多元回归分析模型识别和数据问题

多元回归分析模型识别和数据问题第一页，共二十九页，2022年，8月28日contentsFunctionalformmisspecificationUsingproxyvariablesMeasurementerrorinvariablesMissingdataandOutlyingobservations第二页，共二十九页，2022年，8月28日Functionalfrommisspecification第三页，共二十九页，2022年，8月28日FunctionalFormWe’veseenthatalinearregressioncanreallyfitnonlinearrelationshipsCanuselogsonRHS,LHSorbothCanusequadraticformsofx’sCanuseinteractionsofx’sHowdoweknowifwe’vegottentherightfunctionalformforourmodel?第四页，共二十九页，2022年，8月28日FunctionalForm(continued)First,useeconomictheorytoguideyouY=AKaLbeuorlnY=lnA+alnK+blnL+uThinkabouttheinterpretationlog(wage)=b0+b1

educ+u,orlog(educ)asindependentvariableDoesitmakemoresenseforxtoaffectyinpercentage(uselogs)orabsoluteterms?Doesitmakemoresenseforthederivativeofx1tovarywithx1(quadratic)orwithx2(interactions)ortobefixed?第五页，共二十九页，2022年，8月28日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~χ22第六页，共二十九页，2022年，8月28日RESETtest,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.第七页，共二十九页，2022年，8月28日Ifthemodelshavethesamedependentvariables,butnonnestedx’scouldstilljustmakeagiantmodelwiththex’sfrombothandtestjointexclusionrestrictionsthatleadtoonemodelortheother.Forexample,wehavetochoosemodelbetweeny=b0+b1x1+b2x2+u(m1)y=b0+b1log(x1)+b2log(x2)+u(m2)Whichmodeltochoose?

Method1:estimateacomprehensivemodely=d0+d1x1+d2x2+

d3log(x1)+d4log(x2)+uH0:d3=0,d4=0forthesecondmodelandH0:d1=0,d1=0forthefirstone.Method2:theDavidson-MackinnontestIf(m1)istrue,thenthefittedvaluesfrom(m2)shouldbeinsignificantin(m1).Thus,totest(m1),wefirstestimate(m2)byOLStoobtainthefittedvalues,ŷ.Thenplugitinto(m1),that’sy=b0+b1x1+b2x2+q

ŷ+uAsignificanttstatisticisarejectionofmodel(m1).NonnestedAlternativesTest第八页，共二十九页，2022年，8月28日Proxyvairables第九页，共二十九页，2022年，8月28日ProxyVariablesWhatifmodelismisspecifiedbecausenodataisavailableonanimportantxvariable?ItmaybepossibletoavoidomittedvariablebiasbyusingaproxyvariableModel:y=b0+b1x1+b2x2+b3x3*+uAproxyvariablemustberelatedtotheunobservablevariable–forexample:x3*=d0+d3x3+v3,where*impliesunobservedNowsupposewejustsubstitutex3forx3*第十页，共二十九页，2022年，8月28日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,errortermx3coefficient第十一页，共二十九页，2022年，8月28日Example:IQasaProxyforAbility(wage2.raw,p297)Modellog(wage)=b0+b1educ+b2exper

+b3abil+uAssumeE(u|educ,exper,abil)=0Butthedataofabilityisnotavailable,wethinkIQmaycorrelatewithability,that’sabil=d0+d1IQ+vAssumeE(v|educ,exper,IQ)=0soweuseIQasaproxyforability.Andtheestimatedmodelislog(wage)=b0*+b1educ+b2exper

