Normal Weighted Inverse Gaussian Distributions and Em Algorithm With Appliactions to Risk Measures and Dependence Modelling
Abstract
High frequencyfinancialdataischaracterizedbynon-normality,asymmetric,leptokurtic
and fat-tailedbehaviour.Thenormaldistributionisinadequateincapturingthese
characteristics. Tothisend,variousflexibledistributionshavebeenproposed.Inthis
Thesis weintroducedanewclassofdistributionsknownasNormalWeightedInverse
Gaussian distributions.
WeightedInverseGaussiandistributionsarespecialcasesofGeneralizedInverseGaussian
(GIG)distributionwhicharerelatedtoInverseGaussian(IG)distribution.Finitemixtures
of thesespecialcasesarealsoweightedInverseGaussian(WIG)distributions.Usingthese
WIG distributionsasmixingdistributionstotheNormalVarianceMeanMixture(NVMM)
weobtainaclassofNormalWeightedInverseGaussian(NWIG)distributions.
The propertiesconsideredforthesemodelsaremean,variance,skewnessandkurtosis.
For dataanalysisweconsiderthreedatasets:RangeResourceCorporation(RRC),Shares
of ChevronCorporation(CVX)ands&p500index.Theperiod3/01/2000to1/07/2013with
702 observationsforeachdatasetisconsidered.Estimationofparametersofthesemodels
areobtainedusingExpectation-Maximization(EM)algorithm.TheEMalgorithmisa
powerfultechniqueformaximumlikelihoodestimationfordatacontainingmissingvalues
or datathatcanbeconsideredascontainingmissingvalues.Themixingoperationcanbe
consideredresponsibleforproducingmissingvalues.
TwoimportantriskmeasuresinliteratureareValueatRisk(VaR)andExpectedShortfall
(ES). InthisworkwehaveobtainedVaRandESfortheNWIGdistributions.Backtesting
of thismeasuresisalsoperformed.
Wehavealsoconsidereddependencemodellingoffinancialreturnsusingcopulas.The
marginal distributionsarebasedonNormalWeightedInverseGaussiandistributions.
Wehighlightthefollowingcontributionstothiswork
1. WehaveconstructedanewclassofWeightedInverseGaussiandistributions.
2. WehaveusedthisclassasamixingdistributiontotheNormalVarianceMeanMixture
to obtainaclassofNormalWeightedInverseGaussiandistributions.
3. All worksonparameterestimationofEMalgorithmatthemaximizationstepisbased
on explicitsolutiontonormalequations.Oftenthisinvolvesnumericaltechniques
which aredifficulttoimplement.Inthiswork,weshowthattheiterativeschemes
arenotnecessarilybasedonexplicitsolutions.Theycanalsobedesignedusing
a representationbasedonthenormalequations.Thissubtleapproachiseasily
programmableandpreservesthemonotonicconvergencepropertyoftheEMalgorithm
with eachiterationincreasingthelikelihood.
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4. Fromthedatasetsused,thisclassofNWIGdistributionsisshowntobeagood
alternativetotheNormalInverseGaussiandistribution.However,onespecialcaseof
the GIGdistributionwhenusedasamixingdistributiontoNVMMoutperformsthe
NIG andonefinitecasewhenusedasamixingdistributionoutperformsallmodels.
5. Using backtestingproceduresitcanbeshownthatthisclassofdistributions,NWIG,
which haveheavytailed,isanalternativecandidateforfinancialriskmanagement.
6. The modelshavealsobeenusedasmarginalsindependencemodellingusingcopulas
approach.
Publisher
University of Nairobi
Rights
Attribution-NonCommercial-NoDerivs 3.0 United StatesUsage Rights
http://creativecommons.org/licenses/by-nc-nd/3.0/us/Collections
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