When Tails Are Heavy: The Benefits of Variance-Targeted, Non-Gaussian, Quasi-Maximum Likelihood Estimation of GARCH Models

Finance and Economics Discussion Series Federal Reserve Board, Washington, D.C. ISSN 1936-2854 (Print) ISSN 2767-3898 (Online) When Tails Are Heavy: The Benefits of Variance-Targeted, Non-Gaussian, Quasi-Maximum Likelihood Estimation of GARCH Models Todd Prono 2025-075 Please cite this paper as: Prono, Todd (2025). “When Tails Are Heavy: The Benefits of Variance-Targeted, Non- Gaussian, Quasi-Maximum Likelihood Estimation of GARCH Models,” Finance and EconomicsDiscussionSeries2025-075. Washington: BoardofGovernorsoftheFederalReserve System, https://doi.org/10.17016/FEDS.2025.075. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

When Tails Are Heavy: The Benefits of Variance-Targeted, Non-Gaussian, Quasi-Maximum Likelihood Estimation of GARCH Models 1 ToddProno2 ThisVersion: July2025 Abstract In heavy-tailed cases, variance targeting the Student’s-t estimator proposed in Bollerslev (1987) for the linear GARCH model is shown to be robust to density misspecification, just like the popular Quasi-Maximum Likelihood Estimator (QMLE). The resulting Variance-Targeted, Non-Gaussian, Quasi-MaximumLikelihoodEstimator(VTNGQMLE)isshowntopossessastablelimit,albeitonethat ishighlynon-Gaussian,withanill-definedvariance. Therateofconvergencetothisnon-standardlimit is slow relative pn and dependent upon unknown parameters. Fortunately, the sub-sample bootstrap isapplicable,givenacarefullyconstructednormalization. Surprisingly,bothMonteCarloexperiments and empirical applications reveal VTNGQMLE to sizably outperform QMLE and other performanceenhancing(relativetoQMLE)alternatives. Inanempiricalapplication,VTNGQMLEisappliedtoVIX (option-impliedvolatilityoftheS&P500Index).TheresultingGARCHvarianceestimatesarethenused to forecast option-implied volatility of volatility (VVIX), thus demonstrating a link between historical volatilityofVIXandrisk-neutralvolatility-of-volatility. Keywords: GARCH, VIX, VVIX, heavy tails, robust estimation, variance forecasting, volatility, volatility-of-volatility. JELcodes: C13,C22,C58. 1TheanalysisandconclusionspresentedhereinarethoseoftheauthoranddonotindicateconcurrencebyeithertheFederalReserveBoardor theFederalReserveSystem.Iowethankstoseminarparticipantsatthe9thInternationalWorkshoponFinancialMarketsandNonlinearDynamics forhelpfulcommentsanddiscussions.IadditionallyowethankstoDongHwanOhfor(many)detaileddiscussionsandreviews. 2FederalReserveBoard;(202)510-2398,todd.a.prono@frb.gov. 1

1 Introduction ThelinearGARCHmodelofBollerslev(1986)remainsaworkhorseforconditionalvolatilitymodellingin financialeconomics,itsapplicationsspanningportfolioformation,derivativepricing,andriskmanagement. Forasequence Y ,themostpopularversionofthismodelstates t f g Y = (cid:27) (cid:15) ; (cid:15) i:i:d:D(0; 1); (1) t t t t (cid:24) (cid:27)2 = !+(cid:11)Y2 +(cid:12)(cid:27)2 ; (2) t t 1 t 1 (cid:0) (cid:0) where D is an unknown distribution with probability density function g. The most common method for estimating(1)and(2)involveslikelihoodmethods, whichrequirespecificationofaproxydensityfunction f,where,itislikelythatf = g. ThispapertreatsDand,therefore,gaslatent. Inordertoincreaseefficiency 6 intheGARCHmodelparameterestimates,ex-anteattemptsaremadetobettermatchtheselectedf withthe heavy-tailedfeaturesofdatacommonlymodeled. Theintentoftheseattempts,however,isnottoidentifyg, and,therefore,achievetheCramer-Raolowerbound. Rather,theintentistoselectanf thatis"closer"tog thanaGaussiandensitybutthat(likeaGaussiandensity)alsomaintainsrobustnessinthemodelparameter estimates,inthe(likely)casewheref = g. Thedesiredresultisanon-GaussianGARCHestimatorthatis 6 robusttodensitymisspecificationandmoreefficientthantheGaussianalternative. By far, the most popular choice for f is the Gaussian density, in which case, the estimator for (1) and (2) is the quasi-maximum likelihood estimator (QMLE). Explaining this popularity is the robustness of QMLE to density misspecification. Early demonstrations of QMLE as a robust estimator include Lee and Hansen (1994) as well as Lumsdaine (1996), with more recent (and more general) demonstrations includingBerkes,Horváth,andKokoszka(2003),FrancqandZakoïan(2004),andStraumannandMikosch (2006). These more recent demonstrations also identify E (cid:15)4 < as (close to) necessary for QMLE to t 1 beasymptoticallynormal. (cid:0) (cid:1) It is well known that while robust, QMLE is not particularly efficient, especially in cases of a heavytailedD. EngleandGonzalez-Rivera(1991)show,forinstance,thatasemi-parametricestimatorfor(1)and (2) bests the efficiency of QMLE by up to 50%. The evidenced wide gap between QMLE and (infeasible) fullmaximumlikelihoodestimationhasencouragedaliteratureonGARCHestimatorsthataimstoimprove upon the efficiency of QMLE, while maintaining robustness. Examples of this literature include Francq et al. (2011a),Fanetal(2014),andPremingerandStorti(2017). Figures5and11showtailindexestimatesfor 2

(cid:15) T fromdailyS&P500(log)returnsandVIXlevels.3 Bytheendofthe,respective,datasamples,point f t gt=1 estimates no longer support E (cid:15)4 < for S&P 500 (log) returns, while there is no evidence supporting t b 1 E (cid:15)4 < for VIX levels. F(cid:0)rom(cid:1) Hall and Yao (2003), when E (cid:15)4 = , QMLE has a non-Gaussian t t 1 1 lim(cid:0)it w(cid:1)ith a reduced rate of convergence (relative to pn). In curr(cid:0)ent(cid:1)times, therefore, the efficiency gap for QMLE (when applied to S&P 500 returns and VIX levels, at least) appears even wider than what the literaturedocuments. In light of the empirical evidence in Figures 5 and 11, selecting f as the (standardized) Student’s-t densityofBollerslev(1987)seemslikeanintuitivelyappealingchoice. Inthecasewhereg is(very)heavy tailedbutf isGaussian,theparameter(cid:11)in(2)hasto,insomesense,workdoubly-hardcontrollingforthe heavy-tailedfeaturesof Y unconditionally. Thatis,whenf isGaussian,(cid:11)istheonlymodelparameter f t gt Z 2 capableofcapturingtheseheavy-tailedeffects,wherethoseeffectssourcetoeither"reactivity"in (cid:27)2 t t Z 2 tothepreviousperiod’sshockortostaticfeaturesof (cid:15) . If,instead,f isthe(standardized)S(cid:8)tude(cid:9)nt’sf t gt Z 2 t density, then the additional degree-of-freedom parameter can capture the static tail features of (cid:15) , f t gt Z 2 allowing(cid:11)tofocusonthedynamicfeaturesof (cid:27)2 . Thetroublewithselectingf asthe(standardized) t t Z 2 Student’s-t density, however, is that the result(cid:8)ing (cid:9)Non-Gaussian, Quasi-Maximum Likelihood Estimator (NGQMLE)isnotrobusttodensitymisspecification(see;e.g.,NeweyandSteigerwald,1997,andFanetal., 2014). Specifically,fromFanetal. (2014,Proposition1),biasinNGQMLEsourcestounder-identification ofthescaleof (cid:15) ,whenf = g.4 f t gt Z 6 2 ThispaperinvestigatesVariance-TargetedNGQMLE(VTNGQMLE)forthemodelof(1)and(2),where f isthe(standardized)Student’s-tdensityofBollerslev(1987).5 Wheng is(relatively)thintailedsuchthat E Y4 < , VTNGQMLE is shown to be biased, just like NGQMLE, whenever f = g. In heavy-tailed t 1 6 cas(cid:0)es w(cid:1)hen E Y4 = , however, the asymptotic limit of VTNGQMLE becomes dominated by propt 1 erties of the sa(cid:0)mple(cid:1)variance (the VT part). Explaining this dominance are different rates of convergence; specifically, the sample variance converges slower than does the likelihood function. As a result, effects from the likelihood function disappear as the sample gets large, rendering VTNGQMLE consistent, even 3Foraregularlyvaryingrandomvariable,thetailindex(cid:19)>0isamomentsupremum;meaning,if(cid:15) isregularlyvarying,then t E (cid:15) p < ifandonlyifp<(cid:19)(see;e.g.,Resnick,1987,foranintroductiontoregularvariation). j4 t I j nthe 1 modelof(1)and(2),scaleoftheinnovationsisgivenby!. Whenf isGaussian,!isidentifiedincaseswheref =g. 6 When f is non-Gaussian, identification of ! is no longer guaranteed in these same cases. Moreover, since (cid:11) can be shown to dependonscale,(potential)lackofidentificationof!alsoimpacts(cid:11). 5SeeEngleandMezrich(1996)fortheinitialproposalofvariance-targetedestimationandFrancqetal.(2011b)foraninvestigationintothetheoreticalpropertiesofVTQMLE. 3

whenf = g, solongas (cid:27)2 ismeanstationary.6,7 Consequently, inheavy-tailedcases, VTNGQMLE 6 t t Z 2 isrobusttodensitymissp(cid:8)ecifi(cid:9)cation,makingitamemberoftheclassofrobustestimatorslikeFrancqetal. (2011a),Fanetal. (2014),andPremingerandStorti(2017). VaynmanandBeare(2014)showthatwhenE Y4 = ,thelimitofVTQMLEisanalogouslydomit 1 nated by properties of the sample variance. This pa(cid:0)per(cid:1)(i) extends that result to a non-Gaussian likelihood, one that produces inconsistent GARCH parameter estimates in the absence of variance targeting, and (ii) considers additional (very) heavy-tailed cases that are empirically relevant. Specifically, the distributional limit of VTNGQMLE is determined in cases where E (cid:27)4 = and E (cid:15)4 = but the distribution of t t 1 1 (cid:27)2 and (cid:15)2 ,respectively,remaininthedoma(cid:0)ino(cid:1)fattractionofa(cid:0)no(cid:1)rmallaw. Additionally,cases t t Z t t Z 2 2 (cid:8)whe(cid:9)re E (cid:27)4 (cid:8)=(cid:9) and E (cid:15)4 = but the distribution of (cid:27)2 and (cid:15)2 , respectively, is in the t 1 t 1 t t Z t t Z 2 2 domaino(cid:0)fatt(cid:1)ractionofasta(cid:0)ble(cid:1)lawarealsoconsidered. Inthi(cid:8)shea(cid:9)viest-tail(cid:8)cas(cid:9)e,thedistributionallimitof VTNGQMLE is shown to jointly depend on extremes from both (cid:27)2 and (cid:15)2 . In the case where t t Z t t Z 2 2 E (cid:27)4 = but E (cid:15)4 < , in contrast, the distributional lim(cid:8)it s(cid:9)ingularly(cid:8)dep(cid:9)ends on extremes from t t 1 1 (cid:27)(cid:0)2 (cid:1) . Consistent(cid:0)wit(cid:1)h the logic stated above favoring NGQMLE over QMLE (bias issues aside), simt t Z 2 (cid:8)ulati(cid:9)on results, while confirming both VTQMLE and VTNGQMLE to be consistent in heavy-tailed cases and more efficient than QMLE, even when f = g, also (strongly) favor VTNGQMLE over VTQMLE, on 6 efficiencygrounds,inthesesamecases. The distorting properties of the sample variance on VTQMLE are considered a cost, since these propertiescanpreventVTQMLEfromachievingaGaussianlimit. Complicatedestimatorsaimedatdampening thetailsof Y are,thus,proposedsothatVTQMLEcanretainsuchalimit(see;e.g.,HillandRenault, f t gt Z 2 2012). Thispaper,incontrast,viewsthedistortingpropertiesofthesamplevarianceasabenefit,sincethose properties enable variance-targeted estimation, generally, and VTNGQMLE, specifically, to be robust to density misspecification. Counter-balancing the non-Gaussian limit of VTNGQMLE in heavy-tailed cases are(iii)beneficialeffectsfromtheStudent’s-tlikelihood(effectsthatareretainedinlarge,thoughstillfinite, samples, owingtoarelativelyslowrateofconvergence), and(iv)QMLEalsohavinganon-Gaussianlimit ofsimilar,qualitativeform,inthesesamecases(seeHallandYao,2003,Theorem2.1). Despite its non-Gaussian limit, Monte Carlo experiments reveal VTNGQMLE to perform surprisingly well against the competing robust estimators of both Fan et al. (2014) (hereafter FAN) and Preminger and Storti (2017) (hereafter LSE) . In fact, VTNGQMLE is shown to outperform both estimators in terms 6Inthiscase,"large"isrelative,inthesensethat,owingtoaslowerrateofconvergence,effectsfromthelikelihoodfunction willtendtoremain,eveninfinitesamplesthatarequite"large,"bystandardconvention. Thistendencyisshowntobeabenefit, notacost,however. 7 Y in(1)and(2)needstobecovariancestationary,meaning (cid:27)2 cannotfollowanIGARCH(1;1)process. f tgt 2 Z t t 2 Z (cid:8) (cid:9) 4

