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    IntroductiontoEconometrics (3rdUpdatedEdition,GlobalEdition)   by   JamesH.StockandMarkW.Watson     

SolutionstoEnd‐of‐ChapterExercises:Chapter4*  (This version August 17, 2014)

            *Limiteddistribution:ForInstructorsOnly.Answerstoallodd‐numbered questionsareprovidedtostudentsonthetextbookwebsite.Ifyoufinderrorsin thesolutions,pleasepassthemalongtousat[email protected].

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1 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

 4.1. (a) The predicted average test score is

  6403  493  25  51705 TestScore (b) The predicted change in classroom average test score is

  TestScore   4 93(24  21)   14 79

(c) Using the formula for ˆ0 in Equation (4.8), we know that the sample average of the test scores across the 50 classrooms is

TestScore ˆ 0  ˆ1  CS  6403  4 93(22 8)  527 9

(d) Use the formula for the standard error of the regression (SER) in Equation (4.19) to get the sum of squared residuals:

SSR  ( n  2) SER2  (50  2) 8 72  3,633.12 Use the formula for R2 in Equation (4.16) to calculate the total sum of squares:

TSS  Sample variance is

3633.12 3633.12   4082.16 1  011 089

TSS 4082.16  83.309 . n 1 49

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2 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.2. (a) Substituting Height = 64, 68, 72 into the given equation, the predicted weights are 187, 203.64, and 220.28 pounds.

 Weight  4 16   Height  4 16  2  8 32 pounds. (b) 

(c) 1 inches = 2.54 cm, and 1 lb = 0.4536 kg. Suppose the regression equation in centimeter – kilogram units is

. The coefficients are ˆ0   79 24  0 4536   35 94 kg, and

 0 7429 kg per cm. The R2 is unit free, so remains at 0.72. The ˆ 1  416  024536 54 SER is now 12.6  0.4536 = 5.7154 kg.

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3 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.3.

(a) The coefficient 9.6 shows the marginal effect of Age on AWE; that is, AWE is expected to increase by $9.6 for each additional year of age. 696.7 is the intercept of the regression line. It determines the overall level of the line. (b) SER is in the same units as the dependent variable (Y, or AWE in this example). Thus SER is measures in dollars per week.

(c) R2 is unit free.

(d) (i) 696.7  9.6  25  $936.7; (ii) 696.7  9.6 45  $1,128.7

(e) No. The oldest worker in the sample is 65 years old. 99 years is far outside the range of the sample data.

(f) No. The distribution of earning is positively skewed and has kurtosis larger than the normal. (g) ˆ0  Y  ˆ1 X , so that Y  ˆ0  ˆ1 X . Thus the sample mean of AWE is 696.7 + 9.6  41.6 = $1,096.06.

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4 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.4.

(a) (R  R f )   ( R m  R f )  u , so that var (R  R f )   2  var( Rm  R f )  var(u )  2   cov(u , Rm  R f ). But cov(u , Rm  R f )  0, thus var( R  Rf )   2  var( Rm  Rf )  var( u). With  > 1, var(R  Rf) > var(Rm  Rf), because var(u)  0.

(b) Yes. Using the expression in (a) var (R  Rf )  var ( Rm  Rf )  (  2 1)  var ( Rm  Rf )  var( u), which will be positive if var(u )  (1   2 )  var ( Rm  Rf ).

(c) Rm  Rf  5.3%  2.0%  3.3%.Thus, the predicted returns are

Rˆ  R f  ˆ(R m  R f )  2.0%  ˆ  3.3% Verizon: 2.0% + 0.0×3.3% = 2.0% Wal-Mart: 2.0% + 0.3×3.3% = 3.0% Kellogg: 2.0% + 0.5×3.3% = 3.7% Waste Management: 2.0% + 0.6×3.3% = 4.0% Googl: 2.0% + 1.0×3.3% = 5.3% Ford Moter Company: 2.0% + 1.3×3.3% = 6.3% Bank of America: 2.0% + 2.2×3.3% = 9.3%

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5 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.5.

(a) ui represents factors other than time that influence the participant’s performance on the test including inherent cognitive ability and aptitude. Some may have better memories than others, and some might be worse.

(b) Because of random assignment ui is independent of Xi. Since ui represents deviations from average E(ui) = 0. Since u and X are independent, E(ui|Xi) =

E(ui) = 0. If E(bˆ0 )  b0 and E (bˆ1)  b1 then the estimated coefficients are unbiased.

(c) Non-compliance is a concern because while the researcher can assign nap times to individual participants, they cannot ensure that participants will sleep for the entire period of time. However, if the sleep compliance is equally good (or bad) across both groups, non-compliance will only reduce the precision of the estimates but will not introduce bias.

