Solutions to Odd-Numbered End-of-Chapter Exercises: Chapter 17

Transcription

1 Itroucto to Ecoometrcs (3 r Upate Eto) by James H. Stock a Mark W. Watso Solutos to O-Numbere E-of-Chapter Exercses: Chapter 7 (Ths erso August 7, 04) 05 Pearso Eucato, Ic.

2 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (a) Suppose there are obseratos. Let b be a arbtrary estmator of β. Ge the estmator b, the sum of square errors for the ge regresso moel s ( Y b ). ˆ β, the restrcte least squares estmator of β, mmzes the sum of square RLS ˆ RLS errors. That s, β satsfes the frst orer coto for the mmzato whch requres the fferetal of the sum of square errors wth respect to b equals zero: ( Y b )( ) 0. Solg for b from the frst orer coto leas to the restrcte least squares estmator ˆ Y. RLS β (b) We show frst that ˆ β RLS s ubase. We ca represet the restrcte least squares estmator ˆ β RLS terms of the regressors a errors: Thus ˆ Y ( β + u) u β. RLS β + ˆ RLS u Eu (,, ) E( β ) β E E, β K + + β where the seco equalty follows by usg the law of terate expectatos, a the thr equalty follows from Eu (, K, ) 0 (cotue o the ext page) 05 Pearso Eucato, Ic.

3 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter 7 7. (cotue) because the obseratos are... a E(u ) 0. (Note, E(u,, ) E(u ) because the obseratos are... Uer assumptos 3 of Key Cocept 7., ˆ β RLS s asymptotcally ormally strbute. The large sample ormal approxmato to the lmtg strbuto ˆ RLS of β follows from coserg ˆ u u β β. RLS Coser frst the umerator whch s the sample aerage of u. By assumpto of Key Cocept 7., has mea zero: Eu ( ) EEu [ ( )] 0. By assumpto, s... By assumpto 3, ar( ) s fte. Let u Usg the cetral lmt theorem, the sample aerage or, the σ σ/. / σ N(0, ) σ u N(0, σ ). For the eomator, s... wth fte seco arace (because has a fte fourth momet), so that by the law of large umbers E( ). p Combg the results o the umerator a the eomator a applyg Slutsky s theorem lea to (cotue o the ext page) 05 Pearso Eucato, Ic.

4 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (cotue) ˆ RLS u ar( u ) ( β βu ) N 0,. E( ) (c) ˆ β RLS s a lear estmator: ˆ Y, where. RLS β ay a The weght a (,, ) epes o,, but ot o Y,, Y. Thus ˆ u β β. RLS + ˆ β RLS s cotoally ubase because E( ˆβ RLS,, E β + u,, β + E u,, β. The fal equalty use the fact that E u,, E(u,, ) 0 because the obseratos are... a E (u ) 0. (cotue o the ext page) 05 Pearso Eucato, Ic.

5 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (cotue) () The cotoal arace of ˆ RLS β, ge,,, s ar( ˆβ RLS,, ) ar β + u,, ( σ u ar(u,, ) ( σ u ). ) (e) The cotoal arace of the OLS estmator ˆβ s Sce ar( ˆ σ β,, ). u K ( ) ( ) + <, the OLS estmator has a larger cotoal arace: ar(,, ) ar( ˆ RLS β K > β, K, ). ˆ RLS The restrcte least squares estmator β s more effcet. (f) Uer assumpto 5 of Key Cocept 7., cotoal o,,, ˆ β RLS s ormally strbute sce t s a weghte aerage of ormally strbute arables u : ˆ u β β. RLS + (cotue o the ext page) 05 Pearso Eucato, Ic.

6 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (cotue) Usg the cotoal mea a cotoal arace of ˆ β RLS ere parts (c) a () respectely, the samplg strbuto of ˆ β RLS, cotoal o,,, s (g) The estmator!β The cotoal arace s ˆ RLS σ u β ~ N β,. Y (β + u ) β + u ar( β!,, ) ar β + u,, ar(u,, ) σ u ( ( ). ) The fferece the cotoal arace of! β a ˆβ RLS s ar(! β,, ) ar( ˆβ RLS,, ) σ u ( ) σ u. I orer to proe ar(! β,, ) ar( ˆβ RLS,, ), we ee to show ( ) or equaletly (cotue o the ext page) 05 Pearso Eucato, Ic.

7 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (cotue) Ths equalty comes rectly by applyg the Cauchy-Schwartz equalty. whch mples ( a b) a b. That s Σ (Σ x ), or ar( β!,, ) ar( ˆβ RLS,, ). Note: because! β s lear a cotoally ubase, the result ar(! β,, ) ar( ˆβ RLS,, ) follows rectly from the Gauss-Marko theorem. 05 Pearso Eucato, Ic.

8 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (a) Usg Equato (7.9), we hae β ( ) ( ˆ β ) ( ) u [( µ ) ( µ )] u ( ) ( µ ) u ( µ ) u ( ) ( ) ( µ ) u ( ) ( ) by efg ( µ )u. (b) The raom arables u,, u are... wth mea µ u 0 a arace 0 σ u. < < By the cetral lmt theorem, u ( µ ) u u σ σ u u N(0, ). The law of large umbers mples p µ,or µ 0. By the cosstecy of sample arace, ( ) Σ coerges probablty to populato arace, ar( ), whch s fte a o-zero. The result the follows from Slutsky s theorem. (c) The raom arable ( µ ) u has fte arace: p ar( ) ar[( µ ) µ ] E [( µ ) u ] E < 4 4 [( µ ) ] E[( u) ]. The equalty follows by applyg the Cauchy-Schwartz equalty, a the seco equalty follows because of the fte fourth momets for (, u ). The fte arace alog wth the fact that has mea zero (by assumpto of Key Cocept 5.) a s... (by assumpto ) mples that the sample aerage satsfes the requremets of the cetral lmt theorem. Thus, (cotue o the ext page) 05 Pearso Eucato, Ic.

9 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (cotue) σ σ satsfes the cetral lmt theorem. () Applyg the cetral lmt theorem, we hae σ N(0, ). Because the sample arace s a cosstet estmator of the populato arace, we hae Usg Slutsky s theorem, ( ) p. ar( ) t σ ( ) σ t N(0,), or equaletly ar( ) N 0,. ( ) [ar( )] Thus ( µ ) u β ( ) ( ) ( ˆ β ) ar( ) N 0, [ar( )] sce the seco term for ( ˆ β β) coerges probablty to zero as show part (b). 05 Pearso Eucato, Ic.

10 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter Because E(W 4 ) [E(W )] + ar(w ), [E(W )] E (W 4 ) <. Thus E(W ) <. 05 Pearso Eucato, Ic.

11 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (a) The jot probablty strbuto fucto of u, u j,, j s f (u, u j,, j ). The cotoal probablty strbuto fucto of u a ge u j a j s f (u, u j, j ). Sce u,,,, are..., f (u, u j, j ) f (u, ). By efto of the cotoal probablty strbuto fucto, we hae f( u, u,, ) f( u, u, ) f( u, ) j j j j j j f( u, ) f( u, ). j j (b) The cotoal probablty strbuto fucto of u a u j ge a j equals f( u, uj,, j) f( u, ) f( uj, j) f( u, uj, j) f( u ) f( uj j). f(, ) f( ) f( ) j j The frst a thr equaltes use the efto of the cotoal probablty strbuto fucto. The seco equalty use the cocluso the from part (a) a the epeece betwee a j. Substtutg f( u, u, ) f( u ) f( u ) j j j j to the efto of the cotoal expectato, we hae E( u u, ) u u f ( u, u, ) u u j j j j j j u u f ( u ) f ( u ) u u j j j j u f ( u ) u u f ( u ) u j j j j Eu ( ) Eu ( ). j j (c) Let Q (,,,, +,, ), so that f (u,, ) f (u, Q). Wrte (cotue o ext page) 05 Pearso Eucato, Ic.

12 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (cotue) f( u, Q) f( u,, Q) f(, Q) f( u, ) f( Q) f( ) f( Q) f( u, ) f( ) f( u ) where the frst equalty uses the efto of the cotoal esty, the seco uses the fact that (u, ) a Q are epeet, a the fal equalty uses the efto of the cotoal esty. The result the follows rectly. () A argumet lke that use (c) mples f( uu, K ) f( uu, ) j j j a the result the follows from part (b). 05 Pearso Eucato, Ic.

13 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter We ee to proe p [( ) ˆ u ( µ ) u ] 0. Usg the etty µ + ( µ ), [( ) ˆ ( ) ] ( ) ˆ u µ u µ u ( µ ) ( µ ) u ˆ + ( ) ( ˆ µ u u ). The efto of u ˆ mples uˆ u + ( ˆ β β ) + ( ˆ β β ) u ( ˆ β β ) Substtutg ths to the expresso for u ( ˆ β β ) + ( ˆ β β )( ˆ β β ). 0 0 Σ [( ) u ( ) u ] yels a ˆ µ seres of terms each of whch ca be wrtte as a b where a 0 a b Σ u where r a s are tegers. For example, r s a ( µ ), a ( ˆ β β ) a so forth. The result the follows from Slutksy s p theorem f r s r s Σ u where s a fte costat. Let w u a ote that w s... The law of large umbers ca the be use for the esre result f Ew ( ). < There are two cases that ee to be aresse. I the frst, both r a s are o-zero. I ths case wrte p Ew E u E Eu r s 4r 4s ( ) ( ) < [ ( )][ ( )] a ths term s fte f r a s are less tha. Ispecto of the terms shows that ths s true. I the seco case, ether r 0 or s 0. I ths case the result follows rectly f the o-zero expoet (r or s) s less tha 4. Ispecto of the terms shows that ths s true. 05 Pearso Eucato, Ic.

14 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter Note: early prtg of the thr eto there was a typographcal error the expresso for µ Y. The correct expresso s µ Y µ Y+ ( σy / σ )( x µ ). (a) Usg the ht a equato (7.38) f Y x( y) σ ( ρ ) Y Y x µ x µ y µ Y y µ Y x µ exp ρ. Y + + ( ρy ) σ σ σy σ Y σ Smplfyg yels the esre expresso. (b) The result follows by otg that f Y x (y) s a ormal esty (see equato (7.36)) wth µ µ T a σ σ. (c) Let b σ Y / σ a a µ Y bµ. Y 05 Pearso Eucato, Ic.

15 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (a) The aswer s proe by equato (3.0) a the scusso followg the equato. The result was also show Exercse 3.0, a the approach use the exercse s scusse part (b). (b) Wrte the regresso moel as Y β0 + β +, where β0 E(β0), β E(β), a u + (β0 β0) + (β β). Notce that E( ) E(u ) + E(β0 β0 ) + E(β β ) 0 because β0 a β are epeet of. Because E( ) 0, the OLS regresso of Y o wll proe cosstet estmates of β0 E(β0) a β E(β). Recall that the weghte least squares estmator s the OLS estmator of Y/σ oto /σ a /σ, where σ θ + θ. Wrte ths regresso as 0 Y / σ β (/ σ ) + β ( / σ ) + / σ. 0 Ths regresso has two regressors, /σ a /σ. Because these regressors epe oly o, E( ) 0 mples that E(/σ (/σ), /σ) 0. Thus, weghte least squares proes a cosstet estmator of β0 E(β0) a β E(β). 05 Pearso Eucato, Ic.

16 Stock/Watso - Itroucto to Ecoometrcs - 3 r Upate Eto - Aswers to Exercses: Chapter (a) Wrte W Z where Z ~ N(0,). From the law of large umber W/. E( Z ) (b) The umerator s N(0,) a the eomator coerges probablty to. The result follows from Slutsky s theorem (equato (7.9)). (c) V/m s strbute χ m / m a the eomator coerges probablty to. The result follows from Slutsky s theorem (equato (7.9)). 05 Pearso Eucato, Ic.