Working Paper/Document de travail2013Which Parametric Model for Condit - PDF document

Working Paper/Document de travail2013Which Parametric Model for ConditionalSkewness?by Bruno Feunou, Mohammad R. JahanParvar and Roméo Tédongap 2 Bank of Canada WorkingPaperSeptemberWhich Parametric Model for ConditionalSkewnessBruno Feunou,Mohammad R. JahanParvarand Roméo TédongapFinancial Markets DepartmentBank of CanadaOttawa, Ontario, Canada K1A 0G9feun@bankofcanada.caOffice of Financial Stability Policy and ResearchFederal Reserve Board of Governorsshington, DC 20551Corresponding author: Mohammad.JahanParvar@frb.govDepartment of FinanceStockholm School of EconomicsStockholm, SwedenRomeo.Tedongap@hhs.se Bank of Canada wor king papers are theoretical or empirical works - in - progress on subjects in economics and finance. The views expressed in this paper are those of the authors.No responsibility for them should be attributed to the Federal Reserve Board of Governors or the ank of Canada. ISSN © 2013Bank of Canada ii AcknowledgementsWe thank two anonymous referees, Chris Adcock (the editor), Torben Andersen, Bruce Hansen, Stanislav Khrapov, Nicola Loperfido, Dilip K. Patro, Akhtar Siddique, Scott Hendry, Glen Keenleyside, seminar participants at Joint Statistical Meetings 2011, Midwest Econometric Group meeting 2011, the Bank of Canada, Wayne State University (mathematics department), OCC, and the MAF 2012 conference for many useful comments. The remaining errors are ours. An earlier version of this paper has been circulated and presented at various seminars and conferences under the title “Regime Switchingin the Conditional Skewness of S&P500 Returns.” iii AbstractThis paper addresses an existing gap in the developing literature on conditional skewness. We develop a simple procedure to evaluate parametric conditional skewness models. This procedure is based on regressing the realized skewness measures on modelimplied conditional skewness values. We find that an asymmetric GARCHtype specification on shape parameters with a skewed generalized error distribution provides the best insample fit for the data, aswell as reasonable predictions of the realized skewness measure. Our empirical findings imply significant asymmetry with respect to positive and negative news in both conditional asymmetry and kurtosis processes.JEL classification: C22, C51, G12, G15Bank classification: Econometric and statistical methodsRésuméLes auteurs élaborent une procédure d’évaluation simple des modèles paramétriques d’asymétrie conditionnelle en vue de combler une lacune de la littérature sur le sujetCette procdure est baséesur la régression de l’asymétrie réalisée sur l’asymétrie conditionnelle. Les auteurs constatent qu’une spécification de type GARCHasymétrique pour les paramètres de forme, couplée à une distribution d’erreurs généralisée asymétrique, offre le meilleur ajustement statistique en échantillonainsi qu’une prévisibilité satisfaisante de la mesure de l’asymétrie réaliséeIls notent une importante asymétrie dans l’effet des bonnes et des mauvaises nouvelles sur le plan tant de la dynamique de l’asymétrie conditionnelle que de celle de l’aplatissement conditionnelClassification JEL : C22, C51, G12, G15Classification de la Banque :Méthodes économétriques et statistiques 1IntroductionTraditionalmodellingofnancialtimeseriescriticallyreliesontheassumptionofconditionalnormalityofreturns.Thisassumptionimpliesthatconditionalskewnessandexcesskurtosisshouldbeequaltozero.However,empiricalevidenceisinsharpcontrasttothisassertion.Unconditionally,thesemomentsprovenottobezero.Moreover,similartothersttwoconditionalmoments,highermomentsdemonstrateconsiderabletimevariationasnotedbyBekaertetal.(1998)andGhyselsetal.(2011).Thus,explicitmodellingofconditionalhighermomentsthatallowsfortimevariation,isnecessarytoavoidmodelmisspecication.SincethepioneeringworkofHansen(1994),anumberofresearchershaveproposedparametricmodelsforconditionalskewness.Examplesofstudiesontheeconomicimportanceofconditionalskew-nessinnancialassetreturns,itseconometricmodellinganditsempiricalapplicationsincludeHarveyandSiddique(1999,2000),Chenetal.(2001),BrannasandNordman(2003),JondeauandRockinger(2003),Patton(2004),Leonetal.(2005),LanneandSaikkonen(2007),GrigolettoandLisi(2009),Wilhelmsson(2009),Ghyselsetal.(2011),DurhamandPark(2013),Conradetal.(2013),andFeunouetal.(2013).Thereexistanumberofstudiesthatfocusonconditionalkurtosis,amongthemBrooksetal.(2005)andGuidolinandTimmermann(2008).Inthispaperwefocusonconditionalskewness,orconditionalasymmetry.Theexistingresearchdoesnottelluswhichparametricconditionalskewnessmodelprovidesabettertforthedata.AsnotedbyKimandWhite(2004),thisispartiallyduetotheextremesensitivityoftraditionalskewnessandkurtosismeasurestooutliers.1Theyproposeseveralrobustmeasuresforskewnessandkurtosis.Inpreviouswork(Feunouetal.2013),wefoundthatconditionalasymmetryinreturnsisrelatedtothe\relativesemi-variance,"denedastheupsidevarianceminusthedownsidevariance.2Weshowedthatmodellingdownsideriskispossiblewhenameasureofskewnessisexplicitlyincorporatedinthemodel.WeusedPearson's(1895)\modeskewness"asthemeasureofchoice.Pearson'smodeskewnessismorerobusttooutliersthantraditionalskewnessmeasures.BuildinguponandexpandingonsuggestionsinKimandWhite(2004)andFeunouetal.(2013),thispaperllstheexistinggapintheliteratureregardingmodeladequacyforparametricconditionalskewnessmodels.Ourobjectiveisthreefold.First,weestablishthroughapropositionthattherelativesemi-variance, 1Bytraditional,wemeanstandardizedthirdandfourthmomentsofarandomvariable.2Wedenethe\upsidevariance"asthevarianceofthereturnsconditionalupontheirrealizationaboveacertainthreshold.Theirvarianceconditionalupontheirrealizationbelowthesamethresholdiscalled\downsidevariance."Basedonthesetwodenitions,wedenethedierencebetweenupsidevarianceanddownsidevarianceasthe\relativesemi-variance."1 dividedbythetotalvariance,isameasureofskewnessthatsatisesthepropertiesproposedbyGroen-eveldandMeeden(1984)foranyreasonableskewnessmeasure.Second,wedevelopanintuitiveandeasy-to-implementmethodfornon-parametricmeasurementofrealizedasymmetry.Third,basedonthismeasureofrealizedasymmetry,wecantestanyparametricmodelforconditionalskewnessandprovideamethodofhowtomodeltheconditionalskewness.Wetestanumberofparametricmodelsofconditionalasymmetrywithvariousfunctionalanddistributionalassumptions.ThetestingprocedureisbasedonMincerandZarnowitz(1969)regressionsandissimilarinspirittothemethodologydevel-opedbyChernov(2007)toclosetherealized-impliedvolatilitypredictiveregressiongap.Wendthatinadditiontoallowingtimevariationintheconditionalasymmetry,weneedtoallowfora\leverageeect,"butalsoforasymmetry-in-asymmetrytoobtainthebestcharacterizationoftheconditionalskewnessdynamics.3ThemostsuccessfulcharacterizationoftheconditionalasymmetrysharesseveralfunctionalfeatureswiththecelebratedexponentialGARCH(EGARCH)modelofNelson(1991).Ghyselsetal.(2011)proposeamethodologyformodellingandestimatingtheconditionalskewnessbasedonamixeddatasampling(MIDAS)methodofvolatilityestimationintroducedandextensivelystudiedbyGhyselsetal.(2005,2006,2007),andBowley's(1920)measureofskewness.OurworkdiersfromGhyselsetal.intwoimportantdimensions.First,weareinterestedinassessingtheadequacyofdierentmodelsofconditionalasymmetry,whileGhyselsetal.focusonasinglemodel.Second,Ghyselsetal.buildtheirmodelofconditionalasymmetrybasedonBowley's(1920)robustcoecientofskewness.Thismeasureisconstructedusingtheinter-quantilerangesoftheseriesinvestigated,whileourmeasureisbasedonthedierencebetweenupsideanddownsidesemi-variances.DurhamandPark(2013)studythecontributionofconditionalskewnessinacontinuous-timeframe-work.Fortractability,theyassumesimpledynamicsbasedonasingleLevyprocessfortheconditionalskewnessintheirestimatedmodels.Wemodeltheconditionalasymmetryindiscretetimeandassumemuchricherdynamics.DurhamandParkestablishthecostofignoringconditionalhighermomentsinmodellingreturnsdynamics.Thus,theirsimplemodelisadequatetomotivatetheirwork.Theeconomicrelevanceofconditionalasymmetryhasbeenestablishedinseveralassetpricingstudies.AspointedoutbyChristoersenetal.(2006),conditionallynon-symmetricreturninnovationsarecriticallyimportant,sinceinoptionpricing,forexample,heteroskedasticityandtheleverageeectalonedonotsucetoexplaintheoptionsmirk.Inthisstudy,wetrytondthemodelthatbestcharacterizesthedynamicsoftheconditionalasymmetryinS&P500returnsinarichsetofmodels.JondeauandRockinger(2003)characterizethemaximalrangeofskewnessandkurtosisforwhich 3Wendthattheevidenceforasymmetry-in-asymmetryitselfisrelativelyweak.Butthe exibilityoeredbyseparatingthecontributionsofpositiveandnegativeshocksimprovesthemodel'sperformancesignicantly.2 adensityexists.TheyclaimthatthegeneralizedStudent-tdistributionspansalargedomaininthismaximalset,andusethisdistributiontomodelinnovationsofaGARCH-typemodelwithconditionalparameters.Theyndtimedependencyoftheasymmetryparameter,butaconstantdegree-of-freedomparameterintheseriestheystudy.Theyprovideevidencethatskewnessisstronglypersistent,butkurtosisismuchlessso.Whilein uencedbyJondeauandRockinger(2003),ourstudydiersfromtheirworkintwoimportantdimensions.First,westudyalargernumberofmodelsanddistributionsthanJondeauandRockinger(2003),andwethereforeconsiderourstudytobemorecomprehensive.Second,sincewecomparenon-nestedmodels,werelyonMincerandZarnowitz's(1969)methodologytoinvestigatetheadequacyofmodels.Therestofthepaperproceedsasfollows.Insection2,weprovidethetheoreticalbackgroundforourstudy.Wediscusstheimplicationsofvariousdistributionalassumptionsinsection3.Section4describesthedierentparametricmodelspecicationsfortheconditionalskewnessthatwetestinourempiricalanalysis.Wereportourempiricalndingsinsection5.Section6concludes.2TheoreticalBackgroundConventionalasymptotictheoryineconometricstypicallyleadstolimitingdistributionsforeconomicvariablesthatareconditionallyGaussianassamplesizeincreases.ExamplesofsuchworkincludeBollerslevetal.