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1、Chapter12Multicollinearity:WhatHappensifExplanatoryVariablesareCorrelatedTheClassicalLinearRegressionModel(CLRMassumptions)A7.1.X2andX3donothaveperfectlinearrelationshipsamongthemselves.A7.3.Homoscedasticity,thatis,thevarianceofu,isconstant:var(ui)=σ2A7.4.Noautocorrelationexistsbetweentheerrorte
2、rmsuianduj:cov(ui,uj)=0i≠jMulticollinearityHeteroscedasticityAutocorrelationOneoftheCLRMassumptionsis:Thereisnoperfectmulticollinearity—noexactlinearrelationshipsamongexplanatoryvariables,Xs,inamultipleregression.Inpractice,onerarelyencountersperfectmulticollinearity,butcasesofnearorveryhighmult
3、icollinearitywhereexplanatoryvariablesareapproximatelylinearlyrelatedfrequentlyariseinmanyapplications.Theobjectsofthischapter:●TheNatureofmulticollinearity;●Ismulticollinearityreallyaproblem?●Thetheoreticalandpracticalconsequencesofmulticollinearity;●Howtodetectmulticollinearity?●Theremedialmea
4、sureswhichcanbeusedtoeliminatemulticollinearity.Incasesofperfectlinearrelationshiporperfectmulticollinearityamongexplanatoryvariables,wecannotobtainuniqueestimatesofallparameters.Andsincewecannotobtaintheiruniqueestimates,wecannotdrawanystatisticalinferences(i.e.,hypothesistesting)aboutthemfroma
5、givensample.12.1:TheNatureofMulticollinearity:TheCaseofPerfectMulticollinearityForExample:TheDemandforWidgetsYx2x3x4(Quantity)(Price,$)(Incomeperweek,$)(Earningperweek,$)491298297.5452296294.9443294293.5394292292.8385290290.2376288289.7347286285.8338284284.6309282281.12910280278.8OriginalModel:Y
6、i=A1+A2X2i+A3X3i+μiTransformation:X3i=300－2X2iYi=A1+A2X2i+A3(300－2X2i)+μi=(A1+300A3)+(A2－2A3)X2i+μi=C1+C2X2i+μiEstimation:gettheOLSestimatorsC1=A1+300A3,C2=A2－2A3,SofromtheestimatorsofC1,C2,wecannotgettheestimatorsofA1,A2andA3Thatis:incasesofperfectmulticollinearity,estimationandhypothesistestin
7、gaboutindividualregressioncoefficientsinamultipleregressionarenotpossible.Wecanjustobtainestimatesofalinearcombinationoftheoriginalcoefficients,butnoteachofthemindividually.12.2TheCaseofNear,orImperfect,orHighMulticollineari
