计量经济学中加合作教学课件

AReviewofProbabilityandAReviewofProbabilityandI.SomePopulation（总体）thesetofallpossiblerandom,orchance,experimente.g.allthestudentsinthis ）:eachunitinthee.g.eachstudentinthissampling（随机抽样）fromapopulatione.g.asampleof20studentsbyrandomsamplingfromthisRandomRandomVariablerv)orStochasticVariable（量）:anumericaldescriptionofthe experiment(denotedbyX,Y,Z,etc,andthevaluestakenbythemaredenotedbyx,y,z,etc).e.g.X:EconometricsgradesofallthestudentsinthiscanconsiderapopulationasarandomCanconsidersampleasasetofsomeobservations（观测值）inthispopulation3(1).Discreterv（离散型随 finite(orcountablyinfinite)sequenceofvalues.Example: eofthrowing(2).Continuousrv（连续型随 anyvalueinsomeintervalofvalues.Example:heightofpeople,mosteconomic4likelihoodthataneventwilloccurProbabilitydistribution(PD)（概率分布）:Athevaluesthatarandomvariablecanassume.:afunctiontodescribeprobabilitydistributionofX,denotedbyf(x).Fordiscreterv:f(x)=P(X= fori=1,2,= forX5P(axb)fExample:ExpectedvalueandvarianceforthenumberofExample:ExpectedvalueandvarianceforthenumberofcarssoldduringadayatDicarloMotorsjTargetofStatistic：Ifapopulationisunknown,getasamplebyrandomsamplingfromitandthenusesampleinformationtoinferinformation(mainlyProbabilitydistribution)ofthisunknownpopulation.Jointpointofsampleandpopulation:Probabilitydistribution(PD),becausePDofasamplefromapopulationisthesameasthatofthispopulationusesamplePDtoinferpopulation7Thenumericalfeaturesofapopulation(orrandomvariable)PDareoftendescribedusingitsprobabilitycovarianceandcorrelationcoefficient(computedfromprobabilitydensityfunction).8--measuresofvariabilityor9(1)ExpectedValue(orexpectation,mean):ameasureofthecentrallocationforrvX,denotedbyE(X)ororxFordiscreteE(X)xjf(xjjE(X)xf(2)Variance:(2)Variance:measureshowfaraparticularvalueoftherandomvariableisfromtheexpectedvalueorE(X).Var(X)E(XFordiscretevariable,givenX{x1,x2,...,xkVar(X)(xj)f(xjForcontinuousvariable,Var(X)(X)2fsd(X)Var(X)012345Var(XVar(X)(xj)f(x)jsd(X)1.25f0123465Cov(X,Y)xyE[(Xx)(YyCov(X,Y)E(XYxYXyxyE(XY)E(xY)E(Xy)E(xyE(XY)xwhichisoftenusedtocalculatePropertiesPropertiesofIfX,Yareindependentrvs,Cov(X,Y)=0becauseE(XY)=E(X)E(Y)ButifCov(X,Y)=0,cannotsayX,YareindependentrvsIfY=X2,YandXarenotindependentButifE(X)=x=0,E(X3)=0Cov(X,Y)=E[(Xx)(X2x2)]=E(X3)=Ifa1,b1,a2,b2areCov(a1X+b1,a2Y+b2)=(3)|Cov(X,Y)|Correlationcoefficient:anumericalmeasureoflinearassociationbetweentwovariablesCorr(X,Y) Cov(X,Y Cov(X,Y)=0Corr(X,Y)=–1Corr(X,Y)+1indicateaperfectpositiveassociation,–1indicatesaperfectnegativeassociationGivenconstantsa1,b1,a2,b2Ifa1a2>0,Corr(a1X+b1,a2Y+b2)=Corr(X,Y)Ifa1a2<0,Corr(a1X+b1,a2Y+b2)=–Corr(X,Y)PropertiesPropertiesofExpectedForanyconstant常数）cE(ccE(cXcE(X)Foranyconstantsaandb,E(aX+b)=aE(X)+If{a1,a2,…,an}areconstantsand{X1,X2,…,Xn}arerandomvariables,thenE(a1X1+a2X2+…+anXn)=a1E(X1)+a2E(X2)+…+EaiXiaE(XIfeacha1,E