Introduction to Econometrics - Pearson Educationwps.pearsoned.co.uk/wps/media/objects/16103/16489878/answersto... · Introduction to Econometrics (3. rd. Updated Edition, Global Edition)

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 12

(This version August 17, 2014)

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

1

pack increase in the retail price is ln(7.00) − ln(6.75) = 0.0364. The expected

percentage change in cigarette demand is −0.94 × 0.0364 × 100% = −3.42%. The

95% confidence interval is (−0.94 ± 1.96 × 0.21) × 0.0364 × 100% = [−4.92%,

−1.92%].

(b) With a 5% reduction in income, the expected percentage change in cigarette

demand is 0.53 × (−0.05) × 100% = −2.65%.

(c) The coefficients on price changes and income will most likely be smaller than

0.94 over the eight-year horizon and larger than 0.94 over the twelve-year

horizon. This is because the long-term elasticity for an addictive good like

cigarettes is larger than the short-term elasticity.

(d) The instrumental variable would stronger if the F-statistic in column (1) was

63.6 instead of 33.6, which means we would have greater confidence in the

inference made in (a).

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12.1. (a) The change in the regressor, ,1995 ,1985ln( cigarettes )− ln( cigarettesP ),i iP from a $0.25 per pack

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

12.3. (a) The estimator 2 211 0 12

ˆ ˆ ˆˆ ( )n TSLS TSLSa i i in Y Xσ β β=−= ∑ − − is not consistent. Write this as

2 211 12

ˆ ˆˆ ˆ( ( )) ,n TSLSa i i i in u X Xσ β=−= ∑ − − where 0 1

ˆ ˆˆ .TSLS TSLSi i iu Y Xβ β= − − Replacing

1̂TSLSβ with β1, as suggested in the question, write this as

2 2 2 2 21 1 11 1 1 1 1 1

ˆ ˆ ˆˆ ( ( )) [ ( ) 2 ( )].n n na i i i i i i i i i i i in n nu X X u X X u X Xσ β β β= = =≈ ∑ − − = ∑ + ∑ − + −

The first term on the right hand side of the equation converges to 2ˆ ,uσ but the

second term converges to something that is non-zero. Thus 2ˆaσ is not consistent.

(b) The estimator 2 211 0 12

ˆ ˆˆ ( )n TSLS TSLSb i i in Y Xσ β β=−= S − − is consistent. Using the same

notation as in (a), we can write 2 211ˆ ,n

b i in uσ =≈ S and this estimator converges in

probability to 2.uσ

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

12.5. (a) Instrument relevance. iZ does not enter the population regression for iX

(b) Z is not a valid instrument. *X̂ 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.

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

©2015 Pearson Education, Ltd.

12.7. (a) Under the null hypothesis of instrument exogeneity, the J statistic is distributed

1as a χ 2 random variable, with a 5% critical value of 3.84. Thus the statistic is not

significant at 95%, and instrument exogeneity E(ui |Z1i, Z2i) = 0 cannot be rejected.

(b) The J test suggests that E(ui |Z1i, Z2i) ≠ 0, so the researcher will need to find

alternative instruments in order to make credible inferences.

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

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.

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