多元回归分析模型识别和数据问题课件

第八章多元回归分析模型识别和数据问题contentsFunctionalformmisspecificationUsingproxyvariablesMeasurementerrorinvariablesMissingdataandOutlyingobservationsFunctionalfrommisspecificationFunctionalForm(continued)First,useeconomictheorytoguideyouY=AKaLbeuorlnY=lnA+alnK+blnL+uThinkabouttheinterpretationlog(wage)=b0+b1

educ+u,orlog(educ)asindependentvariableDoesitmakemoresenseforxtoaffectyinpercentage(uselogs)orabsoluteterms?Doesitmakemoresenseforthederivativeofx1tovarywithx1(quadratic)orwithx2(interactions)ortobefixed?FunctionalForm(continued)Wealreadyknowhowtotestjointexclusionrestrictionstoseeifhigherordertermsorinteractionsbelonginthemodellog(wage)=b0+

b1

educ+b2

exper+b3tenure+ulog(wage)=b0+

b1

educ+b2

exper+b3tenure+b4educ2+b5exper2+b6tenure2+b7educ•tenure+uItcanbetedioustoaddandtestextraterms,plusmayfindasquaretermmatterswhenreallyusinglogswouldbeevenbetterAtestoffunctionalformisRamsey’sregressionspecificationerrortest(RESET)Firstestimatelog(wage)=b0+

b1

educ+b2

exper+b3tenure+uGetfittedvalueŷ(log(wâge)

ofaboveequation)Then,considertheexpandedequationlog(wage)=b0+

b1

educ+b2

exper+b3tenure+d4ŷ2+d5ŷ3+uRESETistheFstatisticfortesingH0:d4=0,d5=0InStata,theRESETtestcommand:ovtestWhetherthemodely=b0+b1x1+…+bkxk+umisspecified?RESETreliesonatricksimilartothespecialformoftheWhitetestInsteadofaddingfunctionsofthex’sdirectly,weaddandtestfunctionsofŷSo,estimatey=b0+b1x1+…+bkxk+d1ŷ2+d1ŷ3+errorandtestH0:d1=0,d2=0usingF~F2,n-k-3orLM~χ22RESETtest,exampleHousingpriceequation(hprice.raw)price=b0+b1

lotsize+b2sqrft+b3bdrms+ulog(price)=b0+b1

log(lotsize)+b2

log(sqrft)+b3bdrms+uRESETtestprocedureEstimatethemodels:regpriceonlotsize,sqrft,bdrms,andgetfittedvalueofprice,ŷandSSRr=300723.806,n=88R2=0.6724Calculateŷ2,ŷ3,andplugthemtotheoriginalequation,andestimateit.Thatis,regpriceonlotsize,sqrft,bdrms,ŷ2,ŷ3,andSSRur=269983.825n=88R2=0.7059SotheFvalue=[(300723.806-269983.825)/2]/(269983.825/82)=4.6682,thep-value=0.012,therefore,wewillrejectthenullhypothesisthatthereisnomisspecification.Inthesameway,wecancalculatethesecondmodelF=[(2.86256385-2.69401081)/2]/(2.69401081/82)=2.565,p-value=0.0835.Sowecan’trejectthenullhypothesisatthe5%significance.ProxyvairablesProxyVariablesWhatifmodelismisspecifiedbecausenodataisavailableonanimportantxvariable?ItmaybepossibletoavoidomittedvariablebiasbyusingaproxyvariableModel:y=b0+b1x1+b2x2+b3x3*+uAproxyvariablemustberelatedtotheunobservablevariable–forexample:x3*=d0+d3x3+v3,where*impliesunobservedNowsupposewejustsubstitutex3forx3*ProxyVariables(continued)y=b0+b1x1+b2x2+b3x3*+ux3*=d0+d3x3+v3Whatdoweneedforforthissolutiontogiveusconsistentestimatesofb1andb2?Assumeuisuncorrelatedwithx1,x2andx3*,x3andv3isuncorrelatedwithx1,x2andx3E(x3*|x1,x2,x3)=E(x3*|x3)=d0+d3x3