(1994)andDavidson(1994).Thus,conditionalskewnessshouldconvergetozeroassamplesizeincreases.However,asGhyselsetal.(2011),Brooksetal.(2005),andJondeauandRockinger(2003)show,conditionalskewnessformanynancialtimeseriesdoesnotvanishinlargesamplesorthroughsamplingathigherfrequencies.Inwhatfollows,weextendtheirndingsanddevelopatestingframeworktocomparedierentparametricmodelsofconditionalskewness.Ingeneral,commonparametricdistributionsconsideredinempiricalworktocharacterizethedis-tributionoflogarithmicreturnsareunimodalandsatisfythefollowingconditions:Var[rjrm]�Var[rjrm],Skew[r]&#x-278;0Var[rjrm]=Var[rjrm],Skew[r]=0Var[rjrm]Var[rjrm],Skew[r]0;(1)wheremisasuitablychosenthreshold.Afewstudiesintheliteratureusedistributionsthatexplicitlyallowforskewnessinreturns;inparticular,theskewedgeneralizedStudent-tdistributionpopularizedbyHansen(1994),theskewedgeneralizederrordistributionofNelson(1991),andthebinormaldistributionappliedtonancialdatainourpreviouswork(Feunouetal.2013)allsatisfyequation(1).Weintroduceanewmeasureofasymmetry,calledtherelativesemi-variance(RSV),denedbythedierencebetween3 theupsidevarianceandthedownsidevariance,whereupsideanddownsidearerelativetoacut-opointequaltothemode.Proposition2.1Lettherandomvariablexfollowaunimodaldistributionwithmodem.Denotetheupsidevarianceas2u=Var[xjxm]andthedownsidevarianceas2d=Var[xjxm].Westandardizetherelativesemi-varianceRSV2u�2dbydividingitbythetotalvariancetoobtainascale-invariantanddimensionlessmeasureforskewnessdenedas (x)=2u�2d 2;(2)where2=Var[x]isthetotalvariance.Thedistributionisright-skewedif2u�2d,andleft-skewedif2u2d.Theproposedskewnessmeasureiscoherent;thatis,itsatisesthethreepropertiesproposedbyGroeneveldandMeeden(1984)thatanyreasonableskewnessmeasureshouldsatisfy.4Thesepropertiesare:(P1)foranya&#x-278;0andb, (x)= (ax+b);(P2)ifxissymmetricallydistributed,then (x)=0;(P3) (�x)=� (x).Proof:Notethat,foranya&#x-278;0andb,themodeofax+bisequaltoam+b.Besidesthetotalvariance,theupsidevarianceandthedownsidevarianceofax+baregivenbyVar[ax+b]=Var[ax]=a2Var[x]=a22;Var[ax+bjax+bam+b]=Var[ax+bjxm]=Var[axjxm]=a2Var[xjxm]=a22uVar[ax+bjax+bam+b]=Var[ax+bjxm]=Var[axjzm]=a2Var[xjxm]=a22d:Thus,theskewnessofax+bisgivenby (ax+b)=Var[ax+bjax+bam+b]�Var[ax+bjax+bam+b] Var[ax+b]=a22u�a22d a22=2u�2d 2= (x):The (x)skewnessmeasurethussatises(P1).5Todemonstratethat (x)satisesthesecondproperty,supposethatxissymmetricandunimodal;thenweknowthatthemodeisequaltothemean.Asaresult,x�missymmetricandunimodalwithmeanzero.Consequently,x�manditsopposite,m�x,havethesamedistribution.Theupsidevarianceofx�misequalto2uandthedownsidevarianceofx�misequalto2d.However,2d=Var[x�mjx�m0]=Var[m�xjx�m0]=Var[m�xjm�x&#x]TJ/;ø 9;&#x.962; Tf;&#x 22.;1 ;� Td;&#x [00;0]: 4Suitabilityof (x)asaskewnessmeasurecriticallydependsonthemeasureofvolatilityusedinmodellingthereturnsprocess.Laterinthepaper,weshowthatourresultsarebasedontheEngleandNg(1993)NGARCHvolatilitymodel.Basedonempiricalresults,wearguethatNGARCHisaperfectlyadequatevolatilitymeasureand,thus,ourconditionalskewnessmeasuresarewellspecied.5Thisresultmeansthatrelativesemi-variance,2u�2d,satises(P1)uptoamultiplicativeconstant.4 So,2disalsotheupsidevarianceofm�x.Sincex�mandm�xhavethesamedistribution,then2u=2dandinconsequence (x)=0.Thisshowsthatourmeasureofskewnesssatises(P2).Todemonstratethat (x)satises(P3),notethatthemodeof�xissimply�m.Theupsidevarianceof�xisthusthedownsidevarianceofx:Var[�xj�x�m]=Var[xj�x�m]=Var[xjxm]=2d:Similarly,wecanshowthatthedownsidevarianceof�xisequalto2u,theupsidevarianceofx.Ontheotherhand,�xandxhavethesametotalvariance2.Consequently,wehave (�x)=Var[�xj�x�m]�Var[�xj�x�m] Var[�x]=2d�2u 2=�2u�2d 2=� (x):Ourskewnessmeasurethussatises(P3).2.1BuildingarealizedskewnessmeasureBasedonourdiscussioninsection2,wepositthatmodellingconditionalskewnessorasymmetryisequivalenttomodellingrelativesemi-variance,RSVt.Theliteratureonmodellingandmeasuringvolatilityinnanceandeconomicsisvast.Itsucestosaythat,foryearsnow,usingrealizedvari-ancefollowingthemethodologyofAndersenetal.(2001,2003)isthestandardmethodformeasuringvolatilityinnancialtimeseries.Asanexample,Chernov's(2007)studyontheadequacyofoption-impliedvolatilityinforecastingfuturevolatilitycruciallydependsonthismethodology.Wemodifythisstandardmethodologyintheliteraturetobuildanon-parametricanddistribution-freemeasureforconditionalasymmetryinreturns.Inarecentpaper,Neuberger(2012)discussesasomewhatsimilarmeasureofrealizedskewness.Weconstructourmeasuresfollowingthecommonpracticeintherealizedvarianceliteraturebysummingupnelysampledsquared-returnrealizationsoveraxedtimeinterval,RVt=ntPj=1r2j;t;wheretherearenthigh-frequencyreturnsinperiodt,rj;tisthejthhigh-frequencyreturninperiodt.WethenconstructtherealizeddownsideandupsidevarianceseriesasRVdt=nt 2ndtntXj=1r2j;tI(rj;tmt)andRVut=nt 2nutntXj=1r2j;tI(rj;tmt);(3)wherendtandnutare,respectively,thenumberofhigh-frequencyreturnsbelowandabovetheconditionalmodeofreturnmtinperiodt,andwhereI(
)denotesanindicatorfunction.Thus,themeasureforrealizedrelativesemi-varianceissimplydenedasRRSVt=RVut�RVdt;(4)5 which,dividedbyrealizedvariance,willdenerealizedskewnessaccordingtoourproposedmeasureforskewnessintroducedinProposition2.1.Realizedvolatilitywillrefertothesquarerootofrealizedvariance.Itisawell-knownfactthatEt[RVt+1]=2t,where2t=Vart[rt+1]istheconditionalvariance,andwherert=ntPj=1rj;tisthereturnofperiodt.Thisisawell-establishedresultbasedonCorollary1inAndersenetal.(2003).WeestablishthatEt[RRSVt+1]=2u;t�2d;t,andthefollowingpropositionanditsproofshowtheveracityofourassertion.Proposition2.2Lettheunidimensionalcontinuous-priceprocessfPtgTt=0,whereT�0,bedenedonacompleteprobabilityspace( ;F;P).LetfFgt2[0;T]Fbeaninformationltration,denedasafamilyofincreasingP-completeandright-continuous-elds.InformationsetFtincludesassetpricesandrelevantstatevariablesthroughtimet.LetRVut+1andRVdt+1bedenedasaboveforthispriceprocess.ThenEtRVut+1�RVdt+12u;t�2d;t:Proof:Seetheappendix.Thus,asimpletestingprocedureconsistsofregressingRVut+1�RVdt+1on2u;t�2d;t,orRVut+1�RVdt+1=0+1�2u;t�2d;t
+"t+1:(5)FollowingthestandardMincerandZarnowitz(1969)methodology,weviewthemodelwiththe0closesttozero,the1closesttooneandthehighestregressionR2asthebettermodelforconditionalskewness.6WehavefollowedBarndor-Nielsenetal.(2010)closelyinourtreatmentofsourcesofconditionalskewness.Thisimpliesthatintheproofoftheaboveproposition,andfollowingBarndor-Nielsenetal.(2010),wehaveshutdownthe\instantaneous"or\high-frequencyleverageeect"toderivethedesiredresults.Inpractice,wehaveshownthatourrealizedrelativesemi-variancesharesmanyfeatureswiththerealizedsemi-variancesstudiedbyBarndor-Nielsenetal.(2010).Wehaveobserved,buthavenotstudiedthisissueindepth,thatindatasampledathighenoughfrequency,conditionalskewnessisdrivenpurelybyjumps.Thisisareasonableassumptionatfrequen-ciessuchas5-minute,15-minute,half-hour,hourly,orevendailysampleddata,sincetheinstantaneousleverageeectisclearlyweakatsuchhighfrequencies.However,asthesamplingfrequencyisloweredto,forexample,monthlyorquarterlyperiods,thisassertionlosespower.Theleverageeectisanimportantcontributortoconditionalskewnessinlowersamplingfrequencies. 