ln2 Review of Asymptotic Theory.pdf

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1、ApEc8212EconometricsAnalysis---Lecture#2ReviewofAsymptoticTheory(Wooldridge,Chap.3)Thislecturereviewsbasicresultsinasymptotictheory,whichmostofyouhaveseenbeforeinsomeform(Chapter4andAppendixDofGreene,2008).Thismaterialisnotveryexciting,butitisveryuseful.Wooldridgedoesnotgiveproofs,b

2、utyoucanseeGreene,andmoregenerallyHalbertWhite(2001,AsymptoticTheoryforEconometricians:RevisedEdition)forproofsandamorerigoroustreatment.I.ConvergenceofDeterministicSequencesLet’sstartwithsomedefinitions:Convergence:Asequenceof(nonrandom)numbers{aN:N=1,2,…}convergestoaif,forallε>0th

3、ereexistsanNεsuchthatifN>Nεthen

4、aN–a

5、<ε.ThisiswrittenasaN→aasN→∞,oraslimaN=a.NBoundedness:Asequenceof(nonrandom)numbers{aN:N=1,2,…}isboundedifandonlyifthereissomeb<∞suchthat

6、aN

7、

8、λnumbersisatmostoforderN,denotedbyO(N),ifλaN/Nisbounded.Asequence{aN}isoforderλλλsmallerthanN,denotedbyo(N),iflimaN/N=0.NThesetwodefinitionsformalizewhatismeantby“thesameorderofmagnitude”anda“smallerorderλofmagnitude”.If{aN}isO(N),thenaNcannotbeλλinfinitelylargerthanNasN→∞.If{aN}i

9、so(N),λthenaNisinfinitelysmallerthanNasN→∞.Bothtypesoforderstatisticscanbeusedwithλ=0,inλλwhichcaseN=1.InthiscaseO(N)(“atmostoforder”)isdenotedasO(1),whichsimplymeansthatλ{aN}isbounded.Similarly,o(N)(“ofordersmallerthan”)isdenotedaso(1)whenλ=0,whichmeansthataNconvergestozero,i.e.lim

10、aN=0↔{aN}iso(1).NλλNotethatforanysequence{aN},o(N)impliesO(N):λλiflimaN/N=0thenaN/Nisbounded.Settingλ=0Nshowsthatif{aN}convergesto0then{aN}isbounded.Finally,forsequencesofvectorsormatrices,theλsequenceisO(N)onlyifalltheelementsinitareλλO(N),andthesameholdsforo(N).2II.Convergence

11、inProbabilityandBoundedinProbability(op(1)andOp(1))Sofarwehavediscussedsequencesofnumbers,butineconometricsweworkwithvariables,soweneedtodefineboundednessandconvergenceofvariables.Thissectionpresentsbasicdefinitionsandresults.Convergenceinprobability:Asequenceofvariables{xN:N=1,2,…}

12、convergesinprobabil