of root-mean-squared and mean-absolute-error in samples as large as 10;000 observations. Moreover, in empirical, out-of-sample forecasting exercises using S&P 500 (log) returns and VIX levels, VTNGQMLE isshowntooutperformbothQMLEandFAN.8 Empirical applications involve forecasting the volatility of S&P 500 (log) returns and VIX levels. The former represents a standard application in financial econometrics. The latter, however, is more nuanced and leverages a characteristic unique to the S&P 500 Index. That is, for the S&P 500 Index, the following threefeaturesaredirectlyobservable: (v)thereturn;(vi)option-impliedvolatilityofthereturn(VIX);(vii) option-impliedvolatilityofthevolatility(VVIX).Usingfeatures(vi)and(vii)robustestimationofthemodel in(1)and(2)onVIXdemonstratesthatthevarianceofVIX(viii)evidencesrichGARCHeffectsand(viv) theseeffectsareusefulatforecastingoption-impliedvolatilityofvolatility(VVIX),thusestablishingalink betweenhistoricalVIXvarianceandrisk-neutralvolatility-of-volatility. 2 Preliminaries Define (cid:22) as a measure on a locally compact, second countable Hausdorff space E, and let M (E) denote + d a collection of Radon measures on E. For R = R ; , consider the bounded set R 0 , [ (cid:0)1 1 n f g where bounded here means bounded away from zeron. Also, B oB R d 0 denotes a Borel (cid:27)-field 2 n f g definedonthisboundedset. Lastly,theunitsphereisdenotedbySd (cid:0) 1(cid:16)= x R (cid:17)d : x = 1 . 2 j j ForanR d-valuedrandomvectorX; n o Definition1 Xismultivariateregularlyvaryingwithtailindex(cid:20) 0; if asequence a 0 2 1 9 f n g ! andanonnull(cid:22) M + R d 0 suchthat (cid:16) (cid:17) 1 2 n f g (cid:16) (cid:17) nP a 1X v (cid:22)( ) as n ; (cid:0)n 2 (cid:1) (cid:0)! (cid:1) ! 1 (cid:0) (cid:1) where"v"denotes"vagueconvergence," (cid:22)(sB) = s (cid:0) (cid:20) 0(cid:22)(B); d s > 0andarelativelycompactB B R+ 0 . 8 2 n f g (cid:16) (cid:17) 8InthecaseofVIXlevels,bothQMLEandFANproduceimplausibleestimates,whiletheestimatesfromVTNGQMLEremain "in-line"witheconomicrationaleandempiricalobservation. 5

0 X = A Inaddition,(cid:14) denotestheDiracmeasureatX;meaning,forsomesetA,(cid:14) (A) = 2 : X X 8 9 1 X A < 2 = d C denotesagenericconstantthatcantake-ondifferentvaluesindifferentplaces. " "denotes(weak)con- (cid:0)! : ; vergenceindistribution. 3 The Model and Background Results Under consideration is the linear GARCH (1;1) model of Bollerslev (1986). Results presented herein can be extended to the general GARCH (p;q) case, where p; q 1 (see; e.g., Vaynman and Beare, 2014). (cid:21) Focusing on the special case of p = q = 1, besides being the most practically relevant, also facilitates the illustrationofkeyconceptsandideas,aswellastheverificationofimportantconditions. Forasequence Y ,anda(cid:27)-algebradefinedforthissequenceasdenotedby(cid:10) , f t gt Z t 2 Y = (cid:27) (cid:15) ; (cid:15) i:i:d:D(0; 1); (3) t t t t (cid:24) (cid:27)2 = ! +(cid:11) Y2 +(cid:12) (cid:27)2 ; (4) t 0 0 t 1 0 t 1 (cid:0) (cid:0) where D is an unknown probability distribution with associated density function g. ! denotes the true 0 value of !; ! any one of a set of possible values, and ! an estimate. Parallel definitions hold for all other parametervalues. b ASSUMPTION3.1. ! > 0; (cid:11) > 0; (cid:12) 0; (cid:11)+(cid:12) < 1: (cid:21) Under Assumption 3.1, the GARCH(1;1) model being considered nests the the ARCH(1) model as a specialcase. Given(3),(4),andAssumption3.1, ! E Y2 = E (cid:27)2 = 0 < ; (5) t t 1 (cid:11) (cid:12) 1 (cid:0) 0(cid:0) 0 (cid:0) (cid:1) (cid:0) (cid:1) where,fornotationalconvenience,E (cid:27)2 = E (cid:27)2 . Asaresult,(4)maybere-writtenas t (cid:0) (cid:1) (cid:0) (cid:1) (cid:27)2 = E (cid:27)2 (1 (cid:11) (cid:12) )+(cid:11) Y2 +(cid:12) (cid:27)2 : (6) t 0 0 0 t 1 0 t 1 (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:1) 6

LetX = (cid:27)2; Y2 0. Againowingto(3)-(4), t t t (cid:16) (cid:17) X = A X +B ; (7) t t t 1 t (cid:0) where (cid:12) (cid:11) ! 0 0 0 A = ; B = ; t t 0 (cid:12) (cid:15)2 (cid:11) (cid:15)2 1 0 ! (cid:15)2 1 0 t 0 t 0 t @ A @ A representsastochasticrecurrenceequation(SRE)(see; e.g., MikoschandSta˘rica˘, 2000, eq. 2.2). Assuch, Definition1isshowntoapply(see;e.g.,MikoschandSta˘rica˘,2000,andBasraketal.,2002). ASSUMPTION3.2. Thesequence (cid:15)2 isregularlyvarying,withtailindex(cid:19) . t t Z 0 2 (cid:8) (cid:9) Under Assumption 3.2, innovations to the GARCH(1;1) model are heavy-tailed, in the sense that the (unknown)distributionfortheseinnovationsbelongstotheFréchetclass,asopposedtothemorecommonly assumed Gumbel class.9 Regardless of whether GARCH(1;1) model innovations are heavy-tailed in a Fréchet-class sense, or (relatively) thin-tailed in a Gumbel-class sense, X will be regularly varying (see t Mikosch,1999,Corollary1.4.40). What follows in the remainder of this section is a summary of select (weakly) dependent and heavytailed limit theory results, upon which later sections are based. This summary draws heavily from Davis andMikosch(1998)andMikoschandSta˘rica˘ (2000),bothofwhich,inturn,relyonresultsfromDavisand Hsing (1995). The intent of this summary is to introduce certain key results; not provide a comprehensive review. Adetailedtreatmentoftheseresults,aswellasadditionalbackgroundinformation,canbefoundin theaforementionedworks. Lemma1 Given Assumptions 3.1 and 3.2, let X be the unique stationary solution for the SRE in (7). t f g Then (i) X is regularly varying with tail index (cid:20) 0 2 (1; (cid:19) 0 ), and (ii) P ( j X j > x) (cid:24) Cx (cid:0) (cid:20) 0 for some C (0; ). 2 1 FromDefinition1,let a satisfy n f g nP ( X > a ) 1; n (8) n j j ! ! 1 9SeeMcNeiletal. (2015,Chapter5)fordefinitionsoftheFrechétandGumbelclassesofdistributions,respectively. Asillustrations,theStudent’s-tdistributionwithafinitedegreeoffreedomisamemberoftheFrechétclass,whiletheNormaldistribution isamemberoftheGumbelclass. 7