6Inpractice,weputlessweightontherstcondition,with0statisticallyindistinguishablefrom0.AsChernov(2007)documentsthisissueinaparallelliterature,pinningdownthecorrectfunctionalformofthestatisticalrelationshipbetweenlatentvariablesisdicult.Thus,wefocusonthemorerobustandtheoreticallymoreimportantrelationshipbetweenourskewnessmeasuresthroughslopeparameters.Thatsaid,wereportresultsforjointtestsfor0=0;1=1inourdiscussionofempiricalndings.6 Figure1showsthepathsfordailyrealizedrelativesemi-varianceandrealizedvarianceforS&P500returnsinthe1980{2010period.Bothseriesareconstructedusing15-minutereturns.Inordertoprovideamoretractablepictureofthebehaviorofrealizedrelativesemi-variance,wethinouttheplotanduseevery22nddatapointinthegure.Westudythecorrelationbetweenrealizedrelativesemi-varianceandrealizedvarianceandvolatility.Wendthatthecorrelationbetweenrealizedvarianceandrelativesemi-varianceis-0.5076,thecorrelationbetweenrealizedvarianceandskewnessis-0.0206,thecorrelationbetweenrealizedvolatilityandrelativesemi-varianceis-0.4475,andnally,thecorrelationbetweenrealizedvolatilityandskewnessis-0.0538.Thus,RRSVandRVarenegativelycorrelated.Itisimmediatelyclearthatspikesinrealizedrelativesemi-variancetypicallyleadtosignicantjumpsinrealizedvariance.Itisalsoclearthatthereisevidenceofclusteringvisibleinrelativesemi-variance,particularlyinthersthalfofthesamplingperiod.Therangeofrealizedrelativesemi-varianceiscomparabletothatreportedinourpreviouswork(Feunouetal.2013).3ModelSpecicationfortheConditionalSkewnessInmodellingthersttwoconditionalmoments,itisnotnecessarytotakeastanceontheparametricdistributionofthereturns.Unlikethersttwoconditionalmomentswherenoassumptionontheparametricdistributionisrequired,modellinghighermomentsrequiresaspecicationofaparametricdistribution.Two exiblefamiliesofdistributionsattractalotofattentionintheliterature.TheyaretheskewedgeneralizedStudent-t(GST)andtheskewedgeneralizederrordistribution(SGED).Wealsostudythebinormaldistribution,whichwerecentlyintroducedtothenanceliterature(Feunouetal.2013).Withoutlossofgenerality,westandardizethesedistributionsbyxingtheirmeantobeequaltozero,andtheirvariancetobeequaltoone.Standardizedreturnsaredenotedbyz.3.1TheskewedgeneralizedStudent-tdistributionHansen(1994)popularizedtheskewedGSTdistribution.ItsdensityisdenedbyfGST(z)=8���&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;&#x]TJ ;� -1;.93; Td;&#x [00;:bc1+1 �2bz+a 1�2�(+1)=2ifz�a=bbc1+1 �2bz+a 1+2�(+1)=2ifz�a=bwhereistheskewnessparameter,representsdegreesoffreedomanda4c�2 �1;b21+32�a2;c�((+1)=2) p (�2)�(=2):Thisdensityisdenedfor21and�11:GSTdensitynestsalargesetofconventionaldensities.Forexample,if=0,Hansen'sGSTdistributionreducestothetraditionalStudent-t7 distribution.WerecallthatthetraditionalStudent-tdistributionisnotskewed.Inaddition,if=1,theStudent-tdistributioncollapsestoanormaldensity.Sincecontrolsskewness,ifispositive,theprobabilitymassconcentratesintherighttail.Ifitisnegative,theprobabilitymassisinthelefttail.ItiswellknownthatthetraditionalStudent-tdistributionwithdegreesoffreedomadmitsallmomentsuptotheth.Therefore,giventherestriction�2,Hansen'sdistributioniswelldenedanditssecondmomentexists.ThethirdandfourthmomentsofthisdistributionaredenedasEz3=�m3�3am2+2a3
=b3;Ez4=�m4�4am3+6a2m2�3a4
=b4;withm2=1+32;m3=16c�1+2
(�2)2 (�1)(�3)if�3;andm4=3�2 �4(1+102+54)if�4:Themodeofthedistributionis�a=b.Thus,therelativesemi-varianceisVar[zjz��a=b]�Var[zjz�a=b]=4 b2"1�4c2(�2)2 (�1)2#:(6)Wendthatusingrelativesemi-varianceasameasureofskewnessinthecontextofgeneralizedStudent-tdistributionadds exibilitytotheanalysis.NotethatforthethirdmomenttoexistforarandomvariablewithskewedGSTdistribution,mustbegreaterthan3.However,weneedonly�2forrelativesemi-variancetoexist,whichisthesameconditionfortheexistenceofthesecondmoment.Thus,itispossibletostudyasymmetryevenwhenthethirdmomentdoesnotexist.3.2TheskewedgeneralizederrordistributionTheprobabilitydensityfunctionfortheSGEDisfSGED(z)=Cexp�jz+j [1+sign(z+)]:WedeneC= 2�1 �1;=�1 1 2�3 �1 2S()�1;=2AS()�1;S()=p 1+32�4A22;andA=�2 �1 �1 2�3 �1 2,where�(
)isthegammafunction.Scalingparametersandaresubjectto�0and�11.Thisdensityfunctionnestsalargesetofconventionaldensities.Forexample,when=0;wehavethegeneralizederrordistribution,asinNelson(1991).When=0and=2;wehavethestandardnormaldistribution;when=0and=1;wehavethedoubleexponentialdistribution;andwhen=0and=1,wehavetheuniformdistributionontheinterval�p 3;p 3:Theparametercontrolstheheightandthetailsofthedensityfunction,andtheskewnessparametercontrolstherateofdescentofthedensityaroundthemode(�).Thethirdandthefourthmoment8 aredenedasEz3=A3�3�3;Ez4=A4�4A3+62+34;whereA3=4�1+2
�(4=)�(1=)�13andA4=�1+102+54
�(5=)�(1=)�14:Themodeofthisdistributionis�.Asaresult,wendthattherelativesemi-varianceisVar[zjz��]�Var[zjz�]=4�1�A2
S()2:(7)3.3ThebinormaldistributionThebinormaldistributionwasintroducedbyGibbonsandMylroie(1973).Itisananalyticallytractabledistributionthataccommodatesempiricallyplausiblevaluesofskewnessandkurtosis,andneststhefa-miliarnormaldistribution.7Inourpreviouswork(Feunouetal.2013),weshowedthataGARCHmodelbasedonthebinormaldistribution,whichwecallBiN-GARCH,isquitesuccessfulincharac-terizingtheelusiverisk-returntrade-ointheU.S.andinternationalindexreturns.OurBiN-GARCHmodelexplicitlylinksthemarketpriceofrisktoconditionalskewness.Theconditionaldensityfunctionofastandardizedbinormaldistribution(SBin),orbinormaldis-tributionwithzeromeanunitvariance,andPearsonmodeskewness,isgivenbyfSBin(z)=Aexp �1 2z+ d2!I(z�)+Aexp �1 2z+ u2!I(z�);whered=�p /8+p 1�(3/8�1)2andu=p /8+p 1�(3/8�1)2,andwhereA=p 2//(d+u).If=0,thend=u=1,andthisdistributioncollapsestothefamiliarstandardnormaldistribution.Wendthat�istheconditionalmode,anduptoamultiplicativeconstant,2dand2uareinterpretedasdownsidevarianceandupsidevariancewithrespecttothemode,respectively.Specically,Var[zjz�]=1�2 2dandVar[zjz�]=1�2 2u:(8)Weconsiderthispropertytobethemostimportantcharacteristicofthebinormaldistribution,givenourobjectives.Theexistenceandpositivityofthequantitiesdanduimposeaboundontheparameter,givenbyjj1.p /2�11:3236.Finally,itistrivialtoshowthattherelativesemi-variance 7SeeBangertetal.(1986),KimberandJeynes(1987),andTothandSzentimrey(1990),amongothers,forexamplesofusingthebinormaldistributionindatamodelling,statisticalanalysisandrobustnessstudies.9 forthestandardizedbinormaldistributionisVar[zjz�]�Var[zjz�]=p 21�2 p 1�(3/8�1)2:(9)3.4Theskewness-kurtosisboundaryLet3=E(z3t)and4=E(z4t)denotethenon-centeredthirdandfourthmomentsofarandomvariablefztg1t=0.Foranydistributiononztwith(�1;1)support,wehave234�1with4&#x-278;0(seeWidder1946,p.134,Theorem12.a;JondeauandRockinger2003).Thisrelationconrmsthat,foragivenlevelofkurtosis,onlyaniterangeofskewnessmaybespanned.Thisisknownastheskewness-kurtosisboundary,anditensurestheexistenceofadensity.Thus,therealchallengewhilemodellingthedistributionofztistogetcloseenoughtothisskewness-kurtosisboundary.Figure2showstheskewness-kurtosisboundaryforHansen'sskewedgeneralizedStudent-t,SGED,andbinormaldistributionsagainstthetheoreticalboundarydiscussedinWidder(1946)andJondeauandRockinger(2003).ItisclearfromthisgurethattheSGEDspansalargerareaofthetheoreticalskewness-kurtosisboundarythandoestheskewedgeneralizedStudent-tdistribution.Thus,weexpectmodelsbasedontheSGEDtooutperformmodelsbasedontheGSTdistribution.Thesharplimitonpermissibleskewnesslevelsinthebinormaldistributionseriouslylimitsitsabilitytospanthetheoreticalskewness-kurtosisboundary.Figures3and4demonstratethecontributionofskewnessandpeakednessparameters,and,re-spectively,togeneratingskewnessandkurtosisinskewedgeneralizedStudent-tdistributionandSGED,respectively.Notethatthepatternsfortheskewnesssurfaceinbothdistributionsareverysimilar.Thedierenceliesinthepermissiblevaluesforandthelevelofskewnessgeneratedbycomparablecombinationsofskewnessandpeakednessparameters.Thatis,theskewedgeneralizedStudent-tseemstogeneratelargervaluesforskewnessincomparisonwithSGED.Thepatternofthekurtosissurface,however,isdierentforthesedistributions.TheskewedgeneralizedStudent-tdistributionsdemonstrateamoreexplosivepatternaswemovetowardthecornersoftheadmissiblesetfor.ThebehaviorofthekurtosissurfaceforalladmissiblevaluesofismoresubduedforSGED.Bothdistributionsshowmildevidenceofasymmetryinkurtosisforlowervaluesof.4ModelSpecicationInthissection,weintroducethefunctionalformsofthemodelsthatwettothedatafortestingpurposes.Werstdiscusshowwemakeourmodelscomparable.Wewanttoestimatetheparameters10 ofinterestwithoutimposingrestrictionsonourestimationprocedure;however,wealsowanttopreservethetheoreticalboundsimposedontheshapeparameters.Weusesign-preservingtransformations.Inparticular,followingHansen(1994)andNelson(1991),becauseofthedierentrestrictionsontheshapeparameters,wemapthetransformedparameterstobeestimatedintothetrueparameters,usingalogisticmappingfortheskewnessandanexponentialmappingforthepeakedness.Thisstepallowsustoestimatethetransformedparameters~and~asfreevalues,andthenrecovertheoriginalparameters.ForthegeneralizedStudent-tdistribution,weusethemappings=�1+2 1+exp�~and=2+exp(~):(10)Thesetransformationsareintuitive.RecallthatskewedgeneralizedStudent-tdistributionrequiresthatjj1.Transformingfollowingequation(10)ensuresthattheseboundsarepreserved,regardlessoftheestimatedvalueof~.Similarly,GSTrequiresthat21.Equation(10)preservestheselimitsfor,regardlessoftheestimatedvalueof~.Fortheskewedgeneralizederrordistribution,wehave=�1+2 1+exp�~and=exp(~);(11)soastomaintaintherestrictionsjj1and&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;0.Finally,forthestandardizedbinormaldistribution,wehave=1 p  2�124�1+2 1+exp�~35:(12)Notethatthebinormaldistributiondoesnothaveadistinctpeakednessparameter.Thetransformationinequation(12)istomaintaintheboundonPearsonmodeskewness,jj1.p /2�11:3236,forthebinormaldistributiondiscussedimmediatelyafterequation(8).Weestimateandcompareresultsfromuptoninespecicationsforskewnessandpeakednessfactors,acrossthethreedistributionsdiscussedinsection3.Atotalof24modelsttothedata.Inthemostbasicmodel,bothofthesefactorsareconstantparameterstobeestimated.Werelaxthisspecicationandallowfortimevariationandfunctionalcomplexityintheseprocesses.Weassumethatthecondi-tionalvarianceprocessforreturnsfollowsanEngleandNg(1993)NGARCHspecication.Thus,theconditionalvarianceprocessfollows2t+1=0+12t(zt+1�)2+22t:(13)Thischoiceoffunctionalformfortheconditionalvarianceallowsfora\leverageeect"inreturns.Weassumethatreturnsfollowrt+1=+tzt+1,wherezt+1jItGST(t;t),zt+1jItSGED(t;t),11 orzt+1jItBiN(t),andwhereItdenotestheinformationsetuptotimet.8Themostbasicmodelthatwestudyassumesconstant~and~.WecallthismodelM0.WerelaxtheassumptionoftimeinvariancefortheskewnessprocessinModel1,butmaintainthatthepeakednessprocessisstillaparametertobeestimated.InmodelM1,weassumethattheskewnessprocessfollowsasymmetricARCH(1)process,~t+1=0+1zt+1:(14)Theassumptionofsymmetryheremeansthatthearrivalofgoodorbadnewsimpactstheskewnessprocesswiththesamemagnitude.Asymmetryinvolatilityisawell-documentedfeatureofnancialdata.Severalstudiesinthe(G)ARCHliteratureaddressthisissue,whichleadstotheleverageeectinnancialdata.Amongthesestudies,wenoteNelson(1991),Glostenetal.(1993),andEngleandNg(1993).JondeauandRockinger(2003)arguethatasymmetryinARCHforskewnessandkurtosisrequiresinvestigation.Hence,westudythisissueinmodelM2,whereweassumeanAsym-ARCHstructureintheskewnessprocess,~,butassumeconstant~.Inthismodel,~follows~t+1=0+1;+zt+1I(zt+1�0)+1;�zt+1I(zt+10);(15)whereI(
)denotesanindicatorfunction.WeallowforthericherGARCH(1,1)dynamicsintheskewnessprocessinmodelM3.Inthismodel,westillassumeconstant~.~follows~t+1=0+1zt+1+2~t:(16)Thismodel,exceptfordistributionalassumptionsandnormalization,isthesamemodelstudiedbyHarveyandSiddique(1999).Tostudythepotentialexistenceofanasymmetry-in-asymmetryorleverageeectintheconditionalskewness,weassumethattheskewnessprocessfollowsanasymmetricGARCHforminmodelM4.Inthismodel,~isstillassumedtobeconstant.Asym-GARCHin~impliesthefollowingfunctionalform:~t+1=0+1;+zt+1I(zt+1&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;0)+1;�zt+1I(zt+10)+2~t:(17)Intheremainingfourmodels,werelaxtheassumptionofconstant~.InmodelM5,weassumea 8Tokeepournotationconsistentwiththerealizedvolatilityliterature,ourtimingconventiondiersslightlyfromthefamiliarGARCHnotation.Throughoutthepaper,thesubscripttonanyvariablemeansthatitisobservedexactlyattimet.InthetraditionalGARCHnotation,thesubscripttintheconditionalvariancemeansthatitisthevarianceofthetimetreturns.Hence,thevarianceisobservedattimet�1.12 symmetricARCHstructureforboth~and~:~t+1=0+1zt+1;(18)~t+1= 0+ 1zt+1:InmodelM6,westudytheimpactofgoodandbadnewsonbothskewnessandpeakednessprocessesbyassuminganasymmetricARCHinboth~and~:~t+1=0+1;+zt+1I(zt+1�0)+1;�zt+1I(zt+10);(19)~t+1= 0+ 1;+zt+1I(zt+1&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;0)+ 1;�zt+1I(zt+10):ModelM7investigatestheimplicationsofassumingaGARCHspecicationforbothskewness,~,andpeakedness,~,processes:~t+1=0+1zt+1+2~t;(20)~t+1= 0+ 1zt+1+ 2~t:AswithM3,thisformulationisverysimilartothemodelinHarveyandSiddique(1999).Finally,westudytheimplicationsofassuminganasymmetricGARCHfunctionalformforboth~and~inmodelM8:~t+1=0+1;+zt+1I(zt+1&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;0)+1;�zt+1I(zt+10)+2~t(21)~t+1= 0+ 1;+zt+1I(zt+1&#x]TJ/;ø 9;&#x.962; Tf;&#x 10.;Ԗ ;� Td;&#x [00;0)+ 1;�zt+1I(zt+10)+ 2~t:5EstimationResults5.1DataWeusedailyStandardandPoor's500(S&P500)indexexcessreturnsfromThomsonReutersDatas-tream.ThedataseriesstartsinJanuary1980andendsinSeptember2010.Table1reportssummarystatisticsofthedata.InPanelA,wereportannualizedreturnmeansandstandarddeviations,inpercentages,incolumnstwoandthree.Wereportunconditionalskewnessincol-umnfour.Weobservenegativeunconditionalskewnessformarketreturns.Thevalueofunconditionalskewnessisnotsmallrelativetotheaveragedailyreturns.Ourdataseemtobehighlyleptokurtotic,sincetheseriesdemonstratessignicantunconditionalexcesskurtosis,asseenincolumnve.There-portedp-valuesofJarqueandBera's(1980)normalitytestimplyasignicantdeparturefromnormalityinthedata.Ourproxyfortherisk-freerateistheyieldofthe3-monthconstant-maturityU.S.Treasurybill,whichweobtainedfromtheFederalReserveBankofSt.LouisFREDIIdatabank.Thecrashof13 October1987,theAsiancrisisof1997,theRussiandefaultof1998andthe2007{09GreatRecessionarerepresentedinthedata.OurintradaydataseriescomesfromOlsenFinancialTechnologiesandistheirlongestavailableone-minutecloselevelS&P500indexpriceseries.ThedataspantheperiodfromFebruary1986toSeptember2010.Toreducethemarketmicrostructureeectinourempiricalresults,weconstructintradayreturnsata15-minutefrequency.TheseresultsappearinPanelBofTable1.5.2DiscussionoftheresultsWereporttwosetsofresultsforeachdistribution.First,wereportestimationresultsforallmodelsstudiedforeachdistribution.WethenreporttheresultsfromrunningtheMincerandZarnowitz(1969)regressionsofrealizedskewnessmeasureonparametricskewnessresults.Wereportestimationresultsforthefullsample(1980{2009)andthreesubsamples,spanning1980{89,1990{99,and2000-09.Formodelselection,wemainlyrelyonempiricalndingsbasedonourfull-sampleestimates.Estimatedresultsbasedonsubsamplesaregenerallyfordemonstrationoftimevariationorrobustnessandplayasecondaryrole.Ineachcase,foreachdistributionandforeachsampleperiod,werstidentifythemodelthatbestcharacterizesthereturns.Ourstrategytoidentifysuchamodelistocomputelikelihoodratio(LR)teststatistics.Amodelisconsideredviablewhen(i)theLRtestrejectsequalgoodnessoftbetweenabaselinenormallydistributedmodelandthemodelwithconditionalskewness,and(ii)theLRtestrejectsequalgoodnessoftbetweeneachmodelandthemodelprecedingit.9Thatis,wecontrolforoverparameterizationofmodelsbycomparingmodelssequentially.TheLRtestisnotapplicablefornon-nestedmodels.Insuchcases,wedonotcomputeLRteststatistics.IftheLRtestdoesnotdierentiatebetweentwomodels'toritisinapplicable,thenwerstlookattheBayesianinformationcriteria(BIC)ofmodels.Ifthisisstillnothelpful,wepreferthelessparameterizedmodeloverthemoreparameterizedmodelwiththesamegoodnessoftandsimilarBIC.Basedontheestimatedparametersofthepreferredparametricmodel,welteroutthemodel-impliedrelativesemi-varianceprocess.Notethat,inourstudy,modelM8nestsallothermodels.Wethenregressrealizedrelativesemi-varianceontomodel-impliedrelativesemi-variance.Next,weturntomodelevaluation.Ourcriteriaforthesuccessofaparametricmodelofskewness(relativesemi-variance)here{indescendingorderofimportance{are:(i)aslopeparameterthatisstatistically 9Thismeansthat,inourtables,thereportedLRteststatisticsarecomputedbyusinglog-likelihoodvaluesfrommodelMxandmodelMx�1.Thatis,theLRstatisticformodelM2isbasedonlog-likelihoodvaluesformodelsM1andM2.Whennecessary,weindicatethatwehaveusedlog-likelihoodvaluesfromnon-sequentialbutnestedmodelstoconstructateststatistic.14 indistinguishablefromunityatreasonablecondencelevels,(ii)thehighestpossibleR2giventhatthepreviousconditionismet,andnally(iii)whethertheslopeandinterceptcoecientsoftheMincer-Zarnowitzregressioninquestionarejointlyequaltooneandzero,respectively,giventhattheprevioustwoconditionsaremet.Thelastcriterionre ectsthedicultyofpinningdownthecorrectfunctionalformofalatentvariable,inourcaseconditionalskewness.Thus,weputlessweightoninterceptestimatesthatarestatisticallynotdierentfromzero.Chernov(2007)documentsthisissueforrealizedandimpliedvolatilityliterature.Tables2and3reporttheestimationresultsformodelsinsection4fortheGSTdistribution.Notethatthebaselinemodel(representedasNinbothtables)isrejectedinfavorofallalternativemodelsentertainedinourstudy,basedonlikelihoodratioteststhatarenotreportedtosavespace.ModelsM0{M8deliverabettertforthedata.Thus,therstcriterionismet.FormodelM0,wendthatestimatedpeakednessparametersarestatisticallydierentfromzeroatthe5percentlevelacrossallsamplesstudied.However,theskewnessparameterisstatisticallysignicantinthefullsampleandthe2000{09subsample.Weconcludethatthereisverystrongevidenceinfavoroftheexistenceofexcesskurtosisandconvincingevidencesupportingskewness,whichgetsstrongerasweusethe21st-centurydata.10ModelsM1{M4showasomewhatsimilarpattern.Whilethesizeoftheestimatedparametersmaydieracrossthesemodelsandacrosssamples,theyareallsignicantlydierentfromzeroatthe5percentlevel.Estimatedautoregressiveparameters,whethersymmetricorasymmetric(1+or1�),andtheGARCH-likeparameter2aregenerallystatisticallydierentfromzero.Inaddition,weobserveevidenceofasymmetry-in-asymmetryinourresults.Thatis,positivereturnsincreasetheskewness{or,inotherwords,pushtheskewnesstowardpositivevalues,andnegativereturnsdecreasetheskewness{orpushskewnesstowardnegativevalues.Thus,positiveandnegativenewshavedierentandoppositeimpactsonskewness.Basedonthesizeoftheestimatedparameters,theimpactofpositivenewsislargerthanthatofnegativenews.ModelsM5toM8inTables2and3reporttheimpactoftimevariationontheconditionalkurtosis.Ingeneral,theGARCH-typeparameter 2andtheARCH-typeparameter 1;+arestatisticallysignif-icantatthe5percentlevel.Weobserveevidenceofanasymmetricimpactingoodandbadnewsforconditionalkurtosismodels.Theestimated 1;�parameterisstatisticallydierentfromzerointhefullsample,butthisisnottrueingeneralforsubsampleresults.Inaddition,themagnitudeofestimated 1;+isgenerallylargerthanthatof 1;�parameters.Thus,itseemsthat,inthiscase,goodnewsis 10PleasenotethatinTables2,3,5,6,and8,weabusethenotationtosavespace.Thatis,1+and 1+representboththecoecientsfortheARCH-liketermsandthecoecientsforpositiveinnovationshocks.15 