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首页 ln3 Review of Linear Models and OLS Estimation

ln3 Review of Linear Models and OLS Estimation.pdf

ln3 Review of Linear Models and…

Sunwhisper
2013-11-01 0人阅读 举报 0 0 暂无简介

简介:本文档为《ln3 Review of Linear Models and OLS Estimationpdf》,可适用于高等教育领域

ApEcEconometricAnalysisII–Lecture#ReviewofLinearModelsandOLSEstimation(Wooldridge,Chapter)ThislecturewillreviewmuchofthematerialyouhadinApec,butinawaythatwillprepareyoufortopicsthatwillbecoveredthissemesterthatyouhavenotseenbeforeIhopeyouappreciatethatitismuchlesstheoreticalthanthematerialinLecturesand,althoughIwilloftenrefertothematerialinthosetwolectures(whichcoveredWooldridge’sChaptersand)IOverviewoftheSingleEquationLinearModelStartwithamodelthatshowsalinearrelationshipbetweenthedependentvariable,y,andtheexplanatoryvariablesinthevectorx:y=ββxβx…βKxKu=x΄βuStrictlyspeaking,thisisamodelofthepopulationfromwhichthedataaredrawn,notamodelofthedataitselfBydefinition,thismodelassumesthatthereisacausal(structural)relationshipinthatchangesinxcausechangesiny,andthatthiscausalrelationshipislinearTheerrorterm“u”inthemodelcanrepresentmanythings,suchasother(“omitted”or“unobserved”)variablesormeasurementerrorinthevariablesthatwedoobserveWewillgettothislaterinthislectureThekeyassumptionregardinguthatisrequiredforconsistentestimationofβusingOLSis:Eu=Cov(xj,u)=,j=,,…,KTheassumptionthatEu=istrivialifxincludesaconstantterm(intercept),sincetheβforthatconstanttermcanalwaysbeadjustedtoensurethatEu=withoutchanginganyoftheotherelementsinβItisthezerocovarianceassumptionthatisthereallyimportantassumptionNotethatEu|x=impliesCov(u,x),butnotviceversaThusEu|x=isastrongerassumptionthanCov(u,x)CombiningthelinearmodelassumptionandtheassumptionthatEu|x=impliesthat:Ey|x=ββxβx…βKxK=x΄βOfcourse,sincexcanincludeinteractionorhigherorderterms(egxcouldbexorx·x),thelinearmodelhasafairamountofflexibilityWooldridgestatesmostofhisresultsusingtheassumptionCov(u,x)=,sinceitisaweakerassumptionthanEu|x=Looselyspeaking(we’lltalkmoreaboutthisinlaterlectures)wecansaythatavariablexjisendogenousifCov(u,xj)≠,andavariableisexogenousifCov(u,xj)=ThusinthislinearmodeltheassumptionthatCov(u,x)=impliesthatallofthevariablesinxareexogenous(forthismodelonly!)EndogeneityofoneormoreofthevariablesinximpliesthatOLSestimatesofβareinconsistentEndogeneitycanariseforthreereasons:OmittedvariablesSupposethatweareinterestedinthecausalimpactonyofboththevariablesinxandanothervariableqThatis,wewanttoknowEy|x,qforawiderangeofxandqButsupposethatwedonothavedataonq,sowecanonlyestimateEy|xThisrelationshipbetweenxandywillnotnecessarilybethesameasthe(causal)relationshipbetweenx(andq)andyrepresentedbyEy|x,qifqiscorrelatedwithoneormoreofthevariablesinxAnexampleofthisisestimatingtheimpactofyearsofschoolingonwages,whereqisunobserved“innateability”MeasurementerrorOurdatamaybe“bad”inthatthexinourdataisnotthe“true”x,whichcanbedenotedbyx*Thisisaveryserious(andoftenignored)problemineconometricsSimultaneityPerhapssomeofthevariablesinxnotonlycauseybutarealsocausedbyySinceyiscausedbyu,andthesexvariablesarecausebyy,thenthesexvariableswillbecorrelatedwithuAninterestingexampleiscrimeratesandthesizeofthepoliceforceWemaybeinterestedinestimatingtheimpactofthesizeofthepoliceforceonthecrimerate,buthighcrimeratesmaycausethegovernmenttoincreasethesizeofthepoliceforceIntoday’slecturewewilldiscussindetailthefirsttwoproblems,butthethirdproblemwillhavetowaituntilwetakeupinstrumentalvariables(startingthisWednesday)andestimationofsystemsofequations(startingnextweek)IIAsymptoticPropertiesofOLSEstimateofβEconomistsandotherresearchersoftenuseOLStoestimateβinthemodely=x΄βuForsimplicity,letthefirstelementinx,x,beaconstanttermWehaveasampleofsizeN:{(yi,xi):i=,,…N}Assumethateachobservationisrandomlydrawnfromthesamepopulation,soanytwoobservations,(yi,xi)and(yj,xj)areindependent(ifi≠j)andidentically(jointly)distributed(iid)randomvariablesForeachobservationwehave:yi=xi΄βuiConsistencyInadditiontotheassumptionthatthemodelislinear,weneedtwomoreassumptionsforconsistency:AssumptionOLS:Exu=AssumptionOLS:Rank(Exx΄)=KSincethefirstelementofxisassumedtobeaconstant,AssumptionOLSimpliesthatEu=YoushouldbeabletoshowthatExu=impliesCov(xj,u)=forallxjinxAssumptionOLSisneededtoensurethattheOLSestimateofβisuniqueIfitdoesnothold,thenoneormoreelementsofxisalinearcombinationoftheothers,whichmeansthattherearemanyβ’sthatgivethesame(conditional)expectedvaluesofyNotethatthisassumptionisequivalenttoassumingthatExx΄isapositivedefiniteK×Kmatrix,andthatthevariancecovariancematrixofthevariablesinx(ieafterremovingtheconstantterm)isnonsingularThesetwoassumptionsimplythatβisidentified,thatisitcanbeexpressed(solvedfor)intermsofthepopulationmoments(egvariancesandcovariances)ofxandyThesolutioncomesfrompremultiplyingy=x΄βubyx,takingexpectations,andrearrangingβ=(Exx΄)ExyToshowconsistencywriteouttheOLSestimateofβ:βˆOLS=Niii'NxxNiiiyNx=βNiii'NxxNiiiuNxOfcourse,βˆOLScanbewrittenas(X΄X)(X΄y)ByCorollary,AssumptionOLSimpliesthatX΄Xisnonsingularwpaandplim((N)Nixixi΄)=A,whereA≡Exx΄ByassumptionOLSwehaveplim(N)Nixiui=Exu=ThenbySlutsky’stheoremwehaveplimβˆOLS=βA×=β:Theorem(consistencyofOLS):UnderassumptionsOLSandOLS,βˆOLSfromarandomsampleofthepopulationmodely=x΄βuisaconsistentestimatorofβNotethatβˆOLSisalsoaconsistentestimateoftheparametersofalinearprojectionofyonx,aslongasassumptionOLSholds,sincethelinearprojectionisdefinedas(Exx΄)Exy,plim((N)Nixixi΄)=(Exx΄)andplim((N)Nixiyi)=ExyAfewotherimportantpoints:Inpractice,youusuallydon’thavetoworryaboutassumptionOLSitisobviouswhenitfails(anditrarelyfails)ItisassumptionOLSthatoftenfailsand,moreperniciously,itisverydifficulttodeterminewhetheritholdsAssumptionsOLSandOLSdonotbythemselvesimplythatβˆOLSisunbiasedToshowthatweneedtheslightlymorerestrictiveassumptionthatEu|x=Theconsistencyresultrequiresonlythatuandxareuncorrelated,notthattheyareindependentIndependencewouldimplythatVaru|xisconstant,butwedon’tneedthisforconsistencyAsymptoticNormalityTheaboveexpressionforβˆOLSimpliesthat:N(βˆOLSβ)=Niii'NxxNiiiuNxTheoremimpliesthatNiii'NxxA=op()thisexpressionconvergesinprobabilitytozeroNotethat{(xiui):i=,,…N}isaniidsequencewithzeromean,andassumethateachelementofxiuihasafinitevariance,thenbytheCentralLimitTheoremwehaveNNixiuidN(,B),whereB≡Euxx΄ByLemma,NNixiui=Op()Thus:N(βˆOLSβ)=ANiiiuNxop()sinceop()·Op()=op()Togofarther,weneedahomoskedasticityassumption:AssumptionOLS:Euxx΄=σExx΄,whereσ=EuAstrongerassumptionisthatEu|x=σ,butitisnotneeded(AlthoughitiseasiertointerpretthanAssumptionOLS)Puttingallofthistogethergives:Theorem(asymptoticnormalityofOLS):UnderassumptionsOLS,OLSandOLS:N(βˆOLSβ)~aN(,σA),whereA≡Exx΄Proof:LemmaandCorollaryimplythatN(βˆOLSβ)~aN(,ABA)ThenAssumptionOLSimpliesthatB=σAInpracticewecanestimateAvar(βˆOLS)asˆ(X΄X),whereˆ=((NK))Ni(yi–xi΄βˆOLS)IftheassumptionsofTheoremhold,thentheusual(OLSbased)formulasforstandarderrorsofβˆOLS,fortstatisticsandforFstatisticsareallasymptoticallyvalidHeteroscedasticityRobustInferenceIfAssumptionOLSfails,βˆOLSisstillaconsistentestimatorofβ,butwecan’tuseˆ(X΄X)toestimateAvar(βˆOLS)ThisisaseriousproblembecauseitisquitepossiblethatAssumptionOLSdoesfailFortunately,itisnothardtogetanestimateofAvar(βˆOLS)underlessrestrictiveassumptions(Analternativeapproachisweightedleastsquares(WLS),butthisrequiresa(parametric)modelforVary|x,whichmaybejustasrestrictiveasAssumptionOLS)ToestimateAvar(βˆOLS)underasetoflessrestrictiveassumptions,gobacktotheasymptoticnormalitydiscussionTheasymptoticvarianceofβˆOLSwithoutAssumptionOLSisAvar(βˆOLS)=ABANOurconsistentestimateofAisN(X΄X),sowejustneedaconsistentestimateofBBythe(weak)lawoflargenumbers(Theorem)wehave:(N)Niuixixi΄pEuxx΄=BWecanreplaceuiwithiuˆ=yi–xi΄βˆOLSThus(N)Niiuˆxixi΄isaconsistentestimatorofBand:Avaˆr(βˆOLS)=(X΄X)(Niiuˆxixi΄)(X΄X)ThestandarderrorsofβˆOLSareheteroscedasticityrobuststandarderrorsTheyarealsocalledWhitestandarderrorsorHuberstandarderrorsThesecanbeusedtoobtaintstatisticsintheusualwayHowever,theusualFtestisinvalidYoushouldusetheheteroscedasticityrobustWaldstatisticinsteadtotesthypothesesoftheformH:Rβ=r(RisQ×K,risQ×)Thatteststatisticis:W=(RβˆOLSr)΄(RVˆR΄)(RβˆOLSr)WhereVˆistheaboveformulaforAvaˆr(βˆOLS)UnderHW~aχQDividingWbyQyieldsan(approximate)FstatisticthatisdistributedasFQ,NKLagrangeMultiplier(Score)TestsIneconometricsthereare“classic”statisticaltests:theWaldtest,theLagrangemultipliertestandthelikelihoodratio(LR)testWesawaverygeneralexpositionoftheWaldtestattheendofLectureNowwewilllookattheLagrangeMultipliertestinthecontextofalinearmodelSupposethatwedividethexvariablesintogroups:y=x΄βx΄βuwherexhasKelementsandxhasKelementsIfAssumptionsOLS,OLSandOLShold,itiseasytotestthehypothesisH:β=usingastandardFtestButwhatifAssumptionOLSdoesn’tholdLet~βbetheestimateofβwhentheconstraintthatβ=isimposedQuestion:Howdoyouestimate~βDefineu~i=yixi΄~βUnderH,u~ishouldbeuncorrelatedwithxibecause,conditionalonx΄β,xshouldhavenoexplanatorypowerforyandthusnoexplanatorypowerforu~Thislackofexplanatorypowercanbetestedbyregressingu~onxandxLetRubetheRfromthisregression(assumethatxcontainstheconstantterm)TheLagrangemultiplier(LM)statisticis:LM=NRuUnderH,LM~aχKNotethattheregressionmustincludexaswellasx,eventhoughbyconstructionxwillnotbecorrelatedwithu~Finally,ifxdoesnotincludeaconstant,thenRushouldbetheuncenteredRTheLMtestjustdescribedrequiresAssumptionOLSIfthatassumptionisincorrect,theprocedurecanbemodifiedasfollowsAftersomealgebra(thiswouldbeagoodhomeworkproblem!)youcanshowthattheLMstatisticis:LM=Niiiu~ˆNr΄Niii'ˆˆN~rrNiiiu~ˆNrwhereσ~=Niu~iandeachirˆisaK×vectorofOLSresidualsfromregressingxonxThisteststatisticisnotrobusttoheteroscedasticitybecausethemiddletermisnotaconsistentestimateoftheasymptoticvarianceofNNiirˆu~iwhenuisheteroscedasticUsingtheHuberWhiteapproachjustdiscussed,theheteroscedasticityrobustLMteststatisticis:LM=Niiiu~ˆNr΄Niiii'ˆˆu~NrrNiiiu~ˆNr=Niiiu~rˆ΄Niiii'ˆˆu~rrNiiiu~rˆ~aKThiscanbeobtainedbyregressing(withoutanintercept)aconstantonrˆ·u~LetSSRbethesumofthesquaredresidualsfromthisregressionThen:LM=N–SSR~aKIIIOLSMethodsforOmittedVariableBiasSupposethereisacausalvariablethatdeterminesybutisunobserved,denotedbyqIfweobservedqwecouldderivethecausalimpactsofxandqonybyobservingEy|x,qforalargenumberofobservationsSupposethatthisrelationshipislinear:Ey|x,q=x΄βγqNotethatβmeasurestheimpactsofeachvariableinxony,holdingtheotherxvariables,andq,constantQuestion:SupposethereareunobservedvariablesisthisverydifferentfromhavesuchvariableWecanrewritetheaboverelationshipas:y=x΄βγqv,Ev|x,q=Wooldridgecallsvthestructuralerror,whilethe“effective”error(duetotheunobservabilityofq)isu≡γqvSotheaboveequationcanberewrittenas:y=x΄βuBychangingtheconstanttermwecansetEq=,andthusEu=Unfortunately,ifqiscorrelatedwithanyoftheregressors,thensoisuUsingOLStoestimateβbyregressingxonywill,ingeneral,giveinconsistentestimatesforalltheelementsofβ(notjusttheonesofthexvariablesthatarecorrelatedwithq)Thisiscalledomittedvariablesinconsistencyor(morecasually)omittedvariablesbiasToseethebias,consideralinearprojectionofqonx(notethatwearenotassumingthatEq|x=Lq|x):q=x΄δr,whereEr=andCov(x,r)=Pluggingthisintothefirstequationforygives:y=x΄(βγδ)vγrByassumption,Evγr=andxisuncorrelatedwithvγrThusaregressionofyonxsatisfiesOLSandwewillfindthat,foranyelementxjofx:plimˆj,OLS=βjγδjSometimespeopleassertthatonlyonevariableinx,callitxK,iscorrelatedwithq,or(moreprecisely)thatalltheelementsofδotherthanδKequalzeroIfthismorepreciseassumptionistrue,then:plim=ˆj,OLS=βjforj≠KplimˆK,OLS=βKγδK=βKγCov(xK,q)Var(xK)Butinthegeneralcaseδjisapartialcorrelationofqandxj,nota“raw”correlation,andingeneralthesewillnotbethesameYetifagoodargumentcanbemadethatallδj’sotherthanδKequalzerothenthe“raw”correlationofxKandqgivesthedirectionofthebiasUsingProxyVariablestoRemoveOmittedVarBiasSometimeswehavedataona“proxy”variablethatissimilartotheomittedvariableqIntuitively,itseemsreasonablethatputtingthatvariableinwillfix,oratleastreduce,theomittedvariableproblemInfact,researchersdothisveryoften,soitisusefultoworkoutthepreciseconditionsneededforthisproceduretosucceedinreducingoreliminatingomittedvariablebiasLetzbethecandidateproxyvariableAddingitwillremoveomittedvariablebiasifconditionshold:Ey|x,q,z=Ey|x,qLq|,x,z=Lq|,zThefirstconditionmeansthatzdoesnothaveanyexplanatorypowerforythatisnotalreadycontainedinxandqInmanycasesthisisareasonableassumptionForexample,inawageregressionandyouhavedataon“true”innateabilitythenyouwouldnotexpectsomekindofIQtesttohaveanyexplanatorypowerbeyondtheimpactofinnateabilityActually,thefirstconditionisalittlestrongerthanneeded,allwereallyneedisthatzisuncorrelatedwithvThesecondrequirementisthat,onceweconditiononz,thereisnocorrelationbetweenqandxIntuitively,addingztothex’sasregressorsinalinearmodelremovesthe(partial)correlationbetweenxandqAnotherwaytothinkofthisisthatqcanbedividedintotwoparts,q=θzr,whereCov(z,r)=andthatallofthecorrelationbetweenqandxcomesfromtheθzpart,sothatCov(x,r)=Ifthesetwoconditionshold,wecaninsertq=θzrintotheequationy=x΄βγqv:y=x′βγθz(γrv)Ifyouaddaconstanttoq=θzrthentheconstantinβwillchange,butnothingelsewillchangeinβSincethe(composite)errorterm(γrv)isuncorrelatedwithbothxandz,byTheoremwecanconsistentlyestimateβ(andγθ)Whatifzisanimperfectproxyinthesensethattherinq=θzriscorrelatedwithxAlinearprojectionofqonxandzwillthengive:q=x′ρθzr*wherer*,whichcomesfromr=x′ρr*,isnotcorrelatedwithxTheOLSestimatefromaregressionofyonxandzwillhaveaplimofβγρYoucouldargue(“hope”)thatρissmallerthanδtojustifytheuseofsuchanimperfectproxyAfinalpointisthataddingproxiesandevenimperfectproxiesmayreduceyourestimatesofthevarianceofβˆOLSTheintuitionhereisthatthatvarianceisdeterminedinpartbythevarianceoftheerrorterm,andifyou“convert”partoftheerrortermintoanobservablevariablethenyoureducethevarianceoftheremainingerrortermModelswithInteractionsinUnobservablesThingsbecomemorecomplicatediftheimpactofanobservedvariabledependsonthevalueofanunobservedvariable,ietheunderlyingmodelhasaninteractiontermbetweenqandoneormorevariablesinxInparticular,assumethatthestructuralmodelis:y=x΄βγqγxKqv,Ev|x,q=IfxKisacontinuousvariablethenthepartialeffectofxKonEy|x,qis:Kxq,|yEx=βKγqWecanneverestimatethisforaparticularq,sinceweneverobserveqHowever,wecanestimatetheaveragepartialeffect(APE)ifwehaveagoodestimateforβKTheAPEissimplythepopulationaverageofthepartialeffect,so:APE=EβKγq|q=βKγEq(=βKifEq=)HowdoweestimateβwhenthereisanunobservableqanditisinteractedwithoneofthexvariablesAssumethatwehaveaproxyvariablezthatmeetstheabovecriteriaEy|x,q,z=Ey|x,qand:Eq|x,z=Eq|z=θz,whereEz=Theseassumptionslawofiteratedexpectationsgive:Ey|x,z=x΄βγθzγθxKzAlloftheseparameterscanbeestimatedconsistentlyusingOLSWooldridgeexplains(p)thateveniftheoriginalstructuralmodelishomoscedasti

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ln3 Review of Linear Models and OLS Estimation

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