IntroductiontoEconometrics (3 U pdatedEdition, Global Edition · 2016. 2. 16. · Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter

©2015 Pearson Education, Ltd.

Introduction to Econometrics (3rd Updated Edition, Global Edition)

by

James H. Stock and Mark W. Watson

Solutions to Odd-Numbered End-of-Chapter Exercises:

Chapter 5(This version August 17, 2014)

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


1

5.1 (a)

(b) Calculate the t-statistic:

The p-value is less than 0.01, so we can reject the null hypothesis at the 5% significance level, and also at the 1% significance level.

(c) The t-statistic is

(d)

The 95% confidence interval for β1 is [–4.93 ± 1.96 × 2.02], that is –8.889 ≤ β1 ≤ –0.9708.

1

1

β̂ − 0 4.93β̂( ) 2.08

−t = = = −SE

2.3702

The p-value for the test H0: β1 = 0 against the alternative β1 ≠ 0 is

-value = 2Φ(− |p t |) = 2Φ(−2.3707) = 2× (0.0043) = 0.0086

1

1

β̂ − (−5.0)=−0.07

= −0.03472.02β̂( )

t =SE

The p-value for the test H0: β1 = –5.0 against the alternative β1 ≠ 5.0 is

-value = 2Φ(− |p t |) = 2Φ(−0.0347) = 2× (0.486) = 0.972

The p-value is larger than 0.10, so we cannot reject the null hypothesis at the 10%, 5% or 1% significance level. Because β1 = –5.0 is not rejected at the 5% level, this value is contained in the 95% confidence interval.

The 90% confidence interval for β0 is [640.3 ± 1.65 × 23.5], that is 601.525 ≤ β0 ≤ 679.075.

Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 5 2_____________________________________________________________________________________________________


5.3. The 95% confidence interval is 2 × [4.16 ± 1.96 × 0.42], that is 6.67 ≤ Weight gain ≤ 9.97 lbs.



5. 5 (a) The estimated gain from being in a small class is 13.9 points. This is equal toapproximately 1/5 of the standard deviation in test scores, a moderate increase.

(b) The t-statistic is 13.92.5 5.56,actt = = which has a p-value of 0.00. Thus the null

hypothesis is rejected at the 5% (and 1%) level.

(c) 13.9 ± 2.58 × 2.5 = 13.9 ± 6.45.



5.7. (a) The t-statistic is 3.21.5 2.13= with a p-value of 0.03; since the p-value is less than

0.05, the null hypothesis is rejected at the 5% level.

(b) 3.2 ± 1.96 × 1.5 = 3.2 ± 2.94

(c) Yes. If Y and X are independent, then β1 = 0; but this null hypothesis was rejectedat the 5% level in part (a).

(d) β1 would be rejected at the 5% level in 5% of the samples; 95% of the

confidence intervals would contain the value β1 = 0.



5.9. (a) β =1n (Y1 +Y2 +!+Yn )

Xso that it is linear function of Y1, Y2, …, Yn.

(b) E(Yi|X1, …, Xn) = β1Xi, thus

E(β |X1,…, Xn ) = E 1X

1n

(Y1 +Y2 +!+Yn )|X1,…, Xn )

= 1X

1nβ1( X1 +!+ Xn ) = β1



5.11. Using the results from 5.10, 0̂ m Y and 1̂ w m Y Y . From Chapter 3, SE( ) mm

sY

n and

2 2

SE( m .ww m

m w

Y Yn n

) s s

Plugging in the numbers,

m

00 1= $565.89β and SE(ˆ ˆ ) = $7.65β ; −β1 = $63.52 and SE(ˆ ˆ ) $8.73.β =



5.13. (a) Yes, this follows from the assumptions in KC 4.3.

(b) Yes, this follows from the assumptions in KC 4.3 and conditional

homoskedasticity

(c) They would be unchanged for the reasons specified in the answers to those

questions.

(d) (a) is unchanged; (b) is no longer true as the errors are not conditionally

homosckesdastic.



5.15. Because the samples are independent, ,1ˆmβ and ,1

ˆwβ are independent. Thus

,1 ,1 ,1 ,1ˆ ˆ ˆ ˆvar ( ) var ( ) var( ).m w m wβ β β β− = + ,1

ˆVar ( )mβ is consistently estimated as

2,1

ˆ[ ( )]mSE β and ,1ˆVar ( )wβ is consistently estimated as 2

,1ˆ[ ( )] ,wSE β so that

,1 ,1ˆ ˆvar( )m wβ β− is consistently estimated by 2 2

,1 ,1ˆ ˆ[ ( )] [ ( )] ,m wSE SEβ β+ and the result

follows by noting the SE is the square root of the estimated variance.

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IntroductiontoEconometrics (3 U pdatedEdition, Global Edition · 2016. 2. 16. · Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter