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1、CHAPTER5SOLUTIONSTOPROBLEMS5.1Writey=+x1+u,andtaketheexpectedvalue:E(y) = +E(x1) +E(u),orµy = +µxsinceE(u)=0,whereµy=E(y)andµx =E(x1).Wecanrewritethisas =µy -µx.Now, = -.Takingtheplimofthiswehaveplim() =plim( -) =plim() –plim()plim() =µy -µx,whereweusethefactthatplim() =µy andplim() =µxby
2、thelawoflargenumbers,andplim() =.WehavealsousedthepartsofPropertyPLIM.2fromAppendixC.5.2Ahighertoleranceofriskmeansmorewillingnesstoinvestinthestockmarket,so >0.Byassumption,fundsandrisktolarepositivelycorrelated.Nowweuseequation(5.5),whered1 >0:plim() = +d1 >,sohasapositiveinconsistency(
3、asymptoticbias).Thismakessense:ifweomitrisktolfromtheregressionanditispositivelycorrelatedwithfunds,someoftheestimatedeffectoffundsisactuallyduetotheeffectofrisktol.5.3Thevariablecigshasnothingclosetoanormaldistributioninthepopulation.Mostpeopledonotsmoke,socigs =0foroverhalfofthepopulati
4、on.Anormallydistributedrandomvariabletakesonnoparticularvaluewithpositiveprobability.Further,thedistributionofcigsisskewed,whereasanormalrandomvariablemustbesymmetricaboutitsmean.5.4Writey=+x +u,andtaketheexpectedvalue:E(y) = +E(x) +E(u),orµy = +µx,sinceE(u) =0,where µy =E(y)andµx =E(x).W
5、ecanrewritethisas =µy -µx.Now, = -.Takingtheplimofthiswehaveplim() =plim( -) =plim() –plim()×plim() =µy -µx,whereweusethefactthatplim() =µyandplim() =µxbythelawoflargenumbers,andplim() =.WehavealsousedthepartsofthePropertyPLIM.2fromAppendixC.CHAPTER6SOLUTIONSTOPROBLEMS6.1Thegeneralityisno
6、tnecessary.Thetstatisticonroe2isonlyabout-.30,whichshowsthatroe2isverystatisticallyinsignificant.Plus,havingthesquaredtermhasonlyaminoreffectontheslopeevenforlargevaluesofroe.(Theapproximateslopeis.0215 -.00016roe,andevenwhenroe =25 –aboutonestandarddeviationabovetheaverageroeinthesample–
7、theslopeis.211,ascomparedwith.215atroe =0.)6.2BydefinitionoftheOLSregressionofc0yionc1xi1,,ckxik,i =2,,n,thesolve49[Weobtainthesefromequations(3.13),wherewepluginthescaleddependentandindependentvariables.]Wenowshowthatif =and =,j =1,…,k,thenthesek +1firstordercondit
