计量经济分析（第六版）答案 A6

NYUfSTERNNEWYORKUNIVERSITYLEONARDN.STERNSCHOOLOFBUSINESS

DepartmentofEconomicsECONOMETRICSIFall2000-Tuesday/Thursday10：00-ll：20ProfessorWilliamGreeneOffice：KMC7-88/-wgreeneOfficeHours：TR,3：00-5：00URLforcoursewebpage：

Phone：212.998.0876HomeEmail：wgreene@/-wgreene/Econometrics/Econometrics.htmAssignment6

AsymptoticsforLeastSquaresandtheClassicalRegressionModelSolutionsLParameterEstimationinaDistributionA.RegressionFunctionsf(x|y)=f(x,y)/f(y)=f(y|x)f(x)/f(y)={0exp[-(p+0)x](px)y/y!}/f(y).Thejointdistributionistheproductoftheconditional,f(y|x)andthemarginal,f(x).Weneedf(y),whichisfoundbyintegratingxoutofthejointdistribution.Thus,f(y)=「噎(P+g(Px)，/y!dxWellskipafewlines.Youcanpull0py/y!outoftheintegral.Whatremainsisf(y)二叩y/y!「e-^xxydx.Theintegralisagammaintegral(seepage178ofyourtext),with(P-1)equaltoy,cequalto1,andaequalto(p+0).We'regoingtoneedthisafewmoretimes,sohereisageneralresult:fe~^xxP~[dx=,whichwewillusefordifferentvaluesofXandP.J。 XPPluggingtermsintothatresultforthegammaintegral,wegetthattheintegralisequaltor(y+l)/(p+0)y+,.Now,r(y+l)=y!whenyisaninteger,whichitishere.Thiscancelsthey!wealreadyhave.Pluggingintheterms,weareleftwith

f(y)Now,weneedf(x|y)sowedividethejointdensitybythismarginal.Justdoingthedivisionandskippingalineofalgebraprovidesf(x|y)二6-f(x|y)二6-（肌0“邛+0）），+iTogettheconditionalmean,wenowneedE[x|y]=fxf{x|y)dx

JO(p+9)E[x|y]=fxf{x|y)dx

JO(p+9)>+1rylJ。e-^xy^dxwhichisagammaintegralagain.Usingtheresultgivenaboveandtheresultthat「(y+2)=(y+l)r(y+l)=(y+l)y!,sowecangetthefactorialstocancel,theendresultsimplifiesto(2)E[x|yl2±L=J_(2)E[x|yl2±L=J_+^1_

