[Stat] Foundations of Econometrics.pdf

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1、Chapter1RegressionModels1.1IntroductionRegressionmodelsformthecoreofthedisciplineofeconometrics.Althougheconometriciansroutinelyestimateawidevarietyofstatisticalmodels,usingmanydifferenttypesofdata,thevastmajorityoftheseareeitherregressionmodelsorcloserelativesofthem.Inthischapter,w

2、eintroducetheconceptofaregressionmodel,discussseveralvarietiesofthem,andintroducetheestima-tionmethodthatismostcommonlyusedwithregressionmodels,namely,leastsquares.Thisestimationmethodisderivedbyusingthemethodofmoments,whichisaverygeneralprincipleofestimationthathasmanyapplications

3、ineconometrics.Themostelementarytypeofregressionmodelisthesimplelinearregressionmodel,whichcanbeexpressedbythefollowingequation:yt=¯1+¯2Xt+ut:(1:01)Thesubscripttisusedtoindextheobservationsofasample.Thetotalnum-berofobservations,alsocalledthesamplesize,willbedenotedbyn.Thus,forasam

4、pleofsizen,thesubscripttrunsfrom1ton.Eachobservationcomprisesanobservationonadependentvariable,writtenasytforobserva-tiont,andanobservationonasingleexplanatoryvariable,orindependentvariable,writtenasXt.Therelation(1.01)linkstheobservationsonthedependentandtheexplana-toryvariablesfo

5、reachobservationintermsoftwounknownparameters,¯1and¯2,andanunobservederrorterm,ut.Thus,ofthefivequantitiesthatappearin(1.01),two,ytandXt,areobserved,andthree,¯1,¯2,andut,arenot.Threeofthem,yt,Xt,andut,arespecifictoobservationt,whiletheothertwo,theparameters,arecommontoallnobservation

6、s.Hereisasimpleexampleofhowaregressionmodellike(1.01)couldariseineconomics.Supposethattheindextisatimeindex,asthenotationsuggests.Eachvalueoftcouldrepresentayear,forinstance.Thenytcouldbehouse-holdconsumptionasmeasuredinyeart,andXtcouldbemeasureddisposableincomeofhouseholdsinthesam

7、eyear.Inthatcase,(1.01)wouldrepresentwhatinelementarymacroeconomicsiscalledaconsumptionfunction.Copyright°c1999,RussellDavidsonandJamesG.MacKinnon34RegressionModelsIfforthemomentweignorethepresenceoftheerrorterms,¯2isthemarginalpropensitytoconsumeoutofdisposableincome,and¯1iswhatis

8、sometimescalledautonomousc