Tinbergen Institute is the graduate school and research institute in e - PDF document

Tinbergen Institute is the graduate school and research institute in economics of Erasmus University Rotterdam, the University of Amsterdam and VU University Amsterdam. More TI discussion papers can be downloaded at http://www.tinbergen.nl Tinbergen Institute has two locations: Tinbergen Institute Amsterdam Gustav Mahlerplein 117 1082 MS Amsterdam The Netherlands Tel.: +31(0)20 525 1600 Tinbergen Institute Rotterdam Burg. Oudlaan 50 3062 PA Rotterdam The Netherlands Tel.: +31(0)10 408 8900 Fax: +31(0)10 408 9031 Duisenberg school of finance is a collaboration of the Dutch financial sector and universities, with the ambition to support innovative research and offer top quality academic education in core areas of finance. DSF research papers can be downloaded at: http://www.dsf.nl/ Duisenberg school of finance Gustav Mahlerplein 117 1082 MS Amsterdam The Netherlands Tel.: +31(0)20 525 8579 TimeVaryingTransitionProbabilitiesforMarkovRegimeSwitchingModelsMarcoBazzi(a),FranciscoBlasques(b)SiemJanKoopman(b;c),AndreLucas(b)(a)UniversityofPadova,Italy(b)VUUniversityAmsterdamandTinbergenInstitute,TheNetherlands(c)CREATES,AarhusUniversity,DenmarkAbstractWeproposeanewMarkovswitchingmodelwithtimevaryingprobabilitiesforthetransitions.Thenoveltyofourmodelisthatthetransitionprobabilitiesevolveovertimebymeansofanobservationdrivenmodel.Theinnovationofthetimevaryingprobabilityisgeneratedbythescoreofthepredictivelikelihoodfunction.Weshowhowthemodeldynamicscanbereadilyinterpreted.WeinvestigatetheperformanceofthemodelinaMonteCarlostudyandshowthatthemodelissuccessfulinestimatingarangeofdierentdynamicpatternsforunobservedregimeswitchingprobabilities.WealsoillustratethenewmethodologyinanempiricalsettingbystudyingthedynamicmeanandvariancebehaviourofU.S.IndustrialProductiongrowth.Wendempiricalevidenceofchangesintheregimeswitchingprobabilities,withmorepersistenceforhighvolatilityregimesintheearlierpartofthesample,andmorepersistenceforlowvolatilityregimesinthelaterpartofthesample.Somekeywords:HiddenMarkovModels;observationdrivenmodels;generalizedautoregressivescoredynamics.JELclassication:C22,C32. Theauthorsthankparticipantsofthe\2014WorkshoponDynamicModelsdrivenbytheScoreofPredictiveLikelihoods",LaLaguna,andseminarparticipantsandVUUniversityAmsterdamforusefulcommentsanddiscussions.BlasquesandLucasthanktheDutchScienceFoundation(NWO,grantVICI453-09-005)fornancialsupport.KoopmanacknowledgessupportfromCREATES,CenterforResearchinEconometricAnalysisofTimeSeries(DNRF78),fundedbytheDanishNationalResearchFoundation.1 1IntroductionMarkovregimeswitchingmodelshavebeenwidelyappliedineconomicsandnance.SincetheseminalapplicationofHamilton(1989)toU.S.realGrossNationalProductgrowthandthewell-knownNBERbusinesscycleclassication,themodelhasbeenadoptedinnumerousotherapplications.Examplesareswitchesinthelevelofatimeseries,switchesinthe(autoregressive)dynamicsofvectortimeseries,switchesinvolatilities,andswitchesinthecorrelationordependencestructurebetweentimeseries;seeHamiltonandRaj(2002)forapartialsurvey.ThekeyattractivefeatureofMarkovswitchingmodelsisthattheconditionaldistributionofatimeseriesdependsonanunderlyinglatentstateorregime,whichcantakeonlyanitenumberofvalues.ThediscretestateevolvesthroughtimeasadiscreteMarkovchainandwecansummarizeitsstatisticalpropertiesbyatransitionprobabilitymatrix.Dieboldetal.(1994)andFilardo(1994)arguethattheassumptionofaconstanttransi-tionprobabilitymatrixforaMarkovswitchingmodelistoorestrictiveformanyempiricalsettings.TheyextendthebasicMarkovswitchingmodeltoallowthetransitionprobabili-tiestovaryovertimeusingobservablecovariates,includingstrictlyexogenousexplanatoryvariablesandlaggedvaluesofthedependentvariable.Althoughthisapproachcanbeusefulandeective,itisnotalwaysclearwhatvariablesorwhichfunctionalspecicationweshouldusefordescribingthedynamicsinthetransitionprobabilities.Ourmaincontributioninthispaperistoproposeanew,dynamicapproachtomodeltimevariationintransitionprobabilitiesinMarkovswitchingmodels.Weletthetransitionprobabilitiesvaryovertimeasspecictransformationsofthelaggedobservations.Henceweadoptanobservationdrivenapproachtotimevaryingparametermodels;seeCox(1981)foradetaileddiscussion.Observationdrivenmodelshavetheadvantagethatthelikeli-hoodistypicallyavailableinclosedformusingapredictionerrordecomposition.Ourmainchallengeistospecifyasuitablefunctionalformtolinkpastobservationstofuturetransi-tionprobabilities.Forthispurpose,weusethescoresofthepredictivelikelihoodfunction.SuchscoredrivendynamicshavebeenintroducedbyCrealetal.(2011,2013)andHarvey(2013).Scoredrivenmodelsencompassmanywell-knowntimeseriesmodelsineconomicsandnance,includingtheARCHmodelofEngle(1982),thegeneralizedARCH(GARCH)modelofBollerslev(1986),theexponentialGARCH(EGARCH)modelofNelson(1991),theautoregressiveconditionalduration(ACD)modelofEngleandRussell(1998),andmanymore.Inaddition,varioussuccessfulapplicationsofscoremodelshaveappearedinthere-centliterature.Forexample,Crealetal.(2011)andLucasetal.(2014)studydynamicvolatilitiesandcorrelationsunderfat-tailsandpossibleskewness;HarveyandLuati(2014)introducenewmodelsfordynamicchangesinlevelsunderfattails;Crealetal.(2014)inves-tigatescore-basedmixedmeasurementdynamicfactormodels;OhandPatton(2013)andDeLiraSalvatierraandPatton(2013)investigatefactorcopulasbasedonscoredynamics;andKoopmanetal.(2012)showthatscoredriventimeseriesmodelshaveasimilarfore-2 castingperformanceascorrectlyspeciednonlinearnon-Gaussianstatespacemodelsoverarangeofmodelspecications.WeshowthatthescorefunctioninourMarkovswitchingmodelhasahighlyintuitiveform.Thescorecombinesallrelevantinnovativeinformationfromtheseparatemodelsassociatedwiththelatentstates.Theupdatesofthetimevaryingparametersarethereforebasedontheprobabilitiesofthestates,givenallinformationuptotimet�1.Inoursimulationexperiments,thenewmodelperformswellandsucceedsincapturingarangeoftimevaryingpatternsfortheunobservedtransitionprobabilities.WeapplyourmodeltostudythemonthlyevolutionofU.S.IndustrialProductiongrowthfromJanuary1919toOctober2013.Weuncoverthreeregimesforthemeanandtworegimesforthevarianceoverthesampleperiodconsidered.Thecorrespondingtransitionprobabilitiesaretimevarying.Inparticular,thehighvolatilityregimeappearstobemuchmorepersistentintheearlierpartofthesamplecomparedtothelaterpart.Theconverseholdsforthelowvolatilityregime.Suchchangesinthedynamicsofthetimeseriesarecapturedinastraightforwardwaywithinourmodel.Moreover,thetofthenewmodeloutperformsthetofseveralcompetingmodels.Asanalcontribution,itisworthwhilementioningthatourmodelalsopresentsaninterestingmixofparameterdriven(Markovswitching)dynamicswithobservationdrivenscoredynamicsforthecorresponding(transitionprobability)parameters.Inparticular,itisinterestingtoseethatscoredrivenmodelscanstillbeadoptedwhenanadditionallteringstep(fortheunobserveddiscretestates)isrequiredtocomputethescoreoftheresultingconditionalobservationdensity.Thisfeatureofthenewdynamicswitchingmodelisinterestinginitsownright.SimilardevelopmentsforalinearGaussianstatespacemodelhavebeenreportedbyCrealetal.