moreimportantthanbadnewsforconditionalkurtosisdynamics.Thelikelihoodteststatistics(henceforthLRstatistics)showthatthemodelsstudieddieronhowwelltheytthedata.Overall,modelM8,whichallowsforasymmetricGARCH-typedynamicsforboth~tand~t,performsthebestbasedonLRtests.Thatis,basedonLRtestscarriedout,M8providesabettertthanmodelsM0toM7inthefullsampleandthe1990{99and2000{09subsamplesforreturns.Table4reportstheresultsfromrunningtheMincer-Zarnowitzpredictiveregressionsintroducedinequation(5),wheretheright-hand-sidevariablesareimpliedskewnessmeasuresfromoneofthemodelsinequations(14)-(21)andtheleft-hand-sidevariableistherealizedskewnessmeasureintroducedinequation(4).TheRRSVmeasureisbasedonhigh-frequencyinformation,whileimpliedskewnessmea-suresarebasedondailydata.NotethatmodelM1providesthehighestR2infull-sampleestimation,buttheestimated1issignicantlydierentfromunity.However,thebestmodelshouldbetheonewiththehighestR2amongallspecicationswith1notsignicantlydierentfromunity.OurresultssuggestthatmodelsM4andM8bothhave1notstatisticallydierentfromone,andthehighestR2of19percentinfull-sampleestimation.Boththesespecicationshavetime-varyingconditionalskew-nessdynamicsfeaturingasymmetry-in-asymmetry,butM8nestsM4throughtheconditionalkurtosisdynamics.Forallmodels,thehypothesis1=1isnotrejectedinthesecondsubsampleexceptforspecicationsM0andM7,whilethejointhypothesis0=0and1=1isnotrejectedinthethirdsubsampleexceptforspecicationsM0,M4andM6.OurpreferredmodelwouldthenbeM8basedonfull-sampleandsubsampleMincer-Zarnowitzpredictiveregressionresults,addedtothefactthatM8issignicantlyfavoredoverM4giventheLRtestsreportedinTables2and3.Tables5and6reporttheestimationresultsformodelsM0{M8whenerrorsareSGED.SimilartowhatisseenintheresultsfortheGSTcase,modelsM0{M8performbetterthanthebaselinemodelNacrosstheboard.ModelsM0{M4demonstratestatisticallysignicantpeakednessparameterestimatesacrossmodelsandsamples.Thestatisticalevidenceinfavorofasymmetry-in-asymmetryisquitestronginthefullsample.Statisticalevidencesupportingasymmetry-in-asymmetryismoremixedinthesubsamples.OurestimationresultsformodelsM5{M8implythat,again,evidenceinsupportofasymmetry-in-asymmetryintheconditionalskewnessisquitestrongforthefullsampleandinthesubsamples.Inaddition,thereissignicantsupportforasymmetryintheconditionalkurtosis,wheregoodnewsreducesfat-tailedness.Thisisnotsurprising.Arrivalof\goodnews"shouldreducemarketuncertainty,andhenceconditionalkurtosis.BasedontheLRstatistics,againmodelM8isthepreferredmodelinthe16 fullsampleandacrossthesubsamples.Itappearsasifallowingforarich,asymmetricparameterizationaddstomodel exibilityandhenceitsperformance.ItisworthnotingthatmodelM8sharesmanyfeatureswiththecelebratedNelson(1991)EGARCHmodel.Wendthatinthe1990{99and2000{09subsamples,modelsM6andM8areindistinguishableusingLRtests.BasedonBIC,modelM6ispreferredtoM8inthesesubsamples.However,asmentionedearlier,webaseourjudgmentmainlyonfull-sampleresults,wheretheLRtestclearlypicksM8overM6andM7.Table7reportstheresultsforMincer-Zarnowitzpredictiveregressionswhereimpliedrelativesemi-varianceestimatesarebasedonmodelswithSGEDerrors.Inthefullsample,modelsM0{M2,M5andM8allhaveslopeparametersthatarestatisticallyindistinguishablefromone.Jointtestsofbothslopeandinterceptparametersbeingdierentfromoneandzero,respectively,cannotberejectedfortheabovemodelsatthe5percentcondencelevelforthefullsample,andmodelsM1andM5providethehighestR2of24percent.ThesespecicationshaveARCH-typetime-varyingconditionalskewnessdynamicsincommon,butM5nestsM1throughtheconditionalkurtosisdynamics.However,neitherofthesetwomodelsispreferredinanyofthesubsamples,sincethehypothesis1=1isrejectedatconventionallevelsofcondence.Again,asmentionedearlier,webaseourjudgmentmainlyonfull-sampleresults,andmodelM1wouldbethepreferredspecication,sinceitismoreparsimoniousandslightlyfavoredoverM5basedonLRtestsandBICvaluesinTables5and6.However,notethatthejointhypothesis0=0and1=1isnotrejectedformodelM8,neitherinthefullsamplewithanR2of14percentnorinthesecondandthirdsubsampleswithR2sof7percentand16percent,respectively.Intherstsubsample,M8isalsothepreferredmodel,withanR2of22percentamongallspecicationswherethehypothesis1=1isnotrejected.ThenM8wouldbethebestmodel,shouldweextendourmodel-selectioncriteriatosubsampleregressionresultsandtheneedforttingtheexcess-returnsdataaswell.WereporttheestimationresultsforthebinormaldistributionmodelinTable8.Sincethestandard-izedbinormaldistributionisaone-parameterdistribution,itdoesnotallowforindependentestimationofconditionalkurtosis.Thus,weestimateonlymodelsM0{M4forthecaseofstandardizedbinormaldistributederrors.Similartothepreviouscases,wendsignicantsupportingevidenceforasymmetry-in-asymmetry.Almostallestimatedparametersarestatisticallydierentfromzero.BasedontheLRstatistics,M3isthepreferredmodelforthefullsample.ItallowsforsymmetricGARCH-typedynamicsintheconditionalskewnessmeasure.Inthe1980{89subsample,M3remainsthemodelofchoice.Inthe1990{99subsample,M4istiedwithM2basedontheLRtest,butM2ispreferredtoM4basedonBIC.Inthe2000{09subsample,nomodelperformsbetterthanM0,whichallowsforconstantand.17 Table9reportsmodelevaluationresultsthroughMincer-Zarnowitzpredictiveregressions,formodelsM0{M4withstandardizedbinormallydistributederrors.Acrossthefullsample,modelsM1,M2andM4haveestimated1parametersthatarestatisticallyindistinguishablefromunity.M1andM2havethehighestR2of25percentforthefullsample.Onceweconsidersubsamples,modelsM1andM4demonstratesimilarperformance;bothmodelsdonotrejectthehypothesis1=1inallsubsamplesexcept1980{89.However,basedonourresultsreportedinTable8,weknowthatmodelM1ispreferredovermodelM4inttingtheexcess-returnsdata.Thus,basedonMincer-Zarnowitzpredictiveregressionconsiderations,wechooseM1,whichallowsforasymmetricARCH-typedynamicsintheconditionalskewness.Note,however,thatalthoughmodelM3hasanestimated1parameterof0.9262thatisstatisticallydierentfromunityinthefullsample,thisvalueremainsquiteclosetoone.Furthermore,1parameterestimatesarenotstatisticallydierentfromoneformodelM3inallsubsamples.Giventheseobservations,onceweaccommodatetheneedforttingtheexcess-returnsdataaswell,wemaybetemptedtochooseM3overM1basedonLRtestingandsimilarpredictiveresults.11Comparingourresultsforthefullsampleacrossthethreedistributionalassumptions,weobservethefollowing:(i)GST-basedestimationresultsimplythatModel8,whichallowsforasymmetryinbothconditionalskewnessandkurtosis,performsbetterthanalternativemodelsstudiedincharac-terizingexcess-returnsdynamicsandinMincer-Zarnowitzpredictiveregressionsofparametricontonon-parametricmeasuresofrelativedownsidesemi-variances.(ii)SGED-basedestimationresultsim-plythatifwewantbothreasonablecharacterizationofexcess-returnsdynamicsandstrongpredictiveresultsformeasuringRRSV,thenourchoiceisM8,whichallowsforasymmetryinbothconditionalskewnessandkurtosis.However,ifweareinterestedonlyinpredictivepowerfortheRRSV,thenM1isthebetterchoice.(iii)Binormaldistribution-basedestimationresults,similartotheSGEDcase,leadustochoosetheparsimoniousM1forpredictivepurposesonly.Oncewedecidetocharacterizeexcess-returnsdynamicsaswell,thenwechoosethemore exibleM3,whichallowsforsymmetricGARCH(1,1)-typedynamicsintheskewnessprocess.Wederiveourtheoreticalframeworkfromtheliteratureonrealizedvolatility.Thus,inourtestingprocedures,wefollowthatliteraturecloselyinthatweareconcernedwithin-samplepredictabilityandnotout-of-sampleforecastability.Thus,itappearsthat,strictlyspeaking,modelM8withSGEDerrorsprovidesestimated1sthatarestatisticallyclosetounity,deliversreasonableR2sandprovidesverygoodin-samplecharacterizationofexcess-returnsdynamics.WeconcludethatamodelthatallowsforasymmetricGARCH(1,1)-typedynamicsandaSGEDistheoverallpreferredmodel.Inaddition,this 1122criticalvaluesat5percentand1percentcondencelevelsare5.99and9.21,respectively,whilethecomputedLRteststatisticisover11.18 modelprovidesstrongstatisticalsupportforasymmetry-in-asymmetryandasymmetryinconditionalkurtosis.Inquiteafewcases,wecouldnotrejectastatisticallysignicantpositivedierencebetweeninterceptparameters0and0.Whiletheupwardbiasobservedacrossmodelsanddistributionsmayseemundesirableatrstglance,wekeepinmindthatthevastliteratureontheinformationcontentofimpliedvolatilityfacesthesametypeofproblem.Chernov(2007)addressesthebiasissuebyaddingasuitablepredictortotheright-handsideofthepredictiveregression.Itispossiblethat,byaddressingthepossibilityofamissingvariable,predictiveregressionsmayimprove.However,sinceM8coupledwithanSGEDworkswellandislargelyfreeofthisbias,itisquitepossiblethatthisbiasisduetodistributionalassumptions.Wewilladdressthisinterestingissueinfutureresearch.6ConclusionsInthispaper,weaddressanimportantgapintherecentandgrowingliteratureonconditionalskew-ness.Inrecentyears,anincreasingnumberofstudieshaveaddressedtheimportanceofmodellinghighermomentsandhowtheirdynamicsaectnancialrisk-managementprocedures.Weproposeamethodologytoassessthemodeladequacyofthegrowingnumberofproposedparametricmeasuresoftheconditionalskewness.Weshowthat,theoretically,conditionalskewnessmatterswhenaseriesdemonstratessignicantrelativesemi-variance.Wethenproposeasimpleandintuitivenon-parametricmeasureofrelativesemi-variance,whichwecallrealizedrelativesemi-variance,inthespiritofthesuccessfulliteratureonrealizedvariance.Weshowthat,undermildregularityconditions,parametricmodelsofconditionalasymmetryapproximatetherealizedrelativesemi-variance.Wealsoshowthatourproposedmeasureandmethodologyhaveclosetiestotheliteratureonrealizedvariance,ontheonehand,andtotherecentdevelopmentsonrealizedsemi-varianceontheother.Wethenstudyseveralparametricmodelsofconditionalskewness.Sincetheperformanceoftheparametricmodelsofskewnesscruciallydependsonthedistributionalassumptions,westudyarangeofrelevantdistributionalmodels,includingtheskewedgeneralizedStudent-tofHansen(1994)andJondeauandRockinger(2003),skewedGED,andbinormaldistributioninourpreviouswork(Feunouetal.2013).Wendstatisticallysignicantevidenceinsupportofasymmetry-in-asymmetry,ortheso-calledleverageeectintheconditionalskewness,andstrongsupportforasymmetric(G)ARCH-typedynamicsforbothskewnessandpeakednessprocessesintheconditionaldistribution.Weconcludethat,basedonourresults,thebestmodelforconditionalskewnessisonethatallowsforasymmetric19 GARCH(1,1)dynamicsandadmitsskewedGEDerrors,appliedtobothskewnessandpeakednessprocessesintheconditionaldistributionofreturns.Thismodelsharesmanyfeatureswiththewell-knownEGARCHmodelofNelson(1991).20 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Appendix:ProofofProposition(2.2)Theset-upissimilartothatofAndersenetal.