(d) (i) 55 0.17 60 65.2; 55 0.17 75  67.75; 55  0.17 90  70.3 (ii) 0.17  5  0.85

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6 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.6. Using E(ui |Xi )  0, we have

E(Yi|X i )  E ( 0  1 X i  ui |X i )  0  1E( X i | X i )  E( ui | Xi )   0  1 Xi

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7 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.7. The expectation of ˆ0 is obtained by taking expectations of both sides of Equation (4.8):    1 n  E ( ˆ0 )  E (Y  ˆ1 X )  E   0  1 X   u i   ˆ1X    n i 1    n 1   0  E(  1  ˆ1) X   E( ui) n i 1  0

where the third equality in the above equation has used the facts that E(ui) = 0 and E[( ˆ1 − 1) X ] = E[(E( ˆ −1)| X ) X ] = because E[(  ˆ ) | X ]  0 (see text 1

1

equation (4.31).)

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1

8 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.8. The only change is that the mean of ˆ0 is now 0  2. An easy way to see this is this is to write the regression model as

Yi  (0  2)  1 X i  (ui  2).

The new regression error is (ui  2) and the new intercept is ( 0  2). All of the assumptions of Key Concept 4.3 hold for this regression model.

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9 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

2 4.9. (a) With ˆ1  0, ˆ0  Y , and Yˆi  ˆ0  Y . Thus ESS  0 and R  0.

(b) If R2  0, then ESS  0, so that Yˆi  Y for all i. But Yˆi  ˆ0  ˆ1 X i , so that

Yˆi  Y for all i, which implies that ˆ1  0, or that Xi is constant for all i. If Xi is constant for all i, then



n i 1

( X i  X )2  0 and ˆ1 is undefined (see equation

(4.7)).

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10 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.10. (a) E(ui|X  0)  0 and E(ui|X  1)  0. (Xi, ui) are i.i.d. so that (Xi, Yi) are i.i.d. (because Yi is a function of Xi and ui). Xi is bounded and so has finite fourth moment; the fourth moment is non-zero because Pr(Xi  0) and Pr(Xi  1) are both non-zero. Following calculations like those exercise 2.13, ui also has nonzero finite fourth moment.

(b) var( X i )  0.2  (1  0.2)  0.16 and  X  0.2. Also

var[( X i  X )ui ]  E[( X i   X )ui ] 2  E[( Xi   X ) ui| X i  0]2  Pr( X i  0)  E[( X i   X )u i| X i  1]2  Pr( X i  1) where the first equality follows because E[(Xi  X)ui]  0, and the second equality follows from the law of iterated expectations. E [( X i   X )u i|X i  0]2  0.22  1, and E [(X i   X )u i|X i  1]2  (1 0.2)2  4.

Putting these results together

2ˆ  1

1 (0.22 1  0.8)  ((1  0.2) 2  4  0.2) 1  21.25 n 0.162 n

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11 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.11. (a) The least squares objective function is respect to b1 yields

 in1 ( Yi b1 Xi )2 b1



n

i 1

(Yi b1 X i )2 . Differentiating with

 2 i 1 Xi (Yi  b1 Xi ). Setting this zero, and n

n

XY solving for the least squares estimator yields ˆ1  i n1 Xi 2i . i 1

n  (b) Following the same steps in (a) yields ˆ1  i 1Xn i (XYi2 4) i 1

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i

i

12 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.12. (a) Write n

ESS   (Yˆi  Y ) 2  i1

n

 (ˆ0  ˆ1 Xi  Y )2  i1

n

[ˆ ( X i1

i

1

 X )]2

2

  ( X i  X )(Yi  Y )   ˆ  ( Xi  X )   . in1 ( Xi  X ) 2 i 1 n

2 1

2

n i 1

This implies 2

 ni1 ( X i  X )(Yi  Y )  ESS R  n  i 1 ( Yi  Y ) 2 ni 1 ( Xi  X ) 2 ni 1 ( Yi  Y ) 2 2

  

1 n 1 1 n 1

  2 n 1 Y Y   ( ) n 1 i 1 i 

 ni1 ( X i  X )( Yi  Y )

 ni 1 ( X i  X )2

2

2

 s    XY   rXY2  s X sY 

(b) This follows from part (a) because rXY  rYX.

(c) Because rXY =

s XY sX2

s XY , rXY s Y s X sY sX

=

n 1 ( X i  X )(Yi  Y )  (n 1) i 1   n 1 2 (X i  X ) (n 1)  i 1

n

 (X

i

 X )(Yi  Y )

i 1

n

 (X

X )

2

i

i 1

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 ˆ1

13 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.13. The answer follows the derivations in Appendix 4.3 in “Large-Sample Normal Distribution of the OLS Estimator.” In particular, the expression for i is now i = (Xi − X) ui, so that var(i) = 3var[(Xi − X)ui], and the term 2 carry through the rest of the calculations.

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14 Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 4 _____________________________________________________________________________________________________

4.14. Because ˆ0  Y  ˆ1 X , Y  ˆ 0   1X . The sample regression line is y  ˆ0  1 x , so that the sample regression line passes through ( X , Y ).

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