Given(8),Lemma1(ii)impliesthat a (Cn)1=(cid:20) 0; (9) n (cid:24) andanalogously,Assumption3.2impliesthat b (Cn)1=(cid:19) 0: (10) n (cid:24) ASSUMPTION3.3. E (cid:27)4 = , but the distribution of (cid:27)2 remains in the domain of attraction of a 1 normallaw. Inthis(cid:0)cas(cid:1)e, H(a) = E (cid:27)4 I (cid:27)2 a ; a = inf a > 0 : nH(a) a2 ; (11) n (cid:2) (cid:20) (cid:20) (cid:0) (cid:0) (cid:1)(cid:1) (cid:8) (cid:9) whereH isslowlyvaryingat . 1 Compared to (9), Assumption 3.3 offers an alternative characterization of a , one that applies in the n borderline case where (cid:20) = 2. This condition heralds from Hall and Yao (2003), as do the following two 0 implications;specifically, a2P (cid:27)2 E (cid:27)2 > a (cid:0) 0; a (12) H(a) ! ! 1 (cid:0)(cid:12) (cid:0) (cid:1)(cid:12) (cid:1) (cid:12) (cid:12) (see;e.g.,Feller,1996,(8.5),p. 303),and aE (cid:27)2 E (cid:27)2 I (cid:27)2 E (cid:27)2 > a (cid:0) (cid:2) (cid:0) 0; a : (13) H(a) ! ! 1 (cid:0)(cid:12) (cid:0) (cid:1)(cid:12) (cid:0)(cid:12) (cid:0) (cid:1)(cid:12) (cid:1)(cid:1) (cid:12) (cid:12) (cid:12) (cid:12) Considerthefollowingsequenceofpointprocessesdefinedfromthenormalizedprocess(X ). t n N n = (cid:14) X =a ; n N; (14) t n 2 t=1 X where a isdefinedin(8)and(9). n f g Lemma2 GivenLemma1andthesequenceofpointprocessesin(14), d 1 1 N N = (cid:14) ; n (cid:0)! PiQij i=1j=1 XX 8

where(iii) 1(cid:14) isaPoissonprocesson 0; withabsolutelycontinuousintensitymeasure i=1 Pi 1 (cid:16) (cid:17) P v(dy) = (cid:13) 0 (cid:20) 0 y (cid:0) (cid:20) 0(cid:0) 1dy; (cid:20) 0 2 1; (cid:19) 0 , and (cid:13) 0 2 ( 0; 1 ], (iv) 1 (cid:14) Qij for i 2 N is an i.i.d. sequence of point processes on j=1 2 (cid:16) (cid:17) P R+ 0 takingvaluesintheset n f g 2 (cid:22) M + R+ 0 : (cid:22)( x : x > 1 ) = 0 and (cid:22)(S) > 0; 2 n f g f j j g n (cid:16) (cid:17)o and(v) 1(cid:14) Pi and 1 (cid:14) Qij fori 2 Naremutuallyindependent. i=1 j=1 P P Remark1 From Basrak et al. (2002, Remark 2.12.), the points P ; Q correspond with the rai ij dial and spherical parts, respectively, of the limiting points X =a (cid:16), where the s(cid:17)pherical part accounts for t n clusteringbehaviorinthelimitingpointprocess. Remark2 Considerthesequenceofpointprocesses n N2 = (cid:14) : n X2=a2 t n t=1 X GivenLemma2andthecontinuousmappingtheorem, N2 d N2 = 1 1 (cid:14) : n (cid:0)! P i 2Q2 ij i=1j=1 XX Inwords,Lemma2detailsaconvergenceresultfrompointprocesstheorythatcanbeusedtoestablish thedistibutionallimitofthevectorsequence n Sk = Xk; k = 1;2; (15) n t t=1 X inthecasewhere l E Xk = for l > 1: t 1 (cid:16) (cid:17) Toillustrate,considerthefunctionT " : M P R 2 + nf 0 g (cid:0)! R 2 suchthat (cid:16) (cid:17) a kSk = T Nk : (cid:0)n n " n (cid:16) (cid:17) 9

GivenLemma2andRemark2, T Nk d T Nk ; n : " n " (cid:0)! ! 1 (cid:16) (cid:17) (cid:16) (cid:17) Inaddition,givenDavisandMikosch(1998,Proposition3.3), T Nk d Sk; " 0; " (cid:0)! ! (cid:16) (cid:17) whereSk isavectorof((cid:20) =k)-stablerandomvariablesexpressedintermsoftheP ’sandQ ’sinLemma 0 i ij 2. Theendresult a kSk d Sk; (16) (cid:0)n n (cid:0)! is a limiting distribution for Sk with an ill-defined variance. The univariate analog to (16) was determined n by Davis and Hsing (1995, Theorem 3.1). Both (16) and its univariate analog factor prominently in the limitingresultsdevelopedinSection5. 4 Estimation Forthepurposeofestimating(3)and(4),assume(potentiallyincorrectly)thattheprobabilitydensityfunctionofD isgivenbyf,where ((cid:17)+1) x2 (cid:0) 2 f 1+ ; (cid:17) > 2; / (cid:17) 2 (cid:18) (cid:0) (cid:19) inwhichcase,thenon-Gaussianlikelihoodbaseduponthestandardizedt -distributionofBollerslev(1987) (cid:17) applies. Inthiscase,let (cid:18) = !; (cid:11); (cid:12); (cid:17) = !; #; (cid:17) = !; (cid:25) : (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) GivenAssumption3.1anditsimplicationin(5),considerthealternativeparametervector v = s2; (cid:25) (cid:16) (cid:17) 10