(2003)(henceforth,ABDL(2003)).Proposition1ofABDL(2003)permitsauniquecanonicaldecompositionofthelogarithmicassetpriceprocessp=(p(t))t2[0;T],p(t)=p(0)+A(t)+M(t);whereAisanitevariationandpredictablemeancomponent,andMisalocalmartingale.Letr(t;h)=p(t)�p(t�h)denotethecontinuouslycompoundedreturnover[t�h;t].Thecumulativereturnsprocessfromt=0onward,r=(r(t))t2[0;T],isthenr(t)r(t;t)=p(t)�p(0)=A(t)+M(t):Furtherassumethatthemeanprocess,fA(s)�A(t)gs2[t;t+h],conditionaloninformationattimetisapredeterminedfunctionover[t;t+h]:AnimmediateresultofthisassumptionistheCorollary1,andhenceequation(6)ofABDL(2003),whichestablishesthatVar(r(t+h;h)jFt)=E�[r;r]t+h�[r;r]tjFt
:Inotherwords,theconditionalvarianceequalstheconditionalexpectationofthequadraticvariationofthereturnsprocess.Denotethemodeofthedistributionofr(t+h;h)conditionalonFtbym(t;h).Leti(t+h;h)=1[r(t+h;h)m(t;h)]denoteanindicatorrandomprocessthattakes1ifthereturnsbetween[t;t+h]aregreaterthanorequaltotheconditionalmodem(t;h):Sincethemeanprocess,fA(s)�A(t)gs2[t;t+h],conditionaloninformationattimet,isapredeter-minedfunctionover[t;t+h];withoutlossofgenerality,weomitAfromthispointonwards.25 Wethushave2u(t;h)Var(r(t+h;h)jFt;i(t+h;h)=1)=Var(M(t+h)�M(t)jFt;i(t+h;h)=1)=Var(M(t+h)jFt;i(t+h;h)=1)=E�M(t+h)2jFt;i(t+h;h)=1
�fE(M(t+h)jFt;i(t+h;h)=1)g2=E�[M;M]t+hjFt;i(t+h;h)=1
�fE(M(t+h)jFt;i(t+h;h)=1)g2=E[M;M]t+hi(t+h;h) (t;h)jFt�EM(t+h)i(t+h;h) (t;h)jFt2;where(t;h)Pr[i(t+h;h)=1jFt]:UsingCorollary3inChapterII.6ofProtter(1992),whereitisshownthatE�M(t+h)2
=E�[M;M]t+h
,wehave2d(t;h)Var(r(t+h;h)jFt;i(t+h;h)=1)=E[M;M]t+h(1�i(t+h;h)) 1�(t;h)jFt�EM(t+h)(1�i(t+h;h)) 1�(t;h)jFt2;and2u(t;h)�2d(t;h)=E[M;M]t+hi(t+h;h) (t;h)�[M;M]t+h(1�i(t+h;h)) 1�(t;h)jFt+EM(t+h)(1�i(t+h;h)) 1�(t;h)jFt2�EM(t+h)i(t+h;h) (t;h)jFt22u(t;h)�2d(t;h)=1 (t;h)(1�(t;h))E�[M;M]t+h(i(t+h;h)�(t;h))jFt
�1 (t;h)(1�(t;h))E(M(t+h)(i(t+h;h)�(t;h))jFt)EM(t+h)(t;h)�(t;h)i(t+h;h)+i(t+h;h)�i(t+h;h)(t;h) (t;h)(1�(t;h))jFt:SinceE[i(t+h;h)jFt]=(t;h),wehave2u(t;h)�2d(t;h)=1 (t;h)(1�(t;h))E��[M;M]t+h�[M;M]t
(i(t+h;h)�(t;h))jFt
�1 (t;h)(1�(t;h))E((M(t+h)�M(t))(i(t+h;h)�(t;h))jFt)EM(t+h)(1�2(t;h)) (t;h)(1�(t;h))(i(t+h;h)�(t;h))+2jFt26 2u(t;h)�2d(t;h)=1 (t;h)(1�(t;h))E��[M;M]t+h�[M;M]t
(i(t+h;h)�(t;h))jFt
�1 (t;h)(1�(t;h))E((M(t+h)�M(t))(i(t+h;h)�(t;h))jFt)(1�2(t;h)) (t;h)(1�(t;h))E((M(t+h)�M(t))(i(t+h;h)�(t;h))jFt)+2M(t)2u(t;h)�2d(t;h)=1 (t;h)(1�(t;h))E��[r;r]t+h�[r;r]t
(i(t+h;h)�(t;h))jFt
�1 (t;h)(1�(t;h))E(r(t+h;h)(i(t+h;h)�(t;h))jFt)(1�2(t;h)) (t;h)(1�(t;h))E(r(t+h;h)(i(t+h;h)�(t;h))jFt)+2M(t)((t;h)(1�(t;h)))�2u(t;h)�2d(t;h)
=E��[r;r]t+h�[r;r]t
(i(t+h;h)�(t;h))jFt
�E(r(t+h;h)(i(t+h;h)�(t;h))jFt)(1�2(t;h)) (t;h)(1�(t;h))E(r(t+h;h)(i(t+h;h)�(t;h))jFt)+2M(t)((t;h)(1�(t;h)))�2u(t;h)�2d(t;h)
=cov�[r;r]t+h�[r;r]t;i(t+h;h)jFt
�cov(r(t+h;h);i(t+h;h)jFt)(1�2(t;h)) (t;h)(1�(t;h))cov(r(t+h;h);i(t+h;h)jFt)+2M(t):Hence,twocomponentsdrivetheconditionalskewness.Therstcomponentistheconditionalcovariancebetweenreturnsandtheindicatorvariablethatshowswhetherthemarkethasanupwardordownwardmovement.Thesecondcomponentistheconditionalcovariancebetweenthequadraticvari-ationofreturnsandthesameindicatorvariable.Fromthispointon,wefocusonthelattercomponent,since,asweshowlaterinthisappendix,itisdrivenbyjumpsinthereturns.Theformercomponentistheskewnessinducedbytheinstantaneouscorrelationbetweenlog-returnsandvolatility:2u(t;h)�2d(t;h)1 (t;h)(1�(t;h))cov�[r;r]t+h�[r;r]t;i(t+h;h)jFt
:27 Lety(t+h;h)�y(1)(t+h;h);y(2)(t+h;h)
0�[r;r]t+h�[r;r]t;i(t+h;h)
0:FromCorollary1ofABDL(2003),wehavecov�[r;r]t+h�[r;r]t;i(t+h;h)jFt
=Ehy(1);y(2)it+h�hy(1);y(2)itjFt:Inotherwords,theconditionalcovariancebetweenquadraticreturnvariationandtheupsidein-dicatorequalstheconditionalexpectationofthequadraticcovariationbetweenthequadraticreturnvariationandtheupsideindicator.Proposition2ofABDL(2003)providesaconsistentestimatorofy(1);y(2)t:Recallthatproposition:Foranincreasingsequenceofrandompartitionsof[0;T],0=m;0m;1


;suchthatsupj1(m;j+1�m;j)!0andsupj1m;j!Tform!1withprobabilityone,wehavelimm!18:Xj1hy(1)(t^m;j)�y(1)(t^m;j�1)ihy(2)(t^m;j)�y(2)(t^m;j�1)i9=;!hy(1);y(2)it;andthuswehavelimm!18:jt+hXj=jt[r;r]m;j�[r;r]m;j�1(i(m;j)�i(m;j�1))9=;!hy(1);y(2)it+h�hy(1);y(2)it;wherejtinfm;jt(j),jt+hsupm;jt+h(j)andi(t)i(t;t)=1[r(t)(t;0)]:Thus,Pjt+hj=jt[r;r]m;j�[r;r]m;j�1(i(m;j)�i(m;j�1))isameasureofrealizedskewness.Wenowsimplifythederivedrealizedskewnessexpression[r;r]m;j�[r;r]m;j�1(p(m;j)�p(m;j�1))2;andwecanshowthati(m;j)�i(m;j�1)=1[p(m;j)�p(m;j�1)0]�1[p(m;j)�p(m;j�1)0];sothatthemeasureofrealizedskewnessbecomesjt+hXj=jt(p(m;j)�p(m;j�1))21[p(m;j)�p(m;j�1)0]�1[p(m;j)�p(m;j�1)0]:Thus,usingBarndor-Nielsenetal.'s(2010)(henceforth,BKS(2010))notation,wehaveshownthattherealizedskewnessisthedierencebetweentheupsiderealizedsemi-variance(RS+)andthedownsiderealizedsemi-variance(RS�):28 RS+(t+h;h)=jt+hXj=jt(p(m;j)�p(m;j�1))21p(m;j)�p(m;j�1)0RS�(t+h;h)=jt+hXj=jt(p(m;j)�p(m;j�1))21p(m;j)�p(m;j�1)0:Torecap,wehave2u(t;h)�2d(t;h)1 (t;h)(1�(t;h))E�RS+(t+h;h)�RS�(t+h;h)jFt
:FurtherusingBKS(2010)results,wecanprovidemoreinsightintothesourcesofconditionalskewness.BKS(2010)showthatRS+(t+h;h)�RS�(t+h;h)!t+hXt(
ps)2[1
ps0�1
ps0];where
psp(s)�p(s�)isthejumpattimes:Hencepositiveskewnessisdrivenbythefactthatpositivejumpamplitudesarehigherthannegativeamplitudes,whereasnegativeskewnessisdrivenbythefactthatnegativejumpamplitudesarehigherthanpositiveamplitudes.29 Figure1:Realizedvolatilityandrealizedrelativesemi-varianceforS&P500returns,1986{2009 Inthisgureweplotrealizedvarianceandrealizedrelativesemi-varianceforS&P500returns,sampledat15-minutefrequency.Toprovideamoreinformativeillustration,weplotdailyrealizedvolatilitiesandtheendofthemonthrealizedrelativesemi-variance(every22ndobservationisplotted)values.Figure2:Skewness-KurtosisBoundaryforSkewedGeneralizedStudent-t,SkewedGEDandBinormalDistributions Thisguredepictstheskewness-kurtosisboundariesforHansen(1994)skewedgeneralizedStudent-tdistributionandskewedGEDagainstthetheoreticalboundaryimpliedbyTheorem12.a,page134ofWidder(1946).Thetheoreticalboundaryisobtainedfrom234�1with4&#x-322;0,where3=E(z3t)and4=E(z4t)arethenon-centeredthirdandfourthmomentsofarandomvariablefztg1t=0.30 1986 1990 1993 1997 2001 2006 2009 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Daily Realized Volatility From 15-minute Returns 1986 1991 1995 1999 2004 2009 -6 -4 -2 0 2 4 6 Daily Realized Skewness From 15 minutes Returns 0 5 10 15 20 25 30 35 -6 -4 -2 0 2 4 6 Kurtosis FrontierBinormal Distribution Skewed Generalized Error Distribution Figure3:ImpactofSkewnessandPeakednessParametersonSkewnessandKurtosisSurfacesforSkewedGeneralizedStudent-tDistribution Thisgureshowsthecontributionoftheskewness,,andpeakednessparameter,,ingeneratingskewness(toppanel)andkurtosis(bottompanel)surfacesfortheskewedgeneralizedStudent-tdistributionofHansen(1994).Therangeofparametersischosensuchthattheydonotviolatetheskewness-kurtosisboundary.Figure4:ImpactofSkewnessandPeakednessParametersonSkewnessandKurtosisSurfacesforSkewedGeneralizedErrorDistribution Thisgureshowsthecontributionoftheskewness,,andpeakednessparameter,,ingeneratingskewness(toppanel)andkurtosis(bottompanel)surfacesfortheskewedgeneralizederrordistribution(SGED).Therangeofparametersischosensuchthattheydonotviolatetheskewness-kurtosisboundary.31 1 0.5 0 0.5 1 3 3.5 4 4.5 5 30 20 10 0 10 20 30  and  1 0.5 0 0.5 1 4 4.2 4.4 4.6 4.8 5 0 50 100 150 200 250  and  1 0.5 0 0.5 1 0 1 2 3 4 5 5 0 5  and  1 0.5 0 0.5 1 0 1 2 3 4 5 0 5 10 15 20 25 30 35 40  and  Table1:SummaryStatisticsoftheData PanelA:DescriptiveStatistics,ExcessReturns ReturnSeries Mean(%)Std.Dev.(%)SkewnessKurtosisJ-Bp-Value S&P500 3.4821.94-1.2431.870.01 PanelB:DescriptiveStatistics,Relativedownsidesemi-variance RSV(%)Std.Dev.