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Application of Asymmetric Laplace Laws in Financial Risk Measures and Time Series Analysis

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IamgratefultomyPh.D.advisor,Dr.AlexTrindade.Thisworkcouldnothavebeenwrittenwithouthimwhonotonlyservedasmysupervisorbutalsoencouragedandchallengedmethroughoutmyacademicprogram.Iamgratefulforhisimmensehelpateverystageofmyresearch,frominitiatingthetopics,solvingproblems,torevisingnumerousdrafts.Hisvaluableinsightsandideasdirectlyandsignicantlycontributedtomydissertation.Iwouldliketothankmycommitteenumbers,Dr.RamonLittell,Dr.RonaldRandles,Dr.ClydeSchooleldandDr.FaridAitSahlia,fortakingthetimetoworkwithme.Thanksgoouttomyhusband,alwaysoeringsupportandlove.Thanksgoouttomyparents,fortakingcareofmeandmybaby.Icouldnothavenishedmydissertationwithouttheirsupport.ThankyoumydearKatieforprovidingmehappinessandinspiration. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 1.1FinancialRiskMeasurementviaVaRandCVaR ............... 13 1.2AsymmetricLaplaceDistribution ....................... 20 2APPROXIMATIONTOTHEDISTRIBUTIONOFMLESOFVARANDCVARUNDERALLAW .................................. 24 2.1MaximumLikelihoodEstimationsofVaRandCVaRunderALDistribution 24 2.2Asymptoticdistributionof~(X)and~(X) ................ 25 2.3ApproximationofFiniteSampleDistribution ................. 27 2.3.1GeneralSaddlepointApproximationtoLStatistics ......... 27 2.3.2ApproximationtotheFirstFourCumulantsofMLEsofVaRandCVaR ................................... 29 2.3.3AssessingtheAccuracyoftheSaddlepointApproximations ..... 33 3APPROXIMATIONTOTHEDISTRIBUTIONOFNONPARAMETRICESTIMATORSOFVARANDCVARUNDERALLAW ............. 38 3.1NonparametricEstimatorsofVaRandCVaR ................ 38 3.2AsymptoticDistributionof^(X)and^(X) ................ 39 3.3ApproximationofFiniteSampleDistribution ................. 39 3.3.1MomentGeneratingFunctionofNonparametricEstimatorsofVaRandCVaR ................................ 39 3.3.2SaddlepointApproximationandLugannani-RiceFormula ...... 46 3.3.3LaplaceApproximationofHypergeometricFunction ......... 47 3.4ComparisonoftheDistributionsofParametricandNonparametricEstimators .................................... 49 3.4.1LargeSampleCase ........................... 49 3.4.2FiniteSampleCase ........................... 50 3.5AnalysisofExchangeRateData ........................ 50 4TIMESERIESARMAANDGARCHMODELSUNDERALNOISE ...... 58 4.1ARMA(p;q)Model ............................... 61 4.2ARMA(p;q)ModelunderALNoise ...................... 63 5

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....... 65 4.2.2FitAR(p)ModelUsingConditionalMaximumLikelihoodEstimation ................................ 70 4.2.3FittinganARMA(p;q)ModelUsingConditionalMaximumLikelihoodEstimation .......................... 73 4.3ARMAModelsDrivenbyGARCHNoise ................... 75 4.3.1ARMAModelDrivenbyGARCHnoise ................ 75 4.3.2ConditionalMaximumLikelihoodEstimationofGARCHmodel .. 76 4.3.3ARMAModelsDrivenbyGARCHALNoise ............. 77 4.4AnalysisRealEstateMutualFundData ................... 78 APPENDIX ASAR(P)MODELWITHMULTIVARIATEALMARGINALDISTRIBUTION 89 A.1SAR(p)ModelwithMultivariateALMarginalDistribution ......... 89 A.1.1SAR(p)Model .............................. 89 A.1.2GeneralizedEstimatorof 90 A.1.3MultivariateAsymmetricLaplaceDistribution ............ 91 A.1.4SaddlepointApproximationtotheEstimatingEquation ....... 99 A.1.5ApproximatetheMomentsofrbyTaylorExpansion ......... 104 REFERENCES ....................................... 109 BIOGRAPHICALSKETCH ................................ 113 6

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Table page 3-1MLEsofthelogreturnsofexchangeratedata ................... 55 4-1FittedmodelparametersofAR(1)modelunderALnoise ............. 72 4-2FittedmodelparametersofAR(2)modelunderALnoise ............. 73 4-3FittedmodelparametersofAR(3)modelunderALnoise ............. 73 4-4FittedvalueofARMA(1,1)modelunderALnoise ................. 75 4-5FittedValueofARMA(1;3)underGaussiannoise ................. 81 4-6FittedparametersofARMA(1;3)drivenbyGARCH(1,1)Gaussiannoise. .... 82 4-7FittedparametersofARMA(2;6)underALnoise ................. 84 4-8FittedparametersofARMA(1;3)drivenbyGARCH(1,1)ALnoise. ....... 86 4-9SummaryofAICcofthefourmethods ....................... 86 7

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Figure page 1-1Heavytaileddistributionvs.normality. ....................... 16 1-2Histogramofthedailylogreturnsofexchangeratedata ............. 17 1-3PDFofAsymmetricLaplacedistribution ...................... 21 2-1CDFsofMLEofVaRwith=0:9 ......................... 34 2-2CDFsofMLEofVaRwith=0:99 ........................ 34 2-3CDFsofMLEofVaRwithn=50and=1 .................... 35 2-4CDFsofMLEofVaRwithn=50and=0:8 ................... 35 2-5CDFsofMLEofCVaRwith=1 ......................... 36 2-6CDFsofMLEofCVaRwith=0:8 ........................ 36 2-7PREsofthesaddlepointapproximateddistributionofMLEofVaR ....... 37 2-8PREsofthesaddlepointapproximateddistributionofMLEofCVaR ...... 37 3-1CDFsofNPEofVaRwith=1 .......................... 48 3-2CDFsofNPEofVaRwith=0:8 ......................... 48 3-3PREsofthesaddlepointapproximateddistributionofNPEofVaR ....... 49 3-4AREofMLEswithrespecttoNPEsofVaRandCVaR .............. 51 3-5SaddlepointapproximateddensityfunctionsofMLEsandNPEs ......... 52 3-6Histogram,boxplotandsampleACFofdailylogreturnsofexchangerate .... 53 3-7NormalQ-Qplotofdailylogreturnsofexchangerate. .............. 54 3-8Histogramofthedailylogreturnswithoutweekends ............... 54 3-9NormalQ-Qplotofdailylogreturnswithoutweekends .............. 55 3-10CondenceellipsesfortheMLEsandNPEsbivariateestimatorsofVaRandCVaR ......................................... 57 4-1SimulatedARMA(1,1)processunderALnoise ................... 64 4-2HistogramofoftheSimulatedARMA(1,1)process ................ 64 4-3DerivedmarginalpdfofAR(1)modelunderALnoise ............... 70 4-4ComparisonofderivedmarginalpdfandsimulatedhistogramofAR(1)model 70 8

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............ 71 4-6ComparisonofderivedmarginalpdfandsimulatedhistogramofARMA(2,2)model ......................................... 71 4-7Dailyvaluesofthemutualfund ........................... 78 4-8Dailyreturnsofthemutualfund .......................... 78 4-9Histogramofthedailyreturns ............................ 79 4-10SampleACFofresidualsofARMA(1,3)modelunderGaussiannoise. ...... 81 4-11SampleACFofabsolutevaluesofresidualsofARMA(1,3)modelunderGaussiannoise. ......................................... 81 4-12SampleACFofsquaresofresidualsofARMA(1,3)modelunderGaussiannoise. 82 4-13SampleACFofresidualsofARMA(1,3)modeldrivenbyGARCH(1,1)Gaussiannoise. ......................................... 83 4-14SampleACFofabsolutevalueofresidualsofARMA(1,3)modeldrivenbyGARCH(1,1)Gaussiannoise ............................. 83 4-15SampleACFofsquaresofresidualsofARMA(1,3)modeldrivenbyGARCH(1,1)Gaussiannoise. .................................... 84 4-16SampleACFofresidualsofARMA(2,6)modelunderALnoise. ......... 85 4-17SampleACFofabsolutevaluesofresidualsofARMA(2,6)modelunderALnoise. 85 4-18SampleACFofsquaresofresidualsofARMA(2,6)modelunderALnoise. ... 86 4-19SampleACFofresidualsofARMA(1,3)modeldrivenGARCH(1,1)ALnoise. 87 4-20SampleACFofabsolutevaluesofresidualsofARMA(1,3)modeldrivenGARCH(1,1)ALnoise ................................ 87 4-21SampleACFofsquaresofresidualsofARMA(1,3)modeldrivenGARCH(1,1)ALnoise. ....................................... 88 A-1Saddlepointapproximatedpdfofburgestimatorwithn=10 ........... 100 A-2Comparisonoftheapproximatedpdfandsimulatedhistogramwithn=10 .... 101 A-3Saddlepointapproximatedpdfofburgestimatorwithn=50 ........... 101 A-4Comparisonoftheapproximatedpdfandsimulatedhistogramwithn=50 .... 102 A-5Saddlepointapproximatedpdfofburgestimator .................. 103 A-6Comparisonoftheapproximatedpdfandsimulatedhistogram .......... 104 9

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........... 106 A-8Comparisonoftheapproximatedpdfandsimulatedhistogramwithn=10 .... 107 A-9Saddlepointapproximatedpdfofburgestimatorwithn=50 ........... 107 A-10Comparisonoftheapproximatedpdfandsimulatedhistogramwithn=50 .... 108 10

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AsymmetricLaplace(AL)lawsareappliedinnancialriskmeasurementandtimeseriesanalysis.Traditionalmethodsonnancialriskmeasuresandtimeseriesanalysisarebasedontheassumptionofnormality.Recentstudiesonnancialdatasuggestthatthenormalityassumptionisusuallyviolated. Explicitexpressionsarederivedformaximumlikelihoodestimators(MLEs)andnonparametricestimators(NPEs)ofnancialriskmeasures,Value-at-Risk(VaR)andConditionalValue-at-Risk(CVaR),underrandomsamplingfromtheAsymmetricLaplacedistribution.Asymptoticdistributionsareestablishedunderverygeneralconditions.Finitesampledistributionsareinvestigatedbymeansofsaddlepointapproximations.AnapplicationofthemethodologyinmodelingcurrencyexchangeratessuggeststhattheALdistributionissuccessfulincapturingthepeakedness,leptokurticityandskewness,inherentinsuchdata. Timeseriesautoregressivemovingaverage(ARMA)modelsdrivenbyAsymmetricLaplacenoiseareconsideredformodelingdependentdata.AssumingALnoise,themodelmarginaldistributionisderivedanalytically.ConditionalmaximumlikelihoodestimationisappliedtotARMAmodelsdrivenbyALnoiseandALgeneralautoregressiveconditionalheteroscedasticity(GARCH)noise.Dailyreturnsofrealestatemutualfunddataarettedbyfourmethods.ModelsunderALnoisehavesubstantiallylower 11

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12

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TheideabehindValue-at-Riskoriginatedfrommeasuringmarketrisk.VaRisthelossthatcanoccuroveragivenperiod,atagivencondencelevel,duetoexposuretomarketrisk.Recently,theideaofVaRhasbeenintroducedintomeasuringcreditrisk. Theimportanceofmeasuringtheriskofnancialassetshaslongbeenrealized.Markowitz(1959)rstintroducedthedenitionandmeasurementofriskinportfolioselection,wherenancialriskwasmeasuredbythevarianceandcovarianceofunderlyingassetprices. Since1990's,Value-at-Riskiswidelyusedbycommercialbanks,assetmanagementcompanies,andregulators.Forexample,theBaselAccordIemploysVaRasthemeasurementforcommercialbanks'marketriskexposure.TheBaselAccordistheinternationalcapitaladequacystandardssetupbyBaselCommitteeonBankingSupervision.The1996amendmentoftheBaselAccordextendsthecapitalrequirementstoincluderisk-basedcapitalforthemarketriskinthetradingbook.UnderthesupervisionoftheBaselCommittee,banksneedtosetuptheirownVaRmodelstocalculatetheirminimumregulatorycapitalformarketrisk.In1997,theSecuritiesandExchangeCommissioninUnitedStatesbeganrequiringnancialinstitutionstoreportValue-at-Riskasanimportantmeasureofthemarketriskexposure. 13

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Looselyspeaking,ifYrepresents"losses"and0<<1,giventimehorizont,theVaRatcondencelevel,VaR(Y),isthelowerboundontheworst(1)100%lossesduringthetimehorizon. letYbeacontinuousreal-valuedrandomvariabledenedonsomeprobabilityspace(;A;P),withcumulativedistributionfunction(cdf)F()andprobabilitydensityfunction(pdf)f().Letand2denotethemeanandvarianceofY,respectively,andbothareassumedtobenite.WiththeunderstandingthatYrepresentsloss,theVaRofYatprobabilitylevelisdenedtobethethquantileofY. VaR(Y)(Y)=F1(): Thecondencelevelandtimehorizontvaryamongdierentbanks,companiesandregulators.Acommonlyusedcondencelevelis99%.Forexample,BaselAccordhassettobe99%andttobe10daysinordertomeasurebanks'marketriskexposure.CommercialbankstypicallyuseovernightValue-at-Risktomeasurenancialriskexposureforthepurposeofinternalsupervisionandriskcontrol,anddisclosetwo-weekValue-at-Risktoinvestorsandregulators. Practically,thereisa'squarerootoftime'rule.Thatis,ifthedailyValue-at-Riskis,thentheValue-at-Riskinthetimehorizonofmdayswillbep 14

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AsanalternativetoVaR,ConditionalValue-at-Risk(CVaR)describestheaverageoftheworst(1)%losses.ThetermofCVaRisdrawnfromRochafellarandUryasev(2000),butsynonymsforitalsoincommonusageinclude:"expectedshortfall"(AcerbiandTasche,2002),and"tail-conditionalexpectation"(Artzneretal.,1999). TheCVaRofYatprobabilitylevel,isthemeanoftherandomvariablethatresultsbytruncatingYatVaR(Y)anddiscardingitslowertail. CVaR(Y)(Y)=E(YjY)=1 1Z1yf(y)dy: AnequivalentdenitionofCVaRintermsofthequantilefunctionofYis 1Z1F1(u)du: Parametricmethodsassumeadistributionforthenancialdata.Underparametricmethods,theVaRatcondencelevelisjustthethquantileofthedistribution.Parametricmethodsdependontheassumptionofadistribution.Thenormaldistributionisthemostcommonlyusedfamilyinnancialriskmeasurement.Butthereareobviousviolationsofnormalityinnancialdata.Financialdataaretypicallyskewedandheavytailed.Anotherproblemwithparametricmethodsisthattheyareinappropriatewhentherearediscontinuouspayosintheportfolio. 15

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Semiparametricmethodsassumethedistributionofonetail,whichmakingnoassumptionabouttheunderlyingdistributionawayfromthetail.Atypicalusedtaildistributionisparetotail. AssumingnormalityonheavytaileddatawillcauseunderestimationofVaRathighcondencelevel,whichwillcausemajorproblemsinriskcontrol. Figure1-1. Heavytaileddistributionvs.normality. 16

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HistogramofthedailylogreturnsofUSD/EURexchangeratefromFeb1,2005toJan31,2006. Fig. 1-2 isthedailylogreturnofUSD/EURexchangerates.Thishistogramindicatesthetypicalpropertiesofnancialdata,heavytailsandskewness.Commonlyusedmeasuresofheavytailsandskewnessarekurtosisandskewness,respectively.Formoredetailedinformationaboutkurtosisandskewness,pleaserefertoSection1.2.Properlyadjusted,thekurtosisofanormaldistributionis0.Andwhenkurtosisisgreaterthan0,thedataareconsideredheavytailed.Theskewnessofanysymmetricdistributionis0.Thekurtosisandskewnessofthedataare2.0119and-0.3777,respectively.Therefore,weconsiderthemtobeheavytailedandskewed. Theoretically,therearemanyexplanationaboutthereasonsforfattails.Twoofthemarewidelyaccepted:Therearesomesignicantdiscontinuouschangesinthenancialdataduetounexpectedchangesinmarketfactors,forexample,marketcrash.Thisisalsocalled'jumps'.Theotherexplanationiscalled'VolatilityClustering',thevolatilityattimetishighlycorrelatedwithvolatilitiesatpasttimes,s
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Statistically,let1and2bethedensityfunctionoftwonormaldistributionswithdierentmeanandvariance.Xisarandomvariablefromamixtureofnormalsif wherePisaBernoullirandomvariablewithsuccessprobabilityp.Underthemixturenormalmodel,wecanchoosepandvariancesoftwonormaldistributionstoachieveagivenkurtosis,andthereforegetafattailedX. Thejump-diusionmodelwasintroducedinthecontextofdierentialequations.Forthejump-diusionmodel,nancialdatais'jumped'byaddinganindependentnormalrandomvariable. TheARVmodelassumes log2t=+log2t1+Zt; where,andareconstants.fZtgiswhitenoise.Thevolatilityattimetisafunctionofvolatilityatt1.Asaconsequence,thevolatilitiesarecorrelated.Avalueofnearzeroimplieslowcorrelation,whileavalueofbetanear1implieshighcorrelation. TheEWMAmodelwasintroducedbytheRiskMetricsGroup.Thevarianceattimetisestimatedas 18

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with>0,whereisa'smoothingconstant'andXt1isthereturnattimet1.Thevolatilityattimetisconsideredasafunctionofthereturnandvolatilityattimet1.Since>0,thevolatilityattimetwillbepositivelycorrelatedwiththevolatilityatt1. GARCHisamoreexibleandgeneralmodeldescribingtheclusteringofvolatility.Bollerslev(1986)introducedtheGARCHprocess.Thevolatilityisestimatedas wherefZtgiswhitenoise.00,j0,j0,j=1;2;:::.UnderaGARCHmodel,thevolatilityattimetdependsonthevolatilityofthepast.FormoredetailedinformationaboutGARCHmodel,pleaserefertoChapter4. Somepotentiallymoreexiblemodelsinclude:ExponentialGARCHmodel,EGARCH,Cross-MarketGARCH,etc. ExponentialPowerDistribution(EPD),alsocalledGeneralizedPowerDistribution(GPD),hasalsobeenusedinnancialriskmeasure.ThedensityofEPDis 2a(1+1=b)exp(1jx=ajb): 19

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TheSymmetricLaplaceDistribution(SLD),alsocalledDoubleExponentialDistribution,hasbeenusedformodelingdatawithheavytails.SeeBalakrishnanandBasu(1995),BainandEngelhardt(1973),Kotz,KozubowskiandPodgoriski(2001).SLDdistributionhasthesameproblemasothersymmetricdistributionswhichdonotallowforanyasymmetry.Therefore,weconsidertheasymmetricformoftheLaplacedistribution,theAsymmetricLaplacedistribution. 20

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1+2expp jyj;ify;2 Figure1-3. AsymmetricLaplacedensitieswith=0,=1,=0:5;0:8;1 Kotzetal.(2001),ch.3,generalizetheessentialpropertiesofAsymmetricLaplacedistribution. 22t2 : 21

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ForanAL(;;)distribution,thecoecientofskewnessis Thecoecientofskewnessisnonzerounless=1.Theabsolutevalueof1isboundedbytwo,andasincreaseswithintheinterval(0;1),thecorrespondingvalueof1decreasesmonotonicallyfrom2to2. Foranrandomvariablewithanitefourthmoment,thekurtosisisdenedas Itisameasureofpeakednessandofheavinessofthetails(properlyadjusted,sothat2=0foranormaldistribution)andisindependentofthescale.If2>0,thedistributionissaidtobeLeptokurtic,withheavytailsandhighpeakedness;itisPlatykurticotherwise. ForanAL(;;)distribution,thekurtosisofALdistributionis (1=2+2)2: 22

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p Therefore,theVaRandCVaRaretheneasilyderived. and p 23

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Inthischapter,wegivetheexplicitexpressionsofthemaximumlikelihoodestimators(MLEs)ofVaRandCVaRunderAsymmetricLaplacedistribution.LargesampleasymptoticdistributionsofMLEsareestablishesviaDeltamethods.FinitesampleapproximationsofMLEsaredevelopedbygeneralsaddlepointapproximation.Finally,theaccuracyoftheapproximationsischeckedviasimulations. Wewillassumethatthelocationparameter,,isknown.BothVaRandCVaRaretranslationinvariantandpositivelyhomogenous(Pug,2000),i.e.(Y)=+(X)and(Y)=+(X).Withoutlossofgenerality,inthischapterwefocusonXAL(0;;),provided=0isknown.Consequently,VaRandCVaRcanbepresentedas p ~=1 4~=p 41 48<:vuut wherex+i=xiI[xi0]andxi=xiI[xi<0]. 24

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~=V2 4;and~=p 4p ThentheMLEsofVaRandCVaRareobtainedbyequivariance, ~(X)=~log[(1+~2)(1)] ~p ~(X)=~(X)+~ or,equivalently, ~(X)=V1+p 2logV1+log(1);~(X)=~(X)+V1+p Kotzetal.(2001),ch.3,provetheconsistencyandasymptoticnormalityof~and~. 2{1 are (i)Stronglyconsistent; (ii)Asymptoticallybivariatenormalwiththeasymptoticcovariancematrix 82375; (iii)Asymptoticallyecient,namely,thisasymptoticcovariancematrixcoincideswiththeinverseoftheFisherinformationmatrix. 25

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Letg:DgRk7!RmbeamapdenedonasubsetofRkanddierentiableat.LetTnberandomvectorstakingtheirvaluesinthedomainofg.Ifp Letg(Tn)=264375, then where Applyingthemultivariatedeltamethodandtheasymptoticnormalitiesof[~;~]inProposition2.1,wegettheasymptoticdistributionofMLEsofVaRandCVaR 2{10 andtheMLEsofVaRandCVaR,[~;~],asinEq. 2{1 ,respectively,wehave

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~2()=2 42(2+2)!;22: 2.3.1GeneralSaddlepointApproximationtoLStatistics Supposethatx1;:::;xnareniidrealvaluedrandomvariableswithdensityf.Un(x1;:::;xn)isarealvaluedstatisticwithpdffnandcdfFn.LetMn(t)=Retxfn(x)dxdenotesthemomentgeneratingfunctionandKn(t)=logMn(t)thecumulantgeneratingfunctionofUn.FurthersupposethatthemomentgeneratingfunctionofMn(t)existsforrealtinsomenonvanishingintervalthatcontainstheorigin. Fourierinversiongives 2Z1Mn(it)eitxdt=n whereisanyrealnumberintheintervalwherethemomentgeneratingfunctionexists. Let 27

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~Rn(T)=nT+n2nT2 wheren,2n,3n,4narethemean,thevariance,andthethirdandfourthcumulantofUn. WeassumethattheEdgeworthexpansionuptoandincludingthetermofordern1forfnexists.Expansionsoftheform willsucetokeepthesameorderintheapproximation. ApplyingthesaddlepointtechniquetotheintegralinEq. 2{12 givesthesaddlepointapproximationoffnwithuniformerroroforderO(n1), 2n2~R00n(^t)o1 2expnn~Rn(^t)n^txo; where^tisthesaddlepointanditisthesolutiontothesaddlepointequation,~R0n(^t)=x;~R0n(),~R00n()denotetherstandsecondderivativesof~Rn(t). Inaddition,ifweapplythesametechniquetoLugannaniandRice(1980)formula,wegetthesaddlepointapproximationtothecdfofUn, ^r1 ^q; where()and()arethestandardnormaldistributionanddensityfunctionswith ^r=sgn^th2nn^tx~Rn(^t)oi1 2^q=^tnn~R00n^to1 2:

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2{17 ,atx=E(Un),thealternateexpressionusedis 2+(72n)1 2~R000n(0)~R00n(0)3 2; Considertheauxiliaryrandomvector whereZ(i)1=X+iandZ(i)2=Xi.Therefore,V1=Zn1andV2=Zn2. AccordingtoKotzetal.(2001),ch.3,Z(i)areindependentandidenticallydistributedas,Z(i)d=0B@ (1+2): Therstfourmomentsof2are 29

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(1t).Therefore,therstfourmomentsofWareEW=1,EW2=2,EW3=6,EW4=24. Since1and2areindependentofW1andW2,respectively,therstfourmomentsofZ1andZ2caneasilybeobtained, ThevariancesofZ1andZ2canberepresentedas VarZ1=22 NotethatEZ(i)1Z(i)2=0sinceatleastoneofZi1andZi2equaltozero.ThecovarianceofZ1,Z2canbeobtainedas Cov(Z1;Z2)=22 OtherthirdandfourthcentralmomentsandcovariancescanalsobededucedfromtherstmomentsofZ1andZ2byapplyingEq. 2{24

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. Letf1(V1;V2)=~(X),1=EZ1=EV1and2=EZ2=EV2.ByapplyingtherstvetermsofTaylorexpansionforthetwovariables,V1andV2,wecanapproximatethemeanof~(X)witherrortermofn3, ~(X)=f1(V1;V2)f1(1;2)+"(V11)@f1 2(V11)2@2f1 3!(V11)3@3f1 4!(V11)4@4f1 Takeexpectationsonbothsides,themeanof~(X)canbeapproximatedas 31

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2VarV1@2f1 6E(V11)3@3f1 24E(V11)4@4f1 SinceV1=Z(n)1andV2=Z(n)2,thisapproximationcanbefurtherexpressedas 2nVaRZ1@2f1 6n2E(Z11)3@3f1 24n3E(Z11)4@4f1 32

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Toapproximatetherstfourcumulantsof~(X),weapplytherelationshipofmomentsandcumulants, Astheresult,theapproximationtothedistributionof~(X)and~(X)canbeobtained. Let^Fsim(r)and^Fsad(r)denotetheestimatesofthetruecdfsofmaximumlikelihoodestimator,obtainedviasimulationsandsaddlepointapproximations,respectively.ThePercentrelativeerror(PRE)ofthecdfsisacommonlyusedtechniquetomeasuretheaccuracyofthesaddlepointestimation.WedenethePREatthequantileras PRE(r)=8><>:^Fsad(r)^Fsim(r) ^Fsim(r)100;^Fsim(r)0:5;(1^Fsad(r))(1^Fsim(r)) 33

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Toassesstheaccuracyofthissaddlepointestimation,weapproximatecdfsofMLEsofVaRandCVaRwithparametersn=50;100;=0;=0:8;1;=1;=0:9;0:99.Wecalculateempiricalcdfsthrough106simulations.PREsarecomputedat10pointsofequaldistancebetweenthe10%90%quantiles. Figure2-1. EstimatedCDFsof~(X),obtainedviasimulationsandsaddlepointapproximations,withn=100,=0;=1;=1;=0:9 Figure2-2. Estimatedcdfsof~(X),obtainedviasimulationsandsaddlepointapproximations,withn=100,=0;=1;=1;=0:99 34

