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1、CHAPTER5LINEARREGRESSIONMODELSWITHDEPENDENTOBSERVATIONSCHAWPTER5LINEARREGRESSIONMODELSITHDEPENDENTOBSERVATIONSKeywords:Ergodicity,Martingaledierencesequence,Randomwalk,Serialcorrelation,Stationarity,Unitroot,Whitenoise.Remark:Theasymptotictheorydevelopedaboveisapplicableforcross-sectionaldata(be
2、-causeofthei.i.d.randomsampleassumption).Whathappensifwehaveatimeseriesdata?MotivationConsiderYt=0+1Yt1+;t;;tisi.i.d.N(0,2):Here,Xt=(1;Yt1).WehaveE(;tjXt)=0butwedonothaveE(;tjX)=E(;tjX1;X2;:::;Xn)=0:Remark:Thei.i.d.assumptionrulesouttimeseriesdata.Mosteconomicandnancialdataaretimeseriesobservati
3、ons.Question:Underwhatconditionswilltheasymptotictheorydevelopedinthepreviouschaptercarryovertolinearregressionmodelswithdependentobservations?BasicConceptsinTimeSeriesQuestion:Whatisatimeseriesprocess?Denition[StochasticTimeSeriesProcess]:AstochastictimeseriesfZtgisasequenceofrandomvariablesorr
4、andomvectorsindexedbytimetandgovernedbysomeprobabilitylaw(;F;P);whereisthesamplespace,Fisa-eld,andPisaprobabilitymeasure,withP:F![0;1]:Remarks:1(i)Moreprecisely,wecanwriteZt=Z(t;!);where!2isabasicoutcomeinsamplespace.Foreach!;wecanobtainasamplepathZ(t;!)ofZtasadeterministicfunctionoftimet:Dieren
5、t!swillgivedierentsamplepaths.(ii)ThedynamicsoffZtgiscompletelydeterminedbythetransitionprobabilityofZt;thatis,theconditionalprobabilityofZtgivenitspasthistory.Randomsample:Considerasubset(orasegment)ofadiscretetimeseriesprocessfZtgfort=1;;n:Thisiscalledatimeseriesrandomsampleofsizen;denotedasZn
6、=fZ1;;Zng0:Anyrealizationofthisrandomsampleisadataset,denotedaszn=fz1;;zng0:Question:WhycanthedynamicsoffZtgbecompletelycapturedbyitstransitionprobability?ConsiderarandomsampleZn:Itiswell-knownfrombasicstatisticscoursesthatthejointprobabilityoftherandomsampleZn;fZn(zn)=fZ1;Z2;:::;Zn(z1;z2;:::;zn
7、);zn2Rn;completelycapturesallthesampleinformationcontainedinZn.WithfZn(zn);wecanobtainthesamplingdistributionofanystatistic(e.g.,samplemean,samplevariance,condenceinterval)thatisbasedonZn:Now,putIt=fZt;Zt1;;Z1g,theinformatio