(%)SkewnessKurtosisJ-Bp-Value S&P500(NP) 1.848.711.2116.360.01 Thetoppanelofthistablereportssummarystatisticsofexcessreturns.Calculationofthereturnsisbasedonsubtractingthedaily3-monthU.S.Treasurybillratefromthelogdierenceofthemarkettotalreturnindex.Meanofexcessreturnsandstandarddeviationsarereportedasannualizedpercentages.Excesskurtosisvaluesarereported.Thecolumntitled\J-Bp-Value"reportsp-valuesofJarqueandBera(1980)testsofnormalityinpercentages.Thebottompanelreportsthecomputedstatisticsoftheobservedrelativesemi-variance(RSV)inthedata.ReportedRSVisbasedonthemeandierencebetweenltereddownsideandupsidesemi-variances.TherstcolumnreportstheannualizedRSV,andthesecondcolumnisthestandarddeviationofthisquantity.Duetoavailabilityofhigh-frequencydataforS&P500returns,wereportthenon-parametricestimateforrelativedownsidesemi-variancefortheUnitedStatesdenotedasS&P500(NP),basedon15-minutereturns.ThesampleperiodisJanuary1980toSeptember2010.Source:ThomsonReutersDatastreamandFREDIIdatabankattheFederalReserveBankofSt.Louis.32 Table2:EstimationResultsfortheConditionalSkewnessDynamicswithGeneralizedStudent-tDistribution(1) NM0M1M2M3M4M5M6M7M8 1980-2009 0-0.1047*-0.1050*-0.1147*-0.0327*-0.0594*-0.1109*-0.1246*-0.0324*-0.0655*(0.0298)(0.0296)(0.0428)(0.0136)(0.0282)(0.0297)(0.0435)(0.0128)(0.0286)1+0.1070*0.1222*0.1080*0.1511*0.1144*0.1211*0.1208*0.1584*(0.0253)(0.0544)(0.0224)(0.0459)(0.0260)(0.0486)(0.0220)(0.0443)1�0.0981*0.0814*0.0840*0.0716*(0.0385)(0.0312)(0.0426)(0.0353)20.6593*0.6590*0.6694*0.6643*(0.0955)(0.0915)(0.0823)(0.0902) 0-1.4635*-1.4464*-1.4450*-1.4660*-1.4663*-1.4762*-1.0870*-0.7289*-0.4997*(0.1159)(0.1364)(0.1368)(0.1365)(0.1361)(0.1567)(0.1594)(0.1737)(0.1947) 1+-0.5805*-1.0077*-0.6848*-0.9800*(0.1898)(0.1191)(0.1283)(0.1092) 1�0.2368*0.1909*(0.0943)(0.0762) 20.5119*0.4309*(0.1097)(0.1158) LogL24639.424837.724846.024846.124852.424853.124849.124867.824861.624878.7LRStat396.60*16.60*0.20NA1.40NA37.40*NA34.20*BIC-3.2494-3.2732-3.2731-3.2720-3.2728-3.2717-3.2724-3.2725-3.2716-3.2715 1980-1989 0-0.0157-0.0129-0.0307-0.0022-0.0133-0.0110-0.0678-0.0018-0.0326(0.0546)(0.0517)(0.0732)(0.0168)(0.0407)(0.0512)(0.0731)(0.0165)(0.0493)1+0.08530.11200.0864*0.10560.07830.14460.0760*0.1282(0.0444)(0.0902)(0.0341)(0.0728)(0.0462)(0.0947)(0.0357)(0.0845)1�0.06700.07670.00730.0481(0.0706)(0.0469)(0.0646)(0.0598)20.6741*0.6702*0.6777*0.6617*(0.1566)(0.1557)(0.1681)(0.1757) 0-1.8280*-1.7839*-1.7848*-1.7891*-1.7887*-1.7515*-1.5049*-0.0407-0.6140(0.2053)(0.2007)(0.2006)(0.2012)(0.2011)(0.2128)(0.2418)(0.0219)(0.4645) 1+0.0591-0.6017*0.0510-0.5835*(0.0941)(0.1869)(0.0294)(0.1680) 1�0.21160.1201(0.1321)(0.1017) 20.9779*0.5330*(0.0121)(0.2576) LogL8190.18308.48310.18310.28312.48312.58310.38313.88314.28316.4LRStat236.60*3.400.20NA0.20NA7.00*NA4.40BIC-3.2268-3.2674-3.2650-3.2619-3.2628-3.2597-3.2620-3.2571-3.2573-3.2519 Thistablereportstheestimatedparametersof~and~asinequation(10),andwhereinnovationsfollowaskewedGSTprocessasinHansen(1994).Nrepresentsthebaselinenormallydistributederrorscase.ModelsM0toM8representequations(14)to(21),respectively.Tosavespace,wereportestimatedvaluesfor1and1+onthesamerow.Estimatedstandarderrorsarereportedinparentheses,belowtheestimatedparameters.\LogL"representsthecomputedloglikelihoodfunction.Wereportlikelihoodratioteststatistics,\LRStat,"withrespecttotheprecedingmodel.\NA"impliesthatthemodelsarenon-nested,andthustheLRteststatisticisnotcomputed.Allmodelsarepreferredtothebenchmarknormalmodel,basedonthelikelihoodratiotest.Thus,wedonotreporttheseteststatistics.*representsrejectionofthenullhypothesisinquestionatthe5%condencelevel.\BIC"representsBayesianinformationcriteriafortheestimatedmodels.33 Table3:EstimationResultsfortheConditionalSkewnessDynamicswithGeneralizedStudent-tDistribution(2) NM0M1M2M3M4M5M6M7M8 1990-1999 0-0.0643-0.0582-0.1636*-0.0090-0.0788-0.0527-0.1666*-0.0022-0.1037(0.0515)(0.0516)(0.0769)(0.0156)(0.0538)(0.0508)(0.0770)(0.0154)(0.0533)1+0.1503*0.3335*0.1488*0.2739*0.1572*0.3567*0.1614*0.3524*(0.0457)(0.1101)(0.0377)(0.0982)(0.0415)(0.1040)(0.0339)(0.0957)1�0.05960.09490.06860.0946(0.0683)(0.0509)(0.0696)(0.0528)20.7029*0.6455*0.6945*0.5963*(0.1141)(0.1341)(0.0990)(0.1210) 0-1.5922*-1.5691*-1.5289*-1.6366*-1.6053*-1.6505*-1.1899*-0.6799*-0.1532(0.2022)(0.2221)(0.2297)(0.2164)(0.2221)(0.2274)(0.2842)(0.1876)(0.2080) 1+-0.7164*-1.1082*-0.7516*-1.1332*(0.2075)(0.2077)(0.1571)(0.1588) 1�0.20260.1884(0.2201)(0.1723) 20.6061*0.6387*(0.0957)(0.0920) LogL8628.38687.38692.88694.68697.68698.88696.68702.78706.68714.3LRStat118.00*11.00*3.60NA2.40NA12.20*NA15.40*BIC-3.3989-3.4161-3.4151-3.4127-3.4139-3.4113-3.4135-3.4097-3.4113-3.4081 2000-2009 0-0.2392*-0.2404*-0.1073-0.1069-0.1131-0.2593*-0.0932-0.0897-0.1127(0.0530)(0.0530)(0.0835)(0.0972)(0.1023)(0.0530)(0.0805)(0.0877)(0.1090)1+0.0762-0.09450.0818-0.09770.0359-0.16220.0570-0.1522(0.0521)(0.0983)(0.0509)(0.1025)(0.0491)(0.0904)(0.0422)(0.0935)1�0.2391*0.2392*0.2398*0.2475*(0.0952)(0.0959)(0.0820)(0.1006)20.5504-0.03060.6375-0.1292(0.3974)(0.2909)(0.3412)(0.2886) 0-0.2152*-0.2614*-0.3293*-0.2537*-0.3307*0.3538*0.2914*-0.0378-0.4276(0.0546)(0.0560)(0.0741)(0.0625)(0.0713)(0.1140)(0.1363)(0.2236)(0.4257) 1+-2.0265*-1.9913*-1.9479*-1.7840*(0.2790)(0.2322)(0.2298)(0.2403) 1�0.7292*-4.0525(0.1026)(3.5040) 20.4158*0.3005(0.0875)(0.1588) LogL7866.47890.57891.77893.87892.17893.97910.27916.27916.57919.5LRStat48.20*2.404.20NA3.60NA12.00*NA6.00*BIC-3.1109-3.1143-3.1116-3.1094-3.1087-3.1063-3.1159-3.1121-3.1122-3.1072 Thistablereportstheestimationresultsforthe1990{99and2000{09subsamples.RefertothenotestoTable2formoreinformation.34 Table4:EvaluatingtheConditionalRelativeDownsideVariancewithGeneralizedStudent-tDistribution M0M1M2M3M4M5M6M7M8 1980-2009 0-3.016E-06-1.16E-05*-1.16E-05*-1.81E-05*-1.72E-05*-1.19E-05*-1.33E-05*-1.96E-05*-1.91E-05*(2.229E-06)(1.86E-06)(1.87E-06)(2.73E-06)(1.94E-06)(1.87E-06)(1.85E-06)(1.94E-06)(1.93E-06)12.4336*1.5751*1.5727*0.9441*1.02501.5274*1.4075*0.8826*0.9819(0.0839)(0.0338)(0.0345)(0.0251)(0.0277)(0.0338)(0.0304)(0.0244)(0.0497) R20.12260.26550.25680.19050.18580.25380.26310.17880.1877JT292.89*309.64*295.44*48.692*79.044*284.57*231.66*125.66*98.71* 1980-1989 0-1.37E-06-8.69E-07-8.90E-08-9.59E-06-6.36E-06-3.08E-07-2.74E-06-1.09E-05-7.90E-06(1.17E-05)(7.96E-06)(8.07E-06)(9.40E-06)(6.64E-06)(5.70E-06)(5.53E-06)(6.72E-06)(6.50E-06)12.0431*1.7049*1.7341*0.8763*1.00701.6928*1.4798*0.8189*0.9870*(0.2120)(0.0534)(0.0556)(0.0414)(0.0468)(0.1086)(0.0450)(0.0402)(0.0395) R20.08680.51060.49880.31450.32150.05430.52540.29800.3460JT24.22*174.24*174.30*9.974*0.938162.75*113.90*22.091*1.585 1990-1999 01.88E-06-1.33E-05*-1.32E-05*-1.34E-05*-1.32E-05*-1.35E-05*-1.38E-05*-1.47E-05*-1.47E-05*(4.33E-06)(3.20E-06)(3.21E-06)(3.05E-06)(2.17E-06)(2.26E-06)(2.27E-06)(2.12E-06)(2.14E-06)13.2425*1.12761.13821.02241.04221.08471.03660.9304*1.0414(0.3154)(0.1386)(0.1387)(0.1015)(0.1061)(0.1355)(0.1352)(0.0964)(0.1005) R20.04030.02560.02600.03880.03690.02480.02280.03570.0318JT50.740*18.136*17.94*19.27*37.18*36.18*37.25*48.48*47.01* 2000-2009 0-1.55E-06-4.05E-06-8.24E-06-7.45E-06-8.15E-06*-3.63E-07-9.52E-06*-2.20E-06-7.14E-06*(5.05E-06)(4.88E-06)(4.72E-06)(4.88E-06)(3.33E-06)(3.53E-06)(3.37E-06)(3.51E-06)(3.37E-06)11.2052*1.08080.95790.94890.96371.07910.8153*1.01920.9872(0.0501)(0.0426)(0.0366)(0.0394)(0.0367)(0.0428)(0.0326)(0.0410)(0.0362) R20.18800.20460.21510.18820.21650.20230.20060.19800.2113JT16.91*4.2964.3794.0126.693*3.42240.18*0.6114.613 ThistablereportsMincer-ZarnowitzregressionresultsfromRVdt+1(h)�RVut+1(h)=0+1(Vart[rt+1jrt+1mt]�Vart[rt+1jrt+1&#x-295;mt]),whereerrortermsintheparametricmodelfollowaskewedStudent-tdistribution.*indicatesrejectionofthenullhypothesisthati=i;i=0;1atthe5%condencelevelorbetter.JTrepresentsthevalueoftheteststatisticbuiltunderthenullhypothesisthat0and1arejointlyequalto0and1,respectively.*andydenotetherejectionofthisnullhypothesisatthe5and10%condencelevels,respectively.Thecriticalvaluesforthistestare5.991and4.605,respectively,basedon2df=2.35 Table5:EstimationResultsfortheConditionalSkewnessDynamicswithSkewedGED(1) NM0M1M2M3M4M5M6M7M8 1980-2009 0-0.1347*-0.1317*-0.1355*-0.0450*-0.0764*-0.1316*-0.1370*-0.0449*-0.0997*(0.0293)(0.0270)(0.0346)(0.0150)(0.0257)(0.0268)(0.0277)(0.0150)(0.0273)1+0.1218*0.1281*0.1234*0.1717*0.1221*0.1244*0.1236*0.1944*(0.0211)(0.0402)(0.0186)(0.0393)(0.0185)(0.0293)(0.0151)(0.0305)1�0.1188*0.0948*0.1195*0.0751*(0.0314)(0.0323)(0.0129)(0.0274)20.6347*0.6226*0.6355*0.5666*(0.0830)(0.0714)(0.0799)(0.0936) 00.3199*0.3215*0.3216*0.3174*0.3168*0.3221*0.4620*0.3184*0.3730*(0.0070)(0.0071)(0.0071)(0.0071)(0.0070)(0.0225)(0.0229)(0.0573)(0.0454) 1+0.0034-0.2414*-0.0024-0.2345*(0.0186)(0.0373)(0.0216)(0.0386) 1�0.1280*0.1268*(0.0177)(0.0216) 2-0.00530.2492*(0.1669)(0.1053) LogL24639.424824.024838.724838.724848.124848.924838.724859.024848.124869.7LRStat369.20*29.40*0.00NA1.60NA40.60*NA43.20*BIC-3.2494-3.2714-3.2722-3.2710-3.2722-3.2711-3.2710-3.2713-3.2698-3.2704 1980-1989 0-0.0767-0.0525-0.0193-0.0202-0.0030-0.0419-0.0958-0.0199-0.0761(0.0513)(0.0505)(0.0695)(0.0176)(0.0294)(0.0382)(0.0724)(0.0144)(0.0392)1+0.1111*0.07170.1146*0.0881*0.0856*0.1085*0.0962*0.1565*(0.0395)(0.0559)(0.0255)(0.0241)(0.0399)(0.0430)(0.0237)(0.0421)1�0.1424*0.1263*0.02790.0334(0.0518)(0.0279)(0.0434)(0.0506)20.5797*0.5741*0.5613*0.5535*(0.1352)(0.1435)(0.1610)(0.2190) 00.3210*0.2508*0.2507*0.2527*0.2510*0.2743*0.3873*0.13480.2560*(0.0113)(0.0090)(0.0089)(0.0090)(0.0090)(0.0377)(0.0499)(0.1313)(0.0941) 1+0.0648*-0.1384*0.0423-0.1223*(0.0290)(0.0662)(0.0259)(0.0645) 1�0.1563*0.1291*(0.0379)(0.0355) 20.50720.4137(0.4725)(0.2611) LogL8190.18290.88295.38295.68299.08299.28298.38303.68301.38305.8LRStat201.40*9.00*0.60NA0.40NA10.60*NA9.00*BIC-3.2268-3.2604-3.2591-3.2561-3.2575-3.2544-3.2572-3.2531-3.2522-3.2478 Thistablereportstheestimatedparametersof~and~asinequation(11),andwhereinnovationsfollowaskewedGED.Nrepresentsthebaselinenormallydistributederrorscase.ModelsM0toM8representequations(14)to(21),respectively.Tosavespace,wereportestimatedvaluesfor1and1+onthesamerow.Estimatedstandarderrorsarereportedinparentheses,belowtheestimatedparameters.