E(X(4)IfX,Yareindependentrvs,E(XY)=Foraconstantc,Var(c)=0,Var(cX)=Ifaandbareconstants,Var(aX+b)=Var(X)=E(X–)2=E(X2+2=E(X2)+E(2)–2E(X)==E(X2)whichisoftenusedtocalculatesd(c)=sd(aX+b)=PropertiesofPropertiesofVar(aX+bY)=a2Var(X)+b2Var(Y)+2abCov(X,Y)IfXandYarenotcorrelated,Cov(X,Y)=0Var(XY)=Var(X)+If{X1,X2,…,Xn}arepairwiseuncorrelatedrandomvariablesand{ai:i=1,2,…,n}areconstants,=a1Var(X1)+a2Var(X2)+…+anVar(Xn)22Varna iiaVar(XiIfeachai1,VarXVar(XniiiitheconditionalexpectedvalueofYonXE(Y|X)(orconditionalexpectation,conditionalmean)fordiscretervY{y1,y2,...ym},E(Y|X)yjfY|X(yjY=a+bX+E(Y|X)=E[(a+bX+cX2)|X]=a+bX+Note:IfXtakesaspecialvaluexi,E(Y|xi)isaPropertiesPropertiesofConditionalExpectedForanyfunctionc(X),E[c(X)|X]=Forfunctiona(X)and=a(X)E(Y|X)+IfX,Yareindependentrvs,E(Y|X)=AspecialIfX,UareindependentrvsandE(U)=0E(U|X)=E(U)=0E[E(Y|X)]=E(Y|X)=E(Y)Cov(X,Y)=IfXdoesnotchangetheexpectedvalueofY,thenXandYmustbeuncorrelatedifXandYarecorrelated,thenE(Y|X)mustdependonX. TheconverseofProperty5isnottrue:ifXandYareuncorrelated,E(Y|X)couldstilldependonX.ifE(X)=0,E(X3)=0,Y=X2,thenCov(X,Y)=ButnowE(Y|X)=X2,E(Y|X)couldstilldependonVar(YX)E{[YE(Y|x)]2|IfX,Yareindependentrvs,Var(Y|X)=meanandstandarddeviation,X~N(,2) e(xu)/thehighestthehighestpointonthenormalcurveisattheThestandarddeviationdeterminesthewidthoftheThetotalareaunderthecurveforthenormalprobabilitydistributionis1FigureB.7:Bell-shapedAnormaldistributionwithameanofzeroandastandarddeviationofone. WhichisaspecialcaseofNormalPDF:(Z)eZPropertiesofPropertiesofNormalIfX~N(,2),then(X)/~N(0,X~N(3650050002)Z=(X-36500)/5000IfX~N(,2),thenaX+b~N(a+b,IfXandYarejointlynormallydistributed,thentheyareindependentifandonlyifCov(X,Y)=0.distributednormalrandomvariableshasanormalFromproperty4,wevariableshasanormalThatis,ifY1,Y2,…,YnareindependentrandomvariablesandeachfollowsnormaldistributionN(,2),Y∼N(,2/tdistributionTheFIV.IV.OtherDefinitionsonGivenasamplex={x1,x2,…,xn}fromapopulationxixn(xix)xixxinxxin(xin)i i i i i ixxnSampleStandardDeviation:thepositivesquarerootofthesamplevariancesExample:samplex={46,54,42,46,xxxxxi1 4654424632s(4644)2(5444)2(4244)2(4644)2(325=s64Example:Example:Calculationsforthesamplecovariancex3,yxxixyiyn1WhichisanunbiasedestimatorofpopulationE(Sxy)=xxiyi(xxi(yi(xix)(yiy2--111--43030904131931--2430-090418182--1500xixyiy99nssxxn1yyn120566rsxys Ch.1TheNatureofEconometricsCh.1TheNatureofEconometricsEconomic1WhatisEconometrics计量经济学abranchofevaluateand hefieldofEducation:effectofschoolspendingonstudentperformance.2MethodsMethodsinEconometricsstatisticalobservationaldata观测数据)--appearinsocial--appearinnatural 1.2Stepsin ysis GivenatopicinrealFormulatethequestionof