Soreallyrunningy=(b0+b3d0)+b1x1+b2x2+b3d3x3+(u+b3v3)andhavejustredefinedintercept,errortermx3coefficientProxyVariables(continued)Withoutoutassumptions,canendupwithbiasedestimatesSayx3*=d0+d1x1+d2x2+d3x3+v3Thenreallyrunningy=(b0+b3d0)+(b1+b3d1)x1+(b2+b3d2)x2+b3d3x3+(u+b3v3)Biaswilldependonsignsofb3anddjThisbiasmaystillbesmallerthanomittedvariablebias,thoughLaggedDependentVariablesWhatifthereareunobservedvariables,andyoucan’tfindreasonableproxyvariables?Maybepossibletoincludealaggeddependentvariabletoaccountforomittedvariablesthatcontributetobothpastandcurrentlevelsofy,thatis,usey-1toexplainy.y=b0+b1x1+b2x2+b3x3*+uy=b0+b1x1+b2x2+b3y-1+uObviously,youmustthinkpastandcurrentyarerelatedforthistomakesenseMeasurementErrorMeasurementErrorinaDependentVariableModely*=b0+b1x1+…+bkxk+uyistheobservablemeasureofy*.Definemeasurementerrorase0=y–y*Thus,reallyestimatingy=b0+b1x1+…+bkxk+u+e0WhenwillOLSproduceunbiasedresults?Ife0andxj,uareuncorrelatedisunbiasedIfE(e0)≠0then

b0willbebiased,thoughWhileunbiased,largervariancesthanwithnomeasurementerrorVar(u+e0)=su2+se2MeasurementErrorinanExplanatoryVariabley=b0+b1x1*+uDefinemeasurementerrorase1=x1–x1*x1isthemeasureofthetruevaluex1*AssumeE(e1)=0,E(y|x1*,x1)=E(y|x1*)Reallyestimatingy=b0+b1x1+(u–b1e1)TheeffectofmeasurementerroronOLSestimatesdependsonourassumptionaboutthecorrelationbetween

e1andx1

SupposeCov(x1,e1)=0OLSremainsunbiased,varianceslargerMeasurementErrorinanExplanatoryVariable(cont)SupposeCov(x1*,e1)=0,knownastheclassicalerrors-in-variablesassumption(CEV),thenCov(x1,e1)=E(x1e1)=E(x1*e1)+E(e12)=0+se2Seeestimatedmodely=b0+b1x1+(u–b1e1)x1iscorrelatedwiththeerrorsoestimateisbiasedMissingdataandOutlyingobservationsMissingData–IsitaProblem?Ifanyobservationismissingdataononeofthevariablesinthemodel,itcan’tbeusedIfdataismissingatrandom,usingasamplerestrictedtoobservationswithnomissingvalueswillbefineAproblemcanariseifthedataismissingsystematically–sayhighincomeindividualsrefusetoprovideincomedataNonrandomSamplesIfthesampleischosenonthebasisofanxvariable,thenestimatesareunbiasedIfthesampleischosenonthebasisoftheyvariable,thenwehavesampleselectionbiasSampleselectioncanbemoresubtleOutlierTest1

StudentizedResidualse(i)=yi–b(i)xi,where

b(i)representtheestimatedregressionslopewhentheithobservationhasbeenomitted.E(e(i))=0Thestudentizedresidualisei*=[yi–b(i)xi]/si(i)Where,si(i)isthestandarderroroftheregressionwithoutobservationi.Ifthestudentizedresidualsthataregreaterthan1.96inabsolutevaluecanberegardedasoutliersandshouldreceivespecialattention.InStata,it’seasytocalculatethestudentizedresiduals,