Basic terminology and basic elements of time series

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1、ECON5103/ECON9103ADVANCEDECONOMETRICS{PANELDATALecturenoteno.4ErikBi¿rn,DepartmentofEconomicsVersionofDecember28,2009BasicterminologyandbasicelementsintimeseriesanalysisApreparationfordynamicmodelingwithpaneldataThislecturenoteisrelatedtothelecturesondynamicpaneldatamodels.Thoseofyou

2、whohaveagoodgraspofbasicelementsofeconometrictime-seriesanalysis(notnecessarilyrelatedtopaneldata)willnotneedit.Thosewho(thinkthey)donotpossessthisknowledge,may¯nditusefulasapreparationtothediscussionofdynamicmodelsforpaneldatainthecourse.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=1½¾2;t=sE(")=0;E("")="tts0;

5、t6=sAlternatively(andsomewhatstronger),2"t»IID(0;¾");whereIIDsigni¯esidentically,independentlydistributed.Stillstronger,2"t»IIN(0;¾");whereIINsigni¯esnormally,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=½ut¡1+"t;where"t»w:n:Autoregressiveprocessoforderp:futg»AR(p):ut=½1ut¡1+½2ut¡2+¢¢¢+½put¡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=½ut+"t+µ"t¡1;where"t»w:n:ARMA(p;q)process:futg»ARMA(p;q):ut=½1ut¡1+½2ut¡2+¢¢¢+½put¡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