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Determine

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Determine

    Determine

    Aug.2005,Volume4,No.8(SerialNo.26)ChinaUSABusinessReview,ISSN1537

    1514,USA

    DeterminetheNumberofRegimesinMarkovSwitchingModel

    QingminHaoTianjinUniversity

    Abstract:ThemajorprobleminMarkovSwitchingModelistodeterminethenumberofregimes.Several

    methodsarecomparedandassociatedwiththeeconomicsensestocarryoutavalidwayforregimespecification.

    Keywords:Markovswitchingcriterionregimespecificationeconomicsenses 1.Introduction

    Economicsystemsoftenexperienceshocksthatshiftfrompresentstatetoanotherstatesuchaseconomic

    recession,financialmarketbubbleandcrash.Thesestatestendtobestochasticanddynamic.Oneofthecommon

    waystoanalyzethesephenomenaisMarkovSwitchingModelwhichprovidesananalyticalframeworktocapture

    suchprobabilisticstatetransitions.Ineconomics,MarkovSwitchingModelhasbeenusedtoinvestigating

    businesscycleanddecomposingaseriesinafinitesequenceofdistinctregimes.MarkovSwitchingModelisable

    tocaptureasymmetriesinthedepthanddurationofbusinesscyclesandfitthedatabeaerthanlinearmodels.The

    processineachregimeislinear.Butcombinationofprocessesproducesanonlinearregime.AsMarkovSwitching

    Modelprovidesinformationaboutthetimingofregimechanges,theyareoftenusedtocreatebusinesscycle

    chronologies,andthesecarlbeusedtomeasuresynchronization.However,aunivariateregimeswitching

    frameworkseemsaninappropriatetoolfordrawinginferencesaboutbusinesscycle.Sothemultivariateregime

    modelshouldbeusedtoanalyzethecycle.TherearemanywaystodeterminethenumberofregimesinMarkov

    SwitchingMode1.Wesummarizesomemethodsheretocomparewitheachotherandearlyoutsomevalidtoolsto

    determinethesuitableregime.

    2.DeterminationoftheNumberofRegimes

    TofindthecorrectspecificationfortheMarkovSwitching(MS)Modelisadifficultjob.Themajorproblemwith

    thedeterminationoftheappropriatespecificationforaMarkovSwitchingModelistodeterminethenumberof

    regimes.So,weuseacombinationofthegeneralandspecificprocedureswithananalysisofadequacyofthe

    specifications.Becausethetestdistributionofthenumberofregimesdoesnothaveastandarddistribution,some

    procedaresusedforthederivationoftheasymptoticdistribution.theyarenotvalidforMSmodeIandrequirethe

    simulationofthedatacontainedinagridofvaluesforthenuisanceparameters,whichwouldmeanatimc-consuming

    simulationforeachspecificationtested.Todeterminethenumberofregimes.weusefourmethodsasfollows.

    2.1InformationCriteria

    ThenormaIresultsinMSmodeIshowthelog

    likelihood,AICandBICinformationcriteria,andthenumber

    ofparameters.restrictionsandnuisanceparametersassociatedwitheachregime.remayestimateMSmodels

    usingMS

,,ARsoftwarebyKrolzigwithanumberofregimesfromhightolow,suchas10to1.andamo

    delwith

    XiangLi(1965-),male,Ph.D.candidateofManagement,SchoolofManagement,TianjinUn

    iversity;Mainresearchfield:

    ManagementSystemEngineering;Address:SchoolofManagement,TiinUniversity,Tianji

    n,China,Postcode:300072.

    "QingminHao(1969-),male,Ph.D.ofManagement,SchoolofManagement,TianjinUniver

    sity;Mainresearchfield:Corporate

    Finance;Address:SchoolofManagement,TianjinUnive~ity,Tianjin,China.Postcode."30

    0072.

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    ofRegimesinMarkovSwitchingModel

    oneregimethatcorrespondstothelinearmode1.ThenumberofautoregressivelagswasselectedthroughAICand

    BICinformationcriteria.Accordingtotheresultsfromthemodelestimation ,

    thenumberofregimesselectedbv

    Akaikeinformationcriteria(AIC)correspondstothemodelwith5regimes,whereasBayesinformationcriteria

    (BIC)included6regimes.

    2.2LikelihoodRatio

    Theasymptoticdistributionofthelikelihoodratiotestbetweenn?landnregimescanbeapproximatedbya

    Chi'squaredistribution,wherethenumberofdegreesoffreedomisgivenbythenumberofnuisanceparameters

    ofthemodelwithnregimesplusthenumberofrestrictionsimposedbyregimenonregimen.1.

Theteststatistics

    iscalculatedinausualfashioninlikelihoodratiotests,LR=2(1og?likelihood(n

    log?likelihood(n.1)),wherenand

    n?1arethemodelswithnandn?lregimes.Thistestisbasedonallapproximationtothecorrectcriticalvaluesand

    thereforeweneedfurthersupportinordertodecideontheoptimalnumberofregimes. 2.3BDSStatistics

    BDSstatisticsisarobusttestusedtodeterminethepresenceofre?maimingstructuresbothinthemeanand

    varianceoftheprocess.BDSstatisticsisefficientincheckingwhethertheproposedspecificationcancapturethe

    wholestructureofn?lregimes.Ifintheresidualsofthemodelwith6regimesisstillsomeuncapturedstructurein

    themeanandvariance.Tocheckwhether7regimesareenoughtocaptureallthestructurepresentedinthemean

    andvariance,theresultsofBDStestshowthatitwasnotpossibletorefutetheresidualsofthemodelasbeingliD,

    whichindicatesthatweshouldworkwith7regimes,thuscaptureallthedependencestructurepresentedinthe

    exchangeratelog?returnseries.

    2.4Kullback?Leibler(KL)Divergence

    InapplyingAIC,whichisanestimateofKLdivergence,wefindthatAICretainstoomanystatesand

    variablesinthemode1.Hence,MarkovSwitchingCriterion(MSC)yieldsamarkedimprovementinstate

    determinationandvariableselectionbecauseitimposesanappropriatepenaltytomitigatetheover?retentionof

    statesintheMarkovchain.AndthepapershowsMSCperformswellinMonteCarlostudieswithsingleand

    multiplestates,smallandlargesamples,andlowandhighnoise.Furthermore,itnotonlyappli

esto

    Markov

    switchingregressionmodel,butalsoperformswellinMarkov?switchingautoregressionmode1.Finally,

    theusefulnessofMSCisillustratedviaapplicationstotheU.S.businesscycleandtheeffectivenessofmedia

    advertising.

    3.Conclusion

    Fromthecomparisonabovewecanrealizethateverymethodhasitsowncharacterstodeterminetheregime,

    sotheimportantwaytocarryoutthenumberofregimemustassociatewitheconomicsensesandeachregime

    mustrepresentitseconomiccharacters.

    References:

    I.Laurini.Portuga1.LongMemosyinR$/US$ExchangeRate.FinanceLabWorkingPapersfromFinanceLab,lbmecBusiness

    School,2003

    2.KrolzigH.M.^RModellingStatisticalInferenceandApplicationtoBusi'nessCycleAnab,sis,LectureNotesin

    EconomicsandMathematicaISystems.Springer,I997

    3.SmithPA.How

    ,,doMarkovSwitchingModelsDescribeBusinessCycles,JournalofAppliedEconometrics,2005,Vo1.2O:

    253-274

    4.AaronSmith.PrasadA.Naik.Chih,LingTsai.Markov-switchingModelSelectionUsingKullback-LeiblerDivergence,ARE

    WorkingPapers.DepartmentofAgricultural&ResourceEconomics,UCD,2005 (EditedbyYanyuandCici)

    81

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