Stock Watson 3U Exercise Solutions Chapter 12 Instructors PDF

Preview

Summary

Download Stock Watson 3U Exercise Solutions Chapter 12 Instructors PDF

Description

!

! ! ! Introduction to Econometrics (3rd Updated Edition) by

James H. Stock and Mark W. Watson

Solutions to End-of-Chapter Exercises: Chapter 12*

(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].

©2015 Pearson Education, Inc.

!

rd

Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 12 1 _____________________________________________________________________________________________________

! 12.1. (a) The change in the regressor, ln(Picigarettes ) − ln(Picigarettes ), from a $0.50 per pack ,1995 ,1985 increase in the retail price is ln(8.00) − ln(7.50) = 0.0645. The expected percentage change in cigarette demand is − 0.94 × 0.0645× 100% = −6 .07%. The 95% confidence interval is (−0.94 ± 1.96 × 0.21) × 0.0645 × 100% = [−8.72%, −3.41%]. (b) With a 2% reduction in income, the expected percentage change in cigarette demand is 0.53 × (− 0.02) × 100% = −1.06%. (c) The regression in column (1) will not provide a reliable answer to the question in (b) when recessions last less than 1 year. The regression in column (1) studies the long-run price and income elasticity. Cigarettes are addictive. The response of demand to an income decrease will be smaller in the short run than in the long run. (d) The instrumental variable would be too weak (irrelevant) if the F-statistic in column (1) was 3.7 instead of 33.7, and we cannot rely on the standard methods for statistical inference. Thus the regression would not provide a reliable answer to the question posed in (a).

©2015 Pearson Education, Inc.

!

rd

Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 12 2 _____________________________________________________________________________________________________

12.2. (a) When there is only one X, we only need to check that the instrument enters the first stage population regression. Since the instrument is Z = X, the regression of X onto Z will have a coefficient of 1.0 on Z, so that the instrument enters the first stage population regression. Key Concept 4.3 implies corr(Xi, ui) = 0, and this implies corr(Zi, ui) = 0. Thus, the instrument is exogenous. (b) Condition 1 is satisfied because there are no W’s. Key Concept 4.3 implies that condition 2 is satisfied because (Xi, Zi, Yi) are i.i.d. draws from their joint distribution. Condition 3 is also satisfied by applying assumption 3 in Key Concept 4.3. Condition 4 is satisfied because of conclusion in part (a). (c) The TSLS estimator is βˆ1TSLS =

sZY sZX

using Equation (10.4) in the text. Since Zi =

Xi, we have

s s = βˆ1OLS . βˆ1TSLS = ZY = XY sZX s2X

©2015 Pearson Education, Inc.

!

rd

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

12.3. (a) The estimator σˆa2 = n−12 ∑in=1 (Yi − βˆ0TSLS − βˆ1TSLS Xˆi )2 is not consistent. Write this as

σˆa2 =

1 n −2

∑in=1( uˆi − βˆ1TSLS ( Xˆi − Xi )) 2 , where uˆi = Yi − βˆ0TSLS − βˆ1TSLS X i . Replacing

βˆ1TSLS with β1, as suggested in the question, write this as σˆa2 ≈ n1 ∑in=1 (ui − β1 (Xˆ i − Xi ))2 = n1 ∑in=1 ui2 + n1 ∑in= 1 [β 12 (Xˆi − Xi )2 + 2ui β 1 (Xˆi − Xi )]. The first term on the right hand side of the equation converges to σˆ 2u , but the second term converges to something that is non-zero. Thus σˆ a2 is not consistent. (b) The estimator σˆb2 =

1 n −2

Σin=1 (Yi − βˆ0TSLS − βˆ1TSLS Xi )2 is consistent. Using the same

notation as in (a), we can write σˆb2 ≈ n1 Σin=1ui2 , and this estimator converges in probability to σ u2 .

©2015 Pearson Education, Inc.