izationQ=f(P,--less4ExampleExample1.2:Wagewage=f(educ,exper,wage:hourlywage,measureproductivityeduc:yearsofformaleducation,exper:yearsofworkingtraining:weeksspentinjobNote:Allthevariablesinthefinalmodelmusthaveavailabledata5--ucontainsunobservedfactorsinthewage=0+1educ+2exper+3training+uucontainsotherunobservedfactorsaffectingwage,background,etc.6y=0+1y=0+1x1+2x2+3x3+uy:wagehere数associatedwithx’srelationshipbetweenyandx’su:containsunobservedfactorsotherthanx’sthataffecty,7CollectdataforeachfactorStatisticalinference统计推断andtests估计和检验)andeconometricThegoalof estimatetheparameters’sinthemodeltesthypotheses假设aboutthese’s--thevaluesandsignsofthe’sdeterminethevalidityofaneconomictheoryandtheeffectsofcertainpolicies.8estimateestimateeconomictesteconomicevaluateandimplement ernmentandbusiness--and/91.3TheStructureofEconomicTypesofCross-SectionaldataTimeSeriesdataPooledCrossSectiondata(omitted)混合数Paneldata(omitted)Cross-sectionalCross-sectional--asampleofindividuals,households,firms,cities,states,countries,oravarietyofotherunits,takenatagivenpointininformationtakenatagivenpointintimeTable1.1:across-sectionaldataseton526workingindividualsfortheyear1976,inabbreviatedinthesample,showingtheorderofeconometricssoftwarepackagesassignanorderingofdatadoesnoteachotherTimeseriesconsistofobservationsonavariableorseveralvariablesforeachtimeperiodover–e.g.dailystockpricesin2004,GDPinthepast20yearsdatafrequency:Daily,weekly,monthly,quarterlyandkeykeyfeature:chronologicalorderingof observationsovertimearenotindependent,butarerelatedtotheirrecenthistoryTrends趋势andseasonalitywillbee.g.GDP,morecomplicated 1.41.4Causality因果关系andNotionofCeterisSimplyestablishingarelationshipbetweenyandxIneconometricmodel,whenstudyingtherelationshipbetweentwovariables,allotherrelevantfactorsmustbeheldfixed.ThenthisrelationshipcanoftenbeconsideredtobeExample1.4:MeasurethereturntoIfaischosenfromthepopulationandgivenanotheryearofeducation,byhowmuchwillhisherwageInthesimplestequation:wage=0+1educ+1:thereturntou:otherfactorsotherthaneducationaffectingwage,suchasworkingexperience,abilityceterisparibus:whenanotheryearofeducationisgiventothe,allotherfactorsotherthaneducationareassumedtobeheldfixed. whenanotheryearofeducationisgiventothe,allotherfactorsotherthaneducationnotbeheld--educationlevelsmaynotbeindependentlyofallotherfactorsaffectingwage:Ineconometricresearch,sometimesitmaybeCh.2:TheSimpleRegressionCh.2:TheSimpleRegressiony=0+1x+12.1DefinitionofTheSimpleRegressionGeneraldefinition:y=f(x,x:independentvariable,exnatoryvariableu:errorNotexactfunctionalrelationship,butstatisticalrelationshipbetweenyandx2ChapterChapter2beginswithsimplelinearregression(SLR)model,whichisnotoftenusedinappliedeconometrics,butisagoodstartpointforfurtherstudylater.3Thesimplelinearregressionifu=0,y=:“Runtheregressionofyonx”or“regressyonNote:inthistextbook,usey,x(notY,X)todenotefactors(variables)inthemodels.4MainlyMainlyyzeslopeparameter1,notintercept0Meansofinterceptwhenx=0,y=notmakesense,sincex0inmayaverageeffectofalltheotherfactorsotherthanxonyButalwaysincludeintercept0exceptincaseof0=0basedonstrongtheory.5Usuallywehavenoallinformationon(x,y)inthepopulation0,1anduareunobservedeveniftheyhavetoestimate0,1usingsampledataxandyfromthe6y=0+1x+uy:gradex:studyhoursperTheteacherisinterestedintheeffectofstudyhoursongrade,holdingotherfactorsfixed.ability,mathematicsbackground,etc.7fromE(u)=0andE(u|x)=averagevalueofunobservedfactorsinudoesdependonxcov(x,u)=Weassumetheunobservedfactors,suchasability,affectinggradeareunrelatedtostudyhoursandhaveanaverageofzerointhepopulationofallstudents.8ProblemProblem--wecanredefinetheinterceptinthisequationtomakeE(ability)=0orE(u)=0truey=0+1x+usupposethatE(u)0=y=(0+0)+1x+(uCallthenewerrore=u0,E(e)=Thenewinterceptis0+0,buttheslopeisstill1.themodelcanalwaysberewrittenwiththesameslope,butanewinterceptanderrorterm,wherethenewerrorhasazeroexpectedvalue.9Ifstudyhoursarechosenindependentlyofotherfeaturesofthestudents,theotherfactorswillnotdependonstudyHowever,ifstudent’sabilityincreasedwithmoretimespentonstudy,thentheexpectedvalueofuchangeswithstudyhours,ZeroConditionalMeanwillnotPopulationPopulationRegressionFunctionGiventheassumptionofZeroE(y|x)=E(0|x)+E(1x|x)+PRF:E(y|x)=0+measureonaverageinthepopulation,howychangewiththechangeinx0and1needtobeestimatedregressiony=0+1x+ [y=E(y|x)+Foreachindividualinthepopulation,howchangewiththechangeinxyincludetwocomponents:systematicpartofy:E(y|x)=0+1x,exinedbyxunsystematicpart,u,notexinedbyx.Example1:-{(xi,yExample1:-{(xi,yi):i=1,2,3,4}arethesampledataofgradesandstudyhoursfor4studentsfromthepopulationofthewholeclass)Givenanyx,thedistributionofPopulationregression2.22.2DerivingtheOrdinaryLeastSquares(OLS)y=0+1x+ThreeestimationmethodstoestimatePRF(and BasicideaofToestimatePRF(0and1)fromainfromthepopulation,thenforeachobservationthissample,itwillbethecaseyi=0+1xi+uiistheerrortermforobservationy1=0+1x1+u1y2=0+1x2+yn=0+1xn+yx48Populationregressionline,sampledatapointsandtheassociatederrortermsy: E(y|x)=0+ u } x:study useˆasestimatorofuseˆasestimatorof,ˆasestimatorof forobservationiinasample,givenxi,we1 1y: 1uˆyyˆyˆˆ 1 SRF:yˆˆ y3 .} û2 } x4 PRFandi PRF:E(yx)E(yxiiE(yx ˆ:estimatorof,ˆ:estimatorofxStudyGrantedthattheSRFisonlyanapproximationofPRF,givenasample,canwedevisearuleoraasThatis,theˆsinSRFshouldbeas“close”aspossibletotruevalues’sintherelevantPRFeventhoughwewillneverknowthe’sandPRF.y: uˆiyiyˆiyiˆˆmin } } StudyOLSmethod:fitasampleregressionlinethroughthesamplepointssuchthatthesumofsquaredresidualsisassmallasthetermofy=0+1xy=0+1x+ “regressyony: x:studyhoursper0and1,thenfittedvaluesofyandTheleast-squarescriterion最小二乘标准nMinuˆ2(yyˆ)2(yˆˆxi1 1yi,xiareknownsampleFirstordercondition(FOC)一阶条件 (ˆˆi2(yˆˆx)ˆ0(yˆˆx) 1 10 (ˆx i2(yˆˆx 1i(yx)x 1 1i thesameasequations(2.14)and(2.15),multipliedbyuˆi0,uˆixiˆyˆ xxyy ˆ x,y xprovidedthatxx0xˆˆ：OLSestimators最小二乘估计量ofSRF：yˆˆˆanduˆyEstimate估计值:aparticularnumericalvalueobtainedbytheestimatorinanspecificsample.yx48EXAMPLEEXAMPLE2.3:CEOSalaryandReturnonstudytherelationshipbetweenthismeasureoffirmperformanceandCEOcompensation,wethesimplesalary=0+1roe+u.x:theaveragereturnequity(roe)fortheCEO’sfirmforthepreviousthreeyears.