(2008)andDelleMonacheandPetrella(2014)Theremainderofthepaperisorganizedasfollows.InSection2webrie ydiscussthemainset-upoftheMarkovswitchingmodelanditsresidualdiagnostics.InSection3weintroducethenewMarkovswitchingmodelwithtimevaryingtransitionprobabilitiesbasedonthescoreofthepredictivelikelihoodfunction.InSection4wediscusssomeofthestatisticalpropertiesofthemodel.InSection5wereporttheresultsofaMonteCarlostudy.InSection6wepresenttheresultsofourempiricalstudyintothedynamicsalientfeaturesofU.S.IndustrialProductiongrowth.Section7concludes.2MarkovswitchingmodelsMarkovswitchingmodelsarewell-knownandwidelyusedinappliedeconometricstudies.WerefertothetextbookofFruhwirth-Schnatter(2006)foranextensiveintroductionanddiscussion.ThetreatmentbelowestablishesthenotationanddiscussessomebasicnotionsofMarkovswitchingmodels.3 Letfyt;t=1;:::;TgdenoteatimeseriesofTunivariateobservations.Weconsiderthetimeseriesfyt;t=1;:::;Tgasasubsetofastochasticprocessfytg.Theprobabilitydistributionofthestochasticprocessytdependsontherealizationsofahiddendiscretestochasticprocesszt.Thestochasticprocessytisdirectlyobservable,whereasztisalatentrandomvariablethatisobservableonlyindirectlythroughitseectontherealizationsofyt.ThehiddenprocessfztgisassumedtobeanirreducibleandaperiodicMarkovchainwithnitestatespacef0;:::;K�1g.ItsstochasticpropertiesaresucientlydescribedbytheKKtransitionmatrix,,whereijisthe(i+1;j+1)elementofandisequaltothetransitionprobabilityfromstateitostatej.Allelementsofarenonnegativeandtheelementsofeachrowsumto1,thatisij=P[zt=jjzt�1=i];K�1Xj=0ij=1;ij0;8i;j2f0;:::;K�1g:(1)Letp(
ji; )beaparametricconditionaldensityindexedbyparametersi2and 2 ,whereiisaregimedependentparameterand isnotregime-specic.Weassumethattherandomvariablesy1;:::;yTareconditionallyindependentgivenz1;:::;zT,withdensitiesytj(zt=i)p(
ji; ):(2)Forthejointstochasticprocessfzt;ytg,theconditionaldensityofytisp(ytj ;It�1)=K�1Xi=0p(ytji; )P(zt=ij ;It�1);(3)whereIt�1=fyt�1;yt�2;:::gistheobservedinformationavailableattimet�1.Allparam-eters and0;:::;K�1areunknownandneedtobeestimated.TheconditionalmeanofytgivenztandIt�1maycontainlagsofytitself.FrancqandRoussignol(1998)andFrancqandZakoan(2001)derivetheconditionsfortheexistenceofanergodicandstationarysolutionforthegeneralclassofMarkovswitchingARMAmodels.Inparticular,theyshowthatglobalstationarityofytdoesnotrequirethestationarityconditionswithineachregimeseparately.Asanexample,considerthecaseK=2foracontinuousvariableytwithconditionaldensityp(
jzt)=N�(1�zt)0+zt1;2
;(4)where0and1arestaticregime-dependentmeans,and2isthecommonvariance.Thelatenttwo-stateprocessfztgisdrivenbythetransitionprobabilitymatrix= 001�001�1111!;(5)4 wherethetransitionprobabilitiessatisfy000;111.Wehavei=ifori=0;1,and =(2;00;11)0.Toevaluateequation(3),werequirethequantitiesP(zt=ij ;It�1)forallt.WecancomputetheseecientlyusingtherecursivelteringapproachofHamilton(1989).AssumingwehaveanexpressionforthelteredprobabilityP(zt�1=ij ;It�1),wecanobtainthepredictiveprobabilitiesP(zt=ij ;It�1)asP(zt=ij ;It�1)=K�1Xk=0ki
P(zt�1=kj ;It�1):(6)Hence,theconditionaldensityofytgivenIt�1isgivenbyp(ytj ;It�1)=K�1Xi=0K�1Xk=0p(ytji; )
ki
P(zt�1=kj ;It�1):(7)Wecanrewritethisexpressionmorecompactlyinmatrixnotation.Denet�1astheK�dimensionalvectorcontainingthelteredprobabilitiesP(zt�1=ij ;It�1)attimet�1andlettbetheK�dimensionalvectorcollectingthedensitiesp(ytji; )attimetfori=0;:::;K�1.Itfollowsthat(7)reducestop(ytj ;It�1)=0t�1t:(8)ThelteredprobabilitiestcanbeupdatedbytheHamiltonrecursiont=�0t�1
t 0t�1t;(9)wheredenotestheHadamardelementbyelementproduct.ThelterneedstobestartedfromanappropriatesetofinitialprobabilitiesP(z0=ij ;I0).ThesmoothedestimatesoftheregimeprobabilitiesP(zt=ij ;IT)canbeobtainedfromthealgorithmofKim(1994).TheHamiltonlterin(9)isimplementedfortheevaluationofthethelog-likelihoodfunctionwhichisnumericallymaximizedwithrespecttotheparametervector(00;:::;0K�1; 0)0usingaquasi-Newtonoptimizationalgorithm.Toavoidlocalmaxima,weconsiderdierentstartingvaluesforthenumericaloptimization.DiagnosticcheckinginMarkovregimeswitchingmodelsissomewhatmorecomplicatedwhencomparedtoothertimeseriesmodelsbecausethetrueresidualsdependonthelatentvariablezt.Hencetheresidualsareunobserved.AstandardsolutionistheuseofgeneralizedresidualswhichhavebeenintroducedbyGourierouxetal.(1987)inthecontextoflatentvariablemodels.TheyhavebeenusedinthecontextofMarkovregimeswitchingmodelsbyTurneretal.(1989),Gray(1996),MaheuandMcCurdy(2000),andKimetal.(2004).GiventhelteredregimeprobabilitiesP(zt=ij ;It�1),fori=0;:::;K�1,letiand2i5 denotetheconditionalmeanandtheconditionalvarianceofytinregimei.Thestandardizedgeneralizedresidualetisdenedaset=K�1Xi=0yt�i iP(zt=ij ;It�1);t=1;:::;T:(10)Alsointhecontextofswitchingmodels,Smith(2008)adoptsthetransformationproposedbyRosenblatt(1952)anddenestheRosenblattresidual~etas~et=�1 K�1Xi=0P(zt=ij ;It�1)��1i(yt�i)
