《计量经济学》ch-03-wooldridg

Chapter3,MultipleRegressionAnalysis:Estimation,Wooldridge:IntroductoryEconometrics:AModernApproach,5eInstructedbyprofessorYuan,Huiping,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2MechanicsandInterpretationofOLS,3.3TheExpectedValueoftheOLSEstimators,3.4TheVarianceoftheOLSEstimators,3.5EfficiencyofOLS:TheGauss-MarkovTheorem,3.1MotivationforMultipleRegression,3.6SomeCommentsontheLanguageofMultipleRegressionAnalysis,Assignments:Promblems7,9,10,11,13,ComputerExercisesC1,C3,C5,C6,C8,TheEnd,Definitionofthemultiplelinearregressionmodel,Dependentvariable,explainedvariable,responsevariable,Independentvariables,explanatoryvariables,regressors,Errorterm,disturbance,unobservables,Intercept,Slopeparameters,Explainsvariableintermsofvariables“,3.1MotivationforMultipleRegression(1/5),CHAPTER3MultipleRegressionAnalysis:Estimation,Chapter,End,MotivationformultipleregressionIncorporatemoreexplanatoryfactorsintothemodelExplicitlyholdfixedotherfactorsthatotherwisewouldbeinAllowformoreflexiblefunctionalformsExample:Wageequation,Hourlywage,Yearsofeducation,Labormarketexperience,Allotherfactors,Nowmeasureseffectofeducationexplicitlyholdingexperiencefixed,CHAPTER3MultipleRegressionAnalysis:Estimation,3.1MotivationforMultipleRegression(2/5),Chapter,End,Example:AveragetestscoresandperstudentspendingPerstudentspendingislikelytobecorrelatedwithaveragefamilyincomeatagivenhighschoolbecauseofschoolfinancingOmittingaveragefamilyincomeinregressionwouldleadtobiasedestimateoftheeffectofspendingonaveragetestscoresInasimpleregressionmodel,effectofperstudentspendingwouldpartlyincludetheeffectoffamilyincomeontestscores,Averagestandardizedtestscoreofschool,Otherfactors,Perstudentspendingatthisschool,Averagefamilyincomeofstudentsatthisschool,CHAPTER3MultipleRegressionAnalysis:Estimation,3.1MotivationforMultipleRegression(3/5),Chapter,End,Example:FamilyincomeandfamilyconsumptionModelhastwoexplanatoryvariables:inomeandincomesquaredConsumptionisexplainedasaquadraticfunctionofincomeOnehastobeverycarefulwheninterpretingthecoefficients:,Familyconsumption,Otherfactors,Familyincome,Familyincomesquared,Byhowmuchdoesconsumptionincreaseifincomeisincreasedbyoneunit?,Dependsonhowmuchincomeisalreadythere,CHAPTER3MultipleRegressionAnalysis:Estimation,3.1MotivationforMultipleRegression(4/5),Chapter,End,Example:CEOsalary,salesandCEOtenureModelassumesaconstantelasticityrelationshipbetweenCEOsalaryandthesalesofhisorherfirmModelassumesaquadraticrelationshipbetweenCEOsalaryandhisorhertenurewiththefirmMeaningoflinear“regressionThemodelhastobelinearintheparameters(notinthevariables),LogofCEOsalary,Logsales,QuadraticfunctionofCEOtenurewithfirm,CHAPTER3MultipleRegressionAnalysis:Estimation,3.1MotivationforMultipleRegression(5/5),Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2MechanicsandInterpretationofOLS,3.2.2InterpretingtheOLSRegressionEquation,3.2.3OLSFittedValuesandResiduals,3.2.1ObtainingtheOLSEstimates,3.2.4A“PartiallingOut”InterpretationofMultipleRegression,3.2.5ComparisonofSimpleandMultipleRegressionEstimates,3.2.6GoodnessofFit,3.2.7RegressionthroughtheOrigin,Chapter,End,OLSEstimationofthemultipleregressionmodelRandomsampleRegressionresidualsMinimizesumofsquaredresiduals,Minimizationwillbecarriedoutbycomputer,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.1ObtainingtheOLSEstimates(1/2),Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.1ObtainingtheOLSEstimates(2/2),Section,Chapter,End,InterpretationofthemultipleregressionmodelThemultiplelinearregressionmodelmanagestoholdthevaluesofotherexplanatoryvariablesfixedevenif,inreality,theyarecorrelatedwiththeexplanatoryvariableunderconsiderationCeterisparibus“-interpretationIthasstilltobeassumedthatunobservedfactorsdonotchangeiftheexplanatoryvariablesarechanged,Byhowmuchdoesthedependentvariablechangeifthej-thindependentvariableisincreasedbyoneunit,holdingallotherindependentvariablesandtheerrortermconstant,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.2InterpretingtheOLSRegressionEquation(1/3),Section,Chapter,End,Example3.1:DeterminantsofcollegeGPAInterpretationHoldingACTfixed,anotherpointonhighschoolgradepointaverageisassociatedwithanother.453pointscollegegradepointaverageOr:IfwecomparetwostudentswiththesameACT,butthehsGPAofstudentAisonepointhigher,wepredictstudentAtohaveacolGPAthatis.453higherthanthatofstudentBHoldinghighschoolgradepointaveragefixed,another10pointsonACTareassociatedwithlessthanonepointoncollegeGPA,Gradepointaverageatcollege,Highschoolgradepointaverage,Achievementtestscore,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.2InterpretingtheOLSRegressionEquation(2/3),Section,Chapter,End,Example3.2:HourlyWageEquationwage1.wf1lslog(wage)ceducexpertenure,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.2InterpretingtheOLSRegressionEquation(3/3),Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.3OLSFittedValuesandResiduals,PropertiesofOLSonanysampleofdataFittedvaluesandresidualsAlgebraicpropertiesofOLSregression,Fittedorpredictedvalues,Residuals,Deviationsfromregressionlinesumuptozero,Correlationsbetweendeviationsandregressorsarezero,Sampleaveragesofyandoftheregressorslieonregressionline,Section,Chapter,End,Onecanshowthattheestimatedcoefficientofanexplanatoryvariableinamultipleregressioncanbeobtainedintwosteps:1)Regresstheexplanatoryvariableonallotherexplanatoryvariables2)Regressontheresidualsfromthisregressionwage1.wf1lslog(wage)ceducexpertenurelseduccexpertenureseriesr1=residlslog(wage)cr1,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.4A“PartiallingOut”InterpretationofMultipleRegression(1/3),Section,Chapter,End,Whydoesthisprocedurework?Theresidualsfromthefirstregressionisthepartoftheexplanatoryvariablethatisuncorrelatedwiththeotherexplanatoryvariableslseduccexpertenureseriesr1=residTheslopecoefficientofthesecondregressionthereforerepresentstheisolatedeffectoftheexplanatoryvariableonthedep.Variablelslog(wage)cr1,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.4A“PartiallingOut”InterpretationofMultipleRegression(2/3),Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.4A“PartiallingOut”InterpretationofMultipleRegression(3/3),Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.5ComparisonofSimpleandMultipleRegressionEstimates,Example3.3Participationin401(k)PensionPlansmrate=theamountthefirmcontributestoaworkersfundforeachdollartheworker;prate=thepercentageofeligibleworkershavinga401(k)account.,Section,Chapter,End,DecompositionoftotalvariationR-squaredAlternativeexpressionforR-squared,NoticethatR-squaredcanonlyincreaseifanotherexplanatoryvariableisaddedtotheregression.Thisalgebraicfactfollowsbecause,bydefinition,thesumofsquaredresidualsneverincreaseswhenadditionalregressorsareaddedtothemodel.,R-squaredisequaltothesquaredcorrelationcoefficientbetweentheactualandthepredictedvalueofthedependentvariable,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.6GoodnessofFit(1/3),Section,Chapter,End,Example:ExplainingarrestrecordsInterpretation:Proportionpriorarrests+0.5!-.075=-7.5arrestsper100menMonthsinprison+12!-.034(12)=-0.408arrestsforgivenmanQuartersemployed+1!-.104=-10.4arrestsper100men,Numberoftimesarrested1986,Proportionpriorarreststhatledtoconviction,Monthsinprison1986,Quartersemployed1986,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.6GoodnessofFit(2/3),Section,Chapter,End,Example:Explainingarrestrecords(cont.)Anadditionalexplanatoryvariableisadded:Interpretation:Averagepriorsentenceincreasesnumberofarrests(?)LimitedadditionalexplanatorypowerasR-squaredincreasesbylittleGeneralremarkonR-squaredEvenifR-squaredissmall(asinthegivenexample),regressionmaystillprovidegoodestimatesofceterisparibuseffects,Averagesentenceinpriorconvictions,R-squaredincreasesonlyslightly,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.6GoodnessofFit(3/3),Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.2.7RegressionthroughtheOrigin,Thedecompositionofthetotalvariationinyusuallydoesnothold.R2mightbenegative.someeconomistsproposetocalculateR2asthesquaredcorrelationcoefficientbetweentheactualandfittedvaluesofy.Thecostofestimatinganinterceptwhenitistrulyzero.,Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3TheExpectedValueoftheOLSEstimators,3.3.2IncludingIrrelevantVariables,3.3.3OmittedVariableBias,3.3.1AssumptionsandUnbiasednessofOLS,Chapter,End,StandardassumptionsforthemultipleregressionmodelAssumptionMLR.1(Linearinparameters)AssumptionMLR.2(Randomsampling),Inthepopulation,therelation-shipbetweenyandtheexpla-natoryvariablesislinear,soisbetweenyanddisturbance.,Thedataisarandomsampledrawnfromthepopulation,Eachdatapointthereforefollowsthepopulationequation,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.1AssumptionsandUnbiasednessofOLS(1/6),Section,Chapter,End,Standardassumptionsforthemultipleregressionmodel(cont.)AssumptionMLR.3(Noperfectcollinearity)RemarksonMLR.3Theassumptiononlyrulesoutperfectcollinearity/correlationbet-weenexplanatoryvariables;imperfectcorrelationisallowedIfanexplanatoryvariableisaperfectlinearcombinationofotherexplanatoryvariablesitissuperfluousandmaybeeliminatedConstantvariablesarealsoruledout(collinearwithintercept)nk+1,Inthesample(andthereforeinthepopulation),noneoftheindependentvariablesisconstantandtherearenoexactrelationshipsamongtheindependentvariables“,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.1AssumptionsandUnbiasednessofOLS(2/6),Section,Chapter,End,Exampleforperfectcollinearity:smallsampleExampleforperfectcollinearity:relationshipsbetweenregressors,Inasmallsample,avgincmayaccidentallybeanexactmultipleofexpend;itwillnotbepossibletodisentangletheirseparateeffectsbecausethereisexactcovariation,EithershareAorshareBwillhavetobedroppedfromtheregressionbecausethereisanexactlinearrelationshipbetweenthem:shareA+shareB=1,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.1AssumptionsandUnbiasednessofOLS(3/6),Section,Chapter,End,Standardassumptionsforthemultipleregressionmodel(cont.)AssumptionMLR.4(Zeroconditionalmean)Inamultipleregressionmodel,thezeroconditionalmeanassumptionismuchmorelikelytoholdbecausefewerthingsendupintheerrorExample:Averagetestscores,Thevalueoftheexplanatoryvariablesmustcontainnoinformationaboutthemeanoftheunobservedfactors,Ifavgincwasnotincludedintheregression,itwouldendupintheerrorterm;itwouldthenbehardtodefendthatexpendisuncorrelatedwiththeerror,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.1AssumptionsandUnbiasednessofOLS(4/6),Section,Chapter,End,Discussionofthezeromeanconditionalassumptioncov(u,xj)=0,j=1,kFunctionalformmisspecification,omittedvariables,measurementerror,andsimultaneousequationscancausecov(u,xj)0.Explanatoryvariablesthatarecorrelatedwiththeerrortermarecalledendogenous;endogeneityisaviolationofassumptionMLR.4Explanatoryvariablesthatareuncorrelatedwiththeerrortermarecalledexogenous;MLR.4holdsifallexplanat.var.areexogenousExogeneityisthekeyassumptionforacausalinterpretationoftheregression,andforunbiasednessoftheOLSestimators,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.1AssumptionsandUnbiasednessofOLS(5/6),Section,Chapter,End,Theorem3.1(UnbiasednessofOLS)Unbiasednessisanaveragepropertyinrepeatedsamples;inagivensample,theestimatesmaystillbefarawayfromthetruevaluesPROOF:,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.1AssumptionsandUnbiasednessofOLS(6/6),Section,Chapter,End,Includingirrelevantvariablesinaregressionmodel,=0inthepopulation,Noproblembecause.,However,includingirrevelantvariablesmayincreasesamplingvariance.,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.2IncludingIrrelevantVariables,Section,Chapter,End,Omittingrelevantvariables:thesimplecaseOmittedvariablebiasConclusion:Allestimatedcoefficientswillbebiased,Ifx1andx2arecorrelated,assumealinearregressionrelationshipbetweenthem,Ifyisonlyregressedonx1thiswillbetheestimatedintercept,Ifyisonlyregressedonx1,thiswillbetheestimatedslopeonx1,errorterm,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.3OmittedVariableBias(1/6),Truemodel(containsx1andx2),Estimatedmodel(x2isomitted),Section,Chapter,End,Example:OmittingabilityinawageequationWhenistherenoomittedvariablebias?Iftheomittedvariableisirrelevantoruncorrelated,Willbothbepositive,Thereturntoeducationwillbeoverestimatedbecause.Itwilllookasifpeoplewithmanyyearsofeducationearnveryhighwages,butthisispartlyduetothefactthatpeoplewithmoreeducationarealsomoreableonaverage.,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.3OmittedVariableBias(2/6),Section,Chapter,End,Omittedvariablebias:moregeneralcasesNogeneralstatementspossibleaboutdirectionofbiasAnalysisasinsimplecaseifoneregressoruncorrelatedwithothersExample:Omittingabilityinawageequation,Truemodel(containsx1,x2andx3),Estimatedmodel(x3isomitted),Ifexperisapproximatelyuncorrelatedwitheducandabil,thenthedirectionoftheomittedvariablebiascanbeasanalyzedinthesimpletwovariablecase.,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.3OmittedVariableBias(3/6),Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.3OmittedVariableBias(4/6),Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.3OmittedVariableBias(5/6),Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.3.3OmittedVariableBias(6/6),Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,3.4TheVarianceoftheOLSEstimators,3.4.1TheComponentsoftheOLSVariances:Multicollinearity,3.4.2VariancesinMisspecifiedModels,Theorem3.2SamplingVariancesoftheOLSSlopeEstimators,3.4.3Estimatings2:StandardErrorsoftheOLSEstimators,AssumptionMLR.5(Homoscedasticity),Chapter,End,Standardassumptionsforthemultipleregressionmodel(cont.)AssumptionMLR.5(Homoscedasticity)AssumptionsMLR.1throughMLR.5arecollectivelyknownastheGauss-Markovassumptions(forcross-sectionalregression).Example:WageequationShorthandnotation,Thevalueoftheexplanatoryvariablesmustcontainnoinformationaboutthevarianceoftheunobservedfactors,Thisassumptionmayalsobehardtojustifyinmanycases,with,Allexplanatoryvariablesarecollectedinarandomvector,CHAPTER3MultipleRegressionAnalysis:Estimation,AssumptionMLR.5(Homoscedasticity),Section,Chapter,End,Theorem3.2(SamplingvariancesofOLSslopeestimators),UnderassumptionsMLR.1MLR.5:,Varianceoftheerrorterm,Totalsamplevariationinexplanatoryvariablexj:,R-squaredfromaregressionofexplanatoryvariablexjonallotherindependentvariables(includingaconstant),CHAPTER3MultipleRegressionAnalysis:Estimation,Theorem3.2SamplingVariancesoftheOLSSlopeEstimators(1/2),Section,Chapter,End,CHAPTER3MultipleRegressionAnalysis:Estimation,PROOF:Considerj=1.,Theorem3.2SamplingVariancesoftheOLSSlopeEstimators(2/2),Section,Chapter,End,ComponentsofOLSVariances,1)Theerrorvariance,Ahigherrorvarianceincreasesthesamplingvariancebecausethereismorenoise“intheequationAlargeerrorvariancenecessarilymakesestimatesimpreciseTheerrorvariancedoesnotdecreasewithsamplesize2)Thetotalsamplevariationintheexplanatoryvariable,MoresamplevariationleadstomorepreciseestimatesTotalsamplevariationautomaticallyincreaseswiththesamplesizeIncreasingthesamplesizeisthusawaytogetmorepreciseestimates,CHAPTER3MultipleRegressionAnalysis:Estimation,3.4.1TheComponentsoftheOLSVariances:Multicollinearity(1/4),Section,Chapter,End,3)Linearrelationshipsamongtheindependentvariables,SamplingvarianceofwillbethehigherthebetterexplanatoryvariablecanbelinearlyexplainedbyotherindependentvariablesTheproblemofalmostlinearlydependentexplanatoryvariablesiscalledmulticollinearity(i.e.forsome),Regressonallotherindependentvariables(includingaconstant),TheR-squaredofthisregressionwillbethehigherthebetterxjcanbelinearlyexplainedbytheotherindependentvariables,CHAPTER3MultipleRegressionAnalysis:Estimation,3.4.1TheComponentsoftheOLSVariances:Multicollinearity(2/4),Section,Chapter,End,Thedifferentexpenditurecategorieswillbestronglycorrelatedbecauseifaschoolhasalotofresourcesitwillspendalotoneverything.Itwillbehardtoestimatethedifferentialeffectsofdifferentexpenditurecategoriesbecauseallexpendituresareeitherhighorlow.Forpreciseestimatesofthedifferentialeffects,onewouldneedinformationaboutsituationswhereexpenditurecategorieschangedifferentially.Asaconsequence,samplingvarianceoftheestimatedeffectswillbelarge.,Anexampleformulticollinearity,Averagestandardizedtestscoreofschool,Expendituresforteachers,Expendituresforin-structionalmaterials,Otherex-penditures,CHAPTER3MultipleRegressionAnalysis:Estimation,3.4.1TheComponentsoftheOLSVariances:Multicollinearity(3/4),Section,Chapter,End,DiscussionofthemulticollinearityproblemIntheaboveexample,itwouldprobablybebettertolumpallexpen-diturecategoriestogetherbecauseeffectscannotbedisentangledInothercases,droppingsomeindependentvariablesmayreducemulticollinearity(butthismayleadtoomittedvariablebias)Onlythesamplingvarianceofthevariablesinvolvedinmulticollinearitywillbeinflated;theestimatesofothereffectsmaybeverypreciseNotethatmulticollinearityisnotaviolationofMLR.3inthestrictsenseMulticollinearitymaybedetectedthroughvarianceinflationfactors“,Asan(arbitrary)ruleofthumb,thevarianceinflationfactorshouldnotbelargerthan10,CHAPTER3MultipleRegressionAnalysis:Estimation,3.4.1TheComponentsoftheOLSVariances:Multicollinearity(4/4),Section,Chapter,End,VariancesinmisspecifiedmodelsThechoiceofwhethertoincludeaparticularvariableinaregressioncanbemadebyanalyzingthetradeoffbetweenbiasandvarianceItmightbethecasethatthelikelyomittedvariablebiasinthemisspecifiedmodel2isovercompensatedbyasmallervariance,Truepopulationmodel,Estimatedmodel1,Estimatedmodel2,CHAPTER3MultipleRegressionAnalysis:Estimation,3.4.2VariancesinMisspecifiedModels(1/2),Section,Chapter,End,Variancesinmisspecifiedmodels(cont.)Case1:Case2:,Conditionalonx1andx2,thevarianceinmodel2isalwayssmallerthanthatinmodel1,Conclusion:Donotincludeirrelevantregressors,Tradeoffbiasandvariance;Caution:biaswillnotvanisheveninlargesamples,CHAPTER3MultipleRegressionAnalysis:Estimation,3.4.2VariancesinMisspecifiedModels(2/2),Section,Chapter,End,EstimatingtheerrorvarianceTheorem3.3(Unbiasedestimatoroftheerrorvariance)UndertheGauss-MarkovassumptionsMLR.1throughMLR.5,CHAPTER3MultipleRegressionAnalysis:Estimati