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Estimatedcdfsofthe~(X),obtainedviasimulationsandsaddlepointapproximations,withn=50,=0;=1;=1;=0:9 Figure2-4. Estimatedcdfsofthe~(X),obtainedviasimulationsandsaddlepointapproximations,withn=50,=0;=1;=0:8;=0:9 Calculationsbasedonafewequispacedpointsbetweenthe10thand90thpercentiles,revealthatPREsfortheMLEsofVaRandCVaRarebetween1%and4:5%,withn=50;100;=0:8;1;=0:9;0:95;0:99.ThisindicatesthatourapproachworkswelltoapproximatethedistributionsofMLEsofVaRandCVaRfromtheAsymmetricLaplacedistribution. 35

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Estimatedcdfsof~(X),obtainedviasimulationsandsaddlepointapproximations,withn=100,=0;=1;=1;=0:9 Figure2-6. Estimatedcdfsof~(X),obtainedviasimulationsandsaddlepointapproximations,withn=100,=0;=1;=0:8;=0:9 Wehavechosen=0:9;0:95;0:99becausetheyarecommonlyusedprobabilitylevelsinnancialriskmeasures.Wechoose=0:8becauseitisclosetothettedvaluesobtainedfromarealdataset,while=1indicatesthesymmetriccase. Also,notefromFig. 2-7 andFig. 2-8 ,thePREsarelessthanzeroforlargevalueofr,whichmeansthatwetendtooverestimate~(X)and~(X)overtherighttails.Theapproximationsarebetterforlargersamplesizes,andalsobetterforsymmetriccases. 36

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Percentrelativeerrors(PREs)ofthesaddlepointapproximationtothedistributionof~(X)withn=100;=0;=1;0:8;=1;=0:9,computedatthesamequantilevalues. Figure2-8. Percentrelativeerrors(PREs)forthesaddlepointapproximationtothedistributionof~(X)withn=100;=0;=1;0:8;=1;=0:9,computedatthesamequantilevalues. 37

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Inthischapter,westudythenonparametricestimatorsofVaRandCVaR.WeapproximatethedistributionsofnonparametricestimatorsofVaRandCVaRusingsaddlepointapproximation.WederivethemomentgeneratingfunctionsofNPEsandthenapproximatethedistributionsofNPEsusingsaddlepointapproximation.ThemomentgeneratingfunctionsofVaRandCVaRaremixturesofhypergeometricfunctionswhichmakesthecalculationmorecomputationalintensive. WeanalyzetheperformanceoftheMLEsandNPEsbycomparingthesaddlepointapproximateddistributions.DailylogreturnsofUSD/EURexchangeratearestudiedassumingIIDALdistribution. ^(Y)=Y(k); wherethek=[n]denoteseitherofthetwointegersclosestton.TheNPEofCVaRisthecorrespondingempiricaltailmean, ^(Y)=1 Notethat^(Y)and^(Y)beinglinearcombinationsoforderstatistics,areknownasL-statistics(DavidandNagaraja,2003). 38

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SincetheNPEofVaRissimplyanorderstatisticofarandomsample,theasymptoticdistributionof^(X)comesfromthestandardresultoftheasymptotictheoryoforderstatistics,forexample,DavidandNagaraja(2003),ch.10.Theasymptoticsof^(X)aremorecomplex.ThisresultwasrstderivedbyStigler(1973)inthecontextofthetrimmedmean. Consistencyandthejointasymptoticdistributionof(^;^)underiidsamplingfromacontinuouscdfFwithpdff,hasrecentlybeenestablishedbyGiurcanuandTrindade(2005)usingthetheoryofestimatingequations.Dene2bethevarianceofthedistributionobtainedbytruncatingthedistributionofYat,i.e., 1Z1(y)2f(y)dy: 3.3.1MomentGeneratingFunctionofNonparametricEstimatorsofVaRandCVaR 39

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(b)(cb)Z10ub1(1u)cb1(1uz)adu; wherea,b,c,zarerealconstantsand()isthegammafunction.Thehypergeometricfunctionconvergesforjzj<1providedc>a+b1. (i)Tn(Y)=+Tn(X); ): andconvergesforalljcj<1,providedb>2. 3{5 40

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(j1)!(nj)!fX(x)[FX(x)]j1[1FX(x)]nj: where 1+2; Thepdfandcdfofthestandardcasecanberepresentedas 1+2"expp I(x<0)!+expp and !#I(x<0)+11 1+2expp ApplyingLemma3.3,thepdfofX(r)canbeexpressedas (r1)!(nr)!fX(x)[FX(x)]r1[1FX(x)]nr: 41

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(r1)!(nr)!Z0etxfX(x)FX(x)(r1)[1FX(x)]nrdx+Z+10etxfX(x)FX(x)(r1)[1FX(x)]nrdxc(n;r)fJ1(t)+J2(t)g: ForJ1(t),applytransformationu=FX(x)I(x<0)=2 I(x<0),then00)=1 1+2exp(p 1+2.x=log[v(1+2)]1 1+2etlog[v(1+2)]1 1+20[v(1+2)]t 1+20(1+2)t 42

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1+2,weget 1+2)(nr+1) 1+2: ApplyingLemma3.1,givesthedesiredresult. If^=Y(k)denotestheNPEofVaR(Y)basedonarandomsampleofsizenfromAL(;;),itsmgfisgivenbyTheorem3.2withr=[n]. LetY(nk+1):::Y(n)bethehighestk,1kn,orderstatisticsunderiidsamplingfromAL(;;).LetSn;k(Y)=Pni=nk+1Y(i)betheirsum.ThenthemgfofSn;k(Y)isthemixtureofhypergeometricfunctions, where p p 1+2;d1(t)=p 43

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(Sn;kjX(nk)=x)d=kXi=1Xi; whereX1;:::;Xkareiidwithpdffx(y)=f(y)I(y>x)=(1F(x)).Therefore,thepdfofSn;kisgivenby withcorrespondingmgf whereMx(t)denotesthemgfofX1,whichdependsonx.Thistechnique,whichonlyworksiftheintegrationcanbeperformedanalytically,hasbeenusedbyAlamandWallenius(1979)forobtainingthedistributionofSn;kinarandomsamplefromagammadistribution.Sincex
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Inthecasex>0,i.e.,0
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n[n]+1. If^=1 2expKn(^t)^tx; where^tisthesaddlepointanditisthesolutiontothesaddlepointequation,K0n(^t)=x;andK0n(),K00n()denotetherstandsecondderivativesofthecumulantgeneratingfunctionKn(t).TherelativeerroroftheapproximationisoforderO(n1). ThesaddlepointapproximationtothecumulativedistributionfunctionofSn,duetoLugannaniandRice(1980),isgivenby ^r1 ^q; where()and()arethestandardnormaldistributionanddensityfunctionswith ^r=sgn^t2^txKn^t1 2^q=^tK00n^t1 2: 3{27 ,atthemeanofthedistribution,i.e.,atx=E(Sn),q=0,sothatthealternateexpressionusedis 46

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2+(72)1 2K000n(0)K00n(0)3 2; whereK000n()isthethirdderivativeofthecumulantgeneratingfunctionKn(t). SincethemgfsofNPEsofVaRandCVaRareavailableinclosedform,thedensityanddistributionfunctionsof^(x)and^(x)canbeapproximatedbyapplyingthesaddlepointapproximation.Thereisaprobleminthatbothofthemgfsaremixturesofhypergeometricfunctions,whicharecomputationallyburdensometoevaluateexplicitlyduetoslowconvergenceofthepowerseriesexpansionsdeningthehypergeometricfunction.AcomputationallymoreecientalternativeistoemployinsteadtheLaplaceapproximationsofthehypergeometricfunctiondevelopedbyButlerandWood(2002). ^F2;1(a;b;c;x)=cc1=2r1=22;1^y aa1^y caca(1x^y)b; where ^y=2a Therefore,byapplyingthisapproximationwecanestimatethedistributionsof^(X)and^(X)underALlaw.Togetamoreaccurateapproximation,wenormalizedthecumulategeneratingfunctionbysubtractingitsapproximatedvalueatx=0. 47

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Figure3-1. Estimatedcdfsof^(X),obtainedviasimulationsandsaddlepointapproximations,respectively,underALlawwithn=50;=0;=1;=1;=0:9 Figure3-2. Estimatedcdfsof^(X),obtainedviasimulationsandsaddlepointapproximations,respectively,underALlawwithn=50;=0;=1;=0:8;=0:9 48

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Percentrelativeerrors(PREs)forthesaddlepointapproximationtothedistribution^(X)underALlawwithn=100;=0;=1;0:8;=1;=0:9,computedatthesamequantilevalues. Ingeneral,wehavegoodestimationforNPEsofVaRandCVaRunderALdistribution.ThePREsforthecdfsof^(X)arelessthan2%andPREsforcdfsof^(X)arebetween2%to14%.Notealso,inFig. 3-3 ,thePREsarelessthanzero,whichindicatesatendencytooverestimate^(X)and^(X). 49

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ARE(~;^)=P(~) whiletheAREfortheNPEsofCVaRis, ARE(~;^)=P(~) where!;isdenedthesameasinEq. 2{10 Fig. 3-4 displaysbothAREsasafunctionof0:5and0<1. 3-5 plotthesaddlepointapproximatedcdfsofMLEsandNPEsandcomparethedistributionswiththetruevaluesofVaRandCVaR. Noteingeneral,theMLEsaremoresymmetricandunbiased,whileNPEsarerightskewedandbiased.AlsotheNPEsaremoreskewedandbiasedastheALdistributionbecomesmoreasymmetric. Weareinterestedinanalyzingthenaturallogarithmofthepriceratiofortwoconsecutivedays,andthedataweretransformedaccordinglytogive365dailylogreturns.Summarystatisticsareasfollows:minimum-1.846E-2,median0E-7,mean2.049E-4,maximum1.325E-2. Fig. 3-6 givesthehistogramofthedata,boxplotsbydayoftheweek,andthesampleACFsofthesquaresandabsolutevaluesofthedata. 50

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AsymptoticrelativeecienciesofthemaximumlikelihoodestimatorswithrespecttothenonparametricestimatorsofVaRandCVaR,underiidsamplingfromthestandardAsymmetricLaplacedistribution.TheAREsareplottedasafunctionofthelefttailprobabilitylevel0:5andtheskewnessparameter0<1. Thehistogramofthelogreturnssuggestshighpeakednessandheavytails.ThenormalQ-Qplotofthedataindicatesaviolationofnormality.ThePearsonChi-squarenormalitytestreportsap-valueof2.2e-16.AndtheKolmogorov-Smirovnormalityreportsap-valueof5.032e-14.Thisisstrongevidencethatthedatadonotfollowanormaldistribution. WeconsiderttingthedailylogreturnsdatausinganAsymmetricLaplacedistribution.ThedottedlinesuperimposedonthehistogramshowsattedAL(0.9679,4.436E-3)densitywithparametersestimatedviamaximumlikelihood.Consequently,the 51

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TheresultingsaddlepointapproximateddensityfunctionsofMLEsandNPEsofVaRandCVaRatn=50,=0;=1;=1;0:8;=0:9.ThetruevaluesofVaRandCVaRareindicatedbythesolidverticalline. MLEofVaR,~(X),equals7.566E-3andtheMLEofCVaR~(X),equals1.08E-2,with=0:95. Ithaslongbeennotedthatthereisa"tradingdayeect"incurrencyexchangeratedata(forexample,McFarlandetal.,1982).TheboxplotsinFig. 3-6 decomposethe365returnsbyday-of-the-week.Theday-of-the-weekcorrespondingtoaparticularreturndenotesthelogoftheratioofthepriceonthatdaytothatofthepreviousday.Byinspectionoftheboxplotswecanseethatthereisahighervolatilityduringtheweekthanontheweekends. Theothertwoplotsarethesampleautocorrelationsforthesquaresandabsolutevaluesofthelogreturns.Althoughthe365returnsappearseriallyuncorrelated,thesampleACFsuggeststhepresenceofadependenceoccurringpreciselyatlagsthataremultiplesof7.Thisisevidenceofthepresenceofday-of-the-weekeects. 52

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DailylogreturnsofUSD/EURexchangeratesfromFeb1,2005toJan31,2006.Thetopplotsshowthedatabothfortheentireperiodandbyday-of-the-week.Thebottomplotsdisplaythesampleautocorrelationfunctionofthesquaresandabsolutevaluesofthedataovertheentireperiod. Webelievethereasonforthelowvolatilityduringweekendsisthelackofinstitutionalinvestmentduringweekends.Mostoftheforeignexchangetradesduringweekendsareover-the-counter(OTC)servicebyretailbanks. Nextweremovedtheweekendsfromthedatatoseewhetherthedailylogreturnsofweekdaysfollowanormaldistribution.Asaconsequence,Monday'slogreturnisthelogratioofMondayandthepreviousFriday'sprice.Fig. 3-8 isthehistogramofdailylogreturnsoftheexchangeratedatewithouttheweekends.Thedataarelesspeakedbutstillhaveheavytails.ThePearsonChi-squarenormalitytestreportsap-valueof2.2e-16.And 53

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NormalQ-Qplotofdailylogreturnsofexchangerate. theKolmogorov-Smirovnormalityreportsap-valueof1.011e-13.Therefore,thedataarenotnormallydistributedevenwhenweremovetheweekends. Figure3-8. Histogramofthedailylogreturnsoftheexchangeratewithoutweekends Onobservingthe"trading-day-eect",wethereforettheAL(;)distributionsviamaximumlikelihoodtoeachoftheday-of-the-weekreturns. ThelogreturnsonWednesday,FridayandSundayareapproximatelysymmetricwithtted~equalto0.9962,1.0218and1.032,respectively.WhilelogreturnsonMonday, 54

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NormalQ-Qplotofdailylogreturnsofexchangeratewithoutweekends. Table3-1: Maximumlikelihoodestimatesandstandarderrors(s.e.)fortheskewnessandscaleparametersasanAL(;)ttothelogreturnsoftheUSD/EURexchangeratesbyweekday,aswellastheMLEsofVaRandCVaRat=0:95. Day~(s.e.)~(s.e.)~0:95(s.e.)~0:95(s.e.) Mon0.897(0.089)9.987E-4(1.389E-4)1.892E-3(0.0025)2.680E-3(0.003)Tue0.878(0.086)5.031E-3(6.940E-4)9.820E-3(0.0131)1.387E-2(0.015)Wed0.996(0.098)5.196E-3(7.206E-4)8.507E-3(0.012)1.220E-2(0.014)Thu1.117(0.110)5.523E-3(7.683E-4)7.645E-3(0.0117)1.114E-2(0.014)Fri1.022(0.100)6.330E-3(8.780E-4)9.991E-3(0.014)1.144E-2(0.017)Sat0.836(0.083)5.685E-3(7.948E-4)1.186E-2(0.002)1.667E-2(0.003)Sun1.032(0.101)1.794E-3(2.489E-4)2.791E-3(0.004)4.021E-3(0.005) TuesdayandSaturdayarerightskewedwithtted~equalto0.8973,0.8782and0.8358,respectively.The~onThursdayis1.1169,indicatingleftskewness. TheMLEsofVaRaremuchlargeronTuesdaytoSaturdaythanonMondayandSunday.Consequently,thesametrendisseenfortheMLEsofCVaR. UsingtheasymptoticcovariancematricesinEq. 2{6 andEq. 3{4 ,itisinterestingtocomparecondenceregionsfortheparametricandnonparametricestimatorsofthebivariateparameter(;).Bothregionstaketheformofellipses;forexample,withacondencelevelof95%,theregionfortheMLEsisgivenby 55

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Whiletheconstructionofthisregionisstraightforwardintheparametriccasebysimply"plugging-in"theMLEsthemselveswherevertheyappearinthecovariancematrix,thatforthenonparametriccaseiscomplicatedbytheneedtoestimatetheinverseofthepdf,aquantitysometimescalledthesparsityfunction.Estimationofthesparsityisnotoriouslydicultandtendstobeshiedawayfrominfavorofotherapproacheswheneveritoccurs(asitfrequentlydoes)innonparametricinference.However,therehasbeenrecentrenewedinterestinthissubjectsincethesparsityfeaturesprominentlyintheasymptoticsofquantileregression(Koenker,2005).UsingthemethodsuggestedinKoenker(2005,Section4.10.1)withtheHallandSheatherbandwidth,andpluggingintheNPEswhenevertheyappearinthecovariancematrix,nonparametriccondenceregionconstructionfor(;)isthereforeafeasibleproposition. WeemployedtheaboveapproachinproducingFig. 3-10 ,whichshowstheresultingcondenceellipsesfortheMLEsandNPEsofthe=0:9righttail(VaR,CVaR)fortheweekdaydistributionoftheUSD/EURexchangeratelogreturnsdata.ThehighcorrelationbetweentheMLEsofVaRandCVaRisreectedintheverynarrowsemiminoraxesofeachrespectiveellipse.ThedownwardbiasintheNPEsisalsoimmediatelyapparent,afactthatconcurswiththebiasnotedinthesaddlepointpdfofFig. 3-5 Theimplicationofthesendingsforthepractitioneristhatitmaybepreferabletocommittoanappropriateparametricmodel,suchastheALlaw,whenattemptingtodrawinferencesfromdataofthisnature.Fromasmallsimulationstudywhichweomitforthesakeofbrevity,wehavealsonotedthattheparametriccondencebandshavevastlysuperiorcoverageprobabilities.

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Maximumlikelihood(circle)andnonparametric(square)bivariateestimatorsof(VaR,CVaR)forthe=0:9tailoftheweekdaydistributionsoftheUSD/EURexchangeratelogreturndata.Thedashedanddottedlinesdelineatetheboundaryof95%condenceellipsesforthemaximumlikelihoodandnonparametricestimators,respectively. 57

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Inthischapter,wedevelopthetimeseriesmodelsunderALnoise.TraditionaltimeseriesARMAmodelsassumeGaussiannoise,whichimpliesthatthemarginaldistributionwillbeGaussianaswell.WehavenoticedbeforethatnancialdataareusuallyheavytailedwhichisnotconsistentwithaGaussiandistribution.ItisthereforereasonabletoextendtheALdistributiontotimeseriesmodels. EarlyworkshaveappliedtheSymmetricLaplacedistributiontoARMAmodels.Theseeortshavefocusedontwodirections:assumemarginalSymmetricLaplacedistribution,forexample,NLAR(1)andNLAR(2)models(DewaldandLewis,1985),NAREX(1)model(Novkovic,1998),orassumeSymmetricLaplacenoise,forexample,DamslethandEl-Shaarawi(1989).DamslethandEl-Shaarawi(1989)haveshownthatthesetworequirementscannotbesimultaneouslyachievedwithintheclassoflineartimeseriesmodels. DewaldandLewis(1985)discusstheNLAR(1)andNLAR(2)modelassumingastandardSymmetricLaplacemarginaldistribution.TheSymmetricLaplacedistributionisaspecialcaseofALdistributionwith=1.ASymmetricLaplacedistributioniscalledastandardSymmetricLaplacedistributionwhen=1and=0. RandomvariableXissaidtobedistributedasSymmetricLaplacedistribution(doubleexponentialdistribution)withlocationparameter<<1andscaleparameter>0,ifitspdfisoftheform 2ejxj ASymmetricLaplacedistributionarecalledstandardLaplacedistributionif=1. NLAR(1)modelstartsbyassumingfXngtobeastationaryprocesswithstandardLaplacemarginaldistribution,0
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Thenthenoisetermcanbederivedas whereLtarei.i.d.standardLaplacerandomvariables.Andp=2 2(1p)ejxj+1 2pejxj; whichisaconvexmixtureofLaplacedensities. Similarly,theNLAR(2)modelassumesstandardLaplacedistributionandappliesthistotheAR(2)model, where0
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TheNAREX(1)modeldiscussedbyNovkovic(1998)assumesthemarginaldistributiontobeaSymmetricLaplacedistributionL(),asdenedinEq. 4{1 ,withscaleparameter, where0p0;p1;p21,p0+p1+p2=1,0<1;2;3<1.AfterworkingwiththecharacteristicfunctionoftheXtandZt,ZtcanbeexpressedasamixtureofsymmetricLaplacedistribution, whereA0,A1andA2arefunctionsof1;2;3;p0;p1;p2. DamslethandEl-Shaarawi(1989)deducethemarginaldistributionofobservationsgeneratedbyanARMAmodelassumingSymmetricLaplacenoise. LetfZtgbeaseriesofi.i.d.SymmetricLaplacedistributedrandomvariableswithscaleparameter.LettheobservedstationarytimeseriesfXtgbegeneratedbytheARMAscheme, (B)Xt=(B)Zt; themarginalpdfXtis 21Xj=0jjjj1exp1 j; 60

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Onobservingtheasymmetrypropertyofnancialdata,itisreasonabletoapplytheAsymmetricLaplacedistributiontotimeseriesmodeling. Denition4.1:(ARMA(p;q)process) wherefZtgWN(0;2)andthepolynomial(11zpzp)and(1+1z++qzq)havenocommonfactors. TheprocessfXtgissaidtobeanARMA(p;q)processwithmeaniffXtgisanARMA(p;q)process. Eq. 4{12 canalsobeexpressedas (B)Xt=(B)Zt; where()and()arethepthandqth-degreepolynomials (z)=11zpzp (z)=1+1z+pzp ThetimeseriesfXtgissaidtobeanautoregressiveprocessoforderp(orAR(p))if(z)1,andamoving-averageprocessoforderq(orMA(q))if(z)1. 61

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1 andP1j=jjj<1.Therefore,wecandene 1 (B)=1Xj=jBj: Applyingtheoperator(B)=1 (B)tobothsidesofEq. 4{13 ,weobtain where(z)=(z)(z)=P1j=jZj. AnARMA(p;q)processfXtgiscausal,oracausalfunctionoffZtg,ifthereexistconstantsfjgsuchthatP1j=0jjj<1and causalityisequivalenttothecondition AnARMA(p;q)processfXtgisinvertibleifthereexistconstantsfjgsuchthatP1j=0jjj<1and (4{19) invertibilityisequivalenttothecondition 62

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LetfZtgbeaseriesofi.i.d.AsymmetricLaplacedistributedrandomvariablesAL(;;),withPDFgivenby (1+2)(expp jzjI[z]!+expp thepolynomial(11zpzp)and(1+1z++qzq)havenocommonfactors. ToensurefZtghaszeromean,werequirethat WesimulatedanARMA(1,1)modelunderALnoisewith>1,whichmeansthattheALdistributionisleftskewed.Consequently,theARMAprocessisskewedtothedownsideofthemean.Therearesomedeepdropsapproximatelyattimet=140,180,300. 63

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SimulatedARMA(1,1)processunderALnoisewith1=0:4,1=0:1,=1:25,=1. Figure4-2. HistogramoftheSimulatedARMA(1,1)processunderALnoisewith1=0:4,1=0:1,=1:25,=1. Thesedropsaremorefrequentanddeepercomparedwiththeupperpoints.Thismodelisgoodtodescribemarketswithunexpectedandsuddenlosses,forexample,amarketcrash.Similarly,ifwesettheskewnessparameter<0,theprocesswillbeskewedtotheupperside. ThehistogramofthefXtgindicatesthedistributionhasheavytails. 64

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22t2i orequivalently, 22t2it; where= WeassumefXtgtobecausal,then Let andassumei6=jfori6=j,sinceZ1;Z2;:::;Znarei.i.d.,thecharacteristicfunctionofUnis 222jt2ijt; whichcanberesolvedintopartialfractions 222jt2ijt=nXj=0ajeijt 222jt2ijt; 65

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Thevalueoftheconstantsajcanbeobtainedasfollows: Lety=1+1 222jt2ijt,solvingthisequation,weget Mathematically,bothplusandminussignsinEq. 4{29 arecorrect.Butwewillseeinthenextstep,thattheywillnotnecessarilymakeourderivedfunctionapdf.Weidentifythecorrectsignsbyplottingthepdfs.Actually,theplussigniscorrectforthepartofpdfbelowthemean,andtheminussigniscorrectforthepartofpdfabovethemean. Settingy=0,weget Eq. 4{28 canbeexpressedas 222jt2ijtnYi=0i6=jeiit 222it2iit=ajeijt 222jt2ijt+nXi=0i6=jaieiit 222it2iit; substitutingyintoEq. 4{31 222if2j(y)iifj(y)=ajeijfj(y) 222if2j(y)iifj(y); 66

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222if2j(y)iifj(y)=ajeijfj(y)+ynXi=0i6=jaieiifj(y) 222if2j(y)iifj(y); settingy=0,weget 222if2j(0)iifj(0): 2Z1'Un(t)eitudt=1 2Z1nYj=0eijt 222jt2ijteitudt=1 2Z1nXj=0ajeijt 222jt2ijteitudt=nXj=0aj"1 2Z1eijt 222jt2ijteitudt#; wherewerecognizeeijt 222jt2ijtasthecharacteristicfuntionofZ0jAL(j;j;jjj). Let0j=j,0j=jand0j=jjj,since=p +p j+q j+jjjp 67

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sign(j)1 sign(j)(1 Ifj0,0j=;Ifj<0,0j=1 Consequently,thepdfofZ0jis BacktoEq. 4{35 2Z1eijt 222t2ijteitudt=nXj=0aj1 2'Z0j(t)eitudt=nXj=0ajp where 222if2j(0)iifj(0): Now, 68

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where 222if2j(0)iifj(0): Tocheckthevalidationofourderivedformula,weplotthemarginalpdfofAR(1)andARMA(2,2)modelsunderALnoise.ThereasonforselectinganAR(1)isbecauseitisatypicalmodelinttingnancialdata.Wecompareourtheoreticalcurvewithsimulateddata.Itturnsoutthatourderivedcurvematchesthesimulateddataperfectly. BelowaresomeremarksconcerningtheplottingofthePDFanddatasimulation. 4{16 ,(B)=1(B)(B)canbeobtainedbyexpanding1(B)usingTaylorseriesexpansion,andthencalculatetheproductof1(B)and(B).Somecomputerprogramswithsymbolicmathematicalfunctions,likeMatlab,providefunctionsforTaylorseriesexpansionandpolynomialconvolution. Fig. 4-3 andFig. 4-5 aretheplotsofthederivedPDFsofAR(1)andARMA(2,2).WeselectedALnoisewith=1,=0:8.Toensureazeromeannoise,ispredetermined 69

