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1、Chapter1RegressionModels1.1IntroductionRegressionmodelsformthecoreofthedisciplineofeconometrics.Althougheconometriciansroutinelyestimateawidevarietyofstatisticalmodels,usingmanydi?erenttypesofdata,thevastmajorityoftheseareeitherregressionmodelsorclosere
2、lativesofthem.Inthischapter,weintroducetheconceptofaregressionmodel,discussseveralvarietiesofthem,andintroducetheestima-tionmethodthatismostcommonlyusedwithregressionmodels,namely,leastsquares.Thisestimationmethodisderivedbyusingthemethodofmoments,which
3、isaverygeneralprincipleofestimationthathasmanyapplicationsineconometrics.Themostelementarytypeofregressionmodelisthesimplelinearregressionmodel,whichcanbeexpressedbythefollowingequation:yt=ˉ1+ˉ2Xt+ut:(1:01)Thesubscripttisusedtoindextheobservationsofasam
4、ple.Thetotalnum-berofobservations,alsocalledthesamplesize,willbedenotedbyn.Thus,forasampleofsizen,thesubscripttrunsfrom1ton.Eachobservationcomprisesanobservationonadependentvariable,writtenasytforobserva-tiont,andanobservationonasingleexplanatoryvariabl
5、e,orindependentvariable,writtenasXt.Therelation(1.01)linkstheobservationsonthedependentandtheexplana-toryvariablesforeachobservationintermsoftwounknownparameters,ˉ1andˉ2,andanunobservederrorterm,ut.Thus,ofthe?vequantitiesthatappearin(1.01),two,ytandXt,a
6、reobserved,andthree,ˉ1,ˉ2,andut,arenot.Threeofthem,yt,Xt,andut,arespeci?ctoobservationt,whiletheothertwo,theparameters,arecommontoallnobservations.Hereisasimpleexampleofhowaregressionmodellike(1.01)couldariseineconomics.Supposethattheindextisatimeindex,
7、asthenotationsuggests.Eachvalueoftcouldrepresentayear,forinstance.Thenytcouldbehouse-holdconsumptionasmeasuredinyeart,andXtcouldbemeasureddisposableincomeofhouseholdsinthesameyear.Inthatcase,(1.01)wouldrepresentwhatinelementarymacroeconomicsiscalledacon
8、sumptionfunction.Copyright°c1999,RussellDavidsonandJamesG.MacKinnon34RegressionModelsIfforthemomentweignorethepresenceoftheerrorterms,ˉ2isthemarginalpropensitytoconsumeoutofdisposableincome,andˉ1iswhatissometimescalledautonomousc
