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

Transcription

1 Introducton to Econometrcs (3 rd Updated Edton) by James H. Stock and Mark W. Watson Solutons to Odd-Numbered End-of-Chapter Exercses: Chapter 1 (Ths verson July 0, 014)

2 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter 1 1 cgarettes cgarettes 1.1. (a) The change n the regressor, ln( P,1995 ) ln( P,1985 ), from a $0.50 per pack ncrease n the retal prce s ln(8.00) ln(7.50) The expected percentage change n cgarette demand s %.07%. The 95% confdence nterval s ( ) % [ 8.7%, 3.41%]. (b) Wth a % reducton n ncome, the expected percentage change n cgarette demand s 0.53 ( 0.0) 100% 1.06%. (c) The regresson n column (1) wll not provde a relable answer to the queston n (b) when recessons last less than 1 year. The regresson n column (1) studes the long-run prce and ncome elastcty. Cgarettes are addctve. The response of demand to an ncome decrease wll be smaller n the short run than n the long run. (d) The nstrumental varable would be too weak (rrelevant) f the F-statstc n column (1) was 3.7 nstead of 33.7, and we cannot rely on the standard methods for statstcal nference. Thus the regresson would not provde a relable answer to the queston posed n (a).

3 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter 1 1 n 1.3. (a) The estmator ˆTSLS ˆTSLS ˆ ˆ a n 1( Y 0 1 X) s not consstent. Wrte ths as 1 n ˆ TSLS ˆ ˆ ˆ a n 1( u 1 ( X X)), where ˆTSLS ˆTSLS uˆ Y 0 1 X. Replacng ˆTSLS 1 wth 1, as suggested n the queston, wrte ths as ˆ ( u ( Xˆ X )) u [ ( Xˆ X ) u ( Xˆ X )]. 1 n 1 n 1 n a n 1 1 n 1 n ˆ u, but the The frst term on the rght hand sde of the equaton converges to second term converges to somethng that s non-zero. Thus ˆa s not consstent. (b) The estmator 1 n ˆTSLS ˆTSLS ˆ b n 1( Y 0 1 X) s consstent. Usng the same notaton as n (a), we can wrte ˆ 1 n u, and ths estmator converges n probablty to u. b n 1

4 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter (a) Instrument relevance. Z does not enter the populaton regresson for * (b) Z s not a vald nstrument. ˆX wll be perfectly collnear wth W. (Alternatvely, the frst stage regresson suffers from perfect multcollnearty.) (c) W s perfectly collnear wth the constant term. (d) Z s not a vald nstrument because t s correlated wth the error term. X

5 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter (a) Under the null hypothess of nstrument exogenety, the J statstc s dstrbuted as a 1 random varable, wth a 1% crtcal value of Thus the statstc s sgnfcant, and nstrument exogenety E(u Z 1, Z ) 0 s rejected. (b) The J test suggests that E(u Z 1, Z ) 0, but doesn t provde evdence about whether the problem s wth Z 1 or Z or both.

6 Stock/Watson - Introducton to Econometrcs - 3 rd Updated Edton - Answers to Exercses: Chapter (a) There are other factors that could affect both the choce to serve n the mltary and annual earnngs. One example could be educaton, although ths could be ncluded n the regresson as a control varable. Another varable s ablty whch s dffcult to measure, and thus dffcult to control for n the regresson. (b) The draft was determned by a natonal lottery so the choce of servng n the mltary was random. Because t was randomly selected, the lottery number s uncorrelated wth ndvdual characterstcs that may affect earnng and hence the nstrument s exogenous. Because t affected the probablty of servng n the mltary, the lottery number s relevant.