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4-4 andFig. 4-6 ,wecomparethederivedPDFswiththesimulatedhistograms. Figure4-3. DerivedmarginalpdfofAR(1)modelunderAsymmetricLaplacenoisewith=1,=0:8,=0:318and=0:75. Figure4-4. Comparisonofderivedmarginalpdf(redline)andsimulatedhistogramofAR(1)model.Thetheoreticpdfhasbeeninatedbynumberofreplicationstomatchthehistogram.=1,=0:8,=0:318and=0:75. Proposition4.1(Jointdistributionof(x1;:::;xn)) Letf(x1;:::;xn)denotethepdfofthejointdistributionoftherstnobservations,X1;:::;Xn,underanAR(p)model.Thenthelikelihoodcanbewrittenas 70

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DerivedmarginalpdfofARMA(2,2)modelunderAsymmetricLaplacenoise.=1,=0:8,=0:318,1=0:7,2=0:2,1=0:4,2=0:2,whichiscausalandinvertible. Figure4-6. Comparisonoftheoreticmarginalpdf(redline)andsimulatedhistogramofARMA(2,2)model.Thetheoreticpdfhasbeeninatedbynumberofreplicationstomatchthehistogram.=1,=0:8,=0:318,1=0:7,2=0:2,1=0:4,2=0:2,whichiscausalandinvertible Thisresultisstraightforwardfromthedenitionoftheconditionalprobability. SinceXtfollowsanAR(p)process,XtjXs;s
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Wethenapplyanumericaloptimizationmethodtomaximizeourconditionallikelihood.TheALparametersarechosenas0=1,and0=2.WeusetheYule-Walker(YW)estimatorasourinitialvalue.Thisisastandardmethodofmomentsestimator. IntheTable. 4-1 ,Table. 4-2 ,andTable. 4-3 ,welistthettedmodelparametersoftherstvesimulations.^ywindicatestheYule-Walkerestimatorofeachsimulation,whichisalsoourinitialvalueof.ThecalculationofMeansandMSEsisbasedon100simulations. Table4-1: FittedmodelparametersofAR(1)modelunderALnoise parameters 0.81-0.31820.7 MLEs simu1 0.77910.7713-0.27520.73980.7106MLEs simu2 0.96520.9990-0.05010.61250.6030MLEs simu3 0.89811.1812-0.17990.63050.5256MLEs simu4 0.79620.9572-0.31120.70870.7042MLEs simu5 0.75151.1285-0.46220.70660.6763 mean(MLEs) 0.80020.9733-0.311820.7045MSE(MLEs) 0.00680.01070.01840.0048 Ingeneral,ourmethodworkswellinttingAR(p)models.MSEsforarelessthan1%,withabettertformodelsoflowerorders.MSEsforALparametersandareslightlyhigher,withanaveragearound0.01. 72

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FittedmodelparametersofAR(2)modelunderALnoise parameters 0.81-0.31820.7-0.1 MLEs simu1 0.67110.9298-0.53840.8011-0.17610.7115-0.1884MLEs simu2 0.67420.7489-0.42830.7489-0.16570.8734-0.3434MLEs simu3 0.90571.0423-0.14620.8545-0.19960.8189-0.2161MLEs simu4 0.85971.0650-0.22860.7099-0.08310.7797-0.1810MLEs simu5 0.69220.8903-0.47370.8246-0.10540.8165-0.2273 mean(MLEs) 0.78900.9747-0.33140.6969-0.1052MSE(MLEs) 0.00780.01430.01900.00920.0064 Table4-3: FittedmodelparametersofAR(3)modelunderALnoise parameters 0.81-0.31820.7-0.20.1 MLEs simu1 0.64850.8390-0.53000.7523-0.23700.23560.6332-0.14320.1601MLEs simu2 0.67250.9841-0.56670.7332-0.25790.10400.7251-0.32870.1245MLEs simu3 0.81310.9301-0.27410.5539-0.16740.11830.6046-0.23390.1907MLEs simu4 0.79721.2948-0.41850.7532-0.23450.07350.6538-0.16760.0978MLEs simu5 0.65330.5996-0.37200.7608-0.32320.15720.6343-0.15730.1010 mean(MLEs) 0.80640.9685-0.30210.7097-0.20890.1003MSE(MLEs) 0.00850.01160.02190.00800.00940.0053 Jointdistributionof(X1;:::;Xn)underanARMA(p;q) .Thejointdistribution(X1;:::;Xn)underanARMA(p;q)canbecalculatedinasimilarwaytoEq. 4{44 ,exceptthatfXtgisafunctionoffXsgandfZsg,s
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4{44 NowXtjXs;spcanbecalculatedas Weapplyanumericaloptimizationmethodtomaximizeourconditionallikelihood.TheALparametersarechosenas0=1and0=2.TheinitialvaluesforandcomefromtheARIMAfunctioninRwithGaussiannoise. InTable. 4-4 ,welistthettedmodelparametersoftherstvesimulation.ThecalculationofMeansandMSEsisbasedon100simulations. 74

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FittedvalueofARMA(1,1)modelunderALnoise parameters 0.81-0.31820.70.5 MLEs simu1 0.73980.9210-0.39850.75980.5045MLEs simu2 0.81500.9835-0.28650.71110.4810MLEs simu3 0.82821.0287-0.27590.71190.4770MLEs simu4 0.83570.9979-0.25460.64190.6056MLEs simu5 0.74151.0194-0.43760.72370.4352 mean(MLEs) 0.80740.9853-0.30350.68880.5004MSE(MLEs) 0.00790.01120.02170.00560.0088 Engle(1982)introducedtheARCH(p)processwherethevolatilitiesaredependentonthepastvolatilities.Bollerslev(1986)introducedageneralizationoftheARCH(p)process,theGARCHprocess. Denition4.5:ARMA(p;q)modeldrivenbyGARCH(u;v)noise where wherehtisthepositivefunctionofZs;s
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Whenthevalueofnislargerelativetop,weignoretherstterm,andbaseourlikelihoodestimationontheremainingpieces.Thisisknownasconditionalmaximumlikelihoodestimation. Therefore,usingthepropertyofalocation-scalefamily,thejointlikelihoodofanARMAmodeldrivenbyGARCHnoisecanbeexpressedas, where()denotesthestandardnormaldensityfunction. ThemeanoffXtg,t,canbecalculatedinthesamewayasforanARMA(p;q)model, Standarddeviationst=p 4{50 andEq. 4{51 withZt=0andht=^2forallt0.^2isthesamplevarianceoffZ1;:::;Ztg. WenowhavethejointlikelihoodfunctionofthefXtg.MLEscanbefoundusingnumericaloptimizationmethods.ForthestandardGaussianGARCH,themodelcanbeeasilyttedusingcomputationalpackages,likeITSM,andtheGARCHtoolboxinMatlab. 76

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4{49 ,exceptthatfetghasanAsymmetricLaplacedistribution. where wherehtisthepositivefunctionofZs;s0,andj;j0,j=1;2;:::and ThejointlikelihoodfunctionoftheGARCHALmodelcanthenbeexpressedas wherefAL()isthedensityfunctionofAL(;;). Thismodelcanbetuseconditionallikelihoodestimation,inthesamewayasinthelastsection,exceptthattheconditionallikelihoodfunctionsofXtjXs;s
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Figure4-7. DailyvaluesofthemutualfundmanagedbyTIAA-CREF.ThedatarangefromJan1,2000toDec31,2006,atotalof1807values. Figure4-8. DailyreturnsofthemutualfundmanagedbyTIAA-CREF.ThedatarangefromJan1,2000toDec31,2006,atotalof1807value. 78

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Figure4-9. HistogramofthedailyreturnsoftherealestatemutualfundmanagedbyTIAA-CREF. ThehistogramofFig. 4-9 indicateshighpeakednesscomparedwithanormaldistribution.Also,thedataareasymmetricwithrightskewness.ThedistributionofthedailyreturnsisclosetotheshapeofanAsymmetricLaplacedistributionaswehavealreadydiscussed.Therefore,weconsiderthistobeagoodexampletoapplyourtimeseriesmodelsto. Weusefourmethodstoanalyzethismutualfunddata:anARMA(p;q)modelunderGaussiannoise,anARMA(p;q)modelunderAsymmetricLaplacenoise,anARMA(p;q)modeldrivenbyGARCHGaussiannoise,andanARMA(p;q)modeldrivenbyGARCHALnoise.Thersttwomodelsarelinearmodels.Theothertwoarenonlinearmodelswhichassumethatthevariancesaredependent. WeplotthesampleACFoftheresidualstocheckthevalidationofthemodels.IfthereisnocorrelationinthesampleACF,wecheckthesampleACFoftheabsolutevaluesandsquaresofresiduals.NocorrelationinthesampleACFoftheresidualsdoesnotnecessarilymeanthattheyareindependent.Ifcorrelationisdetectedintheabsolutevaluesorsquaresofresiduals,thatsuggeststheyaredependent. 79

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AIC=2logL(^)+2k: AICc=AIC+2k(k+1) AICusesthe2ktermasthepenaltyforaddingmoreparametersintothemodel.Usually,themodelwithminimumAICvalueischosenasthebestmodeltottothedata. Intimeseriesmodelselection,weusetheAICccriterion.AICcistheempiricalcorrectionforsmallsamplesizes.SinceAICcconvergestoAICasngetslarge,AICcshouldbeemployedregardlessofsamplesize.Inourcase,sincethesamplesize1807islargeenough,AICisveryclosetoAICc. Whenlookingattherstplot,thesampleACFoftheresiduals,sampleACFsareoutoftheboundaryatlags11,20and22.Therearealsofourothersclosetotheboundary. 80

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FittedValueofARMA(1;3)underGaussiannoise parameters ^1^1^2^3 0.98-0.8636-0.0505-0.0715s.e. 1.968E-65.545E-49.405E-45.707E-4 Figure4-10. SampleACFofresidualsofARMA(1,3)modelunderGaussiannoise. Figure4-11. SampleACFofabsolutevaluesofresidualsofARMA(1,3)modelunderGaussiannoise. 81

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SampleACFofsquaresofresidualsofARMA(1,3)modelunderGaussiannoise. WhenanalyzingthesampleACFofabsolutevaluesandsquaresofresiduals,wefoundanobviouscorrelationintheresiduals.Therefore,itisnecessarytoconsiderttingaGARCHmodel. WettheARMA(p;q)modeldrivenbyGARCHGaussiannoise,asdenedinSection4.3.TheGARCHmodelisttedusingthematlabGARCHtoolbox.WesearchforthebestGARCHmodeluptotheorderofARMA(7,7)andGARCH(2,2).ThebestmodelisanARMA(1,3)drivenbyGARCH(1,1)noise.Theloglikelihoodofthettedmodelis1135.9,withAICcequalto-2263.8. Table4-6: FittedparametersofARMA(1;3)drivenbyGARCH(1,1)Gaussiannoise. parameters ^1^1^2^3^1^1 0.998-0.9070.005-0.0830.4840.353s.e. 1.47E-33.237E-23.718E-22.589E-22.468E-22.217E-2 WenowconsiderttinganARMA(p;q)modelunderALnoise,asdiscussedinSection4.2.Themodelisttedusingthemethodofconditionalmaximumlikelihood 82

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SampleACFofresidualsofARMA(1,3)modeldrivenbyGARCH(1,1)Gaussiannoise. Figure4-14. SampleACFofabsolutevaluesofresidualsofARMA(1,3)modeldrivenbyGARCH(1,1)Gaussiannoise. estimationdescribedinsection4.2.Empirically,wedidanumericaloptimizationinMatlab,usingtheSimplexmethod.TheinitialvaluesoftheARMAparametersaregivenbyttingaregularARMAmodelunderGaussiannoise.Theinitialvaluesofandarebothsetto1. WesearchforthebestARMA(p;q)uptotheorderofARMA(7,7).ThebestmodelisanARMA(2,6)withloglikelihood=1439.8,AICc=-2859. 83

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SampleACFofsquaresofresidualsofARMA(1,3)modeldrivenbyGARCH(1,1)Gaussiannoise. Lehmann(1983)indicatesthatmaximumlikelihoodestimatorsofARMA(p;q)processesareapproximatelynormallydistributedwithvarianceatleastassmallasthoseofotherasymptoticallynormallydistributedestimators.BrockwellandDavis(2002)provethatthelarge-sampledistributionofmaximumlikelihoodestimatorsofARMAcoecientsisthesameforfZtgIID(0;2),regardlessofwhetherornotfZtgisGaussian.Therefore,wecanusethestandarderrorsofMLEsfromGaussiannoiseasthestandarderrorsoftheMLEsinourmodelbasedonALnoise. Thestandarddeviationofandareobtainedbybootstrapping.Fordetailedinformationaboutbootstrappingontimeseriesmodels,pleaserefertoShumwayandStoer(2000). Table4-7: FittedparametersofARMA(2;6)underALnoise ^1^2^1^2^3^4^5^6^^ 0.35260.5423-0.2989-0.4524-0.0889-0.06100.01920.01420.9190.129s.e. 1.12E-11.11E-11.11E-18.45E-21.23E-31.75E-39.9E-45.72E-40.2270.019 84

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SampleACFofresidualsofARMA(2,6)modelunderALnoise. Figure4-17. SampleACFofabsolutevaluesofresidualsofARMA(2,6)modelunderALnoise. 85

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SampleACFofsquaresofresidualsofARMA(2,6)modelunderALnoise. WehavesearcheduptotheorderofARMA(7,7).ThebestmodelisanARMA(1,2)drivenbyGARCH(1,1)noise,withloglikelihoodandAICcequalto1458.7and-2971.3,respectively. Table4-8: FittedparametersofARMA(1;3)drivenbyGARCH(1,1)ALnoise. ^1^1^2^3^1^1^^ 0.112480.977-0.922-0.0400.3350.3420.8230.112s.e. 1.472E-33.240E-23.712E-22.589E-22.468E-22.217E-20.01650.344 SummaryofAICcofthefourmethods ModelNoise AICc ARMA(1,3)Gaussian -1981ARMA(2,6)AL -2859ARMA(1,3)GARCHGaussian -2263.8ARMA(1,2)GARCHAL -2971.3 AccordingtoTable. 4-9 ,theAICcvaluesofourmodelsbasedonALnoisearemuchlowerthanmodelsbasedontraditionalGaussiannoise.ThisisanindicationthatmodelsassumingALnoiseprovideabettertforthedailyreturnsofthemutualfund.TheassumptionofALnoiseprovidesabetterdescriptionofthedata. Comparingthersttwolinearmodels,theARMA(p,q)underALnoisehasamuchlowervalueofAICc.Similarlyforthetwononlinearmodels.Anevenmoreattractive 86

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SampleACFofresidualsofARMA(1,3)modeldrivenGARCH(1,1)ALnoise. Figure4-20. SampleACFofabsolutevalueofresidualsofARMA(1,3)modeldrivenGARCH(1,1)ALnoise. featureoftheseresultsisthatourlinearmodelhasabettertthanthenonlinearGARCHmodel. Itisawellknownfactthatthevolatilityofnancialdataishighlycorrelated.ARMAmodelsassumeconstantvariance,whichleadstocorrelatedresiduals.OurARMAmodelunderALnoisecannotremoveallthecorrelationintheresiduals.ButithassignicantlyimprovedthelikelihoodandAICc.AndtheGARCHmodelunderALnoisehasasignicantlybettertthantheGARCHmodelunderGaussiannoise. 87

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SampleACFofsquaresofresidualsofARMA(1,3)modeldrivenGARCH(1,1)ALnoise. 88

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InChapter2,wehaveanalyzedthe"tradingdayeect",thatthevolatilitiesoflogreturnsofexchangeratesarehigherinthemiddleoftheweekandlowerduringtheweekends.AndsampleACFindicatesthatthelogreturnsarecorrelatedatthelagof7ormultipleof7.SAR(p)modelsassumethecoecientsoftherstp1lagsofaAR(p)arezero.Therefore,fXtg'sarecorrelatedatthelagpandamultipleofp. NowwestartfromassumingthatthemarginaldistributionofatimeseriesmodelhasamultivariateAsymmetricLaplacedistributionandapproximatethedistributionofthegeneralizedestimatorofusingsaddlepointapproximation.WehavetriedthreemethodstoapproximatethePDFsofthegeneralizedestimator.Sincetherearerelativelylargedeviationsbetweenourapproximationsandthesimulateddata,wearrangethispartasappendix. A.1.1SAR(p)Model withjj<1.ItfollowsthatXtisstationarywithautocovariancefunction(ACVF)atlagh0, (A{2) 89

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NowwedenetheSAR(p)withmultivariateAsymmetricLaplacemarginaldistribution.LetfYtmgtobethezero-meanprocess,t=:::;1;0;1;:::,withmultivariateAsymmetricLaplacedistribution, Therefore,arealizationY=[Y1;:::;Yn]0fromthismodelhasamultivariateAsymmetricLaplacedistribution,YAL(m;n).WedonotspecifythedistributionofZthere.ItdoesnotfollowanAsymmetricLaplacedistribution,aswehavestatedearlierinthischapter. ^h=1 denoterespectivelythesample-basedestimatesoftheACVFatlagh,andthetruncatedACVFatlag0obtainedbyomittingtherstandlastpobservation.Foranygivennonnegativeconstantsc1andc2,wedenethegeneralizedestimatorof, ^c1;c2=Pnt=1+pxtxtp cT1+T2+c2T3: Someofthemorecommonestimatorsofand2,canbeshowntobethefollowingspecialcasesofEquation(2.11)(Brockwell,Dahlhaus,andTrindade,2005): LeastSquares, ^LS=S T1+T2=^1;0; Yule-Walker, 90

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T1+T2+T3=^1;1; Burg, ^BG=2^p T1=2+T2+T3=2=^1=2;1=2: WedenethennmatrixAtobezeroeverywhere,exceptwhenjijj=p,inwhichcaseitisequalto1=2.Similarly,deneB(c1;c2)tobethennidentitymatrix,withtherst(last)pdiagonalelementsmultipliedbyc1(c2).IfX=[X1;:::;Xn]0isarealizationfrommodel(2.9),wecanthenexpressthegeneralizedestimator^c1;c2asaratioofquadraticformsinnormalrandomvariables, ^c1;c2=X0AX X0B(c1;c2)XP Q(c1;c2): (t)=1 1+1 2t0tim0t; wherem2Rdandisaddnonnegativedenitesymmetricmatrix. WeusethenotationofALd(m;)todenotethedistributionofY,andwriteYALd(m;).Ifthematrixispositive-denite,thedistributionistrulyd-dimensionalandhasaprobabilitydensityfunction.Otherwise,itisdegenerateandtheprobabilitymassofthedistributionisconcentratedinalinearpropersubspaceofthed-dimensionalspace. Form=0thedistributionALd(0;)reducestothesymmetricmultivariateLaplacedistributionLd(). 91

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2zzzd=2dz: LetYALd(m;),eachcomponentYiofYadmitstherepresentation whereXiisithcomponentofXNd(0;).Wisanexponentiallydistributedrandomvariablewithmean1,independentofXi. Letiitobetheithdiagonalelementof,andiitobe(i;j)thelementof.ThenXiN(0;ii)andE(XiXJ)=ij. Byapplyingtherepresentationinproposition1.2,wecanderivethemeanvectorandcovariancematrixofmultivariateALdistribution. WehaveEYi=mi.sinceE(XiXj)=ijandEW2=2,wehave Cov(Yi;Yj)=E(YiYj)EYiEYj=2mimj+ijmimj=mimj+ij:

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and Thegeneralizedestimatorof,^c1;c2,isaratiooftwoquadraticforms.Undernormalcase,themomentgeneratingfunctionofthequadraticformscanbederivedexplicitly.ButundermultivariateAsymmetricLaplacecase,sincethepdfofdistributionisinacomplicatedformwithafunctionofintegration,wecannotderivethemomentgeneratingfunctionofthequadraticformdirectly. WeconsidertousethegeneralizedsaddlepointapproximationmethoddevelopedbyEastonandRonchetti(1986),aswedidinChapter2.Toapplythismethod,weneedtoderivetherstfourcumulantsofthequadraticformsandthenapproximatethecumulativegeneratingfunctionusingEdgeworthexpansion.DetailofthisgeneralizedsaddlepointapproximationmethodisavailableinChapter2. WewillnowderivetheexpressionforEP,EQ,E(P2),E(PQ),E(Q2),E(P3),E(P2Q),E(PQ2),E(Q3),E(P4),E(P3Q),E(P2Q2),E(PQ3),E(Q4),whichwillbeusedtodevelopthecumulantsinthenextsection. TheexpressionforEPandEQisquitestraightforward.LetcolumnYnhasmeanandvariance-covariancematrix,andGisnnmatrix.then 93

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McCullaph(1987,Ch3)speciestheexpressionofgeneralizedmomentsusingtensormethod, Thenwehavethethirdmoments, Thefourthmoments,

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whereai;jandbi;jarethe(i;j)componentsofmatrixAandB.WenowapplytherepresentationofmultivariateALdistributioninCorollary(1.1),theproductofYicanbecalculateas, whereX=[Xi;:::;Xn]hasmultivariatenormaldistribution,XN(0;n).WisanexponentialdistributionwithmeanzerowithEWp=p!. Thetermsfollowedbynumbersinparenthesisindicatethesummationofallthepermutationinthattype.Forexample, (mimjmkXl)[4]=mimjmkXl+mimjXkml+miXjmkml+Ximjmkml: WeknowfornormallydistributedXwithmeanzero,theoddmomentsandcumulantshavethevaluesofzero.Therefore,thetermswithEXi,EXiXjXkequaltozero. TocalculateEXiXjXkXl,weintroducethenotationbyMcCullaph(1987,Ch3). 95

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Thisexpansionimplicitlydenesallthemomentsi;ij;ijkandcumulantsi;i;j;i;j;kofX.Wecanalsoexpressi;ij;ijkwiththeexpectationformswhichwearemorefamiliarwith, (A{23) and Applyingtensormethod,wecanexpressEXiXjXkXlas 96

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IncaseofYhassymmetricmultivariateLaplacedistribution,mi=0.Eq. A{25 reducestoEYiYjYkYl=2(ijkl+ikjl+ilkj). SimilartoEq. A{25 ,weget (A{26) and ThehigherorderofmomentsofYcanbecalculatedas 97

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Incaseofm=0,thesetwoequationsreduceto Tosimplifyourcalculation,wewillconcentrateinthecasem=0.ThecalculationoftherstfourmomentsofPandQiscomputationallyintensive.Butitcanbedoneusingsomecomputerprograms,evenitisverytime-consuming. Wetrythreemethodstoapproximatethedistributionof^c1;c2. 98

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Since(r)isamonotonicallydecreasingfunctioninrforeveryrealizationY,wehave^c1;c2r,(r)0,whichleadstothedeviceP(^c1;c2r)=P((r)0). LetK(r)(T)bethecumulativegeneratingfunctionof(r).wecannotderiveK(r)(T)directly,aswehavestatedbefore.ButwecanapproximateK(r)(T)usingEdgeworthexpansion.Thecumulativegeneratingfunctionof(r),K(r)(T),canbeapproximatedas, wheren,2n,3n,4narethemean,thevariance,andthethirdandfourthcumulantof(r).Therstfourmomentsof(r)canbederivedfromtherstfourmomentsofPandQ, Therelationshipofcumulantsandmomentsareavailableinchapter2. AccordingtoDaniels(1983),thesaddlepointapproximationtothepdfof^c1;c2atris, 2f_K(r)(T0)=T0g2K00(r)(T0)1 2expnK(r)(T0); 99

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Tocheckthevalidationofourmethod,weapproximatetheSAR(3)model,with=0:5,m=0.Thevariance-covariancematrixisdenedasEq. A{2 with=1.WechoosetheBurgestimator,i.e.,c1=0:5,c2=0:5.Samplesizen=10and50. FigureA-1. ApproximatedpdfofburgestimatorinSAR(3)modelwithSymmetricLaplacemarginaldistributionusingsaddlepointapproximationtotheequations.=0:5,m=0,=1,c1=0:5,c2=0:5,n=10. Obviously,thisisnotagoodapproximation.Ourapproximatedpdfhasmuchhigherpeakcomparedwiththesimulateddata.Andtherearesomediscontinuouspointaroundthemodeinourapproximatedpdf. Onethingtonoticeisthattheapproximatedpdfwithsamplesize50aremuchclosertothesimulatedhistogramthantheonewithsamplesize10.Thisisagoodsign,sinceweareexpectingtheapproximatedpdfconvergetotherealvaluewhensamplesizegettinglarger.Wedidnottrytheapproximationwithlargersamplesizesincewearemoreinterestedinthesmallsampleapproximation. 100

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Approximatedpdfandsimulatedhistogramusingsaddlepointapproximationtotheequations.Wehaveinatedthepdfbynumberofreplicationtomatchthehistogram.Thehistogramisbasedon10;000simulations.=0:5,m=0,=1,c1=0:5,c2=0:5,n=10. FigureA-3. ApproximatedpdfofburgestimatorinSAR(3)modelwithSymmetricLaplacemarginaldistributionusingsaddlepointapproximationtotheequations.=0:5,m=0,=1,c1=0:5,c2=0:5,n=50. Lieberman(1994)developtheLaplaceapproximationtothemomentsofaratiooftwoquadraticforms.LetXben1randomvectorwithdensityfunctionf(x).LetmatricesFandGbenonstochasticnn,FsymmetricandGpositivedenite.Denethestatistic 101

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Comparisonoftheapproximatedpdfandsimulatedhistogramusingsaddlepointapproximationtotheequations.Wehaveinatedthepdfbynumberofreplicationtomatchthehistogram.Thehistogramisbasedon10;000simulations.=0:5,m=0,=1,c1=0:5,c2=0:5,n=50. underconsiderationis X0GX: TheLaplaceapproximationforthekthmomentofris [E(X0GX)]k: ThisistheLaplaceapproximationuptotheorderofO(n1).Theapproximationtothehigherorderofexpansionis where 2E[(X0FX)k]2 bn2=k(k+1) 2k2 23E[(X0FX)k]3+k12 8E[(X0FX)k]22

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Specically, Weapproximatedthemomentsofratior=Y0AY Y0BYuptotheorderofO(n3).Tomakeitcomparable,weusedthesameSAR(3)modelandsameparametersaswedidinlastsection.=0:5,m=0.Thevariance-covariancematrixisdenedasEq. A{2 with=1.WechoosetheBurgestimator,i.e.,c1=0:5,c2=0:5.Samplesizen=10. Buttheapproximatedmomentsofrarenotclosetothesimulatedvalueunder100;000simulations.Toourunderstanding,thisproblemliesinthatthesmallsamplesize.WecanseethetrendofconvergewhenweincreasethesizeofY.Butsinceweareconsideringthesmallsamplesize,thisisnotagoodmethod. FigureA-5. SaddlepointapproximatedpdfofburgestimatorinSAR(3)modelwithSymmetricLaplacemarginaldistribution.WeapproximatedmomentsofrusingLaplaceexpansion.=0:5,m=0,=1,c1=0:5,c2=0:5,n=10. 103