suchthat c(v)+(cid:11)Y2 +(cid:12)(cid:27)2 ((cid:29)) if 1 t n (cid:27)2 t ((cid:29)) = 8 s2 t (cid:0) 1 t (cid:0) 1 if t (cid:20) 1 (cid:20) 9 ; < (cid:20) = wherec(v) = s2(1 (cid:11) (cid:12)):. Then,for ; (cid:0) (cid:0) n 1 (cid:27)2 = Y2; n n t t=1 X theloglikelihoodfunctionunderconsiderationbisgivenby n logL (cid:27)2; (cid:25) = l (cid:27)2; (cid:25) n n t n (cid:16) (cid:17) X t=1 (cid:16) (cid:17) b b where l t (cid:27)2 n ; (cid:25) = logf (cid:17) (cid:15) t j (cid:10) t (cid:0) 1 ; (cid:16) (cid:17) (cid:0) (cid:1) asdefinedinBollerslev(1987,eq. 1),anbd(cid:25) isthesolutionto n b logL (cid:27)2; (cid:25) = argmaxlogL (cid:27)2; (cid:25) ; (17) n n n n n (cid:25) (cid:5) (cid:16) (cid:17) 2 (cid:16) (cid:17) b b b a Variance-Targeted, Non-Gaussian, Quasi-Maximum Likelihood Estimator (VTNGQMLE) for # . The 0 FOCfrom(17)is n @ @l ((cid:29) ) 0 = logL (v ) = t n : (18) @(cid:25) n n @(cid:25) t=1 X b Let b v = (cid:27)2; # ; (cid:17) ; 0 0 0 0 (cid:16) (cid:17) where(cid:27)2 = E (cid:27)2 ,and(cid:17) isinterpretedasa"pseudo"truthe.10 0 0 (cid:0) (cid:1) ASSUMPTION4.1. e Q(!;(cid:17)) = ln!+E lnf ! 0 (cid:15) t; (cid:17) (cid:0) ! h (cid:16) (cid:17)i hasauniquemaximumateither! and (cid:17) whenf = g or! and (cid:17) whenf = g. 0 0 0 0 6 Assumption 4.1 is a generic identification condition for the scale aned shape of f. It is the same as Newey and Steigerwald (1997, Assumption 2.4). When f = g, this assumption holds naturally. When f = g,Assumption4.1followsfromidentificationofthescaleandshapeparametersofg. 6 10Inthe(likely)casewheref =g,thereisno"true"(cid:17).Nevertheless,inthiscase,(cid:17)convergestosomething,andthatsomething 6 isdefinedasa"pseudo"truth. b 11

Takingafirst-orderTaylorExpansionof(18)around(cid:29) = (cid:29) produces 0 n @l ((cid:29) ) @2l ((cid:29) ) 0 = t 0 + t i ((cid:29) (cid:29) ) (19) @(cid:25) @(cid:25)@(cid:29) n (cid:0) 0 t=1(cid:26) 0 (cid:27) X n @l ((cid:29) ) @2l ((cid:29) ) b @2l ((cid:29) ) @2l ((cid:29) ) = t 0 + t i (cid:27)2 (cid:27)2 + t i # # + t i ((cid:17) (cid:17) ) @(cid:25) @(cid:25)@(cid:27)2 n (cid:0) 0 @(cid:25)@# n (cid:0) 0 @(cid:25)@(cid:17) n (cid:0) 0 0 X t=1(cid:26) (cid:0) (cid:1) (cid:16) (cid:17) (cid:27) b b b e where(cid:29) liesonthelinesegmentbetween(cid:29) and(cid:29) . Letting i n 0 1 n @2l ((cid:29) ) 1 n @ b2l ((cid:29) ) 1 n @2l ((cid:29) ) 1 n @l ((cid:29) ) J = t i ; K = t i ; M = t i ; Z = t 0 ; n n @(cid:25)@# n n @(cid:25)@(cid:27)2 n n @(cid:25)@(cid:17) n n @(cid:25) 0 X t=1 X t=1 X t=1 t=X1 (19)becomes 0 = nZ +J n # # +K n (cid:27)2 (cid:27)2 +M n((cid:17) (cid:17) ): (20) n n n 0 n n 0 n n 0 (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) (cid:0) (cid:1) ASSUMPTION4.2. b b b e @2l ((cid:29) ) @2l ((cid:29) ) @2l ((cid:29) ) J a:s: J E t 0 ; K a:s: K E t 0 ; M a:s: M E t 0 : n ! (cid:17) @(cid:25)@# n ! (cid:17) @(cid:25)@(cid:27)2 n ! (cid:17) @(cid:25)@(cid:17) 0 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) When(cid:17) = ,Assumption4.2followsfromVaynmanandBeare(2014,Lemma1). GivenAssumption 1 4.2,rearranging(20)andsubstitutingpopulationmomentsforsamplemomentsproduces na 1 # # = J 1 Kna 1 (cid:27)2 (cid:27)2 +Mna 1((cid:17) (cid:17) )+na 1Z : (21) (cid:0)n n 0 (cid:0) (cid:0)n n 0 (cid:0)n n 0 (cid:0)n n (cid:0) (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) (cid:8) (cid:0) (cid:1) (cid:9) b b b e 5 Asymptotics This section considers the large-sample implications of (21) in six cases ranging from (relatively) thintailedto(very)heavy-tailed. Topreviewtheresults,thelarge-samplepropertiesofVTNGQMLElookvery differentdependingonwhetherathin-tailedorthick-tailedcaseapplies. 5.1 Case1: (cid:20) > 2;(cid:17) = . 0 0 1 Inthiscase,thelikelihoodusedinestimationisGaussian,sothesecondtermontheright-hand-sideof(21) e dropsout. Inaddition,a 1 = n 1=2 sothat (cid:0)n (cid:0) pn (cid:27)2 (cid:27)2 d N 0;V ; (22) n (cid:0) 0 (cid:0)! (cid:27)2 (cid:0) (cid:1) (cid:0) (cid:1) b 12

byaCLTforweaklydependentdata,and d pnZ N (0;V ); (23) n Z (cid:0)! by,forinstance,HallandYao(2003,Theorem2.1(a)). Moreover, d pn # # N (0;V ); (24) n 0 # (cid:0) (cid:0)! (cid:16) (cid:17) b by Francq et al. (2011, Theorem 1.1); in which case, given (22) and (23), V is seen to depend upon both # V and V . Moreover, given Francq et al. (2011, Corollary 2), the VTQMLE cannot be asymptotically (cid:27)2 Z moreefficientthanQMLE. 5.1.1 Case2: (cid:20) > 2;(cid:17) 2; . 0 0 2 1 (cid:16) (cid:17) Inthiscase,allthreetermesontheright-hand-sideof(21)matter. a (cid:0)n 1 = n (cid:0) 1=2 continuestohold,asdoboth (22)and(23),exceptthatthelatternowfollowsfromFanetal. (2014,Theorem2). GivenAssumption4.1, wecanpositthat d pn((cid:17) (cid:17) ) N 0;V : (25) n (cid:0) 0 (cid:0)! (cid:17) (cid:16) (cid:17) In (25), Fan et al. (2014, Section 5.3) estabblishesethe rate of conveergence as pn and the limit as Gaussian, bothsolongas(cid:19) > 1. 0 Wecanfurtherpositthat :: d pn # +C # N 0;V ; (26) n n 0 # (cid:0) (cid:0)! (cid:16)(cid:16) (cid:17) (cid:17) (cid:16) (cid:17) b b p using(22)andresultsfromFanetal. (2014,Theorem2). In(26),however,andincontrastto(24),C n (cid:0)! C = 0,thusrendering# from(17)generallyinconsistent. Thepresenceofanon-asymptotically-vanishing 6 n b C follows from Fan et al. (2014, Proposition 1). Specifically, for our chosen f, VTNGQMLE fails to n b identifythescaleofthetruemodelinnovations,wheneverf = g(see,additionally,NeweyandSteigerwald, b 6 1997). Thisissueofunder-identificationimpacts# ,generally,becauseitimpacts(cid:11) ,specifically. Consen n quently, # from VTNGQMLE is inconsistent because the scale of the GARCH(1;1) model’s innovations n b b isnotidentified. b It is well known that NGQMLE is inconsistent whenever f = g. It turns out that in this case, VT- 6 NGQMLEinheritsthissameundesirableproperty. 13