(roeisdefinedintermsofnet easapercentageofcommonequity.)209CEOsfortheyear1990;thesedatawereobtainedfromBusinessWeek(5/6/91).UsingOLSmethodand(2.17)and(2.19),getsalˆary=963.191+18.501Ifroe=0,thepredictedsalaryistheinterceptsalˆary=WeWewillneverknowthePRF,sowecannotlhowclosetheSRFistothePRF.Anothersampleofdatawillgiveadifferentregressionline,whichmayormaynotbeclosertothePRF.YoucanusecomputersoftwarepackagetomakeestimationwithoutcomputingthembyhandThismodelmaybetoosimple,sincetherearemanyotherfactorsthataffectsalarybesidesroe.SRFbasedontwodifferent roe,MoreMoreyiyˆiuˆian Canproveyy2yˆy2 SSE SSTthetotalsumofsquaresSSE:theexinedsumofsquares解释平方SSR:theresidualsumofsquaresyiyTotalyi yˆiy Goodness-of-FitGoodness-of-FitR-squared(coefficientofdetermination判定系数):thefractionofthesamplevariationinycanbeR2=SSE/SST=1–0R2IfR2=1,allsamplepointsareonsampleregressionlineIfR2=0,poorfitofExample2.3:CEOSalaryandReturnonsalˆary=963.191+18.501n=209,R2=roeexinsonlyabout1.3%ofthevariationinsalariesforthissampleof209CEOs.Inthesocialsciences,lowR2inregressionequationsarenot mon,especiallyforcross-sectionalysis,anddon’tmeantheseSRFsarecannotuseR2tojudgewhetheraneconometricysisissuccessfulornotSmallexnatorypowermaybecausedbyotherfactors(inu)thatinfluencesalary. 2.5ExpectedValueandVarianceofOLS2.5ExpectedValueandVarianceofOLS and1thatappearinthepopulationstudypropertiesofthedistributionsofˆand Givenaparticularsample,they particularvalues.forsimplelinearregression.SLR.1:linearinparametersasy=0+1x+uSLR.2:userandomsampling,{(xi,yi):i=1,2,…,n},fromthepopulationmodelyi=0+1xi+SLR.3:E(u|x)=0E(ui|xi)=SLR.4:x’sarenotconstants. xixGiventheabove4 EourestimatoriscenteredaroundthetrueparameterE(ˆ)isE(ˆ|x),E(ˆ)isE(ˆ|x) UnbiasednessTheOLSestimatorsof1and0are E(ˆ),E(ˆ)Proofofunbiasednessdependsonour4assumptions–ifanyassumptionfails,thenOLSestimatorsisnotnecessarilyRememberunbiasednessisadescriptionoftheestimator–inagivensamplewemaybe“near”or“far”fromthetrueparameterˆ,ˆ 0 VarianceofVarianceoftheOLS--MeasurehowthesamplingdistributionofourestimatorspreadoutaroundthetrueparameteranadditionalassumptionVar(u|x)=2(Homoskedasticity同方差CanproveVar(u|x)=Var(u)--Var(u|x)isalsotheunconditionalvariance,calledtheerrorvariance误差项方差,thesquarerootoftheerrorvarianceiscalledthestandarddeviationoftheerrorterm误差项标准差alsohave:E(y|x)=0+Var(y|x)=Var(u|x)= 2 Var1(xix)2 SSTx(xix (xxisdˆ xVarˆisVarˆ|x,VarˆisVarˆ|x VarianceofOLSSlopeEstimatorˆ1Thelargertheerrorvariance,2,thelargerthevarianceoftheslopeestimatorThelargerthevariabilityinthexi,thesmallerthevarianceoftheslopeestimatorAsaresult,alargersamplesizeshoulddecreasethevarianceoftheslopeestimateProblem:theerrorvariance2isunknownbecausewedon’tobservetheerrortermsuiGivenSLR.1-ˆ21uˆ2SSR/n