!;(11)wheredenotesthecumulativedistributionfunctionofastandardnormalwiththecorre-spondinginversefunction�1.IfytisgeneratedbythedistributionimpliedbytheMarkovswitchingmodel,thentheRosenblattresidual~etisstandardnormallydistributed.Further-more,Smith(2008)showsinanextensiveMonteCarlostudythatLjung-BoxtestsbasedontheRosenblatttransformationhavegoodnite-samplepropertiesforthediagnosticcheckingofserialcorrelationinthecontextofMarkovregimeswitchingmodels.3TimevaryingtransitionprobabilitiesIntheprevioussectionweconsideredthetransitionprobabilitymatrixtobeconstantovertime.Dieboldetal.(1994)andFilardo(1994)argueforhavingtimevaryingtransitionprobabilitiest.Theyproposetolettheelementsoftbefunctionsofpastvaluesofthedependentvariableytandofexogenousvariables.TheHamiltonlterandKimsmoothercaneasilybegeneralizedtohandlesuchcasesoftimevaryingt.Akeychallengeistospecifyanappropriateandparsimoniousfunctionthatlinksthelaggeddependentvariablestofuturetransitionprobabilities.Forthespecicationofthedynamicsoft,weadoptthegeneralizedautoregressivescoredynamicsofCrealetal.(2013);similardynamicscoremodelshavebeenproposedbyCrealetal.(2011)andHarvey(2013).WeprovidethedetailsofthescoredrivenmodelfortimevaryingtransitionprobabilitiesintheMarkovregimeswitchingmodel.Thenewdynamicmodelisparsimoniousandtheupdatingmechanismishighlyintuitive.Eachprobabilityupdateisbasedontheweightingofthelikelihoodinformationp(
ji; )in(2)foreachseparateregimei.3.1DynamicsdrivenbythescoreofpredictivelikelihoodTheparametervector containsboththetransitionprobabilitiesaswellasotherparameterscapturingtheshapeoftheconditionaldistributionsp(ytj ;It�1).Withaslightabuseofnotation,wesplit intoadynamicparameterftthatweusetocapturethedynamic6 transitionprobabilities,andanewstaticparameter thatgathersallremainingstaticparametersinthemodel,aswellassomenewstaticparametersthatgovernthetransitiondynamicsofft.Forexample,inthetwo-stateexampleofSection2wemaychooseft=(f00;t;f11;t)0withf00;t=logit(00;t)andf11;t=logit(11;t),wherelogit(00;t)=log(00;t)�log(1�00;t),andlog(
)referstothenaturallogarithm.Atthesametime,weset =(2;!;A;B),where!,A,andBaredenedbelowinequation(12).Fortheremainderofthispaper,wedenotetheconditionalobservationdensitybyp(ytjft; ;It�1).IntheframeworkofCrealetal.(2013),thedynamicprocessesfortheparametersaredrivenbyinformationcontainedinthescoreoftheconditionalobservationdensityp(ytjft; ;It�1)withrespecttoft.ThemainchallengeinthecontextofMarkovswitchingmodelsisthattheconditionalobservationdensityisitselfamixtureofdensitiesusingthelatentmixingvariablezt.Therefore,theshapeofourconditionalobservationdensityasgivenbyequation(3)issomewhatinvolved.Theupdatingequationforthetimevaryingparameterftbasedonthescoreofthepredictivedensityisgivenbyft+1=!+Ast+Bft;st=St
rt;rt=@ @ftlogp(ytjft; ;It�1);(12)where!isavectorofconstants,AandBarecoecientmatrices,andstisthescaledscoreofthepredictiveobservationdensitywithrespecttoftusingthescalingmatrixSt.Theupdatingequation(12)canbeviewedasasteepestascentorNewtonstepforftusingthelogconditionaldensityattimetasitscriterionfunction.AninterestingchoiceforSt,asrecognizedbyCrealetal.(2013),isthesquarerootmatrixoftheinverseFisherinformationmatrix.ThisparticularchoiceofStaccountsforthecurvatureofrtasafunctionofft.Also,forthischoiceofStandundercorrectmodelspecication,thescaledscorefunctionsthasaunitvariance;seealsoSection4.3.2Timevaryingtransitionprobabilities:thecaseof2statesWerstconsiderthetwo-stateMarkovregimeswitchingmodel,K=2.Weletthetransitionprobabilities00;tand11;tvaryovertimewhilethetworemainingprobabilitiesaresetto01;t=1�00;tand10;t=1�11;tasin(5).Wespecifythetransitionprobabilitiesasii;t=ii+(1�2ii)exp(�fii;t)=(1+exp(�fii;t));i=0;1;wheref00;tandf11;taretheonlytwoelementsinthetimevaryingparametervectorft,andwherethetwoparameters0ii0:5,fori=0;1,canbesetbytheeconometriciantolimittherangeoverwhichii;tcanvary.IntheapplicationinSection6,wesetwesetii=0,fori=0;1,suchthatii;tcantakeanyvalueintheinterval(0;1).WeprefertoworkwithaparsimoniousmodelspecicationandthereforewetypicallyhavediagonalmatricesforAandBin(12).Theupdatingequationsforthetimevarying7 parameterftisgivenbyequation(12)wherethescalingissettoSt=I�0:5t�1whereIt�1isthe22Fisherinformationmatrixcorrespondingtothe21scorevectorrtdenedin(12).Thescorevectorfortheconditionaldensityin(7)takestheformrt=p(ytj0; )�p(ytj1; ) p(ytj ;It�1)g�ft; ;It�1
;(13)g�ft; ;It�1
= P[zt�1=0j ;It�1]
(1�200)00;t(1�00;t)�P[zt�1=1j ;It�1]
(1�211)11;t(1�11;t)!:(14)Thisexpressionhasahighlyintuitiveform.Therstfactorin(13)isthedierenceinthelikelihoodofytgivenzt=0versuszt=1.Thedierenceisscaledbythetotallikelihoodoftheobservationgivenallthestaticparameters.Ifthelikelihoodofytgivenzt=0isrelativelylargecomparedtothatforzt=1,weexpectf00;ttoriseandf11;ttodecrease.Thisispreciselywhathappensinequations(13)and(14).Themagnitudesofthestepsaredeterminedbytheconditionalprobabilitiesofbeinginregimezt�1=0orzt�1=1,respectively,attimet�1.Theremainingfactors(1�2ii)ii;t(1�ii;t),fori=0;1,areduetothelogitparameterization.Inparticular,ifwearealmostcertainofbeinginregimezt�1=0attimet�1,thatisP[zt�1=0j ;It�1]1,thenwetakealargestepwithf00;tbutwedonotmovef11;tbymuch.Obviously,ifwearealmostcertainofbeinginregimezt�1=0,ytcanonlylearnussomethingabout00;t.Wedonotlearnmuchabout11;tinthiscase.Theconverseholdsifwearealmostcertainofbeinginregimezt�1=1attimet�1,inwhichcasewecanonlylearnaboutf11;t=logit(11;t).Theweightsforthelteredprobabilitiesinthevectorg(ft; ;It�1)in(13)takesaccountofthis.TheconditionalFisherinformationmatrixbasedon(13)issingularbydesign.Thevectorg(ft; ;It�1)ontheright-handsideof(13)isIt�1-measurableandhencetheexpectationofitsouterproductremainsofrank1.Therefore,wescalethescorebyasquarerootMoore-Penrosepseudo-inverse1oftheconditionalFisherinformationmatrix.Wehavest=Gt[p(ytj0; )�p(ytj1; )]=p(ytj ;It�1) q R1�1[p(ytj0; )�p(ytj1; )]2=p(ytj ;It�1)dyt;(15)withGt=g�ft; ;It�1
= g�ft; ;It�1
,andwheretheintegralhasnoclosedformingeneralandiscomputednumerically,forexampleusingGauss-Hermitequadraturemethods.AnalternativetotheanalyticMoore-Penrosepseudo-inverseisanumericalpseudoinverse;forexample,wecouldusetheTikhonovregularizedmatrixinverseasgivenbyI1=2t�1=�I+(1�)It�1
�1=2,withunitmatrixIandxedscalar01.For!0theTikhonovinversecollapsestotheMoore-Penrosepseudo-inverse. 1Ifx2Rnisavector,thentheMoore-Penrosepseudo-inverseofxx0isgivenbykxk�4xx0,anditssquarerootbykxk�3xx0,askxk�3xx0kxk�3xx0=kxk�4xx0.Asg(ft; ;It�1)isIt�1-measurable,scalingthescorebythesquarerootMoore-Penrosepseudo-inverseoftheconditionalFisherinformationmatrixyeldsanexpressionproportionaltokg(ft; ;It�1)k�3g(ft; ;It�1)g(ft; ;It�1)0g(ft; ;It�1)=kg(ft; ;It�1)k�1g(ft; ;It�1).8 3.3Timevaryingtransitionprobabilities:thecaseofKstatesWecaneasilygeneralizethetwo-regimemodeltoKregimes.Toenforcethatalltransitionprobabilitiesarenon-negativeandsumtoone(row-wise),weusethemultinomiallogitspecication.Givenasetofvaluesfor0ij0:5,wesetij;t=ij+(1�2ij)exp(fij;t)"1+K�1Xj=1exp(fij;t)#�1;i;K�1;t=1�K�1Xj=1ij;t(ij);(16)fori=0;:::;K�1andj=0;:::;K�2.Thetimevaryingparametersfij;t,correspondingtothetimevaryingtransitionprobabilitiesij;t,arecollectedintheK(K�1)1vectorft.Thevectorftissubjecttotheupdatingequation(12).Theingredientsforthescaledscorevectorintheupdatingequation(12)aregivenbyrt=J0trt;It�1=E[J0rtrt0Jt];rt=@logp(ytj ;It�1) @vec()0=t t�1 p(ytj ;It�1);where istheKroneckerproductandtheelementsofJt=@vec(t)=@f0taregivenby@ij;t @fi0j0;t=8�&#x]TJ ;� -2;
