the box jenkins program a case study

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1、ChapterTheBox–JenkinsProgram:ACaseStudy7ThischapterdealswiththepracticalapplicationoftheBox{JenkinsProgramtotheDonauwoerthData,consistingof7300dischargemea-surementsfromtheDonauriveratDonauwoerth,speciedincubiccentimeterpersecondandtakenonbehalfoftheB

2、avarianStateOf-ceForEnvironmentbetweenJanuary1st,1985andDecember31st,2004.Forthepurposeofstudying,thedatahavebeenkindlymadeavailabletotheUniversityofW•urzburg.AsintroducedinSection2.3,theBox{JenkinsmethodologycanbeappliedtospecifyadequateARMA(p;q)-mod

3、elYt=a1Yt1+


+apYtp+"t+b1"t1+


+bq"tq;t2ZfortheDonauwoerthdatainordertoforecastfuturevaluesofthetimeseries.Inshort,theoriginaltimeserieswillbeadjustedtorepresentapossiblerealizationofsuchamodel.BasedontheidenticationmethodsMINIC,SCANandESACF,app

4、ropriatepairsoforders(p;q)arechoseninSection7.5andthecorrespondingmodelcoecientsa1;:::;apandb1;:::;bqaredetermined.Finally,itisdemonstratedbyDiagnosticCheckinginSection7.6thattheresultingmodelisadequateandforecastsbasedonthismodelareexecutedintheconcl

5、udingSection7.7.Yet,beforestartingtheprograminSection7.4,sometheoreticalpreparationshavetobecarriedout.IntherstSection7.1weintro-ducethegeneraldenitionofthepartialautocorrelationleadingtotheLevinson{Durbin-Algorithm,whichwillbeneededforDiagnosticChec

6、king.InordertoverifywhetherpureAR(p)-orMA(q)-modelsmightbeappropriatecandidatesforexplainingthesample,wede-riveinSection7.2and7.3asymptoticnormalbehaviorsofsuitableestimatorsofthepartialandgeneralautocorrelations.224TheBox–JenkinsProgram:ACaseStudy7.1P

7、artialCorrelationandLevinson–DurbinRecursionIngeneral,thecorrelationbetweentworandomvariablesisoftenduetothefactthatbothvariablesarecorrelatedwithothervariables.Therefore,thelinearin uenceofthesevariablesisremovedtoreceivethepartialcorrelation.PartialC

8、orrelationThepartialcorrelationoftwosquareintegrable,realvaluedrandomvariablesXandY,holdingtherandomvariablesZ1;:::;Zm;m2N,xed,isdenedasCorr(XX^Z1;:::;Zm;YY^Z1;:::;Zm)Cov(XX^Z1;:::;Zm;YY^Z1;:::;Zm)=;(Var(XX^Z;:::;Z))1=2(Var(YY^Z