n2:degree dom(df)in(giventhe2restrictionsofOLSFOCsonuˆi x Similarly,cancomputeseˆ02.42.4UnitsofMeasurementandFunctionalEffectsofchangingunitofmeasureonOLSestimatorsExample2.3:salary=0+1roe+u. .1roedecroedecroe R2doesn'tchangeinboth unitsof therateofchangemeasuredinunitsofthe1unitsofInoriginalcase,ˆ1Ifxchangesbyoneunit,ychangesby$1000oIfx=0,yˆ=963.191i1000=Incase1,ˆ18,1Ifxchangesbyoneunit,1%,ychangesby$18,501Ifx=0,yˆ=$963,191Incase2,ˆ1Ifxchangesbyoneunit,ychangesby$10001850.10.01=$18,501Ifx=0,yˆ=963.191i1000=$963,191Thetworesultsareidenticalintheireffectsofxonyy=0+1x+y=0+1logx+logy=0+1x+or%y= TheMeaningof“Linear”--Linearityinthe’sappearwithapowerorindexof1onlyandismultipliedordividedbyanyother--mayormaynotbelinearinthey=0+1x2+y0+1xu,y0+1/0xu,y0+01xy=0+1 yisalinearfunctionof--xappearswithapowerorindexof1onlyandisnotmultipliedordividedbyanyothervariable.y0+1x,y0+1(x/z),y0+1(xByredefiningyandx,wecanturnsomemodelsy=0+1Modellinearin Modellinearin Note:LRM=linearregressionmodelNLRM=nonlinearregressionmodelExampleExample2.10:log-level(semi-Whenu=lety=log(wage),x=educ,alinearmodel,stillusen526,R2Here0doesnotmakeAnincreasingreturnoneducwithu=Example2.11:log-log(constant-CEOsalaryandfirmsalary/salary=1(or%salary=1%Ifsalesby1%,salaryby1lety=log(salary),x=log(sales),alinearmodel,stilluseOLSn209,R2Ch.3MultipleCh.3Multiple 1y=0+1x1+2x2+...kxk+thekfactorsthataffecty:xj(thek+1parameter:j(j=0,1,2,...,k)intercept0andslopeparameters1toku:errorterm,unobservable--allfactorsinuareuncorrelatedwithx’s(ceteris2MLR:MLR:y=0+1x1+2x2+...kxk+--morerealisticandflexiblethanSLR:y=0+1x1+Example:theeffectofeducationonhourlywage=0+1educ+2exper+ morerealisticthanSLR:wage=0+1educ+--morevariationinyisexIfuwage=1educ+2Example:aquadraticcons=0+1inc+2inc2+ cons=1inc+22inc--MLRmodelallowsformuchmoreflexibilityinfunctional3PRFandSRF(parallelwiththoseofPRF:E(y|x)=0+1x1+2x2+...SRF:yˆˆˆxˆx...ˆ 1 2 kuˆyyˆy(ˆˆxˆx...ˆx 1 2 kTerminology:“Regressyonx1,x2,…xk”or“runregressionofyonx1,x2,…xk”4uˆyyˆy(ˆˆxˆx...ˆx 1 2 kOLS:minuˆ2(yˆˆxˆx...ˆx 1 2i kFOCs 5n(yˆˆxˆx...ˆx) 1 2i kin1x(yˆˆxˆx...ˆx)i1 1 2i kx(yˆˆxˆx...ˆx) i2 1 2i k(yˆˆxˆx...ˆx)i1 1 2i ki1,2,...,nobservationj0,1,2,...,kdistinguishdifferentxand601 Note:nk7uy1x12x2...kxk holdingx2,...,xkfixed,y eachjmeasuresapartialeffect(ceterisparibus)ofxjony.1 2 k8thetotalsumofsquares(SST):yyithe inedsumofsquares(SSE):yˆyitheresidualsumofsquares(SSR):iSSTSSER2SSE/SST1 0R21thefractionofthetotalsamplevariationofythatisexinedbythemodel9Whenanotherxisaddedtoamodel,R2canneverdecreaseandusuallywillincrease(becausewhenanotherxisadded,SSRdoesnotincreaseandSSTdoesnotchange).Soitisnotagoodwaytocomparemodelsanddecidewhetheronexorseveralx’sshouldbeaddedtoamodelornot.