.51; Td;&#x [00;:(1�2ij)ij;t(1�ij;t);fori=i0^j=j0;�(1�2ij)ij;tij0;t;fori=i0^j6=j0;0;otherwise:;fori;i0=0;:::;K�1andj;j0=0;:::;K�2.4StatisticalpropertiesInthissectionwestudythestochasticpropertiesoftheestimateddynamictransitionprob-abilitiesinourscoredrivenMarkovswitchingmodel.Inparticular,weanalyzethebehavioroftheestimatedtimevaryingparameterasafunctionofpastobservationsy1;:::;yt�1,pa-rametervector ,andinitialpointf1.Wewritetheprocessasf~ftgwith~ft:=~ft( ;f1),fort=1;:::;T.WefollowBlasquesetal.(2012),whousethestationarityandergodicity(SE)conditionsformulatedbyBougerol(1993)andStraumannandMikosch(2006)forgeneralstochasticrecurrenceequations.Dene~Xt=(~f0t;~0t)0asthestackedvectoroflteredtimevaryingparameters~ftandlteredprobabilities~tasdenedin(9).Wedene~t=~t( ;1)forsomeinitialpoint1.Ourstochasticrecurrenceequationforthelteredprocessf~Xtgnowtakestheform~Xt+1=H(~Xt;yt; ),where~Xt+1="~t+1~ft+1#=H(~Xt;yt; ):="(~t;~ft;yt; )!+As(~t;~ft;yt; )+B~ft#;wheres(~t;~ft;yt; )isthescaledscoredenedin(15)and(~t;~ft;yt; )isthefractiondenedin(9)fortherecursion~t+1=(~ft;~t;yt; ):=�0~t�1
t=~0t�1t.The9 followingpropositionstatessucientconditionsforthelteredprocessf~Xt( ;X1)gwithinitializationatX1:=(01;f01)0toconvergealmostsurelyandexponentiallyfast(e.a.s)toauniquelimitSEprocessf~Xt( )g.2Proposition1.LetfytgbeSE,withijin(16)satisfyingij�0forallpairs(i;j),andassumethatforevery 2 (i)Elog+ H�1;f1;y1; 
1;(ii)Elnsup(f;) _H�f;;y1; 
0;where_H(X;y1; )=@H(X;y1; )=@XdenotestheJacobianfunctionofHw.r.t.X.Thenf~Xt( ;X1)gconvergese.a.s.toauniqueSEprocessf~Xt( )g,forevery 2 ,thatisk~Xt( ;X1)�~X( )ke:a:s:!0ast!18 2 .Proof.TheassumptionthatfytgisSEimpliesthatftgin(9)isSE.TogetherwiththecontinuityofH(andtheresultingmeasurabilityw.r.t.theBorel-algebra),itfollowsthatfH(
;
;yt; )gisSEforevery byKrengel(1985,Proposition4.3).ConditionC1inBougerol(1993,Theorem3.1)isimmediatelyimpliedbyassumption(i)forevery 2 .ConditionC2inBougerol(1993,Theorem3.1)isimplied,forevery 2 ,bycondition(ii).Asaresult,forevery 2 ,f~Xt( ;f1)gconvergesalmostsurelytoanSEprocessf~X( )g.Uniquenessande.a.s.convergenceisobtainedbyStraumannandMikosch(2006,Theorem2.8). Proposition1eectivelydenesthosecombinationsof,AandBforwhichwecanensurethatthelteredsequencef~Xt( ;f1)gconvergese.a.s.toanSElimitforagivenSEdatasequencefytg.Weemphasizethatanite!isrequiredforcondition(i)toholdsince!enterstheHfunction.However,!playsnoroleinthecontractioncondition(ii)asitdoesnotin uencetheJacobian_H.Numericalexperiments(notreportedhere)suggestthatstabilityisachievedbyawiderangeofcombinationsoftheparameters,AandB,whereisavectorcontainingallij.Inparticular,thesetofstablecombinations(A,B)becomeslargerforhighervaluesof.Thismechanismisintuitivebecausetheentriesofthevectorboundtheelementsofthevectorandawayfromzeroandone.Asaresult,�0ensuresthatthedenominatorin(9)isboundedawayfromzeroandhencethesequenceftgbecomesmorestable.Proposition1isessentialincharacterizingthestochasticpropertiesofthelteredtimevaryingparameters.ItdoesnotonlyallowustohavefurtherinsightsintothenatureofthelteredestimatesftintheMonteCarlostudyofSection5,butitalsoenablesustointerprettheparameterestimatesoftheGASmodel.TheSEnatureofthelteredsequenceisalsoanimportantingredientinobtainingproofsofconsistencyandasymptoticnormalityofthemaximumlikelihoodestimatorthatrelyonanapplicationoflawsoflargenumbersandcentrallimittheorems;seeBlasquesetal.(2014)formoredetails. 2Wesaythatarandomsequencef~Xtgconvergese.a.s.toanotherrandomsequencef~Xtgifthereisaconstantc�1suchthatctk~Xt�~Xtka:s:!0;seeStraumannandMikosch(2006)forfurtherdetails.10 Table1:Simulationpatternsfor00;tand11;t Model00;t11;t 1.Constant0:950:852.SlowSine0:5+0:45cos(4t=T)0:5�0:45cos(4t=T)3.Sine0:5+0:45cos(8t=T)0:5�0:45cos(8t=T)4.FastSine0:5+0:45cos(20t=T)0:5�0:45cos(20t=T)5.Break0:21fT=2g+0:81ftT=2g0:81fT=2g+0:21ftT=2g 5MonteCarlostudy5.1DesignofthesimulationstudyToinvestigatetheperformanceofourestimationprocedurefortheMarkovregimeswitchingmodelwithtimevaryingtransitionprobabilities,weconsideraMonteCarlostudyforthetworegimemodel(4).Thetworegimesconsistoftwonormaldistributionswithcommonvariance2=0:5andmeans0=�1and1=1.Weset00=11=0andconsider5dierentformsoftimevariationforthetransitionprobabilities00;tand11;t.ThepatternsaresummarizedinTable1andrangefromaconstantsetoftransitionprobabilities,viaslowandfastcontinuouslychangingtransitionprobabilitiestoanincidentalstructuralbreakinthemiddleofthesample.Weinvestigatetherobustnessofourestimationprocedureinthesedierentsettings.Forallofthedatagenerationprocessesconsidered,ourregimeswitchingmodelwithtimevaryingparametersisclearlymisspecied.InourMonteCarlostudyweconsiderthreedierentsamplesizes:T=250;500;1000.ThenumberofMonteCarloreplicationsissetto100.WeadoptthemodelwithK=2statesasdescribedindetailinSections3.1and3.2.Foreachdatageneratingprocessandsamplesize,weestimatethestaticparameters usingthemethodofmaximumlikelihood.Giventheestimatedvaluesforthestaticparameters,wecomputethelteredparameters^00;tand^11;tusingtheupdatingequationsin(12).WecomparetheresultstothosefortheMarkovswitchingmodelwithtimeinvarianttransitionprobabilities.5.2ThesimulationresultsInTable2,wepresentMonteCarloaveragesofthemaximizedlog-likelihoodvalue,agoodness-of-tstatisticandaforecastprecisionmeasure.OurstatisticformodeltisthecorrectedAkaikeInformationCriterion(AICc).TheAICcistheoriginalAICofAkaike(1973)butwithastrongernitesamplepenalty;seeHurvichandTsai(1991).Ourmeasureofforecastprecisionisthemeansquaredone-stepaheadforecasterror(MSFE).TheMSFEusesthepastobservationsy1;:::;yt�1tomakeaforecastofyt,usingthestaticparameter11 Table2:SimulationresultsIWehavesimulated100timeseriesfromeachdatagenerationprocess(DGP)listedinTable1andforsamplesizesT=250;500;1000.Thestaticparametersareestimatedbythemethodofmaximumlikelihood,bothfortheMarkovregimeswitchingmodelwithstatic(Static )andwithtimevaryingtransitionprobabilities(TV ;ft).Inthelattercase,theunderlyingtimevaryingparametersareupdatedusingequation(12).Wereportthesampleaveragesforthe100simulatedseriesofthemaximizedlog-likelihoodvalue(LogLik),thecorrectedAkaikeInformationCriterion(AICc)andthemeansquarederroroftheone-stepaheadforecastofyt(MSFE).Theone-stepaheadforecasterrorsarecomputedwithinthesamplethatisusedforparameterestimation. LogLikAICcMSFEDGPTStaticTV ;ftStatic TV ;ftStatic TV ;ft Constant250-300.299-293.037610.859604.8711.6951.583500-617.392-611.9471244.9101242.2711.6701.6271000-1255.957-1252.0462521.9752522.2761.6831.668 SlowSine250-355.189-344.200720.639707.1962.3732.240500-730.993-704.0581472.1101426.4932.2122.1001000-1483.296-1412.0452976.6522842.2742.2192.148 Sine250-354.946-346.154720.153711.1052.3892.237500-732.544-711.5641475.2131441.5072.2432.1741000-1485.593-1426.2652981.2482870.7142.2372.152 FastSine250-356.440-347.479723.141713.7542.6022.323500-733.108-720.6501476.3411459.6782.3582.2771000-1487.584-1449.2172985.2292916.6192.2162.212 Break250-356.643-343.721723.547706.2392.4382.197500-735.293-706.5441480.7111431.4672.3842.1371000-1493.767-1429.8122997.5962877.8092.3722.104 estimatesobtainedfromtheentiresampley1;:::;yT.Thecriterionthereforemeasurestheforecastprecisionofthemodelforthemaximumlikelihoodestimateof inthestaticmodelandforthemaximumlikelihoodestimateof inthetimevaryingparametermodel.WelearnfromTable2thattheaveragemaximizedlog-likelihoodvaluesareuniformlyhigherforthemodelwithtimevaryingtransitionprobabilitiescomparedtothemodelwithconstanttransitionprobabilities.Thisresultisnotverysurprisingsincethetimevaryingmodelisadynamicgeneralizationofthestaticmodelandhasmoreparameters.AmoreconvincingresultisthatthetimevaryingmodelproducesoverallsubstantiallysmallerAICcvaluesthantheconstantmodel.Asexpected,theonlyexceptionisthedatageneratingprocesswithconstanttransitionprobabilities.Fortimeseriessimulatedwithtimevaryingtransitionprobabilitieschangegraduallyorareofastructuralbreaktype,themodelwithtimevaryingtransitionprobabilitiesperformssubstantiallybetterthanthestaticmodel.Finally,whenwefocusontheforecastprecisionmeasureMSFE,wealsoconcludethatthe12 Table3:SimulationresultsIIWehavesimulated100timeseriesfromeachdatagenerationprocess(DGP)listedinTable1andforsamplesizesT=250;500;1000.Thestaticparametersareestimatedbythemethodofmaximumlikelihood,bothfortheMarkovregimeswitchingmodelwithstatic(Static)andwithtimevaryingtransitionprobabilities(TV).Inthelattercase,theunderlyingtimevaryingparametersareupdatedusingequation(12).Wereportthesampleaveragesforthe100simulatedseriesofthemeansquarederror(MSE)andthemeanabsoluteerror(MAE)oftheone-stepaheadforecastoftwotransitionprobabilities00and11.Theone-stepaheadforecasterrorsarecomputedwithinthesamplethatisusedforparameterestimation. MSEMAEMSEMAE0011 DGPTStaticTVStaticTVStaticTVStaticTV Constant2500.0000.0100.0150.0460.0060.0370.0490.1365000.0000.0050.0120.0320.0020.0150.0320.08910000.0000.0020.0070.0190.0010.0060.0210.057 SlowSine2500.1890.0870.3560.2230.1970.0970.3620.2385000.2040.0730.3690.1940.2080.0850.3720.21210000.2020.0550.3670.1600.2050.0550.3690.161 Sine2500.1840.1080.3500.2550.1900.1220.3550.2755000.2000.0840.3650.2190.2040.0940.3670.23410000.1990.0650.3640.1820.2010.0680.3650.190 FastSine2500.1590.1640.3290.3260.1550.1670.3260.3285000.1820.1140.3480.2650.1860.1190.3500.27410000.1990.0790.3630.2160.2010.0860.3650.229 Break2500.1770.1010.3410.2200.1700.0940.3420.2215000.1750.0820.3390.1810.1720.0790.3400.18810000.1740.0760.3370.1620.1720.0770.3380.171 Markovswitchingmodelwithtimevaryingtransitionprobabilitiesconvincinglyoutperformsitsstaticcounterpart.Nextweverifytheprecisionofthelteredtransitionprobabilityestimatesforthestaticmodelwith00and11,andforthetimevaryingmodelwith00;tand11;t.InaMonteCarlostudy,thetransitionprobabilitiesaresimulatedaspartofthedatagenerationprocess.Henceweareabletocomparetruetransitionprobabilitieswiththeirlteredestimatesandcomputethemeansquarederror(MSE)andthemeanabsoluteerror(MAE)statistics.InTable3wepresenttheMonteCarloaveragesofthesetwostatistics,forthedierentdatagenerationprocesses,thedierentsamplessizesandthetwomodels:thestaticmodel(static)andthetimevaryingmodel(TV).TheresultsinTable3providestrongevidencethattheMarkovregimeswitchingmodelwithtimevaryingtransitionprobabilitiesissuccessfulinproducingaccuratelteredesti-13 matesoftheprobabilitiesforalldierenttimevaryingpatterns.ItisonlyfortheseriesthataresimulatedfromamodelwithconstanttransitionprobabilitiesthattheMSEandMAEstatisticsforthestaticmodelaresmallerthanthoseforthetimevaryingmodel.Forthiscase,however,theabsolutevalueofallstatisticsareofanorderofmagnitudesmallerthanforalltheotherdatageneratingprocesses.Forconstanttransitionprobabilities,thedierencesbetweenthestaticandtimevaryingmodelcanthusbequaliedassmall.WealsoobservethatforincreasingsamplesizesT,theMSEandMAEstatisticsmostlydecreaseforthetimevaryingmodel,whilethisdoesnotoccurforthestaticmodel.Toputtheseobservationsinsomeperspective,wenoticethatthesinusoidpatternshavethesamenumberofswingsovertheentiresamplefordierentsamplesizes.Therefore,thechangeinthetransitionproba-bilitiesgetssmallerperunitoftimeasTincreases.Itfollowsthattheupdatingequation(12)forthetimevaryingmodelcantrackthetruetransitionprobabilitymoreaccuratelyasTincreases.Thismechanismdoesnotaecttheinaccuracyoftheestimatesobtainedfromthestaticmodel.6AnempiricalstudyofU.S.IndustrialProductionMarkovregimeswitchingmodelsarepopularinempiricalstudiesofmacroeconomictimeseries.Wethereforeillustrateournewmethodologyfortimevaryingtransitionprobabilitiesinanempiricalstudyconcerningakeyvariableformacroeconomicpolicy,U.S.IndustrialProduction(IP).ThetimeseriesforIPisobtainedfromtheFederalReserveBankofSt.Louiseconomicdatabase(FRED);wehavemonthlyseasonallyadjustedobservationsfromJanuary1919toOctober2013,T=1137.WeanalyzethepercentagegrowthofIP(log-dierences100)andconsidertheresultingseriesasourytvariable.Figure1presentsboththeIPindexandtheIPpercentagelogdierencesyt.6.1ThreemodelspecicationsThetimeseriesofIPgrowthisrelativelylong.Inspectingthetimeseriesplotofyt,weanticipatethattheremaybepossiblechangesinboththemeanandvarianceoftheseriesovertime.Toremainwithinaparsimoniousmodellingframework,weadoptthemodelofDoornik(2013).Doornikanalyzesaquarterlytimeseriesofpost-warU.S.grossdomesticproductgrowthbymeansofaMarkovswitchingmean-variancecomponentmodel.Inourspecicsetting,weconsideramodelwiththreeregimesforthemean(m=0;1;2)andtworegimesforthevariance(v=0;1).Thethreeregimesforthemeanmayrepresentrecession,stableandgrowthperiodsinU.S.production.Eachregimeforthemeanconsistsofaconstantandpmlaggeddependentvariablesforyt,withm=0;1;2.Theconstantandthepmautoregressivecoecientsarecollectivelysubjecttotheregimetowhichtheybelong.14 