!

rd

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

12.4. Using Xˆ i =πˆ 0 +πˆ1 Zi , we have Xˆ = πˆ 0 +πˆ 1Z and n

n

= ∑( Xˆ i − Xˆ )(Yi − Y ) = πˆ1 ∑( Z i − Z )( Yi − Y ) = πˆ1 s ZY, sXY ˆ i= 1

i= 1

n

n

s2Xˆ = ∑( Xˆ i − Xˆ )2 = πˆ12 ∑( Z i − Z )2 = πˆ12 s 2Z. i= 1

i= 1

Using the formula for the OLS estimator in Key Concept 4.2, we have

πˆ1 =

sZX . sZ2

Thus the TSLS estimator

βˆ1TSLS =

sXY ˆ s

2 Xˆ

=

πˆ1 sZY sZY = = πˆ 12s Z2 πˆ 1sZ2

©2015 Pearson Education, Inc.

!

sZY s ZY . 2 = × sZ s ZX

sZX sZ2

rd

Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 12 5 _____________________________________________________________________________________________________

12.5. (a) Instrument relevance. Zi does not enter the population regression for X i (b) Z is not a valid instrument. Xˆ * will be perfectly collinear with W. (Alternatively, the first stage regression suffers from perfect multicollinearity.) (c) W is perfectly collinear with the constant term. (d) Z is not a valid instrument because it is correlated with the error term.

©2015 Pearson Education, Inc.

!

rd

Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 12 6 _____________________________________________________________________________________________________

12.6. Use R 2 to compute the homoskedasitic-only F statistic as 2

0.05 FHomoskedasitcOnly = − R2 /−k − = 0.95/98 = 5.16 with 100 observations in which case we 1 R /T k 1

conclude that the instrument may be week. With 500 observations the FHomoskedasitcOnly = 26.2 so the instrument is not weak.

©2015 Pearson Education, Inc.

!

rd

Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 12 7 _____________________________________________________________________________________________________

12.7. (a) Under the null hypothesis of instrument exogeneity, the J statistic is distributed as a χ12 random variable, with a 1% critical value of 6.63. Thus the statistic is significant, and instrument exogeneity E(ui |Z1i, Z2i) = 0 is rejected. (b) The J test suggests that E(ui |Z1i, Z2i) ≠ 0, but doesn’t provide evidence about whether the problem is with Z1 or Z2 or both.

©2015 Pearson Education, Inc.

!

rd

Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 12 8 _____________________________________________________________________________________________________

12.8. (a) Solving for P yields P =

γ 0 −β 0 β1

d

−

s

+ u i β1 u i ; thus Cov( P, us ) =

−σ

2

us

β1

(b) Because Cov(P,u) ≠ 0, the OLS estimator is inconsistent (see (6.1)). (c) We need a instrumental variable, something that is correlated with P but uncorrelated with us. In this case Q can serve as the instrument, because demand is completely inelastic (so that Q is not affected by shifts in supply). γ0 can be estimated by OLS (equivalently as the sample mean of Qi).

©2015 Pearson Education, Inc.

!

rd

Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 12 9 _____________________________________________________________________________________________________

12.9. (a) There are other factors that could affect both the choice to serve in the military and annual earnings. One example could be education, although this could be included in the regression as a control variable. Another variable is “ability” which is difficult to measure, and thus difficult to control for in the regression. (b) The draft was determined by a national lottery so the choice of serving in the military was random. Because it was randomly selected, the lottery number is uncorrelated with individual characteristics that may affect earning and hence the instrument is exogenous. Because it affected the probability of serving in the military, the lottery number is relevant.

©2015 Pearson Education, Inc.

!

rd

Stock/Watson - Introduction to Econometrics - 3 Updated Edition - Answers to Exercises: Chapter 12 10 _____________________________________________________________________________________________________

12.10.

Cov( Zi , Yi ) Cov( Zi , β 0 + β 1Xi + β 2Wi + ui ) β 1Cov (Zi , Xi ) + β 2Cov (Zi ,Wi ) βˆTSLS = = = Cov( Zi , X i ) Cov( Z i , X i ) Cov (Z i , X i )

(a) If Cov( Zi , Wi ) = 0 the IV estimator is consistent.

(b) If Cov( Zi , Wi ) ≠ 0 the IV estimator is not consistent.

©2015 Pearson Education, Inc.

!...