Solutions To Odd-Numbered End-of-Chapter Exercises: Chapter 12

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Introduction to Econometrics (3rd Updated Edition)byJames H. Stock and Mark W. WatsonSolutions to Odd-Numbered End-of-Chapter Exercises:Chapter 12(This version August 17, 2014) 2015 Pearson Education, Inc.

Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 121cigarettescigarettes12.1. (a) The change in the regressor, ln( Pi ,1995) ln( Pi ,1985), from a 0.50 per packincrease in the retail price is ln(8.00) ln(7.50) 0.0645. The expectedpercentage change in cigarette demand is 0.94 0.0645 100% 6.07%. The95% 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 cigarettedemand is 0.53 ( 0.02) 100% 1.06%.(c) The regression in column (1) will not provide a reliable answer to the questionin (b) when recessions last less than 1 year. The regression in column (1) studiesthe long-run price and income elasticity. Cigarettes are addictive. The responseof demand to an income decrease will be smaller in the short run than in thelong run.(d) The instrumental variable would be too weak (irrelevant) if the F-statistic incolumn (1) was 3.7 instead of 33.7, and we cannot rely on the standard methodsfor statistical inference. Thus the regression would not provide a reliable answerto the question posed in (a). 2015 Pearson Education, Inc.

Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 12212.3. (a) The estimator σˆ a2 n 1 2 in 1 (Yi βˆ0TSLS βˆ1TSLS Xˆ i )2 is not consistent. Write this asσˆ a2 n 1 2 in 1 (uˆi βˆ1TSLS ( Xˆ i X i ))2 , where uˆi Yi βˆ0TSLS βˆ1TSLS X i . Replacingβˆ1TSLS with β1, as suggested in the question, write this asσˆ a2 1n in 1 (ui β1 ( Xˆ i X i ))2 1n in 1 ui2 1n in 1[β12 ( Xˆ i X i )2 2ui β1 ( Xˆ i X i )].The first term on the right hand side of the equation converges to σˆ u2 , but thesecond term converges to something that is non-zero. Thus σˆ a2 is not consistent.(b) The estimator σˆb2 n 1 2 Σin 1 (Yi βˆ0TSLS βˆ1TSLS X i )2 is consistent. Using the samenotation as in (a), we can write σˆ b2 1n Σin 1ui2 , and this estimator converges inprobability to σ u2 . 2015 Pearson Education, Inc.

Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 12312.5. (a) Instrument relevance. Z i does not enter the population regression for X i(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. 2015 Pearson Education, Inc.

Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 12412.7. (a) Under the null hypothesis of instrument exogeneity, the J statistic is distributedas a χ12 random variable, with a 1% critical value of 6.63. Thus the statistic issignificant, 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 aboutwhether the problem is with Z1 or Z2 or both. 2015 Pearson Education, Inc.

Stock/Watson - Introduction to Econometrics - 3rd Updated Edition - Answers to Exercises: Chapter 12512.9. (a) There are other factors that could affect both the choice to serve in the militaryand annual earnings. One example could be education, although this could beincluded 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 themilitary was random. Because it was randomly selected, the lottery number isuncorrelated with individual characteristics that may affect earning and hence theinstrument is exogenous. Because it affected the probability of serving in themilitary, the lottery number is relevant. 2015 Pearson Education, Inc.

Solutions to Odd-Numbered End-of-Chapter Exercises: Chapter 12 . Chapter 12 _ 2015 Pearson Education, Inc. ! 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. .

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