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ComparisonoftheapproximatedpdfusingLaplaceapproximationandsimulatedhistogram.TheratioofmomentsareapproximatedbyLaplaceapproximation.Weinatedthepdfbynumberofreplicationtomatchthehistogram.Thehistogramisbasedon10;000simulations.=0:5,m=0,=1,c1=0:5,c2=0:5,n=10. Y0BYbyTaylorseriesexpansion.Thenusegeneralizedsaddlepointapproximation. Letf1(P;Q)=P Q,weexpandf1(P;Q)usingTaylorseriesexpansionatthemeanofP,EP,andmeanofQ,EQ.Wewillonlystatetherstthreetermsoftheexpansion.Forhigherorderofexpansion,pleaserefertotheformulainChapter2. 2(PEP)2@2f1 3!(PEP)3@3f1 104

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2VarP@2f1 6E(PEP)3@3f1 WeusethesimilarformulatocalculatethemomentsofPandQ.TheonlyexceptionisthatthecrosstermsofPandQdonotequaltozeroanymore.Therefore,wehavemorecomplicatedformula,

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Weapproximatedmomentsofr=Y0AY Y0BYusingtherstfourtermsofTaylorexpansion.Tomakeitcomparable,weusedthesameSAR(3)modelandsameparametersaswedidinlasttwosection.=0:5,m=0.Thevariance-covariancematrixisdenedasEq. A{2 with=1.WechoosetheBurgestimator,i.e.,c1=0:5,c2=0:5.Samplesizen=10. FigureA-7. SaddlepointapproximatedpdfofburgestimatorofinSAR(3)modelwithSymmetricLaplacemarginaldistribution.WeapproximatethemomentsofrusingTaylorseriesexpansion.=0:5,m=0,=1,c1=0:5,c2=0:5,n=10. Ourapproximationisnotsuccessfulthistime.WehaveusedthesamemethodtoapproximatethepdfofMLEsofVaRandCVaRunderAsymmetricLaplacedistributioninChapter2.AndtheapproximationareveryaccurateaccordingtoourcalculatedPREs. Thismethodisnotworkingnow.Probablythereasonisthatthevalueofhigherordermomentsofquadraticformsareverylarge.Forexample,thefourthmomentsofPandQaretotheorderof108.Underthissituation,theapproximationtothemomentsofrdoesnotconverge. 106

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Comparisonoftheapproximatedpdfandsimulatedhistogram.ThemomentsareapproximatedbyTaylorexpansion.Weinatedthepdfbynumberofreplicationtomatchthehistogram.Thehistogramisbasedon10;000simulations.=0:5,m=0,=1,c1=0:5,c2=0:5,n=10. FigureA-9. SaddlepointapproximatedpdfofburgestimatorofinSAR(3)modelwithSymmetricLaplacemarginaldistribution.WeapproximatedthemomentsofrusingTaylorseriesexpansion.=0:5,m=0,=1,c1=0:5,c2=0:5,n=50. 107

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Comparisonoftheapproximatedpdfandsimulatedhistogram.ThemomentsareapproximatedbyTaylorexpansion.Wehaveinatedthepdfbynumberofreplicationtomatchthehistogram.Thehistogramisbasedon10;000simulations.=0:5,m=0,=1,c1=0:5,c2=0:5,n=50. 108

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PAGE 113

YunZhuwasborninChengdu,China.ShereceivedherPh.D.degreeinstatisticsfromUniversityofFlorida.Shereceivedherbachelor'sinnancewithconcentrationininternationalnancefromtheSouthwesternUniversityofFinanceandEconomicsinChina.Sheisinterestedinapplyingstatisticalmethodologiesinnancialmodeling. 113


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APPLICATION OF ASYMMETRIC LAPLACE LAWS IN FINANCIAL RISK
MEASURES AND TIME SERIES ANALYSIS



















By

YUN ZHU


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

































2007 Yun Zhu



































To Xi 1''..i- Katie, and my parents









ACKNOWLEDGMENTS

I am grateful to my Ph.D. advisor, Dr. Alex Trindade. This work could not have

been written without him who not only served as my supervisor but also encouraged and

challenged me throughout my academic program. I am grateful for his immense help

at every stage of my research, from initiating the topics, solving problems, to revising

numerous drafts. His valuable insights and ideas directly and significantly contributed to

my dissertation.

I would like to thank my committee numbers, Dr. Ramon Littell, Dr. Ronald

Randles, Dr. Clyde Schoolfield and Dr. Farid AitSahlia, for taking the time to work with

me.

Thanks go out to my husband, albv--i offering support and love. Thanks go out to my

parents, for taking care of me and my baby. I could not have finished my dissertation

without their support. Thank you my dear Katie for providing me happiness and

inspiration.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 4

LIST OF TABLES ....................... ............. 7

LIST OF FIGURES .................................... 8

ABSTRACT . . . . . . . . . . 11

CHAPTER

1 INTRODUCTION ...................... .......... 13

1.1 Financial Risk Measurement via VaR and CVaR .............. .13
1.2 Asymmetric Laplace Distribution ................ ....... 20

2 APPROXIMATION TO THE DISTRIBUTION OF MLES OF VAR AND CVAR
UNDER AL LAW ................................. 24

2.1 Maximum Likelihood Estimations of VaR and CVaR under AL Distribution 24
2.2 Asymptotic distribution of o,(X) and 0,(X) ...... ........ 25
2.3 Approximation of Finite Sample Distribution ....... ........ 27
2.3.1 General Saddlepoint Approximation to L Statistics ........ 27
2.3.2 Approximation to the First Four Cumulants of MLEs of VaR and
CVaR .............. ............ 29
2.3.3 Assessing the Accuracy of the Saddlepoint Approximations . 33

3 APPROXIMATION TO THE DISTRIBUTION OF NONPARAMETRIC
ESTIMATORS OF VAR AND CVAR UNDER AL LAW . . 38

3.1 Nonparametric Estimators of VaR and CVaR ............ .. 38
3.2 Asymptotic Distribution of Q X(X) and a(X) .............. 39
3.3 Approximation of Finite Sample Distribution . . ...... 39
3.3.1 Moment Generating Function of Nonparametric Estimators of VaR
and CVaR .... ........ . ............. 39
3.3.2 Saddlepoint Approximation and Lugannani-Rice Formula . 46
3.3.3 Laplace Approximation of Hypergeometric Function . ... 47
3.4 Comparison of the Distributions of Parametric and Nonparametric
Estim ators .................. ................. .. 49
3.4.1 Large Sample Case .................. ........ .. 49
3.4.2 Finite Sample Case .................. ........ .. 50
3.5 Analysis of Exchange Rate Data .................. ..... .. 50

4 TIME SERIES ARMA AND GARCH MODELS UNDER AL NOISE ...... 58

4.1 ARMA (p,q) Model ........... .. .... ....... 61
4.2 ARMA(p,q) Model under AL Noise ................. . .. 63









4.2.1 Marginal Distribution of ARMA Model under AL Noise .......
4.2.2 Fit AR(p) Model Using Conditional Maximum Likelihood
E stim ation . . . . . . . .
4.2.3 Fitting an ARMA(p, q) Model Using Conditional Maximum
Likelihood Estim ation .. ......................
4.3 ARMA Models Driven by GARCH Noise .. ................
4.3.1 ARMA Model Driven by GARCH noise .. .............
4.3.2 Conditional Maximum Likelihood Estimation of GARCH model .
4.3.3 ARMA Models Driven by GARCH AL Noise .. ...........
4.4 Analysis Real Estate Mutual Fund Data .. ...............


APPENDIX


A SAR(P) MODEL WITH MULTIVARIATE AL MARGINAL DISTRIBUTION

A.1 SAR(p) Model with Multivariate AL Marginal Distribution .........
A.1.1 SAR(p) M odel ..............................
A.1.2 Generalized Estimator of ) .......................
A.1.3 Multivariate Asymmetric Laplace Distribution ............
A.1.4 Saddlepoint Approximation to the Estimating Equation . .
A.1.5 Approximate the Moments of r by Taylor Expansion . . .

REFERENCES ...... ............. .................. .

BIOGRAPHICAL SKETCH . ......................... . .


89

89
89
90
91
99
104

109

113









LIST OF TABLES


Tabl

3-1

4-1

4-2

4-3

4-4

4-5

4-6

4-7

4-8

4-9


e

MLEs of the log returns of exchange rate data . ...

Fitted model parameters of AR(1) model under AL noise .

Fitted model parameters of AR(2) model under AL noise .

Fitted model parameters of AR(3) model under AL noise .

Fitted value of ARMA(1,1) model under AL noise . .

Fitted Value of ARMA(1, 3) under Gaussian noise . .

Fitted parameters of ARMA(1, 3) driven by GARCH(1,1) Gau

Fitted parameters of ARMA(2, 6) under AL noise . .

Fitted parameters of ARMA(1, 3) driven by GARCH(1,1) AL

Summary of AICc of the four methods . ........


ssian noise .



noise . .


page

55

72

73

73

75

81

82

84

86

86









LIST OF FIGURES


Figure page

1-1 Heavy tailed distribution vs. normality. .................. .... 16

1-2 Histogram of the daily log returns of exchange rate data ............ .17

1-3 PDF of Asymmetric Laplace distribution .................. ..... 21

2-1 CDFs of MLE of VaR with a = 0.9 .............. ...... 34

2-2 CDFs of MLE of VaR with a = 0.99 ............... .... 34

2-3 CDFs of MLE of VaR with n 50 and = 1 . ......... . 35

2-4 CDFs of MLE of VaR with n 50 and K = 0.8 ................... 35

2-5 CDFs of MLE of CVaR with = 1 ................. ...... 36

2-6 CDFs of MLE of CVaR with K = 0.8 ............... ..... 36

2-7 PREs of the saddlepoint approximated distribution of MLE of VaR ...... ..37

2-8 PREs of the saddlepoint approximated distribution of MLE of CVaR . 37

3-1 CDFs of NPE of VaR with K = 1 .................. ...... .. 48

3-2 CDFs of NPE of VaR with K = 0.8 ................ ..... 48

3-3 PREs of the saddlepoint approximated distribution of NPE of VaR ...... ..49

3-4 ARE of MLEs with respect to NPEs of VaR and CVaR . . ..... 51

3-5 Saddlepoint approximated density functions of MLEs and NPEs . ... 52

3-6 Histogram, boxplot and sample ACF of daily log returns of exchange rate . 53

3-7 Normal Q-Q plot of daily log returns of exchange rate. ............. .54

3-8 Histogram of the daily log returns without weekends ............. 54

3-9 Normal Q-Q plot of daily log returns without weekends . . ..... 55

3-10 Confidence ellipses for the MLEs and NPEs bivariate estimators of VaR and
CVaR .................................. ...... 57

4-1 Simulated ARMA(1,1) process under AL noise .... . ... 64

4-2 Histogram of of the Simulated ARMA(1,1) process .............. 64

4-3 Derived marginal pdf of AR(1) model under AL noise ............. 70

4-4 Comparison of derived marginal pdf and simulated histogram of AR(1) model .70









4-5 Derived marginal pdf of ARMA(2,2) model under AL noise . ..... 71

4-6 Comparison of derived marginal pdf and simulated histogram of ARMA(2,2)
model ........ .. . . ................ .. 71

4-7 Daily values of the mutual fund ............... ....... .. 78

4-8 Daily returns of the mutual fund ............... ...... .. 78

4-9 Histogram of the daily returns .................. ......... .. 79

4-10 Sample ACF of residuals of ARMA(1,3) model under Gaussian noise. ..... ..81

4-11 Sample ACF of absolute values of residuals of ARMA(1,3) model under Gaussian
noise. ................... ...... ...... .... .. 81

4-12 Sample ACF of squares of residuals of ARMA(1,3) model under Gaussian noise. 82

4-13 Sample ACF of residuals of ARMA(1,3) model driven by GARCH(1,1) Gaussian
noise. ...... .... .... .................... 83

4-14 Sample ACF of absolute value of residuals of ARMA(1,3) model driven by
GARCH(1,1) Gaussian noise ............ . . ..... 83

4-15 Sample ACF of squares of residuals of ARMA(1,3) model driven by GARCH(1,1)
Gaussian noise ............... ............... .. 84

4-16 Sample ACF of residuals of ARMA(2,6) model under AL noise. . ... 85

4-17 Sample ACF of absolute values of residuals of ARMA(2,6) model under AL noise. 85

4-18 Sample ACF of squares of residuals of ARMA(2,6) model under AL noise. 86

4-19 Sample ACF of residuals of ARMA(1,3) model driven GARCH(1,1) AL noise. .87

4-20 Sample ACF of absolute values of residuals of ARMA(1,3) model driven
GARCH(1,1) AL noise ................. .... ........ 87

4-21 Sample ACF of squares of residuals of ARMA(1,3) model driven GARCH(1,1)
AL noise ..................... .. ...... ......... .88

A-i Saddlepoint approximated pdf of burg estimator with n10 . .... 100

A-2 Comparison of the approximated pdf and simulated histogram with n=10 101

A-3 Saddlepoint approximated pdf of burg estimator with n=50 . . ... 101

A-4 Comparison of the approximated pdf and simulated histogram with n=50 . 102

A-5 Saddlepoint approximated pdf of burg estimator ................. .103

A-6 Comparison of the approximated pdf and simulated histogram . ... 104









A-7 Saddlepoint approximated pdf of burg estimator with n10 . .... 106

A-8 Comparison of the approximated pdf and simulated histogram with n=10 . 107

A-9 Saddlepoint approximated pdf of burg estimator with n=50 . .... 107

A-10 Comparison of the approximated pdf and simulated histogram with n=50 . 108









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

APPLICATION OF ASYMMETRIC LAPLACE LAWS IN FINANCIAL RISK
MEASURES AND TIME SERIES ANALYSIS

By
Yun Zhu

May 2007

('C! i: Alex Trindade
Major: Statistics

Asymmetric Laplace (AL) laws are applied in financial risk measurement and time

series analysis. Traditional methods on financial risk measures and time series analysis are

based on the assumption of normality. Recent studies on financial data -ii--.- -1 that the

normality assumption is usually violated.

Explicit expressions are derived for maximum likelihood estimators (MLEs) and

nonparametric estimators (NPEs) of financial risk measures, Value-at-Risk (VaR) and

Conditional Value-at-Risk (CVaR), under random sampling from the Asymmetric Laplace

distribution. Asymptotic distributions are established under very general conditions.

Finite sample distributions are investigated by means of saddlepoint approximations. An

application of the methodology in modeling currency exchange rates -, -.--- -I that the

AL distribution is successful in capturing the peakedness, leptokurticity and skewness,

inherent in such data.

Time series autoregressive moving average (ARMA) models driven by Asymmetric

Laplace noise are considered for modeling dependent data. Assuming AL noise, the model

marginal distribution is derived analytically. Conditional maximum likelihood estimation

is applied to fit ARMA models driven by AL noise and AL general autoregressive

conditional heteroscedasticity (GARCH) noise. Daily returns of real estate mutual

fund data are fitted by four methods. Models under AL noise have substantially lower









Bias-corrected Akaike Information Criterion (AICc), indicating much better fit for the real

financial data.









CHAPTER 1
INTRODUCTION

1.1 Financial Risk Measurement via VaR and CVaR

Value-at-Risk (VaR) has become one of the most important risk measures in modern

financial risk management. In general, financial risk comes from three parts, market

risk, credit risk, and operational risk. Market risk is defined as the uncertainty due to

the changes in financial asset prices such as interest rates, foreign exchange rates, equity

prices, and commodity prices; Credit risk comes from the losses associated with the default

(or credit downgrade) of an obligor; Operational risk is related with operational failures.

The idea behind Value-at-Risk originated from measuring market risk. VaR is the loss

that can occur over a given period, at a given confidence level, due to exposure to market

risk. Recently, the idea of VaR has been introduced into measuring credit risk.

The importance of measuring the risk of financial assets has long been realized.

Markowitz (1959) first introduced the definition and measurement of risk in portfolio

selection, where financial risk was measured by the variance and covariance of underlying

asset prices.

Since 1990's, Value-at-Risk is widely used by commercial banks, asset management

companies, and regulators. For example, the Basel Accord I employs VaR as the

measurement for commercial banks' market risk exposure. The Basel Accord is the

international capital adequacy standards set up by Basel Committee on Banking

Supervision. The 1996 amendment of the Basel Accord extends the capital requirements

to include risk-based capital for the market risk in the trading book. Under the supervision

of the Basel Committee, banks need to set up their own VaR models to calculate their

minimum regulatory capital for market risk. In 1997, the Securities and Exchange

Commission in United States began requiring financial institutions to report Value-at-Risk

as an important measure of the market risk exposure.









The concept of Value-at-Risk is also widely used by financial institutions and asset

managers. For example, in asset management companies, Value-at-Risk is used to set

position limits for traders; In commercial banks, Value-at-Risk is used to calculate market

risk exposure of their assets and is used in capital allocation.

Loosely -I' i .i:i- if Y represents "1, .-- and 0 < a < 1, given time horizon t, the

VaR at confidence level a, VaR (Y), is the lower bound on the worst (1 a)l10I'i losses

during the time horizon.

let Y be a continuous real-valued random variable defined on some probability

space (Q, A, P), with cumulative distribution function (cdf) F(-) and probability density

function (pdf) f(.). Let p and a2 denote the mean and variance of Y, respectively, and

both are assumed to be finite. With the understanding that Y represents loss, the VaR of

Y at probability level a is defined to be the ath quantile of Y.

Definition 1.1 (VaR).


VaR,(Y) (Y) = F-l(). (1-1)


The confidence level a and time horizon t vary among different banks, companies

and regulators. A commonly used confidence level is 9' I' For example, Basel Accord has

set a to be 9'I' and t to be 10 di d- in order to measure banks' market risk exposure.

Commercial banks typically use overnight Value-at-Risk to measure financial risk

exposure for the purpose of internal supervision and risk control, and disclose two-week

Value-at-Risk to investors and regulators.

Practically, there is a 'square root of time' rule. That is, if the daily Value-at-Risk

is 4, then the Value-at-Risk in the time horizon of m di,- will be r/m. This result is

based on the assumption that daily returns are independent. Under this assumption,

Value-at-Risk under different time horizons can easily be transformed by multiplying or

dividing by square root of div-









One 1i i i" criticism of VaR is that it is not a coherent measure, i.e., VaR is not

sub-additive. This means that in general the VaR of a portfolio can exceed the sum of the

stand-alone VaRs of its components. In addition, VaR provides no information about the

loss beyond itself.

As an alternative to VaR, Conditional Value-at-Risk (CVaR) describes the average

of the worst (1 a) losses. The term of CVaR is drawn from Rochafellar and Uryasev

(2000), but synonyms for it also in common usage include: "expected shortfall" (Acerbi

and Tasche, 2002), and "tail-conditional expectation" (Artzner et al., 1999).

The CVaR of Y at probability level a, is the mean of the random variable that results

by truncating Y at VaR,(Y) and discarding its lower tail.

Definition 1.2 (CVaR).


CVaR,(Y) 0(Y) E(YY > ) 1 a y f (y)dy. (1-2)


An equivalent definition of CVaR in terms of the quantile function of Y is

1Y3)
(Y) =- t F-(u)du. (1-3)

Methodologies for Estimating VaR. There are various methodologies for

estimating Value-at-Risk. Most of them fall into three categories: parametric methods,

nonparametric methods and semi-parametric methods.

Parametric methods assume a distribution for the financial data. Under parametric

methods, the VaR at confidence level a is just the ath quantile of the distribution.

Parametric methods depend on the assumption of a distribution. The normal distribution

is the most commonly used family in financial risk measurement. But there are obvious

violations of normality in financial data. Financial data are typically skewed and heavy

tailed. Another problem with parametric methods is that they are inappropriate when

there are discontinuous p li-offs in the portfolio.










Nonparametric methods, also called historical methods, use empirical quantiles

of the data to estimate VaR. Nonparametric estimation makes no assumptions about

the distribution. As a result, it is flexible enough to deal with data with heavy tails.

Nonparametric estimation is based on information from the past. If there is a permanent

change in 1ni, i i market factors, for example, changes in regulations, nonparametric

estimation will underestimate or overestimate VaR.

Semiparametric methods assume the distribution of one tail, which making no

assumption about the underlying distribution away from the tail. A typical used tail

distribution is pareto tail.

Fat tails and Skewness. Tod-,- the presence of heavy tails of financial data

is a well-accepted fact. The central limit theorem (CLT) is not valid here because a

key assumption behind the central limit theorem is that data that go into a sum are

statistically independent. Independence is not a proper description of financial data. And

financial data are not symmetric about the mean in most cases. In financial data, one

tail represent profit and the other represents loss. Consequently, we can not treat them

equally.

Assuming normality on heavy tailed data will cause underestimation of VaR at high

confidence level, which will cause i j ri" problems in risk control.
heavy tailed and normal distribution
04
heavy-talled
norma I


-10 -8 -6 -4 -2 0 2 4 6 8 10

Figure 1-1. Heavy tailed distribution vs. normality.










Daily log return of USD/EUR
90
80
70
60


240





-002 -0015 -001 -0005 0 0005 001 0015 002

Figure 1-2. Histogram of the daily log returns of USD/EUR exchange rate from Feb
1,2005 to Jan 31, 2006.


Fig. 1-2 is the daily log return of USD/EUR exchange rates. This histogram indicates

the typical properties of financial data, heavy tails and skewness. Commonly used

measures of heavy tails and skewness are kurtosis and skewness, respectively. For more

detailed information about kurtosis and skewness, please refer to Section 1.2. Properly

adjusted, the kurtosis of a normal distribution is 0. And when kurtosis is greater than 0,

the data are considered heavy tailed. The skewness of any symmetric distribution is 0.

The kurtosis and skewness of the data are 2.0119 and -0.3777, respectively. Therefore, we

consider them to be heavy tailed and skewed.

Explanation and solution.

Theoretically, there are many explanation about the reasons for fat tails. Two of them

are widely accepted: There are some significant discontinuous changes in the financial

data due to unexpected changes in market factors, for example, market crash. This is also

called 'jumps'. The other explanation is called 'Volatility Clustering', the volatility at time

t is highly correlated with volatilities at past time s, s < t.

The Mixture normal model and Jump-diffusion model are developed to explain data

with jumps. The mixture normal model assumes the financial data come from a random

mixture of two different normal distributions. One of these normal distributions comes









from an ordinary market situation, and the other comes from a market with higher

volatility.

Statistically, let 0i and Q2 be the density function of two normal distributions with

different mean and variance. X is a random variable from a mixture of normals if


fx(xP) PI()+ (1- P)2(x), (1 4)

where P is a Bernoulli random variable with success probability p. Under the mixture

normal model, we can choose p and variances of two normal distributions to achieve a

given kurtosis, and therefore get a fat tailed X.

The jump-diffusion model was introduced in the context of differential equations. For

the jump-diffusion model, financial data is 'jumped' by adding an independent normal

random variable.

Auto-Regressive VJ/1,il7.lh/ model (ARV), El''.'u,' /,:l.all; Weighted Moving Average

(EWMA) and Generalized Autoregressive Conditional Hete, .. :/l. (GARCH)

models are developed upon observing the clustering of the volatility. ARV and EWMA

are intuitive methods. GARCH models are more flexible to describe time series processes

with correlated volatility. Consequently, GARCH models are commonly used in financial

industries.

The ARV model assumes


logo,2 = a + O3log,2_1 + yZ, (1t-5)


where c, 3 and 7 are constants. {Zt} is white noise. The volatility at time t is a

function of volatility at t 1. As a consequence, the volatilities are correlated. A value of

3 near zero implies low correlation, while a value of beta near 1 implies high correlation.

The EWMA model was introduced by the RiskMetrics Group. The variance at time t

is estimated as












2 = (1 )X2 + 2


(1-6)


with Q > 0, where Q is a 'smoothing constant' and Xt-1 is the return at time t 1.

The volatility at time t is considered as a function of the return and volatility at time

t 1. Since Q > 0, the volatility at time t will be positively correlated with the volatility

at t- 1.

GARCH is a more flexible and general model describing the clustering of volatility.

Bollerslev (1986) introduced the GARCH process. The volatility is estimated as
u V
at = ao + ajZ i+ t3-j_, (1-7)
i=1 j=1

where {Zt} is white noise. ao > 0, aj > 0, /j > 0, j = 2,.... Under a GARCH

model, the volatility at time t depends on the volatility of the past. For more detailed

information about GARCH model, please refer to C!i lpter 4.

Some potentially more flexible models include: Exponential GARCH model,

EGARCH Cross-Market GARCH, etc.

Distributions for fat tails and skewness. Various families of distribution

have been introduced to describe the two characteristics of financial data: fat tails and

skewness. A very intuitive one is the t distribution. But t distributions can not capture

the skewness of financial data. Fernandez and Steel (1998), Kuester and Mittnik (2006)

and Patton (2004) introduced a generalized t-distribution and a skewed t-distribution. In

general, those densities are not log-concave.

Exponential Power Distribution (EPD), also called Generalized Power Distribution

(GPD), has also been used in financial risk measure. The density of EPD is

1
f(x)= 2p(lb)exp( -x/ab). (1 8)
2aF(1 + 1/b)









For b = 1 this reduces to the Laplace Distribution. For b = 2, it has the same

form as a normal distribution with a = V2a. EPD is a flexible distribution but does

not allow for any .-i-mmetry in the data. Komunjer (2006) introduces the Asymmetric

Power Distribution (APD) for estimating Expected Shortfall. Some of the properties of

the APD distribution have been studied by Fernandez, Osiewalski and Steel (1995), Ayebo

and Kozubowski (2003). Komunjer (2006) gives no result about the distribution of the

estimators of risk measures nor confidence intervals.

The Symmetric Laplace Distribution (SLD), also called Double Exponential

Distribution, has been used for modeling data with heavy tails. See Balakrishnan and

Basu (1995), Bain and Engelhardt (1973), Kotz Kozubowski and Podgoriski (2001). SLD

distribution has the same problem as other symmetric distributions which do not allow for

any .- i-:iiiii ry. Therefore, we consider the .i-,mmetric form of the Laplace distribution,

the Asymmetric Laplace distribution.

1.2 Asymmetric Laplace Distribution

One of the intents of this work is to apply Asymmetric Laplace law to financial

data. The A-;,,,iiiiii, Laplace (AL) distribution, introduced by Kotz el al. (2001), is a

generalization of the Symmetric Laplace distribution (Double Exponential distribution).The

AL distribution demonstrates flexibility in fitting data with heavy tails and skewness,

which make it a promising candidate for financial data modeling.

Definition 1.3 (The Asymmetric Laplace distribution). Random variable Y is

said to be distributed as Asymmetric Laplace distribution with location parameter 0, scale

parameter 7 > 0, and skewness parameter K > 0, Y ~ AL(0, K, r), if its pdf is of the form


f(y) (1-9)
ex(1p x2, y exp ( -Y 01 if y < 8,










or, the distribution function of Y is the form


{ 1- ( exp(-' 0\, if y > 0,
F(y) 2ex (1 10)
12 exp -j 0), if < 0.


