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M1_Eco_OLS

c20052005
2013-01-04 0人阅读 举报 0 0 暂无简介

简介:本文档为《M1_Eco_OLSpdf》,可适用于经济金融领域

TheOLSmodelIntroductoryEconometrics(MEcoPsyco)MargheritaComolaEconometricsEconometricsisbaseduponthedevelopmentofstatisticalmethodsfor:)estimatingeconomicrelationships)testingeconomictheoriesand)evaluatingpolicies:Imostlyobservationaldata,fewcontrolledexperimentIfromtheeconomicmodeltotheeconometricmodel(throughmathematicalstatistics)NonexperimentaldatacrosssectionIsampleofunitsatagivenpointintimeIcensusvsrandomsampleIexperimentalvsnonexperimentaldataIquasiexperimentaldatapooledcrosssectionIrepeatingcrosssectionrandomsamplesItoanalyzechangesacrossyearstimeseriesIobserveafewvariablesovertimeIcorrelationofdataacrosstimepaneldataIsamecrosssectionalunits:atimeseriesforeachmemberofthepanelTherelationshipofinterestImaginewehaveobservationsonindividualsfirmscountries,andwewanttoidentifycauseandeffect,inordertomakepredictionaboutchangesincircumstancespolicy:ReturnstoschoolingIimportanttoevaluatechangesincompulsoryattendancelaws,orincostsofattendingcollegeIeducationiscorrelatedwithunobservedcharacteristicsIidealexperiment:randomlydistributeeducationColonialinstitutionsandeconomicgrowthImoredemocracyimpliesmoreeconomicgrowth(Acemoglu,JohnsonandRobinson,)Iidealexperiment:randomlyassigndifferentgovernmentstructurestoformercoloniesQuestionsWewanttostudyhowy(dependentvariable)varieswithx(independentvariable,regressor)howdoweallowforotherfactorstoaffectyfunctionalrelationshipbetweenyandxcausalrelationshipbetweenyandxThebivariateregressionmodelThebivariateregressionmodelAssumewehavearandomsamplefromapopulation{yi,xii=,,n}andwewanttoestimatehowy(egsavings)variesintermsofx(egincome)Thelinearregressionmodelisy=ββxu()wherewewanttoestimatetheparametersβˆandβˆ:Iuisanerrortermordisturbance(unobservedcharacteristics):E(u)=IlinearinparametersInote:wecaneliminatetheconstant!ZeroconditionalmeanassumptionAssumezeroconditionalmean(exogeneityofx):E(u|x)=E(u)=Istrongerthansaying“uandxareuncorrelated”:foranyvalueofx,wecanobtaintheaveragevalueofuforthatsliceofpopulation,andthisshouldnotdependonxIregressingwageoneducation:wehavetoassumethataveragelevelofabilityisthesameregardlessofyearsofeducation(cannotbechecked)Iftheassumptionholdstheestimatorinunbiased:E(βˆ)=βE(βˆ)=βTheordinaryleastsquared(OLS)estimatorFromtheassumptionsabove,weobtainthefollowingformulas:βˆ=∑n=(xi−x¯)(yi−y¯)∑n=(xi−x¯)βˆ=y¯−βˆx¯Iproof:pag,basicWooldridgeMinimizingSSRIfyoudefinethefittedvalueyˆi=βˆoβˆxiandtheresidualuˆi=yi−yˆi(note:therearenofthem),youcanprovethatβˆandβˆminimizethesumofsquaredresiduals:n∑=(uˆi)=n∑=(yi−yˆi)=n∑=(yi−βˆo−βˆxi)OncewehavecomputedβˆandβˆweformtheOLSregressionlineyˆi=βˆoβˆxiwhichisalsosaidsampleregressionfunctionbecauseitistheestimatedversionofthe(fixed,yerunknown)populationregressionfunctionE(yi|xi)=βoβxiTheOLSlinearfitFittedvaluesandresidualsMechanicsofOLSForeachsample,youobtainβˆandβˆ,andwhiththoseyoucancomputethefittedvaluesandresidualspereachindividual:IBydefinition,eachfittedvalueisonOLSregressionlineIyi=yˆiuˆiThereforetheresidualisthedifferencebetweenyianditsfittedvalue:ifpositivenegative,theregressionlineunderpredictsoverpredictsyiI∑uˆi=(OLSFOC:wepicktheβˆandβˆinordertomakethistrue)Icov(uˆ,x)=(OLSFOC)Ithepoint(x,y)isontheregressionlineStatisticalpropertiesofthebivariateOLSIfthefollowingassumptionsareverified:linearinparametersrandomsamplezeroconditionalmeansamplevariationintheregressorThenOLSareunbiasedInterpretationofcoefficientsIβˆ=∆yˆ∆xorequivalently∆yˆ=βˆ∆xIholdingotherrelevantfactorsfixed(ceterisparibus)Examples:Iwageandeducation:ˆwage=−eduIcampaignexpenditures:ˆvoteA=shareAImportanceoftheE(ui|xi)=assumptionIfthezeroconditionalmeanassumptionfails,theOLSestimatorisbiasedThe“spuriouscorrelation”problemisverycommoninnonexperimentaldataExample:math=ββlnchprguImath:oftenthgradersreceivingapassingscoreonastandardizedmathexamIlnchprg:ofstudentsparticipatingtothefederallunchprogramIusingdataonMichiganhighschoolsfor–,wegetβˆ=−Ipovertyrate,schoolqualityandresourcesarecontainedinuGoodnessofFitYoucanshowthat(pgW)n∑=(yi−y¯)︸︷︷︸TotSumSquared(SST)=n∑i=(yˆi−y¯)︸︷︷︸ExplainedSumSquared(SSE)n∑i=uˆi︸︷︷︸SumSquaredResiduals(SSR)Thefractionofthesamplevariationinythatisexplainedbyx’sisR=SSESST=−SSRSSTIbestfitatR=,nofitatR=IitneverdecreasesifweaddanotherregressorIasmallRdoesnotimplyabadestimateNonlinearfunctionsAlinearfunctionissuchthat∆yˆ=βˆ∆xIconstantmarginaleffect(notrealistic)Fornonlinearfunctions,∆yˆdependsontheinitialvalueofx:Iquadraticfunctions:y=ββxβxuIparabolicorUshapeIdiminishingmarginaleffectsIwageandyearsofexperienceInaturallogarithms:y=log(x)ThelogarithmThelogarithmofanumberistheexponentbywhichanotherfixedvalue,thebase,hastoberaisedtoproducethatnumberIthelogarithmoftobaseis(log()=),becauseistothepower:=××=Iifx=by,thenyisthelogarithmofxtobaseb,andiswritteny=logb(x)NaturalLogarithmInwhatfollowsweusenaturallogarithms,thatis,inbase:Idiminishingmarginalreturns:theslopeapproacheszerobutneverreachitIsomeusefulspecificalgebraicpropertieslog(a·b)=log(a)log(b)log(ab)=log(a)−log(b)log(ab)=b·log(a)ThelogTransformationIVERYimportantproperty:forsmallchangesinx,∆log(x)=log(x)−log(x)≈(x−x)x=∆xx∆log(x)≈∆xx=∆xIExample:x=,x=Therefore∆x=Notethatlog()−log()=IExample:x=,x=Therefore∆x=Notethatlog()−log()=ElasticityTheelasticityofywithrespecttoxisthepercentagechangeinywhenxinsreasesby:∆y∆x=∆y∆xxyIforalinearfunctiony=ββxu,theelasticitydependsonx:∆y∆x=βxy=βxββxConstantElasticityModelForanonlinearfunctionlog(y)=ββlog(x)u,theelasticityisconstant:∆log(y)=β∆log(x)∆log(y)=(β)∆log(x)∆y=β∆x∆y∆x≈∆log(y)∆log(x)=βIlog(q)=−log(p)SemiElasticityModelForalinearfunctionlog(y)=ββxu,youget∆log(y)=β∆x∆log(y)=(β)∆x∆y=(β)∆xForexample:log(wage)=edu∆wage=()∆edu=eduOnemoreyearofeducationincreaseswagebyWhydowelikelogsLogsareusuallytakenforlargeintegervalues(dollars,sales,headcountsnotfor:years,orpercentages):easyinterpretationnorescalingissueswheny>theyoftenprovidebetterfittothenormalityassumptionnarrowtherangeofoutcomes(mitigateoutliers)InterpretationofcoefficientsAssumewageismeasuredindollarsperhours,andeducinyearsIlevellevel:wage=ββeducu→βˆ=meansthathourlywageincreasesbycentsforeachadditionalyearofeducation(constantreturnstoeducation)Iloglevel:log(wage)=ββeducu→βˆ=meansthatforeachadditionalyearofeducationhourlywageincreasesbypercent(constantpercentageeffect,increasingreturnstoeducation)Iloglog:log(wage)=ββlog(educ)u→βˆ=meansthathourlywageincreasesbypercentforanincreaseofpercentineducation(constantelasticity)Ilinearinparametersonly!InterpretationoflogcoefficientsThesamplingvarianceunderhomoskedasticityIfweassumehomoskedasticity:var(u|x)=σ,thatthesamplingvariancetakestheformvar(βˆ)=σ∑(xi−x)However,σneedstobeestimatedusingtheresiduals:σˆ=n−n∑i=uˆiNote:√σˆiscalledthestandarderroroftheregression,√var(βˆ)isthestandarderrorofβˆetcExample:HomoskedasticityinwageequationsWhileaveragewageincreaseswitheducation,itsvariabilitydoesnot!Example:HeteroskedasticityinwageequationsIsn’tthatmorereasonableThemultivariateregressionmodelThemultivariateregressionmodelNowwecantakemorevariablesoutoftheerrorterm:wage=ββeducβexperuIstillinterestedintheβcoefficient,nowexperienceCANbecorrelatedwitheducationIE(u|educ,exper)=→exogenousexplanatoryvariables:theexpectedvalueofuisthesameforallcombinationsofeducandexperience(reasonable)IyoustillminimizeSSRtoestimateβˆ,βˆandβˆ(butnowitwrites(X′X)−(X′Y))Ipartialeffectinterpretation(ceterisparibus)Whyβˆchangesdependingonwhetherweincludeornotincludeexperbecauseeducandexperarecorrelatedinthesample!Onemoreassumption:NoperfectcollinearityIyoushouldn’tbeabletoexpressoneregressorasalinearfunctionoftheotherIHiddenperfectcollinearity:log(cons)=ββlog(inc$)βlog(incEuros)ulog(cons)=ββlog(inc)βlog(inc)ulog(cons)=ββincAβincBβTotIncuIbutsomecollinearityisnotaproblemReadingregressionresultsEffectsofdatascalingRegressinginfantweightatbirth(inouncesvspounds)oncigarettesmoking(inunitsvspacks)andincome($):Howtointerpretquadratictermswage=ββeducβeducuIestimationisthesame,butinterpretationdiffer∆wage∆educ≈ββeducIthemarginaleffectdependsonthelevelofeducIzeroconditionalmean:E(u|educ)=Regressionswithdummyvariables()Dummyorindicatorvariable:female={ifmaleiffemale()wage=βδfemaleβeducationuInterpretation:δisthedifferenceinhourlywagebetweenfemalesandmales,giventhesameamountofeducation(ameasureofdiscrimination)δ=E(wage|female,educ)−E(wage|male,educ)NBdon’tputbothgroups,thisisredundant(perfectcollinearity)!Regressionswithdummyvariables()Regressionswithcategoricalvariablesrace=ifWhiteifAsianifHispanicifBlack()Includeasmanydummiesasnumberofcategories:wage=ββeducδ(race=)δ(race=)δ(race=)uwhere(race=)isadummyequaltoiftherespondentisAsianandotherwise,andδmeasurestheeffectofbeingAsianratherthanWhiteonwagesRegressionswithinteractionterms()Example:effectofsquarefootageandnumberofbedroomsonrealestatepricesprice=ββsqrftβbdrmsβsqrft∗bdrmsu∆price∆bdrms=ββsqrftwhichmustbeevaluatedatinterestingvaluesofsqrftExample:testingwhetherdiscriminationagainstwomenhasdecreasedpost(vs)wage=ββeducβfemaleβ(−)βfemale∗(−)uIβisfemalediscriminationIβisoverallchangebetweenperiodsIβisthewomenspecificchangebetweenperiodsRegressionswithinteractionterms()Example:effectofbeingamarriedfemaleonwagewage=ββfemaleβmarriedβfemale∗marriedβeduuItheinterceptformarriedmenisββItheinterceptformarriedwomanisββββRegressionswithinteractionterms()ThevarianceofβˆjunderhomoskedasticityLetusbelievethatthevarianceofuisconstantconditionalonx(var(y|x,,xk)=σ)ICAREFUL:varianceofu,NOTexpectedvalue!Wecanprovethatthevarianceofβˆjisvar(βˆj)=σSSTj(−Rj)whereSSTj=∑ni=(xij−x¯j)andRjisthetheRsquaredfromregressingxjonallotherindependentvariablesNotethreeblocks:ItheerrorvarianceIthevariabilityinxj’s:alargersamplesizeorverydifferentvaluesforxjdecreasethevariance!ItheproportionofthetotalvariationinxjthatcanbeexplainedbytheotherindependentvariablesBestvariancewhenRjiszero(butneverhappens),worseisperfectcollinearityThestandarderrorofβˆjHowdowecomputethat,inpracticeItheerrorvariancecanbeestimatedfromtheresiduals:σˆ=∑uˆi(n−k−)=SSR(n−k−)︸︷︷︸dofIthestandarderrorofβˆj,ascomputedbyStata,isse(βˆj)=√var(βˆj)StatisticalpropertiesofthemultivariateOLSIfthefollowingassumptionsareverified:linearinparametersrandomsamplezeroconditionalmean:E(u|x,,xk)=⇒E(y|x,,xk)=ββxβkxknoperfectcollinearityhomoskedasticity:var(u|x)=σThen:Ivar(βˆj)hastheformaboveandE(σˆ)=σItheOLSaretheBLUEs:bestlinearunbiasedestimators(GaussMarkovtheorem)Readingregressionresults:miscedascontinuousReadingregressionresults:miscedascategoricalComparingmodels:adjustedRTheadjustedR,orR¯R¯=−SSR(n−k−)SST(n−)Whichimposesapenaltyforeachindependentvariableadded:IcanincreaseordecreaseifweaddanotherindependentvariableItoseewhetheravariablebelongstothemodelOmittedvariablebias:recapweassumethatinterpretationisceterisparibus,butthisisNOThowwehavecollectedthedata!Ifthetruemodeliswage=ββeducβabilityuButweestimatewage=ββeducvwhere:v=βabilityuweobtainanestimatedcoefficientβ˜suchthatE(β˜)=ββδ˜︸︷︷︸biaswherethebiasdependson:)theeffectofabilityonwage)thecorrelationbetweenabilityandeducation:˜ability=δ˜δ˜educOmittedvariablebias:recapTosummarize,ifweexcludearelevantxtheOLScoefficientarebiased,unless:ItheomittedvariablehasnoimpactonyorItheomittedvariableisNOTcorrelatedwiththeincludedregressor

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