+b3*IQ+u*Resultslog(wâge)=5.503+0.078

educ+0.0198exper

(biasedestimate)(0.112)(0.007)(0.003)n=935R2=0.1309

log(wâge)=5.198+0.057educ+0.0195exper

+0.0058IQ(0.122)(0.007)(0.003)(0.001)n=935R2=0.1622(efficientestimate)第十二页，共二十九页，2022年，8月28日ProxyVariables(continued)Withoutoutassumptions,canendupwithbiasedestimatesSayx3*=d0+d1x1+d2x2+d3x3+v3Thenreallyrunningy=(b0+b3d0)+(b1+b3d1)x1+(b2+b3d2)x2+b3d3x3+(u+b3v3)Biaswilldependonsignsofb3anddjThisbiasmaystillbesmallerthanomittedvariablebias,though第十三页，共二十九页，2022年，8月28日LaggedDependentVariablesWhatifthereareunobservedvariables,andyoucan’tfindreasonableproxyvariables?Maybepossibletoincludealaggeddependentvariabletoaccountforomittedvariablesthatcontributetobothpastandcurrentlevelsofy,thatis,usey-1toexplainy.y=b0+b1x1+b2x2+b3x3*+uy=b0+b1x1+b2x2+b3y-1+uObviously,youmustthinkpastandcurrentyarerelatedforthistomakesense第十四页，共二十九页，2022年，8月28日MeasurementError第十五页，共二十九页，2022年，8月28日MeasurementErrorSometimeswehavethevariablewewant,butwethinkitismeasuredwitherrorExamples:Asurveyaskshowmanyhoursdidyouworkoverthelastyear,orhowmanyweeksyouusedchildcarewhenyourchildwasyoungMeasurementerrorinydifferentfrommeasurementerrorinx第十六页，共二十九页，2022年，8月28日MeasurementErrorinaDependentVariableModely*=b0+b1x1+…+bkxk+uyistheobservablemeasureofy*.Definemeasurementerrorase0=y–y*Thus,reallyestimatingy=b0+b1x1+…+bkxk+u+e0WhenwillOLSproduceunbiasedresults?Ife0andxj,uareuncorrelatedisunbiasedIfE(e0)≠0then

b0willbebiased,thoughWhileunbiased,largervariancesthanwithnomeasurementerrorVar(u+e0)=su2+se2第十七页，共二十九页，2022年，8月28日MeasurementErrorinanExplanatoryVariabley=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,varianceslarger第十八页，共二十九页，2022年，8月28日MeasurementErrorinanExplanatoryVariable(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)x1iscorrelatedwiththeerrorsoestimateisbiased第十九页，共二十九页，2022年，8月28日MeasurementErrorinanExplanatoryVariable(cont)NoticethatthemultiplicativeerrorisjustVar(x1*)/Var(x1)SinceVar(x1*)/Var(x1)<1,theestimateisbiasedtowardzero–calledattenuationbiasIt’smorecomplicatedwithamultipleregression,butcanstillexpectattenuationbiaswithclassicalerrorsinvariables第二十页，共二十九页，2022年，8月28日MissingdataandOutlyingobservations第二十一页，共二十九页，2022年，8月28日MissingData–IsitaProblem?Ifanyobservationismissingdataononeofthevariablesinthemodel,itcan’tbeusedIfdataismissingatrandom,usingasamplerestrictedtoobservationswithnomissingvalueswillbefineAproblemcanariseifthedataismissingsystematically–sayhighincomeindividualsrefusetoprovideincomedata第二十二页，共二十九页，2022年，8月28日NonrandomSamplesIfthesampleischosenonthebasisofanxvariable,thenestimatesareunbiasedIfthesampleischosenonthebasisoftheyvariable,thenwehavesampleselectionbiasSampleselectioncanbemoresubtle第二十三页，共二十九页，2022年，8月28日OutliersSometimesanindividualobservationcanbeverydifferentfromtheothers,andcanhavealargeeffectontheoutcomeSometimesthisoutlierwillsimplybedotoerrorsindataentry–onereasonwhylookingatsummarystatisticsisimportantSometimestheobservationwilljusttrulybeverydifferentfromtheothers第二十四页，共二十九页，2022年，8月28日OutlierTest1

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,youcanusethefollowingcommandafterregressionPredictrstud,rstudente(i)=yi–b(i)xixi第二十五页，共二十九页，2022年，8月28日OutlierinaModelofPublicSpending,(HR,Ex7.3)In“prdata\ex73.txt”exp=-45.698+3.234aid+0.00019inc-0.597popPredictrstud,rstudent/*calculatethestudentizedresiduals*/The7thand14thobservationmaybeoutlier.Omit7thobservationandestimateagainexp=-7.08+2.365aid+0.00018inc-0.426popDummyvariablemethodtocalculatestudentizedresidualsGend7