--evenifsuchx’sarenotrelatedtoy,theycanstillincreaseR2Tomakesuchdecision,statisticaltestsaremore SquaredcorrelationcoefficientyiyyyˆyˆR2 2yy2yˆyˆ Example3.1and3.4:ModelofCollegecolˆGPA1.290.453hsGPA0.0094ACTn141,R20.176ACT:achievementtestscoreInterceptdoesn’tmakesensehsGPAandACTtogetherexinabout17.6%ofthevariationincolGPAmanyotherfactors,suchasfamilybackground,forcollege,contributetoastudent’scollegey1x1y1x12x2"kxkunointerceptSRF:yxx"x(3.30)1 2 kstilluseOLSmethod,butformularsareR21SSR/SST,SSRuˆ2(yxx...x 1 2i kR2canbeR2 Ifthereisnostrongprioriexpectationthat0=0,weshouldincludetheintercept,orelseOLSwillbebiasedreturnandtheBetacoefficient(CAPM).ER-rf=1(ERm-rf)+ER:expectedrateofreturnonsomerf: rateofERm:expectedrateofreturnonthemarket1:theBetacoefficient,ameasureofsystematicrisk,ER–rf:expectedriskpremiumonsecurityERm–rf:expectedmarketrisk3.33.3TheExpectedValueoftheOLS--parallelto2.5onSLRmodelAssumptionsofMLRMLR.1linearinModel:y=0+1x1+2x2+…+kxk+witharandomsampleofsizen,{(xi1,xi2,…,xik,yi)foreachobservationi(i=1,2,…,n)yi=0+1xi1+2xi2+…+kxik+Noneofthex’sisconstant,andtherearenoexactlinearrelationshipsamongthem01x1i2x2i...kxki0notallconstant's=0E(u|x1,x2,…xk)=implyingthatallx’sareThereasonsthatMLR.3doesnotanyofx’s.Theorem3.1:UnbiasedofOLSUnderMLR.1-4:E(ˆj) j0,1,...,meansthatthesamplingdistributionofOLSestimatorsiscenteredaroundthetrueparameter3.43.4VarianceoftheOLSVar(u|x1,x2,…,xk)=Var(y|x1,x2,…,xk)=GiventheGauss-MarkovAssumptionsMLR.1-5 sdˆSST1R21 j SSTjxijxjR2:theR2fromregressingxonallother R:x1= +2R:x1= +2x2+3x3+...+kxk+ 1 3 kR:x=+x+x+x+...+x+ 1 2 3 k-1k-v:errortermintheauxiliary’s:parametersintheauxiliaryAlargeVar(ˆ)meansalessprecisejlargerconfidenceintervalsandlessaccuratehypothesesNotknow2,becauseui’sareunobservable.useûiasestimatorofuiuˆiyiyˆiyiˆˆxi1ˆxi2...ˆ idegree dom(df):nk1n#of'Theorem3.3:unbiasedestimationof2UndertheGauss-MarkovAssumptionsMLR.1- j j SSTj1j12ˆandseˆarenotvalidestimatorofandsdˆ Varˆ j If2,var(ˆ)jmore“noise”intheequation(alarger2)makesitmoredifficulttoestimatethepartialeffectofanyxy,whichisreflectedinhighervariancesoftheOLSslopeestimators;Foragiveny,thereisonlyonewaytoreducetheerrorvariance:addmorex’stotheequation(takesomefactorsoutofu).;Thisisnotalwayspossible,norisitalways(2)TheTotalSampleVariationinxj,Ifthetotalvariationinxj,thatisSSTj,var(ˆ)jeverythingelsebeingequal,prefertohaveasmuchsamplevariationinxjaspossible;Giveny=0+1x1+2x2+...+kxk+Rj:R-squaredfromanauxiliaryregressionofxjonotherx’s,measuringtheproportionofthetotalvariationinxjthatcanbe inedbytheotherR2:x= +x+x+...+x+ 2 3 kR2:x2=0+1 +3x3+...+kxk+2R2:x=+x+x+x+...