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1、ECON5103/ECON9103ADVANCEDECONOMETRICS{PANELDATALecturenoteno.4ErikBi?rn,DepartmentofEconomicsVersionofDecember28,2009BasicterminologyandbasicelementsintimeseriesanalysisApreparationfordynamicmodelingwithpaneldataThislecturenoteisrelatedtothelecturesondynamicpaneldatamodels.Thoseofyou
2、whohaveagoodgraspofbasicelementsofeconometrictime-seriesanalysis(notnecessarilyrelatedtopaneldata)willnotneedit.Thosewho(thinkthey)donotpossessthisknowledge,mayˉnditusefulasapreparationtothediscussionofdynamicmodelsforpaneldatainthecourse.Rememberthatpaneldatacoverthetimedimension,he
3、nceelementsiftime-seriesanalysisareimportantingredi-ents.Autoregressivemodelsforpaneldatahavereceivedsubstantialinterestintheliteratureformorethanthirtyyears.A.BASICTERMINOLOGYConsideratimeseriesofavariableX:fXg=f￠￠￠X?1;X0;X1;￠￠￠;XT;￠￠￠g;whereXtisitsvalueinperiodt.IftheXt'sarestochas
4、ticvariables,wecallfXgastochastic(random)process.A(modelofa)univariatestochasticprocessinvolvesonlyonevariable.A(modelofa)multivariatestochasticprocessinvolvesmorethanonevariable.B.BASIC(MODELSOF)UNIVARIATESTOCHASTICPROCESSES1Whitenoiseprocess:f"gT?w.n.:tt=1??2;t=sE(")=0;E("")="tts0;
5、t6=sAlternatively(andsomewhatstronger),2"t?IID(0;?");whereIIDsigniˉesidentically,independentlydistributed.Stillstronger,2"t?IIN(0;?");whereIINsigniˉesnormally,independentlydistributed.Randomwalkprocess:futg?r.w.:ut=ut?1+"t;where"t(=￠ut)?w:n:Hence:Xtfutg?r:w:=)f￠utg?w:n:=)ut=u0+"s:s=1
6、Autoregressiveprocessoforder1:futg?AR(1):ut=?ut?1+"t;where"t?w:n:Autoregressiveprocessoforderp:futg?AR(p):ut=?1ut?1+?2ut?2+￠￠￠+?put?p+"t;where"t?w:n:Movingaverageprocessoforder1:futg?MA(1):ut="t+μ"t?1;where"t?w:n:Movingaverageprocessoforderq:futg?MA(q):ut="t+μ1"t?1+￠￠￠+μq"t?q;where"t
7、?w:n:ImportantabbreviationsAR=Auto-RegressiveMA=MovingAverage2ARMA=Auto-Regressive/MovingAverageARMA(1;1)process:futg?ARMA(1;1):ut=?ut+"t+μ"t?1;where"t?w:n:ARMA(p;q)process:futg?ARMA(p;q):ut=?1ut?1+?2ut?2+￠￠￠+?put?p+"t+μ1"t?1+￠￠￠+μq"t?q;where"t?w:n:Specialcases:1.AR(p)=ARMA(p;0),2.MA
8、(q)=ARMA(0;q