Figure1:U.S.IndustrialProduction(monthly,seasonallyadjusted)andIPpercentagegrowth(log-dierences100)Hence,incaseofthreeregimesforthemean,wehave3+P3m=1pmcoecients.Thetworegimesforthevariancemaysimplydistinguishperiodsoflowandhighvolatility.Hencethenumberofcoecientsforthevariancepartequals2.ModelI:staticspecicationLetfztgandfztgdenotethehiddenprocessesthatdeterminethemeanandthevarianceforthedensityofyt,respectively.Wehaveytj(zt=m;zt=v;It�1)N�m;t;2v
;m=0;1;2;v=0;1;(17)withthethreemeanequationsm;t=0;m+1;myt�1+:::+pm;myt�pm;m=0;1;2;(18)where0;misanintercept,1;m;:::;pm;mareautoregressivecoecients,forpm2N+,and21and22arethetwovariances.Thetransitionprobabilitiesforthemeanandvariancearecollectedinthematricesand,respectively,whicharegivenby=0B@00011�00�0110111�10�1120211�20�211CA;= 001�001�1111!:(19)15 WefollowDoornik(2013)inspecifyingthetransitionprobabilitymatrixforthe32=6regimesas= ;wherethe36probabilitiesinareafunctionof6meanand2varianceprobabilities.TheconditionaldensityofytgivenIt�1canbeexpressedintermsofthelteredprobabilitiesasin(8).Wehavep(ytj ;It�1)=266664P[zt�1=0;zt�1=0j ;It�1]P[zt�1=1;zt�1=0j ;It�1]...P[zt�1=2;zt�1=1j ;It�1]3777750( )266664p(yt;0;t;20;It�1)p(yt;1;t;20;It�1)...p(yt;2;t;21;It�1)377775=0t�1( )t:(20)Nextweconsidertwotimevaryingextensionsofthevarianceregimesinthestaticmodel.Afterexperimentationwithtimevaryingparametersforthemeanregimes,wehaveconcludedthatinourempiricalsettingonlytheintroductionoftimevaryingtransitionprobabilitiesforthevarianceregimescanimprovethetofthestaticmodel.Wenoticethatoncetheprobabilitiesinaretimevarying,allprobabilitiesin= aretimevarying.ModelII:timevaryingvarianceprobabilitiesasafunctionofjyt�1jInmodelII,weconstructabenchmarkinthespiritofDieboldetal.(1994)andFilardo(1994).Wespecifythetimevaryingtransitionprobabilitiesforthevarianceregimesintasalogistictransformationofthelaggeddependentvariable,thatisvv;t=exp(gvv;t) 1+exp(gvv;t);gvv;t=c0;v+c1;vjyt�1j;v=0;1;(21)wherec0;visaninterceptandc1;visaxedcoecient,forv=0;1.Thefourccoecientsareestimatedbythemethodofmaximumlikelihood,jointlywiththeothercoecients.ModelIII:timevaryingvarianceprobabilitiesasafunctionofthescoreInthisspecicationoftimevaryingtransitionprobabilitiesforthevarianceregimes,weadopttheframeworkofSection3.2.Empirically,itturnsoutthereisnoneedtoshrinktherangeofii;texante.Wethereforesetii=0andspecifythetimevaryingmatrixtasvv;t=exp(fvv;t) 1+exp(fvv;t);v=0;1;ft=(f00;t;f11;t)0;whereftisupdatedovertimeasin(12).Theresultingconditionaldensityforytisgivenbyp(ytj ;It�1)=0t�1(t )t.TheregimeprobabilitystructureofDoornik(2013)is16 morerestrictedthantheoneforthegeneralK-regimeMarkovswitchingmodelinSection3.3.Wethereforehavedierentexpressionsforthescorevectorandscalingmatrix.Thescorevectorisgivenbyrt=(r00;t;r11;t)0wherervv;t=vv;t(1�vv;t) p(ytj ;It�1)0t�1@t @vv;t t;v=0;1;witht�1andtdenedimplicitlyin(20).TocomputetheconditionalFisherInformationE[rtr0t],whereEiswithrespecttop(ytj ;It�1),weevaluate4numericalintegrals,foreverytimetandforeachvaluefortheparametervector ,byaGauss-Hermitemethod.6.2Parameterestimates,modeltandresidualdiagnosticsTable4presentstheparameterestimatesforthethreemodelspecications.Forallmodelswehavetakenpm=3in(18),withm=0;1;2.Wehaveexperimentedwithothervaluesofpmbutthischoiceprovidesanadequatet.Theparameterestimatesareobtainedbynumericallymaximizingthelog-likelihoodfunctionwithrespecttothestaticparametervector or .TheassociatedstandarderrorsareobtainedbynumericallyinvertingtheHessianmatrixatthemaximizedlog-likelihoodvalue.Thesetsofestimatedcoecientsforthethreemeanregimesareverysimilaracrossthethreemodelspecications.Theintroductionoftimevaryingvariancetransitionprobabilitiesdoesnotappeartoaectthemeanspecicationmuch.Thecoecient0;mdeterminestheinterpretationofaregime.Wecanlearnfromourestimationresultsthatregimem=0correspondstolowIPgrowth,m=1representsrecessionandm=2identieshighgrowthinIP.Theautoregressivecoecients1;m;:::;3;mshowthatin"normalyears"IPgrowthispersistent,whileduringrecessionsIPgrowthissubjecttopersistentcyclicaldynamics.PeriodsofhighIPgrowthareveryshortlivedgiventhestrongnegativelyestimatedautoregressivecoecientsform=2.Theestimatedtransitionprobabilitiesrevealthetypicalsituationinregimeswitchingmodelsthatonceweareinarecessionorlowgrowthregime,itismostlikelythatweremaininthisstate.Itisonlyforthehighgrowthregimethatitismorelikelytomovetoalowgrowthregimewhiletheprobabilitytostay,22=1�20�21,isestimatedaround0:35.Forthevarianceregimes,themodelsclearlydistinguishbetweenalow(approximately0:3)andahigh(approximately5:5to6:0)varianceregime.Themagnitudesofthesevariancesareagaincomparableacrossthedierentmodels.ForModelI,bothvarianceregimesarehighlypersistentwithprobabilities00and11bothestimatedcloseto1.WealsolearnfromTable4thatthemodelthasimprovedafterintroducingtimevaryingvariancetransitionprobabilities,bothforModelIIandModelIII.Themaximizedlog-likelihoodvalueshaveincreasedby8and17basispointsatthecostofanadditional2and4parametersfortherespectiveModelsIIandIII.ThecorrectedAkaikeinformationcriterionclearlypointstoModelIIIasthebestcomprimisetomodeltandaparsimoniousmodelspecication.17 Thetimevaryingtransitionprobabilitiesforthelowvolatility(v=0)andhighvolatility(v=1)regimesinModelIIIarehighlypersistentprocessesovertime:theestimatesforthediagonalelementsofBareveryclosetounity.FromModelIIwemayconcludethatonlythetransitionprobabilityofalowvolatilityregimeistimevarying:theestimateofc1;1isnotsignicant.ForModelIIIwendsomewhatstrongerevidencethatbothtransitionprobabilitiesaretimevarying:theestimatesofbothdiagonalelementsofAaresignicantattheusuallevelof5%.Finally,Table4presentsdiagnosticteststatisticsforthegeneralizedandRosenblattresidualswhichwehavediscussedinSection2.Thep-valuesforthewell-knownJarque-Bera2normalitytestandtheLjung-Box2serialcorrelationtest,fortheresidualsandthesquaredresiduals,indicatethatallmodelsarecapableofdescribingthesalientfeaturesinIPgrowth.TherearesomedierencesbetweenthestatisticsforthegeneralizedandRosenblattresiduals,buttheyaresmallandhavenobearingsonthemainconclusions.TheJarque-BeratestmayindicatethattheIPgrowthtimeseriesissubjecttoafewoutlyingobservations.6.3Signalextraction:regimeandtransitionprobabilitiesInFigure2wepresentthesmoothedestimatesofprobabilitiesformeanandvarianceregimesandthelteredestimatesof(timevarying)transitionprobabilitiesforModelsI,IIandIII.ModelIIappearsunabletocapturethedynamicsinthetransitionprobabilities.WehavelearnedfromTable4thattheestimateofcoecientc1;1isnotsignicant;itisalsore ectedinFigure2withthetimeseriesplotofthelteredprobabilityestimatesforthehighvarianceregimethatisalmostconstantovertime.Ontheotherhand,thelteredprobabilityestimatesforthelowvolatilityregimearehighlyerratic.ThelteredprobabilitiesforModelIIIshowanentirelydierentpattern.Boththelowandhighvolatilitytransitionprobabilitiesevolvegraduallyovertime.Inparticular,thepersistenceofthelowvolatilityregimeappearstohavegoneupovertime,withvaluesaround0.7intheearlypartofthesample,andvaluescloseto1inthesecondhalfofthesample.Theconverseholdsforthehighvolatilityregime.Thepersistenceprobability11iscloseto1uptothe1940s.Afterthat,theprobabilitydecreasessubstantiallytovaluesaround0.5,andslowlyrisestowardstheendofthesampleagain.ThepatternforthelteredprobabilitiesisconsistentwiththeempiricalpatterninthedatainFigure1.Intheearlierpartofthesample,highvolatilitylevelsarepredominant.Towardsthemiddleofthesample,largevolatilitiesareincidentalandshort-lived,whereastowardstheendofthesampleduringtheyearsofthenancialcrisis,U.S.debtceilingcrisis,andtheEuropeansovereigndebtcrisis,highervolatilitylevelsappeartoclusteragainmore.TheempiricalpatternsarealsocorroboratedbytheparameterestimatesinTable4.Inparticular,theparameterestimatesforthediagonalelementsofBarebothcloseto1;itsuggeststhatthedynamictransitionprobabilitiesevolvegraduallyovertime.Theestimates18 Table4:Parameterestimates,modeltandresidualdiagnosticsInthersttwopanelswereportthemaximumlikelihoodestimateswithstandarderrorsinparanthesesbelow,forModelsI,IIandIII.Intherstpaneltheparameterestimatesforthemeanm;tin(17)arereportedforeachregimem=0;1;2:theintercept0;m,theautoregressivecoecients1;m;:::;3;m,andthetransitionprobabilitiesmj,forj=0;1,inof(19).Inthesecondpanelthetworegimevarianceestimatesfor2varereported.ThevariancetransitionprobabilityforModelIvvisestimateddirectlywhilewehavevv=logit�1(xv)forModelII(xv=c0;v)andforModelIII(xv=!v=(1�Bvv)),forv=0;1,where!vandBvvarethe(v+1)thelementsofvector!anddiagonalmatrixBin(12),respectively.ThetimevaryingvarianceprobabilitiesaredeterminedinModelIIbyc1;v,andinModelIIIbyAvvandBvvwhicharethe(v+1)thdiagonalelementsofAandBin(12),respectively,forv=0;1.Inthethirdpanelwereportmodeltstatistics:Fit(1)isnumberofstaticparameters;Fit(2)ismaximizedlog-likelihoodvalue;Fit(3)isAICc,seeSection5.2.Wefurtherreportthep-valuesoftheresidualdiagnostic(RD)teststatisticsforthegeneralized(et)andRosenblatt'sresiduals(~et):RD(1)isJarque-Beranormality2(2)test;RD(2)isLjung-Boxserialcorrelation2(6)test;RD(3)isasRD(2)forsquaredresiduals. ModelIModelIIModelIII m=0m=1m=2m=0m=1m=2m=0m=1m=20;m0.076-0.2120.8460.068-0.2770.8370.043-0.1280.737(0.038)(0.182)(0.168)(0.037)(0.140)(0.172)(0.033)(0.095)(0.128)1;m0.3161.121-0.6090.3271.126-0.6210.3511.087-0.479(0.050)(0.096)(0.135)(0.050)(0.088)(0.130)(0.040)(0.079)(0.107)2;m0.212-0.569-0.3950.220-0.526-0.4840.234-0.537-0.221(0.050)(0.146)(0.123)(0.040)(0.139)(0.086)(0.037)(0.108)(0.086)3;m0.1050.0760.0390.1130.0110.1330.1120.0360.103(0.037)(0.107)(0.124)(0.035)(0.115)(0.101)(0.030)(0.065)(0.092)m00.9090.1110.5770.9090.1130.5920.8640.1450.576(0.041)(0.103)(0.140)(0.032)(0.072)(0.142)(0.036)(0.070)(0.175)m10.0160.8580.0550.0140.8580.0730.0210.8420.048(0.019)(0.092)(0.052)(0.012)(0.064)(0.062)(0.015)(0.059)(0.055) v=0v=1v=0v=1v=0v=12v0.3365.5790.3515.8660.3175.920(0.025)(0.629)(0.026)(0.691)(0.023)(0.541)vv0.9800.9470.9960.8830.8860.702(0.007)(0.018)(0.003)(0.053)(0.080)(0.200)c1;v-1.8990.108(0.419)(0.209)Avv0.1320.148(0.058)(0.074)Bvv0.9980.989(0.003)(0.011) i=1i=2i=3i=1i=2i=3i=1i=2i=3Fit(i)22-1642333024-1634331726-16253302RD(i)et0.0650.7720.5560.0320.9680.6590.0190.7920.924RD(i)~et0.0110.4090.6480.0120.8300.7380.0070.6320.676 19 Figure2:Smoothedprobabilityestimatesfortherecessionregimeinthemeanandforthehighvarianceregime.Filteredtransitionprobabilityestimatesforthelowandhighvarianceregimes.Intherstgraph,theverticalgrayareasindicaterecessionsaccordingtotheNBERbusinesscycleclassications.20 ofbothdiagonalelementsofAhavethecorrectsignandleadtoparameterchangesthatincreasethelocaltofthemodelintermsoflog-likelihood.Finally,wepresentthesmoothedestimatesofztinthetoppanelsofFigure2,togetherwiththeNBERbusinesscycleclassications.WemayconcludethatallmodelsresultinhighersmoothedrecessionprobabilitiesintheNBERclassiedperiods.Themodeltforamodelwithtimevaryingtransitionprobabilitiesforthevarianceregimes(ModelIIorIII)istypicallyhigherthanthestaticModelI.Fromthesmoothedprobabilitiesforthehighvarianceregime,mostofthehighvarianceregimeislocatedinthersthalfofthesample.Thesecondepisodeofhighvarianceisduringthenancialcrisis,withtheintermediateperiodhavingpredominantlyalowlevelofvolatility.Wenoticethatsome,butnotall,NBERrecessionscorrespondtoperiodsofhighvolatility.Thissupportstheuseofourcurrentframeworkwithseparateregimesforthe(conditional)meansandforthevariances.7ConclusionWehaveintroducedanewmethodologyfortimevaryingtransitionprobabilitiesinMarkovswitchingmodels.Wehaveshownthattheuseofthescoreofthepredictivelikelihoodandthegeneralizedautoregressivescore(GAS)modellingframeworkofCrealetal.