\LogL"representsthecomputedloglikelihoodfunction.Wereportlikelihoodratioteststatistics,\LRStat,"withrespecttotheprecedingmodel.\NA"impliesthatthemodelsarenon-nested,andthustheLRteststatisticisnotcomputed.Allmodelsarepreferredtothebenchmarknormalmodel,basedonthelikelihoodratiotest.Thus,wedonotreporttheseteststatistics.*representsrejectionofthenullhypothesisinquestionatthe5%condencelevel.\BIC"representsBayesianinformationcriteriafortheestimatedmodels.36 Table6:EstimationResultsfortheConditionalSkewnessDynamicswithSkewedGED(2) NM0M1M2M3M4M5M6M7M8 1990-1999 0-0.0767-0.0845*-0.1917*-0.0121-0.0977*-0.0827-0.1934*-0.0123-0.1486*(0.0518)(0.0462)(0.0571)(0.0182)(0.0435)(0.0469)(0.0650)(0.0097)(0.0347)1+0.1631*0.3541*0.1614*0.3099*0.1632*0.3862*0.1607*0.4361*(0.0387)(0.0571)(0.0374)(0.0722)(0.0346)(0.0673)(0.0142)(0.0385)1�0.07570.1046*0.1095*0.1175*(0.0582)(0.0381)(0.0456)(0.0282)20.7051*0.5881*0.7141*0.4931*(0.1130)(0.1464)(0.0617)(0.0656) 00.3210*0.3154*0.3218*0.2896*0.2987*0.3054*0.4211*0.5355*0.2802*(0.0123)(0.0121)(0.0124)(0.0116)(0.0117)(0.0390)(0.0512)(0.0743)(0.0592) 1+-0.0412-0.2250*-0.0459*-0.2925*(0.0388)(0.0692)(0.0212)(0.0609) 1�0.07910.1180*(0.0624)(0.0428) 2-0.8665*0.5614*(0.0555)(0.1392) LogL8628.38684.18692.98695.48699.58701.18693.48699.58702.28710.0LRStat111.60*17.60*5.00NA3.20NA12.20*NA15.60*BIC-3.3989-3.4148-3.4151-3.4131-3.4147-3.4122-3.4123-3.4085-3.4096-3.4064 2000-2009 0-0.2635*-0.2622*-0.1409-0.1136-0.1457-0.2552*-0.1326*-0.0878-0.1361(0.0492)(0.0455)(0.0767)(0.0951)(0.0924)(0.0508)(0.0661)(0.0650)(0.1137)1+0.0798-0.06950.0857-0.07130.0536-0.1378*0.0779-0.1522*(0.0451)(0.0791)(0.0478)(0.0617)(0.0668)(0.0337)(0.0615)(0.0531)1�0.2265*0.2271*0.1903*0.2181*(0.0997)(0.0922)(0.0734)(0.0667)20.5586-0.02390.6498*-0.1111(0.3577)(0.2454)(0.2383)(0.1505) 00.4713*0.4646*0.4589*0.4655*0.4588*0.4678*0.6886*0.2559*0.6867*(0.0199)(0.0197)(0.0193)(0.0197)(0.0193)(0.0418)(0.0403)(0.1258)(0.0939) 1+-0.1262*-0.4590*-0.1286*-0.4716*(0.0515)(0.0624)(0.0474)(0.0640) 1�0.2288*0.2476*(0.0297)(0.0277) 20.44270.0302(0.2648)(0.1816) LogL7866.47893.47895.07896.97895.67896.97898.17919.57899.97919.6LRStat54.00*3.203.80NA2.60NA42.80*NA39.40*BIC-3.1109-3.1154-3.1130-3.1106-3.1101-3.1075-3.1111-3.1134-3.1056-3.1072 Thistablereportstheestimationresultsforthe1990{99and2000{09subsamples.RefertothenotestoTable5formoreinformation.37 Table7:EvaluatingtheConditionalRelativeDownsideVariancewithSkewedGED M0M1M2M3M4M5M6M7M8 1980-2009 0-2.99E-06-2.36E-06-2.44E-06-9.72E-06*-7.80E-06*-2.41E-064.17E-06*-9.80E-06*2.02E-06(3.25E-06)(2.79E-06)(2.80E-06)(2.80E-06)(2.03E-06)(1.97E-06)(2.09E-06)(1.98E-06)(2.26E-06)11.00330.97020.96140.7091*0.6974*0.96940.6680*0.7071*0.9604(0.0380)(0.0226)(0.0226)(0.0186)(0.0191)(0.0225)(0.0167)(0.0186)(0.0423) R20.10370.23520.23200.19480.18110.23590.21000.19420.1376JT0.8562.4683.698256.62*265.82*3.343399.22*273.80*0.886 1980-1989 0-1.65E-061.61E-05y1.63E-05y1.21E-066.81E-061.59E-05*3.54E-05*1.07E-063.91E-05*(1.23E-05)(8.85E-06)(8.90E-06)(9.65E-06)(6.99E-06)(6.24E-06)(6.75E-06)(6.84E-06)(1.17e-05)10.845251.2084*1.2036*0.7324*0.7736*1.2065*0.9007*0.99281.0314(0.1094)(0.0447)(0.0450)(0.0357)(0.0391)(0.0444)(0.0358)(0.0357)(0.0311) R20.05770.42850.42260.30130.28600.43060.39300.29960.2199JT2.02125.07*23.80*56.20*34.49*28.16*32.19*0.06612.18* 1990-1999 02.58E-06-7.73E-06*-7.66E-06*-8.75E-06*-7.50E-06*-7.80E-06*-4.78E-06-8.76E-06*-1.94E-06(4.61E-06)(3.58E-06)(3.58E-06)(3.31E-06)(3.43E-06)(5.05E-06)(4.13E-06)(3.31E-06)(4.59E-06)11.3910y0.7908y0.7889y0.7286*0.7109*0.7880*0.95370.7280*0.9849(0.2051)(0.1210)(0.1209)(0.097)(0.0970)(0.1210)(0.0950)(0.0970)(0.0690) R20.03530.03230.03270.04320.04130.03250.02590.04320.0244JT3.9467.636*7.622*14.87*13.74*7.820*1.57514.95*0.228 2000-2009 0-1.35E-06-1.17E-05*-1.19E-05*-1.56E-05*-1.58E-05*-1.18E-05*-6.39E-06-1.57E-05*-8.05E-06(5.24E-06)(4.89E-06)(4.90E-06)(4.85E-06)(4.92E-06)(4.89E-06)(5.13E-06)(4.85E-06)(5.34E-06)11.09320.7952*0.7859*0.6909*0.6314*0.7942*1.05120.91560.9531(0.0705)(0.0500)(0.0500)(0.0460)(0.0440)(0.0500)(0.0517)(0.0465)(0.0520) R20.16120.16540.16360.15490.14140.16530.15350.15480.1606JT1.81422.22*24.14*56.33*80.56*22.49*3.50513.89*3.085 ThistablereportsMincer-ZarnowitzregressionresultsfromRVdt+1(h)�RVut+1(h)=0+1(Vart[rt+1jrt+1mt]�Vart[rt+1jrt+1&#x-295;mt]),whereerrortermsintheparametricmodelfollowaskewedStudent-tdistribution.*indicatesrejectionofthenullhypothesisthati=i;i=0;1atthe5%condencelevelorbetter.JTrepresentsthevalueoftheteststatisticbuiltunderthenullhypothesisthat0and1arejointlyequalto0and1,respectively.*andydenotetherejectionofthisnullhypothesisatthe5and10%condencelevels,respectively.Thecriticalvaluesforthistestare5.991and4.605,respectively,basedon2df=2.38 Table8:EstimationResultsfortheConditionalSkewnessDynamicswithBinormalDistribu-tion NM0M1M2M3M4 1980-2009 0-0.2435*-0.1853*-0.1989*-0.0955*-0.1127*(0.0329)(0.0299)(0.0438)(0.0237)(0.0380)1+0.2055*0.2342*0.1819*0.2143*(0.0272)(0.0729)(0.0252)(0.0607)1�0.1948*0.1687*(0.0375)(0.0338)20.4187*0.4128*(0.0960)(0.0953) LogL24639.424667.024692.324692.424697.824698.1LRStat55.20*50.60*0.20NA0.60BIC-3.2494-3.2530-3.2552-3.2540-3.2547-3.2536 1980-1989 0-0.2022*-0.1182*-0.0595-0.0689*0.0071(0.0543)(0.0482)(0.0766)(0.0306)(0.0543)1+0.2709*0.15560.2291*0.0922(0.0436)(0.1211)(0.0400)(0.0910)1�0.3230*0.2883*(0.0758)(0.0633)20.3750*0.4270*(0.1203)(0.1180) LogL8190.18197.58213.78214.28216.78218.0LRStat14.80*32.40*1.00NA2.60BIC-3.2268-3.2296-3.2330-3.2301-3.2311-3.2285 1990-1999 0-0.2211*-0.1529*-0.3460*-0.0864-0.2925*(0.0579)(0.0524)(0.0799)(0.0446)(0.0880)1+0.2079*0.6721*0.1861*0.5903*(0.0440)(0.1424)(0.0432)(0.1516)1�0.06260.0683(0.0712)(0.0658)20.36430.1669(0.2108)(0.1573) LogL8628.38635.78644.78649.88646.08650.4LRStat14.80*18.00*10.20*NA8.80*BIC-3.3989-3.4018-3.4023-3.4012-3.3997-3.3983 2000-2009 0-0.3209*-0.3029*-0.1736-0.1140-0.1826(0.0618)(0.0593)(0.0947)(0.0855)(0.1288)1+0.0881-0.12790.0919-0.1307(0.0640)(0.1402)(0.0556)(0.1451)1�0.2270*0.2279*(0.1026)(0.1034)20.5806*-0.0325(0.2780)(0.3113) LogL7866.47879.67880.67882.17881.37882.1LRStat26.40*2.003.00NA1.60BIC-3.1109-3.1162-3.1135-3.1109-3.1106-3.1078 Thistablereportstheestimatedparametersof~,where~=�ln[p =2�1+1]andinnovationsfollowabinormaldistribution.Nrepresentsthebaselinenormallydistributederrorscase.ModelsM0toM4representequations(14)to(17),respectively.Tosavespace,wereportestimatedvaluesfor1and1+onthesamerow.Estimatedstandarderrorsarereportedinparentheses,belowtheestimatedparameters.\LogL"representsthecomputedloglikelihoodfunction.Wereportlikelihoodratioteststatistics,\LRStat,"withrespecttotheprecedingmodel.\NA"impliesthatthemodelsarenon-nested,andthustheLRteststatisticisnotcomputed.Allmodelsarepreferredtothebenchmarknormalmodel,basedonthelikelihoodratiotest.Thus,wedonotreporttheseteststatistics.*representsrejectionofthenullhypothesisinquestionatthe5%condencelevel.\BIC"representsBayesianinformationcriteriafortheestimatedmodels.39 Table9:EvaluatingtheConditionalRelativeDownsideVarianceforBinormalDistribution M0M1M2M3M4 1980-2009 0-2.30E-06-1.06E-05*-1.09E-05*-1.36E-05*-1.35E-05*(3.13E-06)(2.66E-06)(2.67E-06)(2.72E-06)(1.93E-06)11.5177*1.13171.11990.9262*0.9371(0.0502)(0.0849)(0.0752)(0.0229)(0.0435) R20.13200.25490.24740.21330.2096JT107.11*18.03*19.19*35.17*50.93* 1980-1989 05.70E-06-9.84E-06-1.05E-05-1.06E-05-1.30E-05(1.11E-05)(8.08E-06)(7.86E-06)(9.04E-06)(6.53E-06)12.0913*1.2928*1.2435*0.98600.7991*(0.1570)(0.0423)(0.0386)(0.0419)(0.0362) R20.15370.48840.51480.36200.3334JT48.58*49.38*41.57*1.48834.86* 1990-1999 09.24E-07-1.44E-05*-1.85E-05*-1.34E-05*-1.79E-05*(4.24E-06)(3.10E-06)(2.93E-06)(3.07E-06)(2.09E-06)12.2048*0.85650.5137*0.94371.1143(0.2150)(0.1029)(0.0806)(0.0975)(0.0856) R20.04010.02680.01580.03590.0182JT31.46*23.52*76.39*19.43*75.47* 2000-2009 0-1.26E-06-3.28E-06-5.06E-06-6.55E-06-4.97E-06(5.05E-06)(4.90E-06)(4.76E-06)4.88E-06(3.36E-06)11.1713*1.07190.99810.93951.0034(0.0485)(0.0424)(0.0374)(0.0385)(0.0374) R20.18880.20370.22170.19190.2231JT12.54*3.3271.1344.2702.192 ThistablereportsMincer-ZarnowitzregressionresultsfromRVdt+1(h)�RVut+1(h)=0+1(Vart[rt+1jrt+1mt]�Vart[rt+1jrt+1&#x-295;mt]),whereerrortermsintheparametricmodelarebinormallydistributed.*indicatesrejectionofthenullhypothesisthati=i;i=0;1atthe5%condencelevelorbetter.JTrepresentsthevalueoftheteststatisticbuiltunderthenullhypothesisthat0and1arejointlyequalto0and1,respectively.*andydenotetherejectionofthisnullhypothesisatthe5and10%condencelevels,respectively.Thecriticalvaluesforthistestare5.991and4.605,respectively,basedon2df=2.40