. Do you prefer the regression in (d) to the regression in (b)? Explain. g. Do you prefer the regression in (d) to the regression in (c)? Explain. h. Plot the regression relation between Age and ln(AHE) from (b), (c),

and (d) for males with a high school diploma. Describe the similarities and differences between the estimated regression functions. Would your answer change if you plotted the regression function for females with college degrees?

i. Run a regression of \n(AHE), on Age, Age2, Female, Bachelor, and the interaction term Female X Bachelor. What does the coefficient on the interaction term measure? Alexis is a 30-year-old female with a bach-elor's degree. What does the regression predict for her value of \n(AHE)'l Jane is a 30-year-old female with a high school degree. What does the regression predict for her value of \n(AHE)1 What is the predicted difference between Alexis's and Jane's earnings? Bob is a 30-year-old male with a bachelor's degree. What does the regression predict for his value of \n(AHE)7 Jim is a 30-year-old male with a high school degree. What does the regression predict for his value of in(AHE)'? What is the predicted difference between Bob's and Jim's earnings?

j . Is the effect of Age on earnings different for men than for women? Specify and estimate a regression that you can use to answer this question.

k. Is the effect of Age on earnings different for high school graduates than for college graduates? Specify and estimate a regression that you can use to answer this question.

1. After running all these regressions (and any others that you want to run), summarize the effect of age on earnings for young workers.

Using the data set TeachingRatings described in Empirical Exercise 4.2, carry out the following exercises. a. Estimate a regression of Course_Eval on Beauty, Intro, OneCredit,

Female, Minority, and NNEnglish. b. Add Age and Age2 to the regression. Is there evidence that Age has a

nonlinear effect on Course_Eval? Is there evidence that Age has any effect on Course EvaH

i & t t


c Modify the regression in (a) so that the effect of Beauty on Course^Eval is different for men and women. Is the male-female dif-ference in the effect of Beauty statistically significant?

d. Professor Smith is a man. He has cosmetic surgery that increases his beauty index from one standard deviation below the average to one standard deviation above the average. What is his value of Beauty before the surgery? After the surgery? Using the regression in (c), construct a 95% confidence for the increase in his course evaluation.

e. Repeat (d) for Professor Jones, who is a woman. E8.3 Use the data set CollegeDistance described in Empirical Exercise 4.3 to

answer the following questions. a. Run a regression of ED on Dist, Fondle, Bytest, Tuition^ Black, Hispanic,

Incomehi. Ownhome, DadColl, MomColl, Cue80. and StwmfgSO. If Dist increases from 2 to 3 (that is, from 20 to 30 miles), how are years of edu-cation expected to change? If Dist increases from 6 to 7 (that is, from 60 to 70 miles), how are years^rf education expected to change?

b. Run a regression of \n(ED) on Dist, Female, Bytest, Tuition, Black, Hispanic, Incomehi, Ownhome, DadColl, MomColl, CueSO, and Stwmfg80. If Dist increases from 2 to 3 (from 20 to 30 miles), how are years of education, expected to change? If Dist increases from 6 to 7 (from 60 to 70 miles), how are years of education expected to change?

c Run a regression of ED on Dist, Dist2, Female, Bytest, Tuition. Black, Hispanic, Incomehi, Ownhome, DadColl, MomColl, CueSO, and StwmfgSO. If Dist increases from 2 to 3 (from 20 to 30 miles), how are years of education expected to change? If Dist increases from 6 to 7 (from 60 to 70 miles), how are years of education expected to change?

d. Do you prefer the regression in (c) to the regression in (a)? Explain. e. Consider a Hispanic female with Tuition = $950, Bytest = 58.

Incomehi = 0, Ownhome _ 0, DadColl 1, MomColl - 1, CueSO = 7.1, and Stwmfg = $10.06. i. Plot the regression relation between Dist and ED from (a) and (c)

for Dist in the range of 0 to 10 (from 0 to 100 miles). Describe the similarities and differences between the estimated regression functions. Would your answer change if

Econ 141

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