0.8
0.7 K0.8
0.65
0

0.
0.2
0.1/ / I \


6 -4 -2 0 2 4 6 8
x

Figure 1-3. Asymmetric Laplace densities with 0 =0, Tr 1, = 0.5, 0.8, 1

Kotz et al. (2001), ch.3, generalize the essential properties of Asymmetric Laplace

distribution.

Proposition 1.1 (Moment generating function of AL distribution). If

Y ~ AL(0,, K, ), then the moment generating function of Y is

ot v/2 v/2
-...( ( = < t < (1-11)
1 722 7- K) 1

Proposition 1.2 (Cumulants of AL distribution). The cumulants of an

AL(0, K, 7) can be stated as


0 + (K- K), ifn 1;

n(Y) < (n- 1) ( 0"), if n > 1 is odd; (1-12)

(n -1)! (t-' + 0) if n is even.
(n t) ")"









The mean and variance of Y, which coincide with the first and second cumulants,

respectively, are

'1 2 7 2 2 2.
p=0+ K<- and a 2 2+ K2) = -0) + 2. (113)


Proposition 1.3 (Coefficients of skewness and kurtosis). For a distribution of

an random variance Y with a finite third moment and standard deviation greater than

zero, the coefficient of skewness is a measure of symmetry defined by

E(Y- EY)3
( E(Y EY)2)3/2 (1

For an AL(0, K, r) distribution, the coefficient of skewness is


<7=2 (1t15)
(1/K2 + 2)3/2

The coefficient of skewness is nonzero unless K = 1. The absolute value of 71 is

bounded by two, and as K increases within the interval (0, oo), the corresponding value of

71 decreases monotonically from 2 to -2.

For an random variable with a finite fourth moment, the kurtosis is defined as

E(Y EY)4
72 (Var(Y))2

It is a measure of peakedness and of heaviness of the tails (properly adjusted, so

that 72 = 0 for a normal distribution) and is independent of the scale. If 72 > 0,

the distribution is said to be Leptokurtic, with heavy tails and high peakedness; it is

plIJ.. a rtic otherwise.

For an AL(0, K, r) distribution, the kurtosis of AL distribution is

72-6- 127)
(1/K2 + K2)2









The AL distribution is leptokurtic and 72 varies from 3 (the least value for the

Symmetric Laplace distribution with K = 1) to 6 (the greatest value attained for the

limiting exponential distribution when K -- 0).

Proposition 1.4 (Quantiles of AL distribution). For an AL(O, K, -) distribution,

the qth quantile, q, is

S0+- log {1q }, for qE (0, ,
=q = I (1-18)
0 log{(l + 2)(1 -q)}, for qe ,1 )

Therefore, the VaR and CVaR are then easily derived.

Proposition 1.5 (VaR and CVaR for AL distribution). Let Y ~ AL(0, K,) ,
2
where 0, 6, r are unknown parameters. Then for (i2) < a < 1, VaR and CVaR can be

obtained as

() 0 '-rlog[(l + K2)(1 a)]


and


) = (Y)+ (1-19)









CHAPTER 2
APPROXIMATION TO THE DISTRIBUTION OF MLES OF VAR AND CVAR UNDER
AL LAW

In this chapter, we give the explicit expressions of the maximum likelihood estimators

(MLEs) of VaR and CVaR under Asymmetric Laplace distribution. Large sample
.i-,i!l ill ic distributions of MLEs are establishes via Delta methods. Finite sample

approximations of MLEs are developed by general saddlepoint approximation. Finally, the

accuracy of the approximations is checked via simulations.

We will assume that the location parameter, 0, is known. Both VaR and CVaR are

translation invariant and positively homogenous (Pflug, 2000), i.e. Q(Y) = 0 + T-(X)

and a(Y) = 0 + T- a(X). Without loss of generality, in this chapter we focus on

X ~ AL(0, K, -), provided 0 = 0 is known. Consequently, VaR and CVaR can be presented

as

S rlog[(1 + )(- a)]
l ( ) --
i{2
M(X) = Q(X) + (2-1)


2.1 Maximum Likelihood Estimations of VaR and CVaR under AL
Distribution

Kotz et al. (2001), ch.3, give explicit expressions of MLEs of K and provided the

value of 0 is known. Consider a random sample XI,..., X, from X ~ AL(0, K, r). Let x(1)

and x(n) be the first and nth order statistics of this random sample. If x(l) < 0 < x(,), the

MLEs of K and T exist and are available in closed form as





v- 2 xE (2-2)
=l 1 i= 1Z \ i= \ i= 1 (2

where x+ = xil[>o] and x = -xlr[








Let x Vi and x x V2, the MLEs of K and 7 can be expressed

equivalently as

4(2-3
(i), and / = (ViV2)( v+V ) (2-3)

Then the MLEs of VaR and CVaR are obtained by equivariance,

Tlog[(1 + 2)(1 a)]
(X) 2

and


() = (X) + (2-4)

or, equivalently,


S= + + ) ogV + log(1 a)]

O(X) (X) (V+ VV (2-5)


2.2 Asymptotic distribution of o,(X) and 0,(X)

Kotz et al. (2001), ch.3, prove the consistency and .,-i-~i:!,ll Iic normality of R and 7.

Proposition 2.1 (Consistency and asymptotic normality of R and 7 ). Let

Xi,...,X, be i.i.d. random sample from distribution AL(O, K, r), where the value of 0 is

known. Then the MLEs of [K, 7], [R, r], given by Eq. 2-1 are

(i) Strongly consistent;

(ii) Asymptotically bivariate normal with the .-i-,iill.1 ic covariance matrix

(1+ 2)2 1-4
8 8S
1-4 -2 (1+6s +4)
8[ 8K2

(iii) Asymptotically efficient, namely, this .,-i-,_i!l ic covariance matrix coincides

with the inverse of the Fisher information matrix.









In our case, the fisher information matrix is


n-2 + 2)-2 1-2
I(r, 7) 4(2-7)
[1--2 -2
TK(1+K2)

Proposition 2.2 (Multivariate delta method ).

Let g : Dg C Rk v- R" be a map defined on a subset of Rk and differentiable at

0. Let T, be random vectors taking their values in the domain of g. If n/(T, 08)

Nk(0, E), then


V(g(T g(0)) Nm(0, g(g) T). (2-8)


Let g(T,) = a

then

[6^ OQ ] [ (l+K2)Wc,--2K2 W.,
O K,2 2 (1+,2) v/
g (2-9)
9(6 9(t (1+K2)(w ,K-1)-2K2 (1-Ka,)
o. 9- T V.2 ,(1+,2) Kv/2

where


W,, log[( + 2)( a)]. (2-10)


Applying the multivariate delta method and the ..i-1ill..1 ic normalities of [, f] in

Proposition 2.1, we get the .i-vl !1,', n1 i. distribution of MLEs of VaR and CVaR

Theorem 2.1 (Asymptotic joint distribution of MLEs of VaR and CVaR)

. Let Y ~ AL(0, K, 7). Define w,, the same as in Eq. 2-10 and the MLEs of VaR and

CVaR, [ ,, 6], as in Eq. 2-1, respectively, we have


( a 0 ) a(2,) a









where,


42 a
2()= 472 [ ,2(W,- 1)2 + 2,w ]

2(a) 7 [(2 +2) -4 + )) 4(2 ,, + 42 + 2]

(, O) 2 ) [( 2 + 2)U,, 2,2] (2-11)

2.3 Approximation of Finite Sample Distribution

2.3.1 General Saddlepoint Approximation to L Statistics

Easton and Ronchetti (1986) derive the saddlepoint approximation for the density of

a general statistic U,.

Suppose that xj,..., x, are n iid real valued random variables with density f.

U,(xl,..., x,) is a real valued statistic with pdf f, and cdf F,. Let .1 (t) = fet f,(x)dx

denotes the moment generating function and K,(t) = logl [ (t) the cumulant generating

function of U,. Further suppose that the moment generating function of .1 (t) exists for

real t in some nonvanishing interval that contains the origin.

Fourier inversion gives


f(x) = (it) e-itxdt

n I I[(nT)e-TxdT
27ri -i
2r- n n(R (T)-Tx)dT, (212)
27i J\-ioc

where A is any real number in the interval where the moment generating function

exists.

Let

R,(T) = Kn(nT)/n, (2-13)









applying Edgeworth approximation, R,(T) can be approximated in terms of the first four
cumulants of U,.

noT2 3n 2T3 4na 3 4
R(T) pT + + + (2-14)
2 6 24

where pn, 4 i, I3 i4n are the mean, the variance, and the third and fourth
cumulant of U,.
We assume that the Edgeworth expansion up to and including the term of order n-1
for fn exists. Expansions of the form

pn p + +a/n + o(n-1)

On = c7/ 1/2 + bl/3/2 + o(-3/2), (2-15)

will suffice to keep the same order in the approximation.

Applying the saddlepoint technique to the integral in Eq. 2-12 gives the saddlepoint
approximation of fT with uniform error of order O(n-1),


f2(x) nI {27R(t)} exp {nRn(i) ntx (2-16)

where t is the saddlepoint and it is the solution to the saddlepoint equation, R(t) =

x; RT(-), RT(-) denote the first and second derivatives of R,(t).
In addition, if we apply the same technique to Lugannani and Rice (1980) formula, we
get the saddlepoint approximation to the cdf of U,

F,(x) () + ) _l{ }, (2-17)

where 4(-) and 0(-) are the standard normal distribution and density functions with

S sgn (t) [2n tx Rn(t)}j~ 2

q= t{nR( t)}I2.









In Eq. 2-17, at x E(U,), the alternate expression used is


1 1 3
F,(x) + (727)-I"' (0)R(0)- 2
2

where R"(-) is the third derivative of the R (t).

2.3.2 Approximation to the First Four Cumulants of MLEs of VaR and CVaR

The first four cumulants of MLEs of VaR and CVaR can not be derived straightforwardly.

To apply Easton and Ronchetti method, we consider to use Taylor expansion to

approximate the cumulants.

Distributions and moments of x+ and x.

Consider the auxiliary random vector

ZW [Z Z ]', i 1,2,...,n, (2 18)


where Z) = X and Z() X-. Therefore, Vi = Z" and V2 = Z".

According to Kotz et al. (2001), ch.3, Z(i) are independent and identically distributed

as,

z(i) T J2
S62,i 2,i
where {W1,4} and {W2,i} are the standard exponential variables. {61,i} and {62,i}

are iid Bernoulli random variables with success probabilities 1/(1 + K2) and K2/(1 + K2),

respectively. W1, W2, and (61,62) are mutually independent. By definition, the first four

moments of 61 are


E61 E E62 = E6 6 = ). (2-19)
I1 + K)

The first four moments of 62 are

E(2
E62 E62 E63 E64 ( (2 20)
2 ( t + K2)









The moment generation function of standard exponential variable is (-t. Therefore,

the first four moments of W are EW = 1, EW2 = 2, EW3 = 6, EW4 = 24.

Since 61 and 62 are independent of W1 and W2, respectively, the first four moments of

Z1 and Z2 can easily be obtained,


T
v (1 + K2)'
37T3



/(1 + 2)'
373 5
V(1 + 2)'


T2
EZ2

1 4(1 + K2)
4 674

(2)4


L22
E^2 = T


The variances of Z1 and Z2


VarZ1


VarZ2


can be represented as

T2/2


Note that EZ) Z =) 0 since at least one of Z' and Z2 equal to

of Z1, Z2 can be obtained as


Cov(Zi, Z2)


(2-22)


zero. The covariance


T2 22
2(1 + K2)2


(2-23)


Other third and fourth central moments and covariances can also be deduced from the

first moments of Z1 and Z2 by applying Eq. 2-24.


E(Z1

E(Z1


E -
E(Z1-

EZ1)2(Z2

Z1)(Z2 -

E(Z2-
E(Z -

E(Z1


EZI)3

- EZ2)

EZ2)2

EZ2)3

EZI)4


EZ3 3EZ,2EZ1 + 2(EZ1)3

-EZ2EZ2 + 2(EZ1)2EZ2

-EZIEZ2 + 2EZ1(EZ2)2

EZ3 3EZEZ2 + 2(EZ2)3

EZ4 4EZ3EZI + 6EZ12(EZ1)2 3(EZ1)4


EZ1

EZ3


EZ2

ELZ


(2-21)









E(Z1

EZ-
E(Z1

E(Z1


-EZI)3(Z2

EZI)2(Z2-


- EZ(Z2
E(Z2


- EZ2)



EZ2)
EZ2 3

EZ2) 4


-EZ EZ2 + 3EZ EZIEZ2 3(EZI)3EZ2

(EZI)2EZ + EZ (EZ)2 3(EZ1)2(EZ2)2

-EZIEZ2 + 3EZIEZ2EZ 3EZ(EZ2)3

EZ2 4EZ2EZ2 + 6EZ (EZ2)2 3(EZ24.


(2-24)


Taylor expansion for a(X) and 73(X).
Let fl(VI, V2) (X), 01 = EZ1 = EVI and 02 = EZ2 = EV2. By applying the first

five terms of Taylor expansion for the two variables, Vi and V2, we can approximate the
mean of a((X) with error term of n-3,


fl(0, f2) + (V 1) +(V2- 2)
a VI 01 V2 v V2 02]
1 2f 02 f,
+ (vl 01) V2 + 21(V 0 )(V- 02)a2
2 V101 a)2 VV V2 ,Vf 02

(+ ( 2- 8 f
S( 2 V22 V202

1 8- 3 f, 0 03 f,
+ ( +)3 3( 01)2(V 82)
V 3 f + (V -22 02) a 2 V
+ 3! 012 02 f VI 2,V2
S 3)2 04 0 )3 4 f ,
+ 3(V- 0) 0 2) 02 ( V2 2) 3


1 =V22 V 2
S V4 f, 4 f,

+ 4(V -V 01 )(V2 2 8-
4 V10 v23 VV2,V2 02



+ (V2 02)2 4 ff o
S 6(VI )2V 22 02
1V2 V22 ^ 2-

* 4(VI 01)(V2 82)3 3

* (V2 02 4
0V24 V2 2


02




02


(2-25)


Take expectations on both sides, the mean of a((X) can be approximated as











E [(X)]=Ef (V, V)

fl (01, 2)
1 2 1 f2\1 2 = v 1
+ VarV+ 2Cov(V, V) + VarV2
2 2 V1 0l +2C v V1 V2 v 01,V2 02 '2 V22 V202
1 3a3 fl 3 fl
+ E(Vi 0 ) v3 + 3E(V 1)V- 2)V 2
1 V V21 va V2 1,V2
2 3 f, 3 f
+ 3E(V1 01)(V2 02)2 + E(V2 02 3
SV1i 1V22 01,V2 V203 2v2
1 (1 4 04 f 04 f1
+ E --(I V + 4E(V1 1)3 2 0- 2 3
24 4 V2 01,V2 002
+ 1 01)2(V 2 4 f
6E(V 01)22 82 )2 V 2
a V2 V2 8i1, V02
84 fl
+ 4E(V1 01)(V2 82)3
0 VI V1 i=v23 1,=2 02
,4 04 f,
+ E(V2 f2 4

(2 26)

Since Vi Z," and V2 Z2 ,( this approximation can be further expressed as

E [Z,(X)]

fl (01, 02)

+ VaRZ 2Cov(Z Z) +VaRZ2o 02

+ E(Z -6 )3 + 3E(Z1 01)2(Z2 82)03
6n2 8 V3 V2 V2 y V3 22
03 fl 1 03 f 1
+ 3E(Z1 01)(Z2 82)2 + E(Z2 0 2 3 2 21
v V V22 1,1; 2 02 a V2 13
1 E] (1 8-..404 fl o4 fl
+ 24 (Z 01)4 f + 4E(Z1 01)3(Z2 82) 0
24n3 0 V4 V3 V2 y y 9
04 fl
+ 6E(Z1 01)2(Z2 82 )2
4 Vf 4 f]0 (2127)
+ 4E(Z1 01)(Z2 82)3 + E(Z2 2) 4 (2-27)
S1a v23 -01,V2 02 ( 27)









Let ki, k2, k3, k4 be the first four cumulants, p/, /2, /[3, /4 be the first four moments

of (X), respectively. Set f2(V, V2) ((X) f)(V1, V2) ((X)), f4(VI, V2)

(o(X)) The same technique can be applied to approximate the second, third and forth
moments as


P2 E (X)
P3 E (X)

P4 E ((X)


SEf,2(V, V2)

E f3(VI, V2)

Ef4((V, V2).


(2-28)


To approximate the first four cumulants of ,((X), we apply

and cumulants,


ki = 11p

k2 = [2 (1i)2

k3 =2(/1)3 32tl/2 + 3

k4 (p)4 + 12()22 3(2 )2


the relationship of moments


(2-29)


4//1 3 + 14.


As the result, the approximation to the distribution of a((X) and 0,(X) can be obtained.

2.3.3 Assessing the Accuracy of the Saddlepoint Approximations

In this section, we compare the saddlepoint approximation to the distribution of

MLEs of VaR and CVaR, with empirical values obtained via simulation.

Let Fsim(r) and Fsad(r) denote the estimates of the true cdfs of maximum likelihood

estimator, obtained via simulations and saddlepoint approximations, respectively. The

Percent relative error (PRE) of the cdfs is a commonly used technique to measure the

accuracy of the saddlepoint estimation. We define the PRE at the quantile r as


PRE(r) {


Fs.d(r)-Fi,,,(r) t p (r) < 0.5,
( 1i- F (r)-(iH)
1- /,()o 100, Fsimr ) > 0.5.
1-F sim \r)


(2-30)












Thus, large absolute values of PRE denote larger discrepancies between the

saddlepoint approximation and the true distribution, while a PRE value of 0 indicates

perfect agreement.


To assess the accuracy of this saddlepoint estimation, we approximate cdfs of MLEs

of VaR and CVaR with parameters n = 50,100, 0 0, = 0.8, 1, = 1, a = 0.9, 0.99. We


calculate empirical cdfs through 106 simulations. PREs are computed at 10 points of equal


distance between the 1('. ~ 91'. quantiles.

n=100, K=1, a=0.9
SPA
09 Simulation

08
07-
06-
S05
04
03


06 08 1 12 14 16 18 2
X

Figure 2-1. Estimated CDFs of t,(X), obtained via simulations and saddlepoint
approximations, with n 100, 0, 7 = 1, = 1, a 0.9



n=100, K=1, a=0.99
SPA


08-

07 Simu n

06-

t05-

04

03

02 -

Ol

15 2 25 3 35 4
X

Figure 2-2. Estimated cdfs of a(X), obtained via simulations and saddlepoint
approximations, with n=100, 0 = 0, r = l, K = l, a = 0.99












n=50, K=1, u=0.9
SPA
9 Simulation

08

07

06

005
04 -

03

02-

01

0 05 1 15 2 25
X

Figure 2-3. Estimated cdfs of the a((X), obtained via simulations and saddlepoint
approximations, with n=50, 0 = 0, 7 = 1, = 1, a = 0.9

n=50, K=0.8, ca=0.9
SSPA
S Simulation


Figure 2-4. Estimated cdfs of the ,,(X), obtained via simulations and saddlepoint
approximations, with n 50, 0 0, = 1 K = 0.8, a = 0.9



Calculations based on a few equispaced points between the 10th and 90th percentiles,

reveal that PREs for the MLEs of VaR and CVaR are between 1 and 4.5'. with


n = 50, 100, = 0.8,1, = 0.9, 0.95, 0.99. This indicates that our approach works well to


approximate the distributions of MLEs of VaR and CVaR from the Asymmetric Laplace

distribution.












n=50, K=1, a=0.9
S-SPA
SSimulation
09

08-

07-

06

-n05-

04

03-

02

01 -

15 2 25 3 35
X


Figure 2-5. Estimated cdfs of 0,(X), obtained via simulations and saddlepoint
approximations, with n 100, 0 =0, = 1, = 1, a 0.9

n=50, K=0.8, c=0.9
S-SPA
Simulation I


Figure 2-6. Estimated cdfs of 0,(X), obtained via simulations and saddlepoint
approximations, with n 100, 0 =0,- = 1, = 0.8, a = 0.9



We have chosen a = 0.9, 0.95, 0.99 because they are commonly used probability


levels in financial risk measures. We choose K = 0.8 because it is close to the fitted values


obtained from a real data set, while K = 1 indicates the symmetric case.


Also, note from Fig. 2-7 and Fig. 2-8, the PREs are less than zero for large value of


r, which means that we tend to overestimate a((X) and <,(X) over the right tails. The


approximations are better for larger sample sizes, and also better for symmetric cases.




























50%


Figure 2-7. Percent relative errors (PREs)
distribution of a((X) with n
at the same quantile values.


50%


Quantile

of the
100,0


saddlepoint approximation to the
S0, K 1,0.8, T 1,a 0.9, computed


95%


Figure 2-8. Percent relative errors (PREs) for the saddlepoint approximation to the
distribution of 4>(X) with n 100, 0 0, K 1, 0.8, 7 1, a 0.9, computed
at the same quantile values.









CHAPTER 3
APPROXIMATION TO THE DISTRIBUTION OF NONPARAMETRIC ESTIMATORS
OF VAR AND CVAR UNDER AL LAW

In this chapter, we study the nonparametric estimators of VaR and CVaR. We

approximate the distributions of nonparametric estimators of VaR and CVaR using

saddlepoint approximation. We derive the moment generating functions of NPEs and then

approximate the distributions of NPEs using saddlepoint approximation. The moment

generating functions of VaR and CVaR are mixtures of hypergeometric functions which

makes the calculation more computational intensive.

We analyze the performance of the MLEs and NPEs by comparing the saddlepoint

approximated distributions. Daily log returns of USD/EUR exchange rate are studied

assuming IID AL distribution.


3.1 Nonparametric Estimators of VaR and CVaR

The Nonparametric approach makes no assumption about the distribution the of

underlying financial data. Consider a random sample Y1,..., Y', let Y(1) < < Y(.)

denote the corresponding order statistics from this random sample, the NPE of VaR is the

ath empirical quantile,


(Y) =Y(k.), (3-1)


where the k, = [na] denotes either of the two integers closest to na. The NPE of

CVaR is the corresponding empirical tail mean,



1 F1
S(Y) 1 ,) (3-2)
r= k

Note that a(Y) and <(Y) being linear combinations of order statistics, are known as

L-statistics (David and N 1,1 i ii 2003).









3.2 Asymptotic Distribution of o,(X) and 0,(X)

Since the NPE of VaR is simply an order statistic of a random sample, the .-i:'!,i,,ll ic

distribution of ti(X) comes from the standard result of the .,-i-! iiii' ic theory of order

statistics, for example, David and N I, ii ija (2003), ch.10. The .-ivmptotics of <,(X)

are more complex. This result was first derived by Stigler (1973) in the context of the

trimmed mean.

Consistency and the joint .i-mptotic distribution of (o, ) under iid sampling

from a continuous cdf F with pdf f, has recently been established by Giurcanu and

Trindade (2005) using t th theory of estimating equations. Define a be the variance of the

distribution obtained by truncating the distribution of Y at 0,, i.e.,

1 (3_
1 a 42
'J)Va j(Y -)af(11)d11. (3-3)


Theorem 3.1 (Asymptotic distribution of NPEs of VaR and CVaR). Under

random sampling from Y with cdf F, pdf f, and finite variance, we have the following

central limit theorem for the NPEs of VaR and CVaR,

( a ta \ a(l-a) a(6.-.)
vOn ) N2} N ^ K (3-4)
a(p0 & ) (c +a(6. -2)2
f(O ) 1--a j


3.3 Approximation of Finite Sample Distribution

3.3.1 Moment Generating Function of Nonparametric Estimators of VaR and
CVaR

In this section, we approximate the distribution of NPEs of VaR and CVaR from iid

random sample of AL distribution. We apply saddlepoint approximation starting from the

moment generating functions (mgfs) of o((X) and a(X).









Trindade and Zhu (2007) derive the mgfs of a(X) and 0,(X). First, we define the

Gauss hypergeometric function F2,1(a, b; c; z) as


F2,1(a,b; c; z) r(c) I b1 1 -b-l( uz)-du, (3 5)
1 (b)F(c b) Jo

where a,b,c,z are real constants and F(.) is the gamma function. The hypergeometric

function converges for |z < 1 provided c > a + b 1.

Lemma 3.1. Let Y1,..., Y, be a random sample from AL(0, K, r), with corresponding

standardizations Xi = (Yi 0)/7 ~ A[(0, K, 1), i = 1,..., n, if
n n
T.(Y) Z= ciY(), and T,(X) jciXi)
i= 1 i= 1
are any L-statistics, then we have the following relations between them, their mgfs,

and their pdfs:

(i) T,(Y) 0 + TT,(X),

(ii)MiT(Y) (t) =etMl(x)(tr),

(iii) f,() ()= / (x)( )-.
Proof:.Straightforward results for location-scale families of distributions.

Lemma 3.2. For any real constants a, b, and c,


B(c;a+1,b+1)= a(1-)bdu 1F(-b, a + 1;a + 2;c), (3-6)
Jo a + 1

and converges for all Icl < 1, provided b > -2.

Proof:. This is just the definition of the Incomplete beta function (Abramowitz and

Stegun, 1972). The connection with the hypergeometric function is easily derived from Eq.

3-5.