+x+ 1 2 3 k-1k-Anintercept0shouldbealwaysincludedintheauxiliaryregression.jxx...x 1 2 kj01X1i2X2i...kXkivixjandotherx'sarecorrelatedinthejProblemProblemofMulticollinearity:usuallyrefertocorrelationbetweentwoormoreofthex’s,nocleardefinition0R2jproblemonsampledataofx,butnotonpopulationIfR2,var(ˆ) 0R2 cons=0+1inc+2inc+inc2isnotanexactlinearfunctionof x3i=5x2iimperfectmulticollinearitybetweenX2andX3x*=x+v=5x wherev=2,0,7,9, 3i 2i rightway:collectmorewrongway: avariablethatbelongsinpopulationmodel,whichcanleadtovariousdegrees.Ifthenumberofx’sincreases,thepossibilityofmulticollinearityincreasestoo.ExpendituresonExpendituresonteachersalaries,instructionalmaterials,athletics,etc.maybehighlycorrelatedsincewealthierschoolstendtospendmoreeverything,andpoorerschoolsspendlessonThisleadstohighR2foreachoftheexpenditure ItisdifficulttoestimatetheeffectofanyparticularexpenditurecategoryonstudentperformanceSuchmulticollinearityproblemscanbemitigatedbycollectingmoredata,orsumthemtogethertostudythetheeffectoftotalexpenditureonstudentExample:y=0+1x1+2x2+3x3+Ifx2andx3arehighlycorrelated,butx1isuncorrelatedwithx2andx3 1ahighdegreeofcorrelationbetweencertainx’scanbeirrelevantastohowwellwecanestimateotherparametersinthemodel.modeleventhoughtheyhavenopartialeffectony(i.e.their’sare0).--underspecifying:excludeaorseveralx’sinthemodeleventhoughtheyhavepartialeffectony(i.e.their’sarenot0)SimpleExample:Truemodel:y=0+1x1+ Wrongmodel:y=0+1x1+2x2+u(2=0)'noeffectontheparameterestimates,andOLSestimatorsremainsunbiasedE(ˆ),E(ˆ),E(ˆ) ButhaveundesirableeffectonthevariancesofOLSVar(ˆ)2/[SST(1R2)],Var()2/ Ifx1andx2arecorrelated,Var()Var(ˆ IfIf2=0,x2doesnothaveapartialeffectony,thenincludingitinthemodelcanonlyexacerbatetheefficientestimatorof1truemodel:y=0+1x1+2x2+u(3.40)ˆ' regressx2onx1:x2 PRF:x201x1 1E 2Onlyif2=0or=0(xandxareuncorrelatedin 2 truetruemodel:yxx...x 1 2 kifexcludexk:y01x12x2...k1xk1 xk01x12x2...k1xk1 'E(j)jk j Onlyifk=0orj=0(xkandxjareuncorrelatedinsample),thenEjjomittedvariablebias:kjEjExample:Example:3.2&3.6ModelofHourlylog(ˆwage)0.5840.083educDataset:526workersinWAGE1.RAW,educ:yearsofeducationexper:yearsoflabormarketexperiencetenure:yearswiththecurrentSimpleExample:truemodel:y=0+1x1+2x2+u(3.40)ˆ' Var1 Var1 Ifx1andx2arecorrelated,VarVarˆ Ifx1andx2areuncorrelated,VarVarˆ If20,x2doeshaveapartialeffectony,thenexcludingitinthemodelresultsinabiasedOLSestimatorbutamoreefficientvarianceestimatorof1 AssumptionsMLR.1-5,theOLSestimatorjnjijlineariny:ˆ w (3.59fromunbiased:E(ˆ)jbest:efficient(smallestvariance)Var(ˆ)Var( E(ˆ)) var(ˆj)jjjjThecriticalquestioninGrantedthattheSRFisonlyanapproximationofthePRF,givenasample,canwedevisearuleoramethodthatwillmakethisapproximationas“close”Thatis,the'sinSRFshouldbeas“close”possibletotruevalues’sintherelevantPRFeventhoughwewillneverknowthe’sandPRF.--UnderMLR.1-5,theOLSestimatorsinSRFshouldbeas“close”aspossibletotruevalues’sintherelevantPRF,thatis,theywillmakethisapproximationas“close”aspossible.Ch.4:Multiple ysis:1 Ch.4:Multiple ysis:1OLSisBLUE.Inordertodostatisticaltest,weneedtoaddanotherMLR.6:uisindependentofx1,x2,…,xkanduisnormallydistributedwithzeromeanandvariance2:u~2