(2013)candrivethedynamicsofthetransitionprobabilitieseectivelyovertime.Thecorrespondingdynamicscaneasilybeinterpretedwhiletheinformationembeddedintheconditionalob-servationdensitiesarefullyincorporated.Wehaveformulatedconditionsfortheestimatedtimevaryingprobabilitiesfromourscoredrivenmodeltoconvergetostationaryandergodicstochasticprocesses.BymeansofanextensiveMonteCarlostudy,wehaveshownthattheourproposedobservationdrivenmodelisabletoadequatelytrackthedynamicpatternsintransitionprobabilities,eveniftheunderlyingdynamicsthemselvesarepossiblymisspecied.Bothfordeterministicstructuralbreaksanddeterministicsinusoidpatterns,ourmodelyieldsalargeimprovementinmodeltcomparedtoamodelwithconstanttransitionprobabilitiesonly.InourempiricalstudyforIndustrialProductiongrowth,wehaveshownthatwecaneectivelyusethemodeldynamicfeaturesinthemeanandvariancesimultaneously.WehavefoundthatourproposedmodeloutperformsboththeMarkovswitchingmodelwithconstantprobabilitiesandwithtransitionprobabilitiesdependingonalaggeddependentvariable.Inparticular,thepatternslteredbyourmodelcanbeeasilyinterpreted,withhigher(lower)persistenceforhigh(low)volatilityregimesinthebeginningofthesamplecomparedtothelaterpartofthesample.Highervolatilitiesappeartore-occuragainattheveryendofthesample,duringthenancialandsovereigndebtcrises.Weconcludethatthemodelcanprovideausefulbenchmarkinsettingswheretransitionprobabilitiesinaregimeswitchingmodelmayvaryovertime.21 ReferencesAkaike,H.(1973).MaximumlikelihoodidenticationofGaussianautoregressivemovingaveragemodels.Biometrika60(2),255{265.Blasques,F.,S.J.Koopman,andA.Lucas(2012).Stationarityandergodicityofunivariategeneralizedautoregressivescoreprocesses.DiscussionPaperTinbergenInstituteTI12-059/4.Blasques,F.,S.J.Koopman,andA.Lucas(2014).Maximumlikelihoodestimationforgeneralizedautoregressivescoremodels.DiscussionPaperTinbergenInstituteTI14-029/III.Bollerslev,T.(1986).Generalizedautoregressiveconditionalheteroskedasticity.Journalofeconometrics31(3),307{327.Bougerol,P.(1993).Kalmanlteringwithrandomcoecientsandcontractions.SIAMJournalonControlandOptimization31(4),942{959.Cox,D.R.(1981).Statisticalanalysisoftimeseries:somerecentdevelopments.Scandina-vianJournalofStatistics8,93{115.Creal,D.,S.J.Koopman,andA.Lucas(2008).Ageneralframeworkforobservationdriventime-varyingparametermodels.DiscussionPaperTinbergenInstituteTI08-108/4.Creal,D.,S.J.Koopman,andA.Lucas(2011).ADynamicMultivariateHeavy-TailedModelforTime-VaryingVolatilitiesandCorrelations.JournalofBusiness&EconomicStatistics29(4),552{563.Creal,D.,S.J.Koopman,andA.Lucas(2013).Generalizedautoregressivescoremodelswithapplications.JournalofAppliedEconometrics28(5),777{795.Creal,D.,B.Schwaab,S.J.Koopman,andA.Lucas(2014).Observationdrivenmixed-measurementdynamicfactormodels.ReviewofEconomicsandStatistics,forthcoming.DeLiraSalvatierra,I.andA.J.Patton(2013).Dynamiccopulamodelsandhighfrequencydata.DukeUniversityDiscussionPaper.DelleMonache,D.andI.Petrella(2014).Ascoredrivenapproachforgaussianstate-spacemodelswithtime-varyingparameter.WorkingPaper,ImperialCollegeLondon.Diebold,F.,J.Lee,andG.Weinbach(1994).RegimeSwitchingwithTime-VaryingTran-sitionProbabilities.InC.Hargreaves(Ed.),NonstationaryTimeSeriesAnalysisandCointegration,pp.283{302.OxfordUniversityPress.22 Doornik,J.(2013).AMarkov-switchingmodelwithcomponentstructureforUSGNP.EconomicsLetters118(2),265{268.Engle,R.F.(1982).AutoregressiveconditionalheteroscedasticitywithestimatesofthevarianceofUnitedKingdomin ation.Econometrica50(4),987{1007.Engle,R.F.andJ.R.Russell(1998).AutoregressiveConditionalDuration:ANewModelforirregularlySpacedTransactionData.Econometrica66(5),1127{1162.Filardo,A.J.(1994).Business-cyclephasesandtheirtransitionaldynamics.JournalofBusiness&EconomicStatistics12(3),299{308.Francq,C.andM.Roussignol(1998).ErgodicityofautoregressiveprocesseswithMarkov-switchingandconsistencyofthemaximum-likelihoodestimator.Statistics:AJournalofTheoreticalandAppliedStatistics32(2),151{173.Francq,C.andJ.-M.Zakoan(2001).Stationarityofmultivariatemarkov{switchingARMAmodels.JournalofEconometrics102(2),339{364.Fruhwirth-Schnatter,S.(2006).FiniteMixtureandMarkovSwitchingModels.Springer.Gourieroux,C.,A.Monfort,E.Renault,andA.Trognon(1987).Generalisedresiduals.JournalofEconometrics34(1),5{32.Gray,S.F.(1996).Modelingtheconditionaldistributionofinterestratesasaregime-switchingprocess.JournalofFinancialEconomics42(1),27{62.Hamilton,J.(1989).ANewApproachtotheEconomicAnalysisofNonstationaryTimeSeriesandtheBusinessCycle.Econometrica57(2),357{384.Hamilton,J.D.andB.Raj(2002).Newdirectionsinbusinesscycleresearchandnancialanalysis.EmpiricalEconomics27(2),149{162.Harvey,A.C.(2013).DynamicModelsforVolatilityandHeavyTails:WithApplicationstoFinancialandEconomicTimeSeries.EconometricSeriesMonographs.CambridgeUniversityPress.Harvey,A.C.andA.Luati(2014).Filteringwithheavytails.JournaloftheAmericanStatisticalAssociation,forthcoming.Hurvich,C.M.andC.-L.Tsai(1991).BiasofthecorrectedAICcriterionforunderttedregressionandtimeseriesmodels.Biometrika78(3),499{509.Kim,C.(1994).DynamiclinearmodelswithMarkov-switching.JournalofEconomet-rics60(1),1{22.23 Kim,C.-J.,J.C.Morley,andC.R.Nelson(2004).Isthereapositiverelationshipbe-tweenstockmarketvolatilityandtheequitypremium?JournalofMoney,CreditandBanking36,339{360.Koopman,S.J.,A.Lucas,andM.Scharth(2012).Predictingtime-varyingparameterswithparameter-drivenandobservation-drivenmodels.TinbergenInstituteDiscussionPapers12-020/4.Krengel,U.(1985).Ergodictheorems.Berlin:DeGruyterstudiesinMathematics.Lucas,A.,B.Schwaab,andX.Zhang(2014).Measuringcreditriskinalargebankingsystem:econometricmodelingandempirics.JournalofBusinessandEconomicStatistics,forthcoming.Maheu,J.M.andT.H.McCurdy(2000).Identifyingbullandbearmarketsinstockreturns.JournalofBusiness&EconomicStatistics18(1),100{112.Nelson,D.B.(1991).ConditionalHeteroskedasticityinAssetReturns:ANewApproach.Econometrica59(2),347{370.Oh,D.H.andA.J.Patton(2013).Time-varyingsystemicrisk:EvidencefromadynamiccopulamodelofCDSspreads.DukeUniversityDiscussionPaper.Rosenblatt,M.(1952).Remarksonamultivariatetransformation.TheAnnalsofMathe-maticalStatistics23(3),470{472.Smith,D.R.(2008).EvaluatingSpecicationTestsforMarkov-SwitchingTime-SeriesMod-els.JournalofTimeSeriesAnalysis29(4),629{652.Straumann,D.andT.Mikosch(2006).Quasi-maximum-likelihoodestimationincondition-allyheteroscedastictimeseries:astochasticrecurrenceequationsapproach.TheAnnalsofStatistics34(5),2449{2495.Turner,C.M.,R.Startz,andC.R.Nelson(1989).AMarkovmodelofheteroskedasticity,risk,andlearninginthestockmarket.JournalofFinancialEconomics25(1),3{22.24 TI 2014-072/III Tinbergen Institute Discussion Paper z University of Padova, Italy; Faculty of Economics and Business Administration, VU University Amsterdam, the Netherlands.