Lemma 3.3 (pdfs of order statistics from iid random sample). Let X(1),...,X(n)

denote the order statistics of a random sample, X1,...,X,, from a continuous population








with cdf Fx(x) and pdf fx(x). Then the pdf of X(j) is


fx() (x) ( )' fxx)[Fx(x)] -[1 Fx(x)]"- (3-7)

Theorem 3.2 (MGF of an order statistic under AL law). Let Y(r) be the rth
order statistic under iid sampling from AL(0, K, r), then the mgf of Y(r) is the mixture of
hypergeometric functions,
2 n-ai
Ay")(t) a2= ) F2,'1 (-ai, b (t); 1 + bi(t); zi) (3-8)

where
K2
a, n r, bl(t) n- al+tKT/ 2 2, z1 K
a2 = r- b2(t) = n a2 tT//2), z2 =-
1 + K2


and is defined for all -oo < t < 00.
Proof:. Since Y(,) is an L-statistic, we can simply consider the standard case
X1,... ,X, ~ iid AL(0, K, 1). Applying Lemma 3.1, the mgf in the general case is

MY( (t)= C Mx(tr). (3-9)

The pdf and cdf of the standard case can be represented as

fx(x) 1 /2 exp I(x < 0) + exp (-2x(x > 0)) (3-10)

and

Fx(x) exp(x < 0) + exp (-/x) I(x > 0). (3-11)
S+x K2 ( K) I 1+ x

Applying Lemma 3.3, the pdf of X(r) can be expressed as

n! (3-12)
fx(, (x) (r f)!(_ )!fx(x)[F (x)]' [1- Fx(x)]-'. (3 12)









The mgf of X(r) can be split into two parts,

/+o
Mx(' (t) J eCfx( (x)dx

*o !
] exfx( (x)dx + x fx() (x) dx

( )!( { J ex fx(x)Fx(x)(r1) [1 F(x)]- dx
(r- )! (n r)! (3 13)

0/+oo
c(n, r)0 JI(t) ) (3-t3)


For Ji(t), apply transformation u

0< u< 2. Also notice, x = F (u)

in all these details, we get


Fx(x)I(x < 0)

log(j u) du


+-e- I(x < 0), then
dFx(x)= fx(x)dx. Plugging


K2




I ( 2 u2 + (1 u)n-rdu
2 /2 uK 2 (1 )-r du.
\^ Jo


222
Set a- +r- t, b -n r and 1+ 2, apply Lemma 3.2,


J, M 72,1 d n2, + r; + r + 1; t
J1+r /2 v/

For J2(t), Let v 1 Fxl(x > 0) = lexp(-V2Kx)I(x > 0), the]

x =log [v(1 + K2)]- dv = d[ Fx(x))] = -fx(x)dx. Therefore,


2 (t)


(3-15)


n 0

- tlg[v(1+ )] V/ (1 v)r'-l -dv
1

S+ [v(1 + K2)]- )r-tn-rd
JO

/ (1 +2) -K(1 -- )r-ldv.


(3-16)


Jl(t)


(3-14)









Set a = n b = r 1 and c = we get


J2(t) J1 r t 2,1 1 r, -r7 1, t-r + 2, t
n v (2-r- +1 t 1+K2
(3-17)

Applying Lemma 3.1, gives the desired result.
Corollary 3.1 (MGF of NPE of VaR under AL law).

If ,a = Y(ka) denotes the NPE of VaR,(Y) based on a random sample of size n from
AL(0, K, 7), its mgf is given by Theorem 3.2 with r = [na].
Theorem 3.3 (MGF of sum of upper order statistics under AL law).

Let Y(n-k+1) < ... < Y(n) be the highest k, 1 < k < n, order statistics under iid

sampling from AL(O, K, 7). Let Sn,k(Y) = in-k+l Y(i) be their sum. Then the mgf of

Sn,k(Y) is the mixture of hypergeometric functions,

n!eto 2
Ms, (Y)(t)= di(t)F2,1(- a, bi(t); 1 + b(t); z (t)), (3-18)
t!a,! i=1

where

(n k)/2 tr
a k, bi(t) zi(t) = 2- ,
v/2 2+ tKT Kv/2 + 2v2/ K
tkr 1
a2 = n-k-1, b2(t)=l+k- z2(t) = ,


di(t) K | Z2- (t) alz (-2 2 (t n-a,
zI(t (tr + 2/K) n al

d2 (d2 (t) Za(t) )al Z2(t)2
( 2 tr b2(

and is defined for -//2/(KT) < t < K/2/r.

Proof:. In light of Lemma 3.1, it suffice to consider the standard case, X1,...,Xn ~

iid AL(0, K, 1), hence the mgf in the general case is


Ms, () (t) =teMs, (x) (tr). (3 19)









Since S,,k(X) S,,k =n-k+ X(i), David and N I, i1 ija (2003), Section 6.5, show

that conditional on X(n-k), S,,k can be written as a sum of k iid random variables. If F

and f denote the cdf and pdf of the distribution of X1, this means that
k
X(3 20)
(Sn,k|X(n-k) x) d X, (3-20)
i= 1

where X,,... X are iid with pdf f(y) = f(y)I(y > x)/(1 F(x)). Therefore, the

pdf of S,,k is given by


fs (Y) = f s,,,x =(''1 )fx( )(x)dx, (3-21)

with corresponding mgf


Ms,,k (t) = JR Yf, (y)dy


JR ty fS k x, kx('l I)fx(, )(x)dxdy
I f eCCfsf,kx(n-k)-L )fx(n-k) (x d
-00 J -o




= [ (t)]k fx( (x)dx, (3-22)
JR

where .3 [(t) denotes the mgf of X*, which depends on x. This technique, which

only works if the integration can be performed analytically, has been used by Alam and

Wallenius (1979) for obtaining the distribution of Sn,k in a random sample from a gamma

distribution. Since x < y, there can be three cases: (i) 0 < x < y, (ii) x < y < 0, (iii)

x < 0 < y. In general,

Kv/exp(-Kv/-(y x)) if 0 < x < y;
n2e if x < y < 0-
fX; (y) 1+,2 (-e (3-23)
we-/ if x < 0 < y.
1tK2(








The mgf of X* can be considered in two cases x > 0 and x < 0.
In the case x > 0, i.e., 0 < x < y,


ml[(t)


f00
I gf* (y)dy
Jx


rooo
JetYKvt2exp(-Kv2(y x))dy
J~ x e jtK2Y

" L ( vt-KV2)y y
t KV
t v2-


If t Kv < 0, e(t-r'2)yY goes to 0,
~ 00o


ml[(t)


Kvze"x (0 e(t-/2)x)
t- V2
,' /2etx


In case x < 0, 3 1*(t) can be split into two parts,


[I, (t)


_J 0
Sef*(y)dy +j etfj(y)dy


1+ ( ) t 2) 00

1+K2 ( t- C t-v'2

1 -e(t+ ) 0-



'2l+ K'2l+v-e'2- (xt + 2 -)) _
ex(325)
(3-25)


Substituting this into Eq. 3-22, and by a series of obvious u-substitutions, the integrands
can be reduced to mixtures of hypergeometric functions, and we obtain eventually the


(3-24)









statement of the theorem, which proves the result. The distribution of 0,(Y) follows

immediately be setting k = n [na] + 1 and t t
n-[na]+l
Corollary 3.2 (MGF of NPE of CVaR under AL law).

If -[]+ [na] Y(i) denotes the NPE of CVaRo(Y) based on a random sample

of size n from AL(0, K, t), its mgf is given by Theorem 3.3 with k = n [na] + 1 and

t = t/(n [na] + 1)

3.3.2 Saddlepoint Approximation and Lugannani-Rice Formula

The saddlepoint approximation was first introduced by Daniels (1954). Let

X1,... ,X, be an iid random sample from a distribution with density f(x) and let

S,n(Xi,... ,Xn) be a real valued statistic with density fn. Let .i (t) and K,(t) be the
moment generating function and cumulant generating function of S,, respectively. For

continuous random variables, the saddlepoint approximation to fn at x is

fn(x) {2nK"(t)} 2 exp {K,(t) tx}, (3-26)

where t is the saddlepoint and it is the solution to the saddlepoint equation, K,(t)

x; and K,'(), K,"(-) denote the first and second derivatives of the cumulant generating

function K,(t). The relative error of the approximation is of order O(n-1).

The saddlepoint approximation to the cumulative distribution function of S., due to

Lugannani and Rice (1980), is given by


Fn(x) () + (r) 1- (3-27)

where )(.-) and 0(-) are the standard normal distribution and density functions with

r sgn (t) [2 {tx K (t)}]

q t{K( t) }.

In Eq. 3-27, at the mean of the distribution, i.e., at x = E(S,), q = 0, so that the

alternate expression used is












1 1 3
F,(x) = +(727)- K"'(0) (0) 2 (3-28)


where K )"'(- is the third derivative of the cumulant generating function K,(t).

Since the mgfs of NPEs of VaR and CVaR are available in closed form, the density

and distribution functions of (x) and ,"(x) can be approximated by applying the

saddlepoint approximation. There is a problem in that both of the mgfs are mixtures of

hypergeometric functions, which are computationally burdensome to evaluate explicitly

due to slow convergence of the power series expansions defining the hypergeometric

function. A computationally more efficient alternative is to employ instead the Laplace

approximations of the hypergeometric function developed by Butler and Wood (2002).

3.3.3 Laplace Approximation of Hypergeometric Function

Butler and Wood(2002) provide a Laplace approximation for the hypergeometric

function,


F2,i(a, b; c; x) = c1/2 -1/2 ( a ( t c-a (1 b ,(3-29)
a \ a

where


r2,1 1_ b( 2 -
a~a c-a (1X2 a c-a

Let g(y) = {alog(y) + (c a)log(1 y) blog(1 xy)}, is the solution of

g'(y) = 0,

= 2a
e2 -4ax(c-b)-e

where e = x(b a) c.

Therefore, by applying this approximation we can estimate the distributions of Q,(X)

and <,(X) under AL law. To get a more accurate approximation, we normalized the

cumulate generating function by subtracting its approximated value at x = 0.










Assessing the Accuracy of the Saddlepoint Approximations

In this section, we apply the same technique to estimate the accuracy of the

saddlepoint approximation to the distribution of Qa(X) and 0,(X). We use the same

parameters for simulations. And the percent relative error are defined the same as in

C'! n=5ter 2.

n=50, K=1, a=0.9


Figure 3-1. Estimated cdfs of (,(X), obtained via simulations and saddlepoint
approximations, respectively, under AL law with
n =50,0 =0, -= 1, 1, a 0.9


n=50, i=0.8, a=0.9


05 1 15 2 25
X


Figure 3-2. Estimated cdfs of Q,(X), obtained via simulations and saddlepoint
approximations, respectively, under AL law with
n = 50, = 0, = 1, = 0.8, a = 0.9











-02-I -
-0 4 -- -
-046

-0 8


-1 2




50% quantile 95%

Figure 3-3. Percent relative errors (PREs)for the saddlepoint approximation to the
distribution ,o(X) under AL law with n 100, 0, \ 1,0.8,7 1, 0.9,




In general, we have good estimation for NPEs of VaR and CVaR under AL

distribution. The PREs for the edfs of so(X) are less than 2 and PREs for edfs of

o,(X) are between o'. to 11 Note also, in Fig. 3-3, the PREs are less than zero, which

indicates a tendency to overestimate ,(X) and 0,(X).

3.4 Comparison of the Distributions of Parametric and Nonparametric
Estimators

In this section we compare the MLEs and NPEs of VaR and CVaR under iid sampling

from an Asymmetric Laplace law. We consider both the large sample case and finite

sample case.

3.4.1 Large Sample Case

In light of the .*i-mptotic normality results of Theorem 2.1 and Theorem 3.1, we

compare the estimators of VaR and CVaR through the .ii-mptotic relative efficiencies

(AREs) of the MLE with respect to the NPE. If X ~ AL(0, K, 1), routine calculations give

for K2/(1 + K2) < a < 1, f(o) = (1 a), and o + c( 0t)2 (1 + a)/(2K2).

In the general case of Y = TX distributed as AL(n, r), these results when substituted

into Theorem 3.1 give E() = (ar2)/[22(1 a)] e (, a), and YE(o) -









[(1 + a)'-2]/[22(1 a)]. Using Theorem 2.1, the ARE for the NPEs of VaR is,


ARE(, ) = E() la [2w + K(t )] (3 30)


while the ARE for the NPEs of CVaR is,




(3-31)


where U,, is defined the same as in Eq. 2-10.

Fig. 3-4 di-p,~,- both AREs as a function of a > 0.5 and 0 < K < 1.

3.4.2 Finite Sample Case

In this section, we compare performance of MLEs and NPEs of VaR and CVaR under

an AL distribution in the finite sample case. Fig. 3-5 plot the saddlepoint approximated

cdfs of MLEs and NPEs and compare the distributions with the true values of VaR and

CVaR.

Note in general, the MLEs are more symmetric and unbiased, while NPEs are right

skewed and biased. Also the NPEs are more skewed and biased as the AL distribution

becomes more .i-viii,, lic.

3.5 Analysis of Exchange Rate Data

In this section, we analyze the exchange rate of the USD to the EUR. The data

consists of the daily average "ask price" of 1 USD in EUR from Jan 31, 2005 to Jan 31,

2006 with 366 data points. The source of the data is from oanda.com.

We are interested in analyzing the natural logarithm of the price ratio for two

consecutive d-iv, and the data were transformed accordingly to give 365 daily log returns.

Summary statistics are as follows: minimum -1.846E-2, median 0E-7, mean 2.049E-4,

maximum 1.325E-2.

Fig. 3-6 gives the histogram of the data, box plots by d-,- of the week, and the

sample ACFs of the squares and absolute values of the data.











ARE of NPE of VaR







o 1 I I I I I
0.5 0.6 0.7 0.8 0.9 1.0




ARE of NPE of CVaR




SI I I I I
0.5 0.6 0.7 0.8 0.9 1.0












Figure 3-4. Asymptotic relative efficiencies of the maximum likelihood estimators with
00< .<1.







The histogram of the log returns -ii:: -'s high peakedness and heavy tails. The

normal Q-Q plot of the data indicates a violation of normaelty. The Pearson Chi-square
normality test reports a p-value of 2.2e-16. And the olmogorov-Smirov normality reports






a p-value of 5.032e-14. This is strong evidence that the data do not follow a normal

distribution.
normal Q-Q plot of the data indicates a violation of normality. The Pearson Chi-square

normality test reports a p-value of 2.2e-16. And the Kolmogorov-Smirov normality reports






We consider fitting the daily log returns data using an Asymmetric Laplace

distribution. The dotted line superimposed on the histogram shows a fitted AL(0.9679,

4.! :1 .-3) density with parameters estimated via maximum likelihood. Consequently, the










VaR for AL(0,1,1) VaR for AL(0,0.8,1)


NPE
MLE
/ \ true value

/ \'
I.

I.I
/ / I


1i


1.5

/I

0.5
/ \\
J "J I


0 1 2 3 0 1 2 3
x x
CVaR for AL(0,1,1) CVaR for AL(0,0.8,1)
2- 2

1.5 1.5

4- 4-
S/ / ./

0.5 \ 0.5 /
/ / \\ / // / \\
// \\ / / .' C

0 1 2 3 4 1 2 3 4
X X

Figure 3-5. The resulting saddlepoint approximated density functions of MLEs and NPEs
of VaR and CVaR at n 50,0 = 0,7= 1, = 1,0.8, a= 0.9. The true values of
VaR and CVaR are indicated by the solid vertical line.


MLE of VaR, a(X), equals 7.566E-3 and the MLE of CVaR 0-(X), equals 1.08E-2, with

a = 0.95.

It has long been noted that there is a "trading d4- effect" in currency exchange

rate data (for example, McFarland et al., 1982). The boxplots in Fig. 3-6 decompose

the 365 returns by di --of-the-week. The di -of-the-week corresponding to a particular

return denotes the log of the ratio of the price on that di- to that of the previous div.

By inspection of the box plots we can see that there is a higher volatility during the week

than on the weekends.

The other two plots are the sample autocorrelations for the squares and absolute

values of the log returns. Although the 365 returns appear serially uncorrelated, the

sample ACF si--:. -i the presence of a dependence occurring precisely at lags that are

multiples of 7. This is evidence of the presence of d4i,-of-the-week effects.


1.5

4-
- 1


0.5

n


2l-










Log returns by day of the week


I I I I I I I
-0.020 -0.010 0.000 0.010


ACF of squared log returns


IL ] L I U_'_ J 11' 1


ACF of absolute log returns


i--i--- .L J---L' 1-
L A 1-'-_-1"-1 1_ L'- I


Figure 3-6. Daily log returns of USD/EUR exchange rates from Feb 1, 2005 to Jan 31,
2006. The top plots show the data both for the entire period and by
d- i-of-the-week. The bottom plots display the sample autocorrelation function
of the squares and absolute values of the data over the entire period.


We believe the reason for the low volatility during weekends is the lack of institutional

investment during weekends. Most of the foreign exchange trades during weekends are

over-the-counter (OTC) service by retail banks.

Next we removed the weekends from the data to see whether the daily log returns

of we -:d. v- follow a normal distribution. As a consequence, Monday's log return is the

log ratio of Monday and the previous Frid-v's price. Fig. 3-8 is the histogram of daily log

returns of the exchange rate date without the weekends. The data are less peaked but still

have heavy tails. The Pearson Chi-square normality test reports a p-value of 2.2e-16. And


SI 0

0o
I I

0 o
a


Mon WedI I
Mon Wed


I I
Sun


Log returns










Normal Q-Q plot of daily log returns


0
C5 -
o
0
0

0
o








lo o
oo




0 0

-3 -2

Figure 3-7. Normal Q-Q plot of daily


-1 0 1 2 3

log returns of exchange rate.


the Kolmogorov-Smirov normality reports a p-value of 1.011e-13. Therefore, the data are

not normally distributed even when we remove the weekends.

Log returns without weekend data


Figure 3-8. Histogram


-0020 -0015 -0010 -0005 0000 0005 0010 0015

of the daily log returns of the exchange rate without weekends


On observing the "trading-d -effect", we therefore fit the AL(K, 7) distributions via

maximum likelihood to each of the di -of-the-week returns.

The log returns on Wedn -div Fli ,, and Sund-idc are approximately symmetric

with fitted i equal to 0.9962,1.0218 and 1.032, respectively. While log returns on Monday,


000 0


rFh


____FF









Normal Q-Q plot of daily log returns without weekend data
Cooo






0 8,



oIC
0/


0 0
-3 -2 -1 0 1 2 3

Figure 3-9. Normal Q-Q plot of daily log returns of exchange rate without weekends.

Table 3-1: Maximum likelihood estimates and standard errors (s.e.) for the skewness and
scale parameters as an AL(t, 7) fit to the log returns of the USD/EUR exchange rates by
week d-4, as well as the MLEs of VaR and CVaR at a 0.95.
Day R (s.e.) r (s.e.) (0.95 (s.e.) (0.95 (s.e.)
Mon 0.897 (0.089) 9.987E-4 (1. ;'E- !) 1.892E-3 (0.0025) 2.680E-3 (0.003)
Tue 0.878 (0.086) 5.031E-3 (6.940E-4) 9.820E-3 (0.0131) 1.387E-2 (0.015)
Wed 0.996 (0.098) 5.196E-3 (7.206E-4) 8.507E-3 (0.012) 1.220E-2 (0.014)
Thu 1.117 (0.110) 5.523E-3 (7.683E-4) 7.645E-3 (0.0117) 1.114E-2 (0.014)
Fri 1.022 (0.100) 6.330E-3 (8.780E-4) 9.991E-3 (0.014) 1.144E-2 (0.017)
Sat 0.836 (0.083) 5.685E-3 (7.948E-4) 1.186E-2 (0.002) 1.667E-2 (0.003)
Sun 1.032 (0.101) 1.794E-3 (2.489E-4) 2.791E-3 (0.004) 4.021E-3 (0.005)


T-i.-di and Saturd-v are right skewed with fitted R equal to 0.8973, 0.8782 and 0.8358,

respectively. The k on Thulrii- is 1.1169, indicating left skewness.

The MLEs of VaR are much larger on T -d ,li to Saturdia than on Monday and

Su-ndii. Consequently, the same trend is seen for the MLEs of CVaR.

Using the .,-i!,illI.1ic covariance matrices in Eq. 2-6 and Eq. 3-4, it is interesting

to compare confidence regions for the parametric and nonparametric estimators of the

bivariate parameter (1, 0,). Both regions take the form of ellipses; for example, with a

confidence level of 95' the region for the MLEs is given by












r<$ol> r<~ol, 1 1 i
{ Z( ) ( X2,0.95


(3-32)


While the construction of this region is straightforward in the parametric case by simply

"pliiLi:__ i _-ii the MLEs themselves wherever they appear in the covariance matrix, that

for the nonparametric case is complicated by the need to estimate the inverse of the pdf, a

quantity sometimes called the -'i ,, ..:/;/ function. Estimation of the sparsity is notoriously

difficult and tends to be shied away from in favor of other approaches whenever it occurs

(as it frequently does) in nonparametric inference. However, there has been recent renewed

interest in this subject since the sparsity features prominently in the asymptotics of quantile

regression (Koenker, 2005). Using the method -,,:_:. -I.1 in Koenker (2005, Section 4.10.1)

with the Hall and Sheather bandwidth, and 1p.1i1::-. in the NPEs whenever they appear in

the covariance matrix, nonparametric confidence region construction for (,, 0,) is therefore a

feasible proposition.

We employed the above approach in producing Fig. 3-10, which shows the resulting

confidence ellipses for the MLEs and NPEs of the a = 0.9 right tail (VaR, CVaR) for the week

day distribution of the USD/EUR exchange rate log returns data. The high correlation between

the MLEs of VaR and CVaR is reflected in the very narrow semiminor axes of each respective

ellipse. The downward bias in the NPEs is also immediately apparent, a fact that concurs with

the bias noted in the saddlepoint pdf of Fig. 3-5.

The implication of these findings for the practitioner is that it may be preferable to

commit to an appropriate parametric model, such as the AL law, when attempting to draw

inferences from data of this nature. From a small simulation study which we omit for the sake of

brevity, we have also noted that the parametric confidence bands have vastly superior coverage

probabilities.







































Monday








5 i

o




-0010 0000 0010

VaR


Friday


I

0 t
o -









0010 0000 0010
VaR
o .



o_ *

-0010 0000 0010

VaR


Tuesday




o $









-0010 0000 0010

VaR


Saturday



0




0
o





-0010 0000 0010

VaR


Wednesday



I











-0010 0000 0010

VaR


Sunday




V a
0 i
o -




-0010 0000 0010

VaR


Sunday




5.-









-0010 0000 0010
VaR


Thursday







.*.'






-0010 0000 0010

VaR


LEGEND
* ML Estimate
S NP Estimate
S- ML Conf Ban
NP Conf Ban


Figure 3-10.


Maximum likelihood (circle) and nonparametric (square) bivariate estimators

of (VaR, CVaR) for the a = 0.9 tail of the week dw distributions of the

USD/EUR exchange rate log return data. The dashed and dotted lines

delineate the boundary of 95'. confidence ellipses for the maximum likelihood

and nonparametric estimators, respectively.









CHAPTER 4
TIME SERIES ARMA AND GARCH MODELS UNDER AL NOISE

In this chapter, we develop the time series models under AL noise. Traditional time

series ARMA models assume Gaussian noise, which implies that the marginal distribution

will be Gaussian as well. We have noticed before that financial data are usually heavy

tailed which is not consistent with a Gaussian distribution. It is therefore reasonable to

extend the AL distribution to time series models.

Early works have applied the Symmetric Laplace distribution to ARMA models.

These efforts have focused on two directions: assume marginal Symmetric Laplace

distribution, for example, NLAR(1) and NLAR(2) models (Dewald and Lewis, 1985),

NAREX(1) model (Novkovi6, 1998), or assume Symmetric Laplace noise, for example,

Damsleth and El-Shaarawi (1989). Damsleth and El-Shaarawi (1989) have shown that

these two requirements cannot be simultaneously achieved within the class of linear time

series models.

Dewald and Lewis (1985) discuss the NLAR(1) and NLAR(2) model assuming a

standard Symmetric Laplace marginal distribution. The Symmetric Laplace distribution is

a special case of AL distribution with K = 1. A Symmetric Laplace distribution is called a

standard Symmetric Laplace distribution when r = 1 and 0 =0.

Random variable X is said to be distributed as Symmetric Laplace distribution

(double exponential distribution) with location parameter -oo < 0 < oo and scale

parameter A > 0, if its pdf is of the form

1 Iz 0
fx(.) = e A -oo < x < o. (4-1)


A Symmetric Laplace distribution are called standard Laplace distribution if A = 1.

NLAR(1) model starts by assuming {X,} to be a stationary process with standard

Laplace marginal distribution, 0 < 101 < 1 and 0 < a < 1,












X Xt-1+Zt, w.p. a, (4-2)
Zt w.p. 1- a.

Then the noise term can be derived as

Z = Lt, w.p. 1 p,
a ^ (4-3)
I1 ~a\Q)\Lt w.p. p,

where Lt are i.i.d. standard Laplace random variables. And p = _(7 2. In

addition, let A = (1 C)-1/211-1, the density function of Zt is

1 1
fz(x) (1 p)e-1 + |ApeXl, (4-4)
2 2

which is a convex mixture of Laplace densities.

Similarly, the NLAR(2) model assumes standard Laplace distribution and applies this

to the AR(2) model,

Xt = 1K'Xt 1 + 2K Xt-2 + Zt, (4-5)


where 0 < I|l < 1, for i = 1,2. {Kt, K7'} is a sequence of i.i.d. discrete bivariate

random variables with distribution,

(1,0), w.p. aI1,

{ ,Ki'}= (0,1), w.p. a2, (4-6)

(0, 0), w.p. 1 a a2,

for t = 0, 1,2,- ; 0 < a < 1 for i = 1, 2 and a + a2 < 1.

Therefore the noise term can be expressed as

Lt, w.p. 1 -p2 -P3,
Z= b2 Lt, w.p. P2, (4 7)

|b3 Lt w.p. p3,









where {Lt} are i.i.d. standard Laplace random variable; p2,p3, b2, b3 are functions of

t, 82 1, 2.

The NAREX(1) model discussed by Novkovi6 (1998) assumes the marginal distribution

to be a Symmetric Laplace distribution L(A), as defined in Eq. 4-1, with scale parameter

A,

OiXt_- + Zt, w.p. po,
Xt = 2Xt- + Zt, w.p. pi, (4-8)

03Xt-1 w.p. p2,

where 0 < po, PI,P2 <- 1, P + P + 2 = 1, 0 < 01, 02, 13 < 1. After working with the

characteristic function of the Xt and Zt, Zt can be expressed as a mixture of symmetric

Laplace distribution,

0, w.p. Ao,

L(A), w.p. A1,9
< (4-9)
L(3A), w.p. A2,
L(A po_2+p1) A3
PO+P )w. PO-1-pi'

where Ao, A1 and A2 are functions of 1, 2, 3, PO, Pi, P2*

Damsleth and El-Shaarawi (1989) deduce the marginal distribution of observations

generated by an ARMA model assuming Symmetric Laplace noise.

Let {Zt} be a series of i.i.d. Symmetric Laplace distributed random variables with

scale parameter A. Let the observed stationary time series {Xt} be generated by the

ARMA scheme,


B(B)Xt = O(B)Zt, (4-10)

the marginal pdf Xt is

1 (1 x
fx,(x) "j I |j I- 1exp A (4-11)
j=0









Where the aj are given by aj = If (1 Il )2
SiJ
On observing the .,-vmmetry property of financial data, it is reasonable to apply the

Asymmetric Laplace distribution to time series modeling.

4.1 ARMA (p,q) Model

Definition 4.1: (ARMA(p,q) process).

{Xt} is an ARMA(p, q) process if {Xt} is stationary and if for every t,

Xt 1Xt-1 OpXt-p = Zt + AiZt-_ + + AqZt-q, (4-12)


where {Zt} ~ WN(0, aO2) and the polynomial (1 lz .. pzP) and (1 + Alz +

+ Aqz') have no common factors.