5.1.2 Case3: (cid:20) = 2;(cid:17) 2; . 0 0 2 1 (cid:16) (cid:17) Inthiscase,applyAssumeption3.3. Inaddition,let n U = a 1 Y2 E Y2 (27) n (cid:0)n t (cid:0) t=1 X(cid:0) (cid:0) (cid:1)(cid:1) n = a 1 (cid:27)2+W E Y2 ; W = (cid:15)2 1 (cid:27)2 (cid:0)n t t t t t (cid:0) (cid:0) (cid:2) t=1 X(cid:0)(cid:0) (cid:1) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1) n n = a 1 W +a 1 (cid:27)2 E (cid:27)2 (cid:0)n t (cid:0)n t (cid:0) t=1 t=1 X X(cid:0) (cid:0) (cid:1)(cid:1) = I(a)+II(a) FollowingnotationfromMikoschandSta˘rica˘ (2000),forarandomvariableX,let n h 1 (cid:0) (cid:13) (h) = X X : n;X n t t+h t=1 X Given(6), n II(a) = a 1 (cid:11) Y2 E Y2 +(cid:12) (cid:27)2 E (cid:27)2 (28) (cid:0)n 0 t 1 0 t 1 (cid:0) (cid:0) (cid:0) (cid:0) t=1 X (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:0) (cid:1)(cid:1) = (cid:11) na 1 (cid:13) (0) E Y2 +(cid:12) na 1 (cid:13) (0) E (cid:27)2 0 (cid:0)n n;Y 0 (cid:0)n n;(cid:27) (cid:0) (cid:0) (cid:11) = 0 (cid:0) na 1 (cid:13) (cid:0)(0) (cid:1)(cid:1)E Y2 (cid:0) (cid:0) (cid:1)(cid:1) 1 (cid:12) (cid:0)n n;Y (cid:0) (cid:18) (cid:0) 0(cid:19) (cid:0) (cid:0) (cid:1)(cid:1) Plugging(28)backinto(27)produces n 1 (cid:12) U n = 1 (cid:11) (cid:0) 0 (cid:12) a (cid:0)n 1 W t (29) (cid:18) (cid:0) 0(cid:0) 0(cid:19) t=1 X Theorem3 GivenAssumptions3.1and3.3, 1 (cid:12) 2 na (cid:0)n 1 (cid:27)2 n (cid:0) (cid:27)2 0 (cid:0) d ! N 0; 1 (cid:11) (cid:0) 0 (cid:12) V W (30) (cid:18) (cid:0) 0(cid:0) 0(cid:19) ! (cid:0) (cid:1) b wherea isgivenby(11),andV isdefinedin(65)oftheAppendix. n W Proof. Unlessotherwisestated,allproofsappearinAppendixA. MotivatedbyresultsinHallandYao(2003),Theorem3establishes(cid:27)2 asasymptoticallynormalinthe n borderline case where (cid:20) = 2. In this case, (23) continues to hold as in Case 2, as does (25). Moreover, 0 b 14

givenAssumption3.3, n1=2a 1 = O(1); (cid:0)n inwhichcase,from(21), na 1 # # = J 1 Kna 1 (cid:27)2 (cid:27)2 +Mpn((cid:17) (cid:17) )+pnZ : (cid:0)n n 0 (cid:0) (cid:0)n n 0 n 0 n (cid:0) (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) (cid:8) (cid:0) (cid:1) (cid:9) b b b e Asaresult,VTNGQMLEremainsaninconsistentestimatorof# ,owingtothe(asymptotic)effectsofZ . n n b 5.2 Case4: (cid:20) 1; 2 ;(cid:19) > 2. 0 2 0 (cid:16) (cid:17) GivenFigure5,thiscasehasbeenempiricallyrelevantforSPX(log)returns,atleast,inthepast Theorem4 GivenLemma1,(9),Assumption4.2,and(21),ifE Y4 = ;(cid:27)2 isinthedomainofattract t 1 tionofa(cid:20) -stablelaw,andE (cid:15)4 < ,then (cid:0) (cid:1) 0 t 1 (cid:0) (cid:1) na 1 # # = J 1Kna 1 (cid:27)2 (cid:27)2 +o (1): (31) (cid:0)n n 0 (cid:0) (cid:0)n n 0 p (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) (cid:0) (cid:1) b b When E Y4 = because (cid:20) < 2, and E (cid:15)4 < (or, equivalently, (cid:19) > 2), the distributional t 1 0 t 1 0 limit of # be(cid:0)com(cid:1)es dominated by the limit of (cid:27)2 (cid:0)(an(cid:1)analogous result is reported in Vaynman and Beare, n n 2014,forVTQMLE).Thisdominancesourcestoaslowerrateofconvergencefor(cid:27)2 comparedtoeither(cid:17) b b n n or the score of the likelihood function. Since (cid:27)2 p (cid:27)2, an effect of this dominance is that VTNGQMLE n (cid:0)! 0 b b becomes a consistent estimator for # . As such, VTNGQMLE is a robust estimator like QMLE and the n b multi-stepestimatorsaimedatimprovingQMLE,likeFanetal. (2014)andPremingerandStorti(2017). b Theorem5 GivenLemma1and(9),ifE Y4 = ; (cid:27)2 isinthedomainofattractionofa(cid:20) -stablelaw, t 1 t 0 andE (cid:15)4 < ,then (cid:0) (cid:1) t 1 1 (cid:12) (cid:0) (cid:1) na (cid:0)n 1 (cid:27)2 n (cid:0) (cid:27)2 0 (cid:0) d ! 1 (cid:11) (cid:0) 0 (cid:12) U (cid:27)2 ; (32) (cid:18) (cid:0) 0(cid:0) 0(cid:19) (cid:0) (cid:1) whereU isthe(cid:20) -stablerandomvarbiablegivenin(72). (cid:27)2 0 Remark3 The method of proof behind Theorem 5 borrows from both Davis and Mikosch (1998) and Mikosch and Sta˘rica˘ (2000). Theorem 5 is also closely related to Vaynman and Beare (2014, Theorem 4). Remark4 Thelimitin(32)relatestotheP ’sandQ ’sinLemma2. i ij 15