p+ep+ep+e=+|iy,where日l/(p+0).(yikes!!)and,remember,E[y|x]=Px.SimpleEstimation:Linearregressionofyonxwithoutaconstantestimatespconsistently.Callthisestimatorh.Linearregressionofxon。'+1)estimates日consistently.Callthislatterestimatorm.Use1/mtoestimatep+0.So,1/m-hestimates0consistently,byvirtueoftheSlutskytheorem.Thisestimatorisnotunbiased.Itisonlyconsistent.TheestimatorofPisunbiased,ascanbeshownusingallourresultsforthelinearregressionmodel.Theestimatorofisalsounbiased.Sinceweareusing1/m,itisnotunbiased,sinceE[l/m]isnot1/E[m].(Seethetext,onJensen'sinequality.)Onemoreapproach...IfE[x]=1/0,thenXisaconsistentestimatorof1/0,and0canbeestimatedconsistentlyusing1/X.Onceagain,thisestimatorwillnotbeunbiased,sinceXisunbiasedfor1/0andthereciprocalfunctionisnotlinear.(SeepartB.)MaximumLikelihoodEstimation:Theloglikelihoodfunctionisthelogoftheproductofthedensities,orthesumofthelogs;Log-L=Z；f-px/+j/logCpx/)-logy；!]=2,[-。为+)Mog(。)+ylog®)-logy!]ThederivativeiseiogL/dp=(-2访)+(1/。)(2必).Equatetozeroanddividebothsidesbyn,sothesolutionisthepforwhich-X+(l/p)y=Q.Thesolutionfallsouteasily.ILAsymptoticEfficiency.A.Forthemaximumlikelihoodestimatorfoundabove,bm=y/x.Theexpectationconditionedonthex'sisE[bm|x]=(l/X)E[j|x]=(l/X)(l/n){E[E/yz]|x}二（1/冗）（1/〃）（2£回比］）二(1/工)(1/〃)(2佃)=p-SinceE[bm|xJ=0,E[bmJ=p.Thismakesitunbiased.Forconsistency,wellusemeansquareconvergence.Themeanobviouslyconvergestop,sinceisequalspforalln.Thevarianceistheexpectedconditionalvarianceplusthevarianceoftheconditionalmean.SincetheconditionalmeanisP，thevarianceoftheconditionalmeaniszero.Tofindthevariance,westartwithbm=y/x=(l/x)(l/n)Eiyi.So,theconditionalvarianceisVar[bm|x]=[(l/x)(l/n)JVar[bm|x]=[(l/X)(l/n)]2xSipXi=p/(nX)Itisobviousthattheconditionalvariancegoestozero,since1/Xconvergesto1/E[x]=0,whichgetsustheconsistencyweneed,bymeansquare.Infact,itispossibletogettheexact,unconditionalvariance.(Though,thisisnotstrictlynecessary;youcouldstophere.)Inassignment7,you'llfindthatEf1/x1isactually0xn/(n-l).Thisgetsustheconsistencythatweneedunconditionally.Forlookingahead,weshouldnotethatthelargesamplevarianceabovewillbehavethesameasAsy.Var[bm]=p/nxwhatever1/Xconvergesto,whichis0,whichleadsustoAsy.Var[b,n]=p0/n.Forleastsquares,wehaveanunbiasednessprooffromourclassworkthatstillapplies,yi=E[yi|x1+&bysimpleconstruction;wisjustthedeviationtomakethisaddup.But,alsobyconstruction,E[Si|Xi]=0.Now,y=xp+8,andourproofofunbiasednessgoesthrough.But,letsdoitthelongway,soweseetheresultinfull.b=(x'y)/(x*x)=2凶yi/EjXi2.E[b|x]=.XjE[yi|xi]/ZiXp='.胱/ZjXi2.Thesumsofsquaredx'scancelafterpispulledoutofthesummation,whichgetsustheunbiasedness.ThevarianceoftheleastsquaresestimatorisnotW/x'xhere,however,sincethisisnotaclassicalregressionmodel.Inordertogetconsistencyofleastsquares,wearegoingtohavetogetitsvariancetogotozero.Thevarianceisgivenintheproblemset:Varfb|x]=(1/SjXi2)x(1/ZjXi2)xpSiXp.It'snothardtoprovethis.Varfb|x]=Var[SiXiyi/SiXi2|x]=[1/XiXi2]2xVar[ZiXjyi]=[1/ZjXi2]2xI,Var[Xiyi|Xi]二[1/ZiXi2]2xZiXi2Var[yi|xi]=[1/ZiXi2]2xI,xrpXi]=P[l/SiXi2]2xZixp=p(l/ZiXi2)x(1/ZjXi2)xZiXi3Toseewherethisgoes,multiplythemiddletermbynanddividethethirdtermbyn-then'scancel.Now,themiddleterm,byvirtueofGreene'snontheoremandtheSlutskytheoremconvergesto1/E[x2],or02/2.ThethirdtermwillconvergetoE[x3]=6/03.So,theproductwillconvergeto3p/0.Thatleavesthefirstterm,whichdoesn'tconvergeatall,sinceitisjustasumofnsquares,whichkeepsgrowing.Thatgetsusconsistency,sincethevariancegoestozero.Theasymptoticvarianceoftheleastsquaresestimatorisgivenbytheexpressionabove,involvingthesums.But,forpurposesofunderstandingit,wecanalsoinsertournowknownresultsforthesummations,andanalyzeAsy.Var[b|x]=(3p/6)x(1/ZiXi2).Goingyetonemorestep,weshouldviewthisasbehavingthesameasAsy.Varfbl=(3p/0)x(1/n)x(n/ZiXi2)Asy.Varfblwhichbehavesthesameas二whichbehavesthesameas二(3p/9)x(1/n)x02/2,so,finally,wewoulduseAsy.Varfb]=3p9/(2n).so,finally,wewoulduseAsy.Varfb]=3p9/(2n).B.Wedidthisabove.B.Wedidthisabovealso.Thebonusinsightisthatthisisadirectapplicationofthatresult.Allofthelegworkforthispartwasdoneabovealso.Theendresultisthatbothestimatorsareconsistentsobothvariancesconvergetozero.But,whenyoutaketheratioofthevarianceoftheleastsquaresestimatortothevarianceoftheMLE,youfindthattheratiois3/2or1.5.ThatmeansthattheMLEis50%moreefficientthantheleastsquaresestimator.Itsabetteruseofthedata.WhathappenedtotheGaussMarkovtheorem?ItdoesnotapplyherebecausetheregressionmodelE[y|x]=Px,Var[y|x]=Pxdoesn'tsatisfytheconditionsofthetheorem.Gauss-Markovrequiresalldisturbancestohavethesamevariance,q2.PartIII.TOC\o"1-5"\h\zRegressionofyonx,noconstantterm.Estimatesp.+ +IOrdinaryleastsquaresregressionWeightingvariable=none|IDep.var.=Y Mean=1.720000000 ,S.D.= 1.837570860 |IModelsize:Observations= 25,Parameters= 1,Deg.Fr.= 24|IResiduals:Sumofsquares=27.73386709 ,Std.Dev.= 1.07498|IFit: R-squared=.657776,AdjustedR-squared= .65778|IModeltest:F[ 1, 24]= 46.13,Probvalue= .00000|IDiagnostic:Log-L= -36.7707,Restricted(b=0)Log-