The process {Xt} is said to be an ARMA(p, q) process with mean p if {Xt p} is an

ARMA(p, q) process.

Eq. 4-12 can also be expressed as


S(B)Xt = A(B)Zt, (4-13)

where )(.-) and A(.) are the pth and qth-degree polynomials


)(Z) =1 -- Iz- .- pz

and


A(z) 1 + Xlz + ApXz


and B is the backward shift operator (B Xt = Xt_, B Z = Zt-, j = 0, ,...).

The time series {Xt} is said to be an autoregressive process of order p (or AR(p)) if

A(z) 1, and a moving-average process of order q (or MA(q)) if 4(z) = 1.

{Xt} is assumed to be stationary, which means that the autoregressive polynomial

)(z) = 1 1z z ., :- / 0 for all complex z with z = 1.










If )(z) / 0 for all z on the unit circle, then there exists 6 > 0 such that


1 00
S- yz for 1 6 < z\ < 1 + 6, (4-14)
J =-00

and Yj _- |Xj < oc. Therefore, we can define

1 0o
S(XjBJ". (4 15)


Applying the operator X(B) = to both sides of Eq. 4 -13, we obtain

00
Xt (B)(B)Xt = X(B)A(B)Zt = I(B)Zt = z_-,, (4-16)
j--00

where '(z) (z)A(z) -= 0, Zj.

Definition 4.2: (Causality).

An ARMA(p, q) process {Xt} is causal, or a causal function of {Zt}, if there exist

constants {uj} such that E0,o ijl < oo and


Xt = Zt -j for all t, (4-17)
j=0

causality is equivalent to the condition


(z) = 1 Iz zP pz / 0 for all z| < 1. (4-18)


Definition 4.3: (Invertibility).

An ARMA(p, q) process {Xt} is invertible if there exist constants {ir} such that

Ej 1o J| < 00 and


Zt = YjXt_j for all t, (419)
j=0

invertibility is equivalent to the condition


A(z) = 1 + Alz + + ApzP / 0 for all zI < 1. (4-20)









4.2 ARMA(p, q) Model under AL Noise

Traditional analysis of ARMA models assumes that the white noise term {Zt} is a

series of i.i.d. normal distributions, which we already know is not appropriate for fitting

financial data. To apply an ARMA model to financial time series data, we introduce the

ARMA model under AL noise.

Definition 4.4: (ARMA(p,q) process under AL noise).

Let {Zt} be a series of i.i.d. Asymmetric Laplace distributed random variables

AL(O, K, r) with PDF given by


fz (z) -exp -- z -O[>o] + exp -' -0 ] ,

{Xt} is an ARMA(p, q) process under AL noise if {Xt} is stationary and if for every t,

Xt 1Xt-1 - OpXt-p = Zt + AiZt-l + + AqZt-q, (4-21)

the polynomial (1 1iz ,) and (1 + A1z + .. + Aqzq) have no common

factors.

To ensure {Zt} has zero mean, we require that

0 K (4-22)

Motivation of our model. Our defined model has the flexibility to describe a

stationary time series process with .,i-iii. ii, : turbulence about the mean. Asymmetry

is an important property of financial data, for example, the cost to buy a call option is

limited but the return can be substantial. An improvement in credit quality brings limited

returns to investors, but in case of defaults or downgrades, the loss would be substantial.

Again, our model has the flexibility to describe data with heavy tails.

We simulated an ARMA(1,1) model under AL noise with K > 1, which means that

the AL distribution is left skewed. Consequently, the ARMA process is skewed to the

downside of the mean. There are some deep drops approximately at time t=140, 180, 300.










ARMA(1,1) Process under AL noise


Time

Figure 4-1. Simulated ARMA(1,1) process under AL noise with k1= 0.4, A = 0.1,
S= 1.25, T- 1.


Histogram of X,


-6 -4 -2 0 2 4 6

Figure 4-2. Histogram of the Simulated ARMA(1,1) process under AL noise with 1= 0.4,
A1 0.1, = 1.25, 7- 1.


These drops are more frequent and deeper compared with the upper points. This model is

good to describe markets with unexpected and sudden losses, for example, a market crash.

Similarly, if we set the skewness parameter K < 0, the process will be skewed to the upper

side.

The histogram of the {Xt} indicates the distribution has heavy tails.









4.2.1 Marginal Distribution of ARMA Model under AL Noise

We derive the pdf of the marginal distribution of an ARMA(p, q) model with

Asymmetric Laplace noise. Similar method have been used by Damsleth and EL-Shaarawi

(1989) in deriving the marginal distribution of an ARMA model with double-exponential

noise. This method is also a special case of Box (1954), where he derives the distribution

of any linear combination of independent X2 variables with even degrees of freedom.

Characteristic function of AL noise. The characteristic function of Zt

A[(O, K,T), is

eiot
z(t) 1+ -2t2 i 1 (4 23)
1)1 7_2_2 = _

or equivalently,

eiot
zc (t) =1 + 2 (4-24)
1 + 1-t2 it

where p = t().

We assume {Xt} to be causal, then


Xt = Zt-j. (4-25)
j=0

Let
n
Un = jZt_- (4-26)
j=0

and assume / j for i / j, since Zi, Z2,..., Z, are i.i.d., the characteristic function

of Un is


Su (t) I 1 2(4-27)
=o 1 + 2b2 I

which can be resolved into partial fractions


) Ci t n i (428)
O 1 + j j=o 1+ -









where the aj are constants not containing t. We will give an explicit expression for aj

in the next section.

Solve for constant aj.

The value of the constants aj can be obtained as follows:

Let y = 1 + 1r2,T2 -2 ipjt, solving this equation, we get


t sign(y)) fj(y). (429)
72bj

Mathematically, both plus and minus signs in Eq. 4-29 are correct. But we will see in

the next step, that they will not necessarily make our derived function a pdf. We identify

the correct signs by plotting the pdfs. Actually, the plus sign is correct for the part of pdf

below the mean, and the minus sign is correct for the part of pdf above the mean.

Setting y = 0, we get

i(p + sign(Q ) p2 + 2r2)
f (0) (4-30)
72bj

Eq. 4-28 can be expressed as

Cioibjt n Ci i oit
1 7+ 2t2 i t i 1 + 2/





substituting y into Eq. 4-31,
i 0




y i o j + t ff (y)- "i,,, f()

i 02









multiplying by y on both sides,


6() n eiobifjw(


n CiOifj (y)
ae' + 1 2 2 2 ) (4 33)
i 0

setting y = 0, we get
n i ifj (0)(4 34)
o 1 + -2 f(0) j (0)
i30

Marginal pdf of Xt Apply the Fourier inversion formula,

1 C"
fu. (u) = u(t)e-itdt

f I- o C n I _____ -e'tdt
2 -- + o 1 2 2
1 f io@bjt _itUdt

a- e-it"dt
2F Jii a+ 1T 2,),,22 it'@j

27 J j=o I +2 t2
1_ 0,+ 2 __1o i0tjt

S= a 1 -it dt (4-35)
0a [27 10_ 1 + JT2a]t2 ifU/)t
j= 0 2 2
ie O t

where we recognize i j2s .t as the characteristic function of Z' ~ AL(IjO ,(jp, ij 1 7).

Let 0' = j0, p'. = jp and ,T = |1;, since K = the skewness parameter

of Z6, K can be obtained as


K v/2 Tj 2
p+ 2 7 +p)12

/+ I/2 I 2





/27 (4-36)
sign(j)p + V '2+T2+ (32









N(t K), therefore


sign( bj) 7(


sign( j) (
v/-2


V2T72
i) 2/2+7( T(I )2


) 1+2
+)+r2
v/ \2K


sign(n j)(1


(4-37)


If j > 0, K'


; If b < 0,


Consequently, the pdf of Z' is


fz (z) =- j exp
7bjj'T(1 + Ki2)


1 j' z


j 01l[> 0] )


+ exp (


V/2
I {J


j 0 I[z<'eO J 1


(438)
(4-38)


Back to Eq. 4-35,


fu, (u)


n
Zaj
j=o
n
Yaj
j=o


2I__, 1+


L27Tri


C iOjt
72a2 t2


tudt


exp VK' U |[
I I bo 7T

b 01 I[<' ]] 01)


j|o j(7(1+k K2)


+ exp


V/2
I )j Kj,


n eioe~ifj (0)
a= 1 + T, ff () f (0)
ino




f(x) = lim fx).
--OO


where


(4-39)


Now,


(4-40)


(4-41)


Note that p


K) + ( + K)


i e-it" dt
ip~jt I









Then we obtain the following expression for the marginal pdf of Xt,
D 2K' f V2K'\
fx (x) j a- exp x[- j0[X>_)o
Y- o jjJ + I K)

+ exp v xI ]1x
where
00 eiOifj (0)
Ca7jj (0 (4 43)
i=o1+ ,- 2 fj2f (0) -"pr fj(0)
i j

To check the validation of our derived formula, we plot the marginal pdf of AR(1) and

ARMA(2,2) models under AL noise. The reason for selecting an AR(1) is because it is a

typical model in fitting financial data. We compare our theoretical curve with simulated

data. It turns out that our derived curve matches the simulated data perfectly.

Below are some remarks concerning the plotting of the PDF and data simulation.

Remark 1: Our derived marginal pdf is an infinite summation. But in practice, this

series converges very fast. The convergence is slower for higher order ARMA models. But

generally, it converges within j = 25. So we actually use a finite summation.

Remark 2: In Eq. 4-16, T(B) -= -1(B)O(B) can be obtained by expanding -'1(B)

using Taylor series expansion, and then calculate the product of 1-'(B) and O(B). Some

computer programs with symbolic mathematical functions, like Matlab, provide functions

for Taylor series expansion and polynomial convolution.

Remark 3: When we generate a simulated realization from an ARMA(p, q)

model with a sample size of n, we use zeros for the unknown values X1,..., Xp in the

ARMA(p, q) recursions, run the recursions well beyond n to n + 500, and then take the last

n values as our sample. By doing this, we diminish the initialization error of the recursions

due to the use of zeros.

Fig. 4-3 and Fig. 4-5 are the plots of the derived PDFs of AR(1) and ARMA(2,2).

We selected AL noise with r 1, K = 0.8. To ensure a zero mean noise, 0 is predetermined











by 0 = -p = -'-( K). In Fig. 4-4 and Fig. 4-6, we compare the derived PDFs with

the simulated histograms.



025-


02


015


01


005


0-
-8

Figure 4-3. Derived marginal
K 1, 0.8,



3000


-6 -4 -2 0 2 4 6 8

pdf of AR(1) model under Asymmetric Laplace noise with
= -0.318 and = 0.75.


2500

2000

1500

1000

500

0
-8 -6 -4 -2 0 2 4 6 8

Figure 4-4. Comparison of derived marginal pdf (red line) and simulated histogram of
AR(1) model. The theoretic pdf has been inflated by number of replications to
match the histogram. r 1 K = 0.8, 0 = -0.318 and Q = 0.75.



4.2.2 Fit AR(p) Model Using Conditional Maximum Likelihood Estimation

Proposition 4.1 (Joint distribution of (xl,... x,)).

Let f(xi,..., x,) denote the pdf of the joint distribution of the first n observations,

X1,... ,X,, under an AR(p) model. Then the likelihood can be written as


f(xil,... X) = f(xI, ... Xp)f(xp+ i|t,t < p) ... f(xnlxt, t < n- 1).


(4-44)










0.14

0.12 /

0.1

0.08

0.06 \

0.04 \

0.02


-15 -10 -5 0 5 10 15

Figure 4-5. Derived marginal pdf of ARMA(2,2) model under Asymmetric Laplace
noise.r 1, K = 0.8, 0 = -0.318, 41 0.7, 2 = 0.2, A1 = 0.4, A2 = 0.2, which
is causal and invertible.

1400

1200

1000

800

600

400

200

0
-20 -15 -10 -5 0 5 10 15 20

Figure 4-6. Comparison of theoretic marginal pdf (red line) and simulated histogram of
ARMA(2,2) model. The theoretic pdf has been inflated by number of
replications to match the histogram. 7 = 1, K = 0.8, 0 = -0.318, = 0.7,
2 = 0.2, AI = 0.4, A2 = 0.2, which is causal and invertible


This result is straightforward from the definition of the conditional probability.

Since Xt follows an AR(p) process, X tlX, s < t, has the same distribution as Zt, but

with a mean equal to ixt-_+- -+ pXt-p. That is, the conditional distribution of Xt given

X, for s < t, is the same as the distribution as Zt but with mean ixtl + + .' ,_p. As

a consequence, we can compute the densities of all the pdf on the right hand side of Eq.

4-44, exept that of the first term, f(xl,... Xp).









We can not derive explicitly the expression for the first term, f(xl,..., xp). When

the value of n is large relative to p, we ignore the first term, and base our likelihood

estimation on the remaining pieces. This is known as conditional maximum likelihood

estimation. For example, this is what is done in fitting GARCH models in finance, since it

is difficult to compute the distribution of f(xl,..., x,).

Fit simulated data. In this section, we simulate data to check the accuracy of our

conditional maximum likelihood estimation method. We simulate data based on AR(1),

AR(2) and AR(3) models with AL noise. Sample size is n = 100. We use the same method

to simulate time series data as we did in the last section. The parameters Oi are chosen to

ensure a causal AR model.

We then apply a numerical optimization method to maximize our conditional

likelihood. The AL parameters are chosen as KO = 1, and To = 2. We use the Yule-Walker

(YW) estimator as our initial value. This is a standard method of moments estimator.

In the Table. 4-1, Table. 4-2, and Table. 4-3, we list the fitted model parameters

of the first five simulations. ... indicates the Yule-Walker estimator of each simulation,

which is also our initial value of 0. The calculation of Means and MSEs is based on 100

simulations.

Table 4-1: Fitted model parameters of AR(1) model under AL noise
parameters K 7 0 1 1yw
True 0.8 1 -0.3182 0.7
MLEs_simul 0.7791 0.7713 -0.2752 0.7398 0.7106
MLEs_simu2 0.9652 0.9990 -0.0501 0.6125 0.6030
MLEs_simu3 0.8981 1.1812 -0.1799 0.6305 0.5256
MLEs_simu4 0.7962 0.9572 -0.3112 0.7087 0.7042
MLEs_simu5 0.7515 1.1285 -0.4622 0.7066 0.6763
mean(MLEs) 0.8002 0.9733 -0.31182 0.7045
MSE(MLEs) 0.0068 0.0107 0.0184 0.0048


In general, our method works well in fitting AR(p) models. MSEs for < are less than

1 with a better fit for models of lower orders. MSEs for AL parameters 7 and 0 are

slightly higher, with an average around 0.01.









Table 4-2: Fitted model parameters of AR(2) model under AL noise
parameters K 7 0 1 42 ~1yw
True 0.8 1 -0.3182 0.7 -0.1
MLEs_simul 0.6711 0.9298 -0.5384 0.8011 -0.1761 0.7115 -0.1884
MLEs_simu2 0.6742 0.7489 -0.4283 0.7489 -0.1657 0.8734 -0.3434
MLEs_simu3 0.9057 1.0423 -0.1462 0.8545 -0.1996 0.8189 -0.2161
MLEs_simu4 0.8597 1.0650 -0.2286 0.7099 -0.0831 0.7797 -0.1810
MLEs_simu5 0.6922 0.8903 -0.4737 0.8246 -0.1054 0.8165 -0.2273
mean(MLEs) 0.7890 0.9747 -0.3314 0.6969 -0.1052
MSE(MLEs) 0.0078 0.0143 0.0190 0.0092 0.0064

Table 4-3: Fitted model parameters of AR(3) model under AL noise
parameters K 0 81 2 3 lyw .
True 0.8 1 -0.3182 0.7 -0.2 0.1
MLEssimul 0.6485 0.8390 -0.5300 0.7523 -0.2370 0.2356 0.6332 -0.1432 0.1601
MLEs_simu2 0.6725 0.9841 -0.5667 0.7332 -0 -17' 0.1040 0.7251 -0.3287 0.1245
MLEs_simu3 0.8131 0.9301 -0.2741 0.5539 -0.1674 0.1183 0.6046 -0.2339 0.1907
MLEssimu4 0.7972 1.2948 -0.4185 0.7532 -0.2345 0.0735 0.6538 -0.1676 0.0978
MLEssimu5 0.6533 0.5996 -0.3720 0.7608 -0.3232 0.1572 0.6343 -0.1573 0.1010
mean(MLEs) 0.8064 0.9685 -0.3021 0.7097 -0.2089 0.1003
MSE(MLEs) 0.0085 0.0116 0.0219 0.0080 0.0094 0.0053


4.2.3 Fitting an ARMA(p, q) Model Using Conditional Maximum Likelihood
Estimation

Joint distribution of (Xi,..., X,) under an ARMA(p, q) The joint distribution

(X1,... ,X,,) under an ARMA(p, q) can be calculated in a similar way to Eq. 4-44, except

that {Xt} is a function of {Xs} and {ZI}, s < t. Therefore the values of {ZJ} have to first

be calculated recursively.

Let {Xt} be an ARMA(p, q) process under AL noise,


Xt 1Xt-1 pXt-p Zt + IZt- + + AqZt-q,


(4-45)


where Zt ~ AL(O, K, 7).

Equivalently, Eq. 4-45 can be expressed as


Xt = O1Xt- + + OpXt-p + XZti + + AqZt-q + Zt.


(4-46)









The joint distribution of (X1,X2,... ,X) can be calculated as in Section 4.2.2. using

Eq. 4-44.

Now XtlXs, s < t has the same AL distribution as Zt, but with a mean equal to

l1Xt-1 + + OfpXt-p + A1Zt-1 + + qZt-q = XPl X-i I+ E Al Zt-j

The value of Zt can be iteratively calculated from Eq. 4-45 with

Zt = Xt iXt-1 OpXt-p Zt-1- AqZt-q. (4-47)


Empirically, we set Z = Z2 = ... = Z 0, using Eq. 4-47, Zt, t > p can be

calculated as


Zp+I = Xp+ Xp- pXl

Zp+2 = Xp+2 lXp+l -...- pX2 A1Zp+

Zp+3 = Xp+3 1Xp+2 ... pX3 AZp+2 A2Zp+l



Zn = X, iX, ... pX-p AIZ,-1 A2Zn-2 -A qZn-q+l

(4-48)


Fitting simulated data. As in the last section, we now simulate data to check the

accuracy of our conditional maximum likelihood estimation method. We simulate data

from ARMA(1,1) models with AL noise and Sample size of n = 100. We use the same

method to simulate time series data as in the last section. The parameters Oi and Aj are

chosen to ensure model causality and invertibility.

We apply a numerical optimization method to maximize our conditional likelihood.

The AL parameters are chosen as to = 1 and To = 2. The initial values for Q and A come

from the ARIMA function in R with Gaussian noise.

In Table. 4-4, we list the fitted model parameters of the first five simulation. The

calculation of Means and MSEs is based on 100 simulations.









of ARMA(1,1) model under AL noise


parameters K 7 08 1 A1
True 0.8 1 -0.3182 0.7 0.5
MLEs_simul 0.7398 0.9210 -0.3985 0.7598 0.5045
MLEs_simu2 0.8150 0.9835 -0 -. 0.7111 0.4810
MLEs_simu3 0.8282 1.0287 -0.2759 0.7119 0.4770
MLEs_simu4 0.8357 0.9979 -0.2546 0.6419 0.6056
MLEssimu5 0.7415 1.0194 -0.4376 0.7237 0.4352
mean(MLEs) 0.8074 0.9853 -0.3035 0.6888 0.5004
MSE(MLEs) 0.0079 0.0112 0.0217 0.0056 0.0088


4.3 ARMA Models Driven by GARCH Noise

When linear models are not appropriate, nonlinear time series models such as

GARCH, bilinear models, autoregressive models with random coefficients, and threshold

models, are possible alternatives. GARCH models were developed on observing that the

volatility of some time series processes are correlated. This is a common situation in

financial time series data. As a result, GARCH models are widely used in finance.

Engle(1982) introduced the ARCH(p) process where the volatilities are dependent on

the past volatilities. Bollerslev (1986) introduced a generalization of the ARCH(p) process,

the GARCH process.

4.3.1 ARMA Model Driven by GARCH noise

Definition 4.5: ARMA(p, q) model driven by GARCH(u, v)noise. {Xt} is an

ARMA(p, q) process driven by GARCH(u, v) noise if

P q
Xt = fXt-, + Aj Z-,, (4-49)
i= l j= l

where


Zt = tet, {et} ~ IID N(0, 1), (4 50)


where ht is the positive function of Z,, s < t, defined by
ui V
ht rao i + Z ZZ + jht-j, (4-51)
i=l j=1


Table 4-4: Fitted value









with 0o > 0, and aj, 3j > 0, j = 1,2,...

4.3.2 Conditional Maximum Likelihood Estimation of GARCH model

The model parameters of a GARCH can be estimated using conditional maximum

likelihood estimation. Let f(xl,..., x,) denote the pdf of the joint distribution of the first

n observations, X1,..., X,, under an AR(p) model. Then the likelihood can be written as


f(x1,...,x,)= f(x1,... ,Xp)f(xp+, Xt,t < p)... f(xIxt,t < n- 1). (4-52)

When the value of n is large relative to p, we ignore the first term, and base our

likelihood estimation on the remaining pieces. This is known as conditional maximum

likelihood estimation.

Therefore, using the property of a location-scale family, the joint likelihood of an

ARMA model driven by GARCH noise can be expressed as,

t (t (4 -53)
L( l,..., l,A ',...,A,,q ao,..., ,/300, v) 1 (4-53)t t
t= p+1

where 0(.) denotes the standard normal density function.

{Zt} can be derived similarly as in Section 4.2.3, with

Zt = Xt lXt-1 .. OpXt-p iAZt-l A. tqZt-q. (4-54)


The mean of {Xt}, pt, can be calculated in the same way as for an ARMA(p, q)

model,
P q
I-It Xt- + AjZt-j. (4-55)
i= 1 j= 1

Standard deviations at = ht, t > 0, can be computed recursively from Eq. 4-50 and

Eq. 4-51 with Zt = 0 and ht = g2 for all t < 0. g2 is the sample variance of {Z1,... Z}.

We now have the joint likelihood function of the {Xt}. MLEs can be found using

numerical optimization methods. For the standard Gaussian GARCH, the model can be

easily fitted using computational packages, like ITSM, and the GARCH toolbox in Matlab.









4.3.3 ARMA Models Driven by GARCH AL Noise

We now introduce AL noise into the GARCH model. We define an ARMA model

driven by AL GARCH noise in exactly the same way as we defined the Gaussian GARCH

model of Eq. 4-49, except that {et} has an Asymmetric Laplace distribution.

Definition 4.6: ARMA (p, q) model driven by GARCH(u, v) AL noise. {Xt}

is an ARMA(p, q) process driven by GARCH(u, v) AL noise if

P q
Xt= OiXt-i + AjZt-j_, (4-56)
i=l j=l

where


Zt = Vhet, {et} ~ IIDAL(0, K, r), (4-57)


where ht is the positive function of Zs, s < t, defined by
u v
ht = ao + aiZt, + i ht-j, (4-58)
i=l j=1

with co > 0, and aj, 3j > 0, j = 1, 2... and


0 (4-59)


The joint likelihood function of the GARCH AL model can then be expressed as


L(,.,,l,..A'qo,...,"',/3o,...,/v) i fAL (xt (4-60)
t= p+l

where fAL(') is the density function of AL(K, r, 0).

This model can be fit use conditional likelihood estimation, in the same way as in the

last section, except that the conditional likelihood functions of XtIXs, s < t are different

now. The joint conditional likelihood function can be maximized by using numerical

optimization methods.












4.4 Analysis Real Estate Mutual Fund Data

In this section, we fit some real financial data to check the usefulness of our model.

We analyze the returns of a real estate mutual fund that is managed by TIAA-CREF. The

data range from Jan 1, 2000 to Dec 31, 2006, a total of 1807 daily values. The reason for

selecting this data set is that the histogram of the data indicates heavy tails and right

skewness.

Values of Mutual Value


260

240

220

200

180

160

140
0 200 400 600 800 1000 1200 1400 1600 1800
Days


Figure 4-7. Daily values
from Jan 1,


of the mutual fund managed by TIAA-CREF. The data range
2000 to Dec 31, 2006, a total of 1807 values.


Daily Returns


lii I. Ill


0


5


0 200 400 600 800 1000 1200 1400 1600 1800
Days

Figure 4-8. Daily returns of the mutual fund managed by TIAA-CREF. The data range
from Jan 1, 2000 to Dec 31, 2006, a total of 1807 value.










Obviously, the data are not stationary. Therefore, we differentiated the data at lag 1.

The resulting data are the daily returns of the mutual fund.

500
450
400
350
300
250
200
150
100
50
0
-1 -0.5 0 0.5 1 1.5 2

Figure 4-9. Histogram of the daily returns of the real estate mutual fund managed by
TIAA-CREF.


The histogram of Fig. 4-9 indicates high peakedness compared with a normal

distribution. Also, the data are .,i-iiiiii. I ii: with right skewness. The distribution of

the daily returns is close to the shape of an Asymmetric Laplace distribution as we have

already discussed. Therefore, we consider this to be a good example to apply our time

series models to.

We use four methods to analyze this mutual fund data: an ARMA(p, q) model under

Gaussian noise, an ARMA(p, q) model under Asymmetric Laplace noise, an ARMA(p, q)

model driven by GARCH Gaussian noise, and an ARMA(p, q) model driven by GARCH

AL noise. The first two models are linear models. The other two are nonlinear models

which assume that the variances are dependent.

We plot the sample ACF of the residuals to check the validation of the models. If

there is no correlation in the sample ACF, we check the sample ACF of the absolute

values and squares of residuals. No correlation in the sample ACF of the residuals does

not necessarily mean that they are independent. If correlation is detected in the absolute

values or squares of residuals, that -ii--.- -I they are dependent.









Since these models have different numbers of parameters, the log-likelihood by itself is

not a good criterion for model selection. We apply the bias-corrected Akaike Information

Criterion (AICc) to evaluate our models.

Definition 4.7: (Akaike Information Criterion (AIC)).For a model based on

parameters 0, let L(O) be the maximized likelihood function, and k the number of free

parameters in the model.The Akaike information criterion (AIC) is defined as


AIC -2log (L()) + 2k. (4-61)


Definition 4.8: (Bias-corrected Akaike Information Criterion (AICc)).For a

model based on parameters 0, let L(0) be the maximized likelihood function, and k the

number of free parameters in the model. The bias-corrected Akaike information criterion

(AICc) is defined as

2k(k + 1)
AICc = AIC + + ) (4-62)
n-k-1

AIC uses the 2k term as the penalty for adding more parameters into the model.

Usually, the model with minimum AIC value is chosen as the best model to fit to the data.

In time series model selection, we use the AICc criterion. AICc is the empirical

correction for small sample sizes. Since AICc converges to AIC as n gets large, AICc

should be employ, 1 regardless of sample size. In our case, since the sample size 1807 is

large enough, AIC is very close to AICc.

Method 1: ARMA(p, q) model under Gaussian noise. This is the traditional

time series model as we stated in Section 4.1. We first fit the data up to the order of

ARMA(7,7) using the autofit function in ITSM2000. It follows that ARMA(1,3) is the

best model with largest log likelihood 995.57, lowest AICc -1981. Below is the fitted

value and standard errors of the ARMA(1,3) from the ARIMA function in the R package.

When looking at the first plot, the sample ACF of the residuals, sample ACFs are out

of the boundary at lags 11, 20 and 22. There are also four others close to the boundary.












Table 4-5: Fitted Value of ARMA(1, 3) under Gaussian noise
parameters <1 A1 A2 A3
MLEs 0.98 -0 <.. -0.0505 -0.0715
s.e. 1.968E-6 5.545E-4 9.405E-4 5.707E-4


Sample ACF of Residuals


'J L' '"-------------------. '

S 5 10 15 20 25 30
0 5 10 15 20 25 30


Lag


Figure 4-10. Sample ACF of residuals of ARMA(1,3) model under Gaussian noise.


Sample ACF of Absolute Value of Residuals


I I I I I I I
0 5 10 15 20 25 30


Lag

Figure 4-11. Sample ACF of absolute values of residuals of ARMA(1,3) model under
Gaussian noise.





81











Sample ACF of Square of Residuals


0


CO
0

(D
O
LLLag
0
0







0 5 10 15 20 25 30
Lag

Figure 4-12. Sample ACF of squares of residuals of ARMA(1,3) model under Gaussian
noise.


When analyzing the sample ACF of absolute values and squares of residuals, we found an

obvious correlation in the residuals. Therefore, it is necessary to consider fitting a GARCH

model.

Method 2: ARMA(p, q) model driven by GARCH Gaussian noise.

We fit the ARMA(p, q) model driven by GARCH Gaussian noise, as defined in Section

4.3. The GARCH model is fitted using the matlab GARCH toolbox. We search for the

best GARCH model up to the order of ARMA(7,7) and GARCH(2,2). The best model

is an ARMA(1,3) driven by GARCH(1,1) noise. The log likelihood of the fitted model is

1135.9, with AICc equal to -2263.8.

Table 4-6: Fitted parameters of ARMA(1, 3) driven by GARCH(1,1) Gaussian noise.
parameters 01 A a 2 3 11
MLEs 0.998 -0.907 0.005 -0.083 0.484 0.353
s.e. 1.47E-3 3.237E-2 3.718E-2 2.589E-2 2.468E-2 2.217E-2



Method 3: ARMA(p, q) model under AL noise.

We now consider fitting an ARMA(p, q) model under AL noise, as discussed in

Section 4.2. The model is fitted using the method of conditional maximum likelihood










Sample ACF of Residuals


Figure 4-13. Sample ACF of residuals of
Gaussian noise.


10 15 20
Lag

ARMA(1,3) model driven by GARCH(1,1)


Sample ACF of Absolute Value of Residuals


08

06

04

02

0

-02


Figure 4-14. Sample ACF of absolute values of residuals of ARMA(1,3) model driven by
GARCH(1,1) Gaussian noise.


estimation described in section 4.2. Empirically, we did a numerical optimization in

Matlab, using the Simplex method. The initial values of the ARMA parameters are given

by fitting a regular ARMA model under Gaussian noise. The initial values of K and 7 are

both set to 1.

We search for the best ARMA(p, q) up to the order of ARMA(7,7). The best model is

an ARMA(2,6) with log likelihood 1439.8, AICc=-2859.


: "` '""'''" --










Sample ACF of Square of Residuals


08-

06-

04-
E
02

0

-02-
0 5 10 15 20
Lag

Figure 4-15. Sample ACF of squares of residuals of ARMA(1,3) model driven by
GARCH(1,1) Gaussian noise.


Lehmann(1983) indicates that maximum likelihood estimators of ARMA(p, q)

processes are approximately normally distributed with variance at least as small as those

of other .i-, iii:! ically normally distributed estimators. Brockwell and Davis (2002)

prove that the large-sample distribution of maximum likelihood estimators of ARMA

coefficients is the same for {Zt} ~ IID(0, a2), regardless of whether or not {Zt} is

Gaussian. Therefore, we can use the standard errors of MLEs from Gaussian noise as the

standard errors of the MLEs in our model based on AL noise.

The standard deviation of K and r are obtained by bootstrapping. For detailed

information about bootstrapping on time series models, please refer to Shumway and

Stoffer (2000).

Table 4-7: Fitted parameters of ARMA(2, 6) under AL noise
Para 1 02 A,1 a2 3 A4 A5 X6 k
MLEs 0.3526 0.5423 -0.2989 -0.4524 -0.0889 -0.0610 0.0192 0.0142 0.919 0.129
s.e. 1.12E-1 1.11E-1 1.11E-1 8.45E-2 1.23E-3 1.75E-3 9.9E-4 5.72E-4 0.227 0.019



Method 4: ARMA(p, q) model driven by GARCH AL noise. We fit an

ARMA(p, q) model driven by GARCH AL noise as defined in Section 4.3. The model is

fitted using conditional maximum likelihood similarly as in method 2, except that the

conditional variances are dependent. The conditional variances can be found recursively.














Sample ACF of Residuals


-------------il'-,-,--.----l- -,------

S 5 I 2I I I
0 5 10 15 20 25 30


Lag

Figure 4-16. Sample ACF of residuals of ARMA(2,6) model under AL noise.


Sample ACF absolute value of Residuals


-- - -r1--- -- l l--- -,-,---
I I I I I I I
0 5 10 15 20 25 30


Lag

Figure 4-17. Sample ACF of absolute values of residuals of ARMA(2,6) model under AL
noise.







85









Sample ACF square of Residuals


0 ---I --,--,, --- ---- -.- r,-- -.-,-, ------ -

0 5 10 15 20 25 30
Lag

Figure 4-18. Sample ACF of squares of residuals of ARMA(2,6) model under AL noise.


We have searched up to the order of ARMA(7,7). The best model is an ARMA(1,2)

driven by GARCH(1,1) noise, with log likelihood and AICc equal to 1458.7 and -2971.3,

respectively.

Table 4-8: Fitted parameters of ARMA(1, 3) driven by GARCH(1,1) AL noise.
parameters 1i A1 A2 A3 a1 1
MLEs 0.11248 0.977 -0.922 -0.040 0.335 0.342 0.823 0.112
s.e. 1.472E-3 3.240E-2 3.712E-2 2.589E-2 2.468E-2 2.217E-2 0.0165 0.344


Table 4-9: Summary of AICc of the four methods
Model Noise AICc
ARMA(1,3) Gaussian -1981
ARMA(2,6) AL -2859
ARMA(1,3) GARCH Gaussian -2263.8
ARMA(1,2) GARCH AL -2971.3


According to Table. 4-9, the AICc values of our models based on AL noise are much

lower than models based on traditional Gaussian noise. This is an indication that models

assuming AL noise provide a better fit for the daily returns of the mutual fund. The

assumption of AL noise provides a better description of the data.

Comparing the first two linear models, the ARMA(p,q) under AL noise has a much

lower value of AICc. Similarly for the two nonlinear models. An even more attractive














0

0
U-
LL
O

0


Figure 4-19. Sample ACF


Sample ACF of Residuals


8

6

4

2

0 T 1 T 1 T

J_


0 5 10 15 20
Lag


of residuals of ARMA(1,3) model driven GARCH(1,1) AL noise.

Sample ACF of Absolute Value of Residuals


-n


Figure 4-20. Sample ACF of absolute value of residuals of ARMA(1,3) model driven
GARCH(1,1) AL noise.



feature of these results is that our linear model has a better fit than the nonlinear GARCH

model.

It is a well known fact that the volatility of financial data is highly correlated.

ARMA models assume constant variance, which leads to correlated residuals. Our ARMA

model under AL noise can not remove all the correlation in the residuals. But it has

significantly improved the likelihood and AICc. And the GARCH model under AL noise

has a significantly better fit than the GARCH model under Gaussian noise.


T T I TT I-- 1 I T 1 1




































Sample ACF of Square of Residuals


8


6


4-


2





S5 Lag10 15 2
Lag


Figure 4-21. Sample ACF of squares
GARCH(1,1) AL noise.


of residuals of ARMA(1,3) model driven









APPENDIX A
SAR(P) MODEL WITH MULTIVARIATE AL MARGINAL DISTRIBUTION

In C'! ipter 2, we have analyzed the "trading day effect", that the volatilities of

log returns of exchange rates are higher in the middle of the week and lower during the

weekends. And sample ACF indicates that the log returns are correlated at the lag of 7 or

multiple of 7. SAR(p) models assume the coefficients of the first p 1 lags of a AR(p) are

zero. Therefore, {Xt}'s are correlated at the lag p and a multiple of p.

Now we start from assuming that the marginal distribution of a time series model has

a multivariate Asymmetric Laplace distribution and approximate the distribution of the

generalized estimator of Q using saddlepoint approximation. We have tried three methods

to approximate the PDFs of the generalized estimator. Since there are relatively large

deviations between our approximations and the simulated data, we arrange this part as

appendix.

A.1 SAR(p) Model with Multivariate AL Marginal Distribution

A.1.1 SAR(p) Model

SAR(p) model is a special case of AR(p) model, where the coefficients of the first

p 1 lags are zero. Therefore, {Xt} are correlated at lag p and a multiple of p. Paige and

Trindade (2003) define the SAR(p) model.

Definition A.1 (SAR(p)model) The zero-mean process {Xt}, t = ..., -1,0,1,...

has SAR(p) model if


Xt = Xt-p + Vt, {Vt} iid N(0, a2), (A-l)

with 1Q1 < 1. It follows that Xt is stationary with autocovariance function (ACVF) at lag

h>0,

2 0 /(1- 2), if h =kp,k = 0,, 2,...,
7 := E(XtXt+h) = (A-2)
0, otherwise.









A realization X = [X1,..., X,]' from the model has a multivariate normal distribution,

X ~ N,,(O, F,), where the (i, j)th entry of covariance matrix F, is just 71Y-jl.

Now we define the SAR(p) with multivariate Asymmetric Laplace marginal

distribution. Let {Yt m} to be the zero-mean process, t ...,-1,0,1,..., with

multivariate Asymmetric Laplace distribution,


Yt = Yt,-p + Zt. (A-3)

Therefore, a realization Y = [Y1,..., Y]' from this model has a multivariate

Asymmetric Laplace distribution, Y ~ AL(m, Fr). We do not specify the distribution of

Zt here. It does not follow an Asymmetric Laplace distribution, as we have stated earlier

in this chapter.

A.1.2 Generalized Estimator of Q

Given observations xl,..., x, from a stationary time series, let
n-h n-p
XtXt+h, and )- 2t (A4)
t=1 t=p+l

denote respectively the sample-based estimates of the ACVF at lag h, and the

truncated ACVF at lag 0 obtained by omitting the first and last p observation. For any

given nonnegative constants cl and c2, we define the generalized estimator of Q,

cc2 t -l+p 2t5t-P S
O2 2 i+ (A-5)
clc E l x + Etp x + c2 =n-p+l X CT1 + T2 + C2T3

Some of the more common estimators of Q and a2, can be shown to be the following

special cases of Equation(2.11) (Brockwell, Dahlhaus, and Trindade, 2005):

Least Squares,

S
LS =-1,0o, (A-6)
TiY + T2

Yule-Walker,











7P3 S
YW = (A-7)
70 TI + T2 +T3

Burg,

2%p S
0BG = 9a (A-8)
o+ +T1/2 + T2 T3/2

We define the n x n matrix A to be zero everywhere, except when Ii j = p, in

which case it is equal to 1/2. Similarly, define B(cl, c2) to be the n x n identity matrix,

with the first (last) p diagonal elements multiplied by c1(c2). If X = [X,..., X,]' is a

realization from model (2.9), we can then express the generalized estimator Qc,,2 as a ratio

of quadratic forms in normal random variables,

X'AX P
ct XB(c, c2)X Q(c, c(A9)

A.1.3 Multivariate Asymmetric Laplace Distribution

Kotz el al. (2001) define the multivariate Asymmetric Laplace distribution.

Definition A.2: (Multivariate Asymmetric Laplace Distribution). A random

vector of Y in Rd is said to have a multivariate Asymmetric Laplace distribution (AL) if

its characteristic function is given by

1
(t) 1(A-10)
1 + 1t'Et im't'

where m E Rd and E is a d x d nonnegative definite symmetric matrix.

We use the notation of ALd(m, E) to denote the distribution of Y, and write Y ~

ALd(m, E). If the matrix E is positive-definite, the distribution is truly d-dimensional and

has a probability density function. Otherwise, it is degenerate and the probability mass of

the distribution is concentrated in a linear proper subspace of the d-dimensional space.

For m = 0 the distribution ALd(0, E) reduces to the symmetric multivariate Laplace

distribution La(E).









Proposition A.1 (Density function of Multivariate AL Distribution). If

Y ~ Ad(m, E), the density function of Y is


g(y) (2)-d/2 -1/2 exp (y zm)'- (y- zm) ) -d/2dz. (A-11)
Jo z /

Proposition A.2 (Presentation of multivariate AL distribution). Let Y ~

ALd(m, E) and let X ~ .d(0, E). Let W be an exponentially distributed random variable
with mean 1, independent of X. Then,

Y d mW + W1/2X. (A-12)

Corollary A.1 (Presentation of component Yi multivariate AL distribution).

Let Y ~ ALd(m, E), each component Yi of Y admits the representation


Yd d m~W + W17/2X, (A-13)

where Xi is ith component of X ~ Ad(0, E). W is an exponentially distributed

random variable with mean 1, independent of Xi.

Let aii to be the ith diagonal element of E, and aii to be (i,j)th element of E. Then

Xi ~ I(0,j o u) and E(XiXj) = jj.

By applying the representation in proposition 1.2, we can derive the mean vector and
covariance matrix of multivariate AL distribution.

We have EYi = mi. since E(XiXj) = rij and EW2 = 2, we have

E(YY) = E [(mTiW + W1/2Xi) (mlW + W1/2Xj)]

= mimjEW2 + E(W)E(XiXj)

= 2mimj + -ij.

Therefore,

Cov(Y, ,) = E(Y3 ) EYEY, = 2mm.i + i mim. = mimj + j.









Proposition A.3 (Mean Vector and covariance matrix). Let Y ~ ALd(m, E),

EY is the mean vector of Y, and Cov(Y) is the covariance matrix of Y, then


E(Y) m, (A-14)


and


Cov(Y) = E + mm'. (A-15)


Saddlepoint Approximation to the PDF of 9c1,c2

Paige and Trindade(2003) approximate the distribution of c1,c,2 by applying

saddlepoint approximation to equations. Now we want to apply the similar method to

approximate the distribution of c 1,c2 under the SAR(p) with multivariate Asymmetric

Laplace distribution.

The generalized estimator of Q, Q1c,c2, is a ratio of two quadratic forms. Under normal

case, the moment generating function of the quadratic forms can be derived explicitly.

But under multivariate Asymmetric Laplace case, since the pdf of distribution is in a

complicated form with a function of integration, we can not derive the moment generating

function of the quadratic form directly.

We consider to use the generalized saddlepoint approximation method developed

by Easton and Ronchetti(1986), as we did in C'!i lpter 2. To apply this method, we need

to derive the first four cumulants of the quadratic forms and then approximate the

cumulative generating function using Edgeworth expansion. Detail of this generalized

saddlepoint approximation method is available in C'!i lpter 2.

We will now derive the expression for EP, EQ, E(P2), E(PQ), E(Q2), E(P3),

E(P2Q), E(PQ2), E(Q3), E(P4), E(P3Q), E(P2Q2), E(PQ3), E(Q4), which will be used

to develop the cumulants in the next section.

The expression for EP and EQ is quite straightforward. Let column Y, has mean p

and variance-covariance matrix E, and G is n x n matrix, then












E(Y'GY) = p'Gp + trace(GE).


McCullaph(1987, Ch3) specifies the expression of generalized moments using tensor

method,


E(P2)

E(PQ)

E(Q2)


E(Y'AY)2 ajak,lEYYjYkYf
i,j,k,l
E(Y'AY)(Y'BY) a ./', EYiYkYi
i,j,k,l
E(Y'BY)2 b .,EYYjYkYl.
i,j,k,l


(A-17)


Then we have the third moments,


E(P3)

E(P2Q)

E(PQ2)

E(Q3)


E(Y'AY)3 i ajak,la,dEY YjYkYYcYd
i,j,k,l,c,d
E(Y'AY)2(Y'BY) ajak,lb,daEYjkYlcYd
i,j,k,l,c,d
E(Y'AY)(Y'BY)2 a .A ,lbdEY YkYlYYd
i,j,k,l,c,d
E(Y'BY)3 b, ., ,lbcdEY YkYYcYd.
i,j,k,l,c,d


(A-18)


The fourth moments,


E(Y'AY)4 a jak,lac, ;. fEYijYkY YdYkYf
i,j,k,l,c,d,e,f
E(Y'AY)3(Y'BY) aijak,ia ,. fEYiYYkYlYcYdYYf
i,j,k,l,c,d,e,f
E(Y'AY)2(Y'BY)2 i ajak,lbc,,,. fEYIYjYkYYcYdYYf
i,j,k,l,c,d,e,f
E(Y'AY)(Y'BY)3 a ,, .b,,,. fEYY YkYycYdY7Yf
i,j,k,l,c,d,e,f


E(P4)

E(P3Q)

E(P2Q2)

E(PQ3)


(A-16)









E(Q4) E= (Y'BY)4 b A, b,,/. frEYYkYjYYYYf,
i,j,k,l,c,d,e,f
(A-19)

where aij and bij are the (i, j) components of matrix A and B. We now apply the

representation of multivariate AL distribution in Corollary(1.1), the product of Y1 can be

calculate as,

EY Yj Yk Y

= E [(miW + W1/2Xi)(nmjW + W1l2Xj)(mkkW + WI/2Xk)( mlW + W1/2Xi)]

E [W4 mimjmkmI + W7/2 (mimjmkX) [4] + W3 (mmjXkXi) [6]

+ W5/2 (miXXkXI) [4] + W2XiXjXkXl]

= mimjmk mlEW4 + mimjmmkEW7/2EXz [4] + mimjEW3EXkXi [6]

+ mTEW5/2EXjXkX [4] + EW2EXiXjXkXi,

(A-20)

where X = [Xi,..., XT,] has multivariate normal distribution, X ~ N(0, Tr). W is an

exponential distribution with mean zero with EW = p!.

The terms followed by numbers in parenthesis indicate the summation of all the

permutation in that type. For example,


(mimjmkXl) [4] = mimmnjkX + mimrjXkml + miXjmnkml + Xmrjmknml.

(A-21)

We know for normally distributed X with mean zero, the odd moments and

cumulants have the values of zero. Therefore, the terms with EXi, EXiXjXk equal to

zero.

To calculate EXiXjXkXi, we introduce the notation by McCullaph(1987, Ch3).









Let X be the random variable whose components are X1,..., Xp with moment

generating function Mx() and cumulative generating function Kx(}), we have the

expansion,


Mx(A ) + + j j/2!+ UjkJ /3! + Ujk, l/4! +...
Kx() = 1 + p + ~i~,j1 /2! + j ,Jk/3! + j Jkl/4! + ...
Kx () = 1+ (r, + (ijr,',d/2!+Ujyk i,j,k / i j k Ki,j,k,l

(A-22)


This expansion implicitly defines all the moments Ki, Kij, Kijk and cumulants

Ki, i'j, 1ijk of X. We can also express ~i, 0ij, 1ijk with the expectation forms which

we are more familiar with,


,i E(X), ,'j E(XiXJ), ijk = E(XiXjXk) (A-23)


and


Ki i Ki i i J

i,j,k -K ijk i ~jk[3] + 2K Kk

i,j,k,l ijkl i jkl [4] j kl[3] + 2Ki K kl[6] 6Ki jkKI.

(A-24)


Applying tensor method, we can express EXiXjXkXi as


EXiXjXkXi

Sijkl

i,j,k,l + ik'kl [4] + K,'1[3] + iKk,~'[6] + kik kkk

i,j Kk,l[3]

jUkcri + oikcjl + crilkj.


Therefore,













= 24 mimjmkml + a"' mj-kl [6] + 2(oij-kl + oikojl + lkJiil)

24 mimjmkml + 6 [mm ,kl + mimkcrjl + mimlojk + mjmkuil + m.m ,-Ti + mkmlij

+ 2(7ijUkl + 7iklcjl + 0l7kil). (A-25)

In case of Y has symmetric multivariate Laplace distribution, mi = 0. Eq. A-25
reduces to EYjYkYi = 2(oijukl + cikcjl + oilukj).
Similar to Eq. A-25, we get

EXiXjXkXIXXd = ki'kklkcd[15] -= Oij^klcd[15] (A-26)

and

EXiXjXkXIXcXdXXf -k,Jlkcdke,k f [105] -= ijUkllcd[ef [105]. (A-27)

The higher order of moments of Y can be calculated as

E YYj YkYYYYd

SE [(riW + W1/2X) + (ImnW + W1/2X) + (nkW + W1/2Xk)

+ (mnW + W1/2X1) + (mnW + W1/2Xe) + (mdW + W1/2Xd)]

[W6miMjMkMIMcMd + W1/2 ijMkMIMclXb[6] + W5m77,77,rmklXcXd[15]

+ W9/2miMjmkXlXcXd[20] + W4m mXkXlXcXd[15]

+ W7/2mniXjXkXlXcXd[6] + W3XiXJXkXlXcXd]

m= mi mj rkmmcmdEW6 + mimjmmlEW5EXcXd [15]

+ mimjEW4EXkXlXcXd[15] + EW3EXiXjXkXlXcXd

= 720 mimjmkmlmcmd + 120iii i" ,mkmlucd[15] + 24mimjTklcd[45] + 67ij-klc7d[15],

(A-28)













= E W8mi7jmkmlmcmdmemf + W15 /2M7i7fjMkMIMcMdMf77deXf [8]

+ WT'mimjmkmimcmdXeXf [28] + W13 2 Tnjmkm1mcX-dXeX [56]

+ W6miMjmkMlXcXaXeXf [70] + W11/2MiMjmkXlxcXdXXXf [56]

+ W5mmjXkXlXcXdXXf [28] + W9/2mXjXkXiXcXdXX Xf [8]

+ W4XXjXkXlXcXaXeXf]

r' ii i iii ,'mcmdmemf EW8 + m77imj7k mmlmcdEW7EXeXf [28]

+mimjmkmlEW6EXcXdXeXf [70] + mimjEW5EXkXjXcXdX,Xf [28]

+ EW4EXiXjXkXIXcXdXeXf

40i220i i ,mkmmcmdmemf + 504mi,, i,.,i, ii,'JmcT-dcd[28]

+720mimjmkmlucd7ef [210] + 12tnn( M ..TklUcdJef [420]

+ 24ijyUkl7-cd-ef [105].

(A-29)

In case of m = 0, these two equations reduce to

EYYjYkYYcYd = 67ijUklUcd[15]

EYYjYkYYcYdYeYf = 24 ijaUklcdef [105].

(A-30)

To simplify our calculation, we will concentrate in the case m = 0. The calculation of

the first four moments of P and Q is computationally intensive. But it can be done using

some computer programs, even it is very time-consuming.

We try three methods to approximate the distribution of








A.1.4 Saddlepoint Approximation to the Estimating Equation

Daniels (1983) extend the saddlepoint method to approximate the distribution of an

estimator defined by an estimating equation. We can define ^ci,c2 as the unique root in r

of the estimating equation


r(r)= P- rQ(c,c2). (A-31)

Since r](r) is a monotonically decreasing function in r for every realization Y, we have

c1,c2 > r # ri(r) > 0, which leads to the device P(ci,c,2 > r) = P(r(r) > 0).
Let K,(r)(T) be the cumulative generating function of rT(r). we can not derive

K,(r) (T) directly, as we have stated before. But we can approximate K,(r) (T) using
Edgeworth expansion. The cumulative generating function of rl(r), K,(r)(T), can be

approximated as,

na2T2 3 n2T3 4n on3T4
K~(,) (T) -pT + +
(r)() = + 6 24 (A32)

where pT O, ,3, I{, are the mean, the variance, and the third and fourth

cumulant of r](r). The first four moments of r](r) can be derived from the first four

moments of P and Q,

E[r(r)] = E(P rQ) = EP rEQ

E[r(r)]2 = E(P rQ)2 = EP2 2rEPQ + r2EQ2

E[r(r)]3 = E(P rQ)3 = EP3 3rEP2Q + 3r2EPQ2 3EQ3

E[r(r)]4 = E(P rQ)4 = EP4 4rEP3Q + 6r2EP2Q2 EPQ3 4EQ4

(A-33)

The relationship of cumulants and moments are available in chapter 2.

According to Daniels(1983), the saddlepoint approximation to the pdf of (ci,c, at r is,


f,(r|) n{K2(r)(To)/To} {2K'(r)(To)} exp {nK,(r)(To)} (A-34)










where K,(r) (T) indicates the first derivative of K,(r) (T) with respect to r, K~(r) (T)

OK,(r)(T)/Or. K ,,(T) and K ()(T) indicate the first and second derivatives of K,(r)(T)

with respect to T, respectively.

To is the saddlepoint solving saddlepoint equation


aK,()(T)/T T=To =: K(r)(To) 0. (A-35)


To check the validation of our method, we approximate the SAR(3) model, with

= 0.5, m = 0. The variance-covariance matrix E is defined as Eq. A-2 with a = 1. We

choose the Burg estimator, i.e.,cl = 0.5, c2 = 0.5. Sample size n = 10 and 50.

4
3.5
3-
2.5
2-
1.5
1 -/ \
0.5 \
0
-0.5
-0.5 0 0.5 1

Figure A-i. Approximated pdf of burg estimator in SAR(3) model with Symmetric
Laplace marginal distribution using saddlepoint approximation to the
equations. = 0.5,m = 0, a = 1, cl = 0.5, c2 = 0.5, n = 10.


Obviously, this is not a good approximation. Our approximated pdf has much higher

peak compared with the simulated data. And there are some discontinuous point around

the mode in our approximated pdf.

One thing to notice is that the approximated pdf with sample size 50 are much

closer to the simulated histogram than the one with sample size 10. This is a good sign,

since we are expecting the approximated pdf converge to the real value when sample size

getting larger. We did not try the approximation with larger sample size since we are more

interested in the small sample approximation.