MODELLING VOLATILITY AND CORRELATIONS WITH A HIDDEN MARKOV DECISION TREE

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Niveau: Supérieur, Doctorat, Bac+8
MODELLING VOLATILITY AND CORRELATIONS WITH A HIDDEN MARKOV DECISION TREE. PHILIPPE CHARLOT GREQAM & Aix-Marseille University Centre de la Vieille Charité 2, rue de la Charité 13236 Marseille cedex 02 France VERY PRELIMINARY DRAFT – COMMENTS ARE MOST WELCOME! ABSTRACT. The goal of the present paper is to present a new multivariate GARCH model with time- varying conditional correlation. Since the seminal work of Bollerslev (1990), conditional correlation models have become a attractive field in economics. Different specifications have been developed to study both empirical findings and practical use like asymmetry, change in regime but also estimation of large correlation matrix (see, e.g. Silvennoinen and Teräsvirta (2009) for a survey of recent advances). Among this field of research, our work focus on change in regime specification based on tree structure. Indeed, tree-structured dynamic correlation models has been developed to analyse volatility and co- volatility asymmetries (see Dellaportas and Vrontos (2007)) or linking the dynamics of the individual volatilities with the dynamics of the correlations (see Audrino and Trojani (2006)). The common ap- proach of these models is to partitioning the space of time series recursively using binary decisions. This can be interpreted as a deterministic decision tree. At the opposite, the approach that we adopt for this paper is developed around the idea of hierarchical architecture with a Markov temporal structure.

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MODELLINGVOLATILITYANDCORRELATIONSWITHAHIDDENMARKOVDECISIONTREE.PHILIPPECHARLOTGREQAM&Aix-MarseilleUniversityCentredelaVieilleCharité2,ruedelaCharité13236Marseillecedex02FranceVERYPRELIMINARYDRAFT–COMMENTSAREMOSTWELCOME!ABSTRACT.ThegoalofthepresentpaperistopresentanewmultivariateGARCHmodelwithtime-varyingconditionalcorrelation.SincetheseminalworkofBollerslev(1990),conditionalcorrelationmodelshavebecomeaattractivefieldineconomics.Differentspecificationshavebeendevelopedtostudybothempiricalfindingsandpracticaluselikeasymmetry,changeinregimebutalsoestimationoflargecorrelationmatrix(see,e.g.SilvennoinenandTeräsvirta(2009)forasurveyofrecentadvances).Amongthisfieldofresearch,ourworkfocusonchangeinregimespecificationbasedontreestructure.Indeed,tree-structureddynamiccorrelationmodelshasbeendevelopedtoanalysevolatilityandco-volatilityasymmetries(seeDellaportasandVrontos(2007))orlinkingthedynamicsoftheindividualvolatilitieswiththedynamicsofthecorrelations(seeAudrinoandTrojani(2006)).Thecommonap-proachofthesemodelsistopartitioningthespaceoftimeseriesrecursivelyusingbinarydecisions.Thiscanbeinterpretedasadeterministicdecisiontree.Attheopposite,theapproachthatweadoptforthispaperisdevelopedaroundtheideaofhierarchicalarchitecturewithaMarkovtemporalstructure.OurmodelisbasedonanextensionofHiddenMarkovModel(HMM)introducedbyJordan,Ghahramani,andSaul(1997).ItisafactorialandcoupledHMM.Hence,ourmodelisbasedisastochasticdecisiontreelikingthedynamicsofunivariatevolatilitywiththedynamicsofthecorrelations.ItcanbeviewasaHMMwhichisbothfactorialanddependentcoupled.Thefactorialdecompositionprovidesafactorizedstatespace.Thisstatespacedecompositionisdoneusingstatedependentandtime-varyingtransitionprobabilitiesgivenaninputvariable.Thetoplevelofthetreecanbeseenasamasterprocessandthefollowinglevelsasslaveprocesses.Theconstraintofalevelonthefollowingisdoneviaacouplingtransitionmatrixwhichproducetheorderedhierarchyofthestructure.AsthelinksbetweendecisionstatesaredrivenwithMarkoviandynamicsandtheswitchfromoneleveltothefollowingisdoneviaacouplingtransitionmatrix,thisarchitecturegivesafullyprobabilisticdecisiontree.Estimationisdoneinonestepusingmaximumlikelihood.Wealsoperformanempiricalanalysisofourmodelusingrealfinancialtimeseries.Resultsshowthatourhiddentree-structuredmodelcanbeaninterestingalternativetodeterministicdecisiontree.Keywords:MultivariateGARCH;Dynamiccorrelations;Regimeswitching;HiddenMarkovDecisionTrees.JELClassification:C32,C51,G1,G0.E-mailaddress:ph.charlot@gmail.com.Date:thisversionjanuary2011(firstdraftapril2010).1