EstablishedinTheorem5isastablelimitfor(cid:27)2 thatishighlynon-Gaussian. Convergencetothisnonn Gaussian limit is also slower than the usual pn rate and dependent upon the tails of (cid:27)2 . In this case, t b while both (cid:27)2 and (cid:15)2 are allowed to be heavy tailed, only the tail properties of the(cid:8)for(cid:9)mer impact the t t limitin(32)(cid:8).11 (cid:9) (cid:8) (cid:9) Given (31) and (32), establishing the limit of # requires a straight-forward application of Slutsky’s n Theoremtoproduce b 1 (cid:12) na (cid:0)n 1 # n (cid:0) # 0 (cid:0) d ! (cid:0) 1 (cid:11) (cid:0) 0 (cid:12) J (cid:0) 1KU (cid:27)2 : (33) (cid:16) (cid:17) (cid:18) (cid:0) 0(cid:0) 0(cid:19) b 5.3 Case5: (cid:20) 1; (cid:19) ;(cid:19) = 2. 0 2 0 0 (cid:16) (cid:17) This second borderline case was first introduced and studied in Hall and Yao (2003). Given Figure 6, this caseappearstobeempiricallyrelevantforSPX(log)returnsincontemporaneoustimes. Foranalyzingthis case,thefollowingConditionisimportant. ASSUMPTION5.1. E (cid:15)4 = ,butthedistributionof (cid:15)2remainsinthedomainofattractionofanormal 1 law. Inthiscase, (cid:0) (cid:1) H(b) = E (cid:15)4 I (cid:15)2 b ; b = inf b > 0 : nH(b) b2 ; n (cid:2) (cid:20) (cid:20) (cid:0) (cid:0) (cid:1)(cid:1) (cid:8) (cid:9) whereH isslowlyvaryingat . 1 Remark5 Assumption 5.1 parallels Assumption 3.3 but for (cid:15)2 and is identical to Hall and Yao (2003, eq. 2.8). Itcontrolstherateoftaildecayinthedistributionof(cid:15)2. In this borderline case, (31) continues to hold, in which case, the asymptotic properties of # remain n dominated by those of (cid:27)2. Establishing the stable limit of (cid:27)2, however, becomes more complicated comn n b pared to Theorem 5, since it is now the case that E (cid:15)4 = . Nevertheless, with the aid of Assumption b b1 5.1anditsassociatedimplications(see12and13,app(cid:0)ro(cid:1)priatelymodifiedfor(cid:15)2), (32)continuestohold,as establishedbythefollowingTheorem. Theorem6 GivenLemma1,(9),andAssumption5.1,ifE Y4 = ,and(cid:27)2isinthedomainofattraction t t 1 ofa(cid:20) -stablelaw,then(32)continuestohold. (cid:0) (cid:1) 0 11Justastheimpactofthelikelihoodfunctionvanishesin(31),theimpactofextremesin (cid:15)2 vanishindeterminingthelimit t of(cid:27)2.ThenotationforthelimitingvariableU emphasizesthissingularimpact. n (cid:27)2 (cid:8) (cid:9) b 16

Under Theorem 6, despite (cid:15)2 being a heavier-tailed process compared to Case 4, its tail properties continue to exercise no effect on the asymptotic limit of (cid:27)2. Moving from Case 1 to Case 2, the rate of n convergencechangesbutthedistributionallimitremains(generally)thesame. MovingfromCase4toCase b 5, in contrast, both the rate of convergence and the distributional limit remain unaltered. Explaining this difference between borderline cases are the dual results that (1) the rate of convergence implied by b is n f g fasterthantherateofconvergenceimpliedby a ,causinganyeffectsrelatedtotheformertovanish,and n f g n (2)extremesin(cid:27)2 continuetodominatetheasymptoticbehaviorof Y2: t t t=1 Animmediateconsequenceof(31)and(32)continuingtoholdiPsthat(33)alsocontinuestohold. 5.4 Case6: (cid:20) 1; (cid:19) ;(cid:19) < 2. 0 2 0 0 (cid:16) (cid:17) Thereisstrongempiricalevidencesupportingthisveryheavy-tailedcaseasbeingrelevantforVIX(seeFigures11and12). Inaddition,empiricalrelevanceofthiscaseevenforSPX(log)returns,incontemporaneous times,cannotbedismissed(seeFigures5and6). Theorem7 Given Lemma 1, (9), (10), (20), and Assumption 4.2, if E Y4 = ; (cid:27)2 is in the domain of t t 1 attractionofa(cid:20) -stablelaw;E (cid:15)4 = ,and(cid:15)2 isinthedomainofat(cid:0)trac(cid:1)tionofa(cid:19) -stablelaw,then 0 t 1 0 (cid:0) (cid:1) na 1b 1 # # = J 1Kna 1b 1 (cid:27)2 (cid:27)2 +o (1): (34) (cid:0)n (cid:0)n n 0 (cid:0) (cid:0)n (cid:0)n n 0 p (cid:0) (cid:0) (cid:0) (cid:16) (cid:17) (cid:0) (cid:1) b b Inthiscase,theanalogto(27)is n U = a 1b 1 Y2 E Y2 ; n (cid:0)n (cid:0)n t (cid:0) t=1 X(cid:0) (cid:0) (cid:1)(cid:1) inwhichcase, n 1 (cid:12) U n = 1 (cid:11) (cid:0) 0 (cid:12) a (cid:0)n 1b (cid:0)n 1 W t ; (35) (cid:18) (cid:0) 0(cid:0) 0(cid:19) t=1 X n followingthestepsoutlinedin(27)–(29). Given(34),analysisofa 1b 1 W thendeterminestheasymp- (cid:0)n (cid:0)n t t=1 toticlimitof# . P n b Theorem8 GivenLemma1,(9)and(10),ifE Y4 = ;(cid:27)2 isinthedomainofattractionofa(cid:20) -stable t 1 t 0 law;E (cid:15)4 = ,and(cid:15)2 isinthedomainofatt(cid:0)racti(cid:1)onofa(cid:19) -stablelaw,then t 1 0 (cid:0) (cid:1) 1 (cid:12) na (cid:0)n 1b (cid:0)n 1 (cid:27)2 n (cid:0) (cid:27)2 0 (cid:0) d ! 1 (cid:11) (cid:0) 0 (cid:12) U (cid:15)2;(cid:27)2 ; (36) (cid:18) (cid:0) 0(cid:0) 0(cid:19) (cid:0) (cid:1) b 17

whereU isthe(cid:20) -stablerandomvariablegivenin(86). (cid:15)2;(cid:27)2 0 UnderTheorem8,andforthefirsttime,tailpropertiesofboth (cid:27)2 and (cid:15)2 matterindeterminingthe t t asymptoticlimitin(36). (cid:8) (cid:9) (cid:8) (cid:9) Remark6 IntheproofofTheorem8,whenestablishingtheasymptoticvarianceofcertainsumsasnegligible,itappearsinsufficienttorelysolelyonthenormalizingconstants a ,sincedoingsoimpliesexplosive n f g (as opposed to dampened) behaviour in the affected sums, as n grows large. Joint reliance on the normalizing constants a and b , however, enables the variance of these affected sums to smoothly vanish. n n f g f g Moreover, theasymptoticlimitoftheremainingsumisseentodependonboth a and b , asopposed n n f g f g tojust a alone. n f g Remark7 Thelimitsin(32)and(36)arenotthesame,buttheyaresimilarinaqualitativesense(see;e.g., Davis and Mikosch, 1998, Remark 3.2). That is, U can be expressed in terms of quantities that are (cid:15)2;(cid:27)2 qualitativelysimilartotheP ’sandQ ’sinLemma2. i ij The limit of VTNGQMLE in (36) appears (qualitatively) similar to the limit of QMLE, as determined by Hall and Yao (2003, Theorem 2.1(c)). Under Case 6, consequently, it is unclear which estimator (VT- NGQLME or QMLE) dominates the other, on efficiency grounds. A similar statement appears to hold true when comparing VTNGQMLE to the multi-step estimator of Preminger and Storti (2017) that assumes E Y2 < (hereafter LSE), since pn asymptotic normality of this estimator also depends on t 1 E (cid:15)4 <(cid:0) ,(cid:1)justasintheQMLEcase.12 Themulti-stepestimatorofFanetal. (2014),ontheotherhand, t 1 (he(cid:0)rea(cid:1)fter FAN) should be more efficient (asymptotically) than VTNGQMLE, since the former should be pnasymptoticallynormal,solongas(cid:19) > 1. 0 Lastly,given(34)and(36),Slutsky’sTheoremestablishes 1 (cid:12) na (cid:0)n 1b (cid:0)n 1 # n (cid:0) # 0 (cid:0) d ! (cid:0) 1 (cid:11) (cid:0) 0 (cid:12) J (cid:0) 1KU (cid:15)2;(cid:27)2 : (37) (cid:16) (cid:17) (cid:18) (cid:0) 0(cid:0) 0(cid:19) b 6 Bootstrap Inference Troubleswiththeresultsin(33)and(37)aretwofold: 12InthecaseofQMLE,E (cid:15)4 < isnecessaryforpnasymptoticnormality.StillinthecaseofQMLE,HallandYao(2003) showthatwhenE (cid:15)4 = ,th t easym 1 poticlimitis(cid:11)-stablewithaslowerrateofconvergencethatdependsuponthetailproperties of (cid:15)2 .Owingtoth t isres 1 ult, (cid:0) its (cid:1) eemsreasonabletoconcludethatwhenE (cid:15)4 = ,LSE,too,wouldhavean(cid:11)-stablelimitand t (cid:0) (cid:1) t 1 arateofconvergenceslowerthanpn.Thisconclusion,however,hasnotbeenformallyestablished. (cid:8) (cid:9) (cid:0) (cid:1) 18

1. thepreciseformofthedistributionallimitsisawkward,renderinghowtoconstructasymptoticconfidencebandsunclear; 2. therateofconvergencedependsupondistributionalcharacteristicsthatareunknown. Owingtothesetwintroubles,itisequallyunclearthepracticalrelevanceof(33)and(37). Tohelpdispel thesetroubles,consider n n 2 (cid:28)2 = n 1 Y4 n 1 Y2 ; (38) n (cid:0) t (cid:0) t (cid:0) ! t=1 t=1 X X whichisanalogoustoafinite-samp b levariancefor(cid:27)2,ifthetruevariancewerewelldefined. n Theorem9 UnderCase6andtheassumptionsofbTheorem8, na 2b 2(cid:28)2 d U ; (39) (cid:0)n (cid:0)n n (cid:15)4;(cid:27)4 (cid:0)! b where U is a ((cid:20) =2)-stable random variable determined by the extremes of both (cid:15)2 and (cid:27)2 (see (cid:15)4;(cid:27)4 0 t t theproofofTheorem9foradditionaldetails). (cid:8) (cid:9) (cid:8) (cid:9) Remark8 HallandYao(2003)considerastatisticanalogousto(38)thatisbasedon(cid:15) ,asopposedtoY . t t DenotetheHallandYao(2003)statistic(cid:28)2 ((cid:15) ),sothatthestatisticin(38)canbedenoted(cid:28)2 (Y ). Because n t n t (cid:15) isi.i.d.,anappropriatelyscaledversionof(cid:28)2 ((cid:15) )canbeshowntohaveastablelimitusingresultsfrom t n t f g b b Feller(1971)andLepageetal. (1981),evenwhenE (cid:15)4 = . Complicatingananalgousdemonstration t b 1 for (cid:28)2 (Y ), in the case where E Y4 = , is depe(cid:0)nd(cid:1)ence in Y that sources to (cid:27)2 . Theorem (9) n t t t t 1 f g establishes a stable distributional(cid:0)limi(cid:1)t for (cid:28)2 (Y ) by relying upon the convergence re(cid:8)sults(cid:9)summarized in n t b Section3. b Corollary10 UnderCase4andtheassumptionsofTheorem5, na 2(cid:28)2 d U ; (40) (cid:0)n n (cid:27)4 (cid:0)! b whereU isa((cid:20) =2)-stablerandomvariabledeterminedbytheextremesof (cid:27)2 . (cid:27)4 0 t (cid:8) (cid:9) Proof. The general method of proof follows the same arguments in the proof of Theorem 5 immediately below(71)andthroughtotheend. Establishing T N2 d U ; " 0 " (cid:27)4 (cid:0)! ! (cid:0) (cid:1) 19

foranappropriatelydefinedT ( )andtheN2 inRemark2followsfromDavisandHsing(1995, Theorem " (cid:1) 3.1(i)). Theorems (8) and (9) demonstrate that individually na 1b 1 (cid:27)2 (cid:27)2 and na 2b 2(cid:28)2 have proper (cid:0)n (cid:0)n n (cid:0) 0 (cid:0)n (cid:0)n n limitingdistributions. Thefollowingtheoremandcorollaryestablish(cid:0)thatthes(cid:1)e(weak)marginalconvergence b b resultsarealsojoint. Theorem11 UnderCase6andtheassumptionsofTheorem8, na 1b 1 (cid:27)2 (cid:27)2 ; na 2b 2(cid:28)2 d U ; U ; (41) (cid:0)n (cid:0)n n (cid:0) 0 (cid:0)n (cid:0)n n (cid:0)! (cid:15)2;(cid:27)2 (cid:15)4;(cid:27)4 (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) b b wherethe,respective,marginallimitsarethosefrom(36)and(39). Corollary12 UnderCase4andtheassumptionsofTheorem5, na 1 (cid:27)2 (cid:27)2 ; na 2(cid:28)2 d U ; U ; (42) (cid:0)n n (cid:0) 0 (cid:0)n n (cid:0)! (cid:27)2 (cid:27)4 (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) b b wherethe,respective,marginallimitsarethosefrom(32)and(40). Proof. ThemethodofprooffollowsthatofTheorem(11)(seeAppendixA).Alternatively,let na 1 (cid:27)2 (cid:27)2 ; na 2(cid:28)2 = U(cid:1)(cid:1) ; V ; (cid:0)n n (cid:0) 0 (cid:0)n n n;" n;" (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) b b and note that U(cid:1)(cid:1) is a special case of U in Vaynman and Beare (2014, eq. 35). Then (42) can be n;" n;" establishedbyfollowingthestepsoutlinedinVaynmanandBeare(2014,proofofTheorem4). WiththeaidofthecontinuousmappingtheoