Introduction to Econometrics (3 rd Updated Edition, Global Edition) Solutions to Odd-Numbered End-of-Chapter Exercises: Chapter 9

Transcription

1 Itroducto to Ecoometrcs (3 rd Udated Edto, Global Edto) by James H. Stock ad Mark W. Watso Solutos to Odd-Numbered Ed-of-Chater Exercses: Chater 9 (Ths verso August 7, 04) 05 Pearso Educato, Ltd.

2 Stock/Watso - Itroducto to Ecoometrcs - 3 rd Udated Edto - Aswers to Exercses: Chater As exlaed the text, otetal threats to exteral valdty arse from dffereces betwee the oulato ad settg studed ad the oulato ad settg of terest. The statstcal results based o West Afrca 000 mght aly to some arts of Ida 000 but ot to the Uted Kgdom 000. I 000, may arts of Ida, artcularly rural Ida, were lagued by hgh malutrto ad oor chld health. I cotrast, the UK had very hgh stadards of ublc health as well as hgher er cata food cosumto, so the dstrbuto of vtams would most lkely ot have a hgh mact o test scores. The results from West Afrca 000 may cotue to aly to the same rego 05 but ths deeds o mrovemets ublc health ad er cata cosumto that may have take lace over ths tme erod. 05 Pearso Educato, Ltd.

3 Stock/Watso - Itroducto to Ecoometrcs - 3 rd Udated Edto - Aswers to Exercses: Chater The key s that the selected samle cotas oly emloyed wome. Cosder two wome, Beth ad Jule. Beth has o chldre; Jule has oe chld. Beth ad Jule are otherwse detcal. Both ca ear $5,000 er year the labor market. Each must comare the $5,000 beeft to the costs of workg. For Beth, the cost of workg s forgoe lesure. For Jule, t s forgoe lesure ad the costs (ecuary ad other) of chld care. If Beth s just o the marg betwee workg the labor market or ot, the Jule, who has a hgher oortuty cost, wll decde ot to work the labor market. Istead, Jule wll work home roducto, carg for chldre, ad so forth. Thus, o average, wome wth chldre who decde to work are wome who ear hgher wages the labor market. 05 Pearso Educato, Ltd.

4 Stock/Watso - Itroducto to Ecoometrcs - 3 rd Udated Edto - Aswers to Exercses: Chater 9 3 γβ 0 γβ 0 γu βv 9.5. (a) Q = +. γ β γ β 0 0 u v ad P β γ = +. γ β γ β γβ γβ EQ ( ) =, γ β (b) 0 0 EP ( ) β0 γ0 = γ β (c) Var( Q) = ( γσ u + βσ v), Var( P) = ( σu + σv),ad γ β γ β Cov( P, Q) = ( γσ u + βσ V) γ β (d) () ˆ Cov( Q, P) γσ + βσ β, u V = Var( P) σu + σv ˆ Cov( P, Q) β0 EQ ( ) EP ( ) Var ( P ) () ˆ σu ( γ β) β 0, σu + σv β > usg the fact that γ > 0 (suly curves sloe u) ad β < 0 (demad curves sloe dow). 05 Pearso Educato, Ltd.

5 Stock/Watso - Itroducto to Ecoometrcs - 3 rd Udated Edto - Aswers to Exercses: Chater (a) False. Correlato betwee the deedet varable ad the regressors reduce the recso of the OLS estmator, but do ot duce bas. (b) False. If the error term exhbts heteroskedastcty, the stadard errors eed to be corrected for ths to roduce ubased estmates of. If heteroskedastcty-robust stadard errors are ot used, the estmates of wll be based. 05 Pearso Educato, Ltd.

6 Stock/Watso - Itroducto to Ecoometrcs - 3 rd Udated Edto - Aswers to Exercses: Chater Both regressos suffer from omtted varable bas so that they wll ot rovde relable estmates of the causal effect of come o test scores. However, the olear regresso (8.8) fts the data well, so that t could be used for forecastg. 05 Pearso Educato, Ltd.

7 Stock/Watso - Itroducto to Ecoometrcs - 3 rd Udated Edto - Aswers to Exercses: Chater There are several reasos for cocer. Here are a few. Iteral cosstecy: To the extet that rce s affected by demad, there may be smultaeous equato bas. Exteral cosstecy: The teret ad troducto of E-jourals may duce mortat chages the market for academc jourals so that the results for 000 may ot be relevat for today s market. 05 Pearso Educato, Ltd.

8 Stock/Watso - Itroducto to Ecoometrcs - 3 rd Udated Edto - Aswers to Exercses: Chater (a) ( ) ˆ )( Y Y. Because all of the s are used (although some are used for the ( ) wrog values of Y j ),, ad ( ). Also, Y Y ( ) u u. Usg these exressos: ˆ 0.8 ) 0.8 ) ( )( u ( ( ) ( ) ) ( ( ) ( u) ( ) ( ) 0.8 ( ) ( ) ( ) 0.8 )( )( u u ( ) where, ad the last equalty uses a orderg of the observatos so that the frst 40 observatos ( 0.8 ) corresod to the correctly measured observatos ( ). As s doe elsewhere the book, we terret as a large samle, so we use the aroxmato of tedg to fty. The soluto rovded here thus shows that these exressos are aroxmately true for large ad hold the lmt that teds to fty. Each of the averages the exresso for ˆ have the followg robablty lmts: ( ), 0.8 ( ) 0.8, ( )( u u)0, ad 0.8 ( ) ( ) 0, where the last result follows because for the scrambled observatos ad j s deedet of for j. Take together, these results mly ˆ 0.8. (cotued o ext age) 05 Pearso Educato, Ltd.

9 Stock/Watso - Itroducto to Ecoometrcs - 3 rd Udated Edto - Aswers to Exercses: Chater (cotued) (b) Because ˆ 0.8, ˆ /0.8, so a cosstet estmator of s the OLS estmator dvded by 0.8. (c) Yes, the estmator based o the frst 40 observatos s better tha the adjusted estmator from art (b). Equato (4.) Key Cocet 4.4 (age 9) mles that the estmator based o the frst 40 observatos has a varace that s var ( ) u var( ˆ (40obs)). 40 var( ) From art (a), the OLS estmator based o all of the observatos has two sources of samlg ( ) ( u u) error. The frst s whch s the usual source that comes from the ( ) omtted factors (u). The secod s 4 )( ( ) ( ), whch s the source that comes from scramblg the data. These two terms are ucorrelated large samles, ad ther resectve large-samle varaces are: ad Thus var ( ) ( u u) var ( ) u ( ) var( ) ( ) 4 ( ) ( ) 0. var. ˆ (obs) var ( ) u 0. var var( ) whch s larger tha the varace of the estmator that oly uses the frst 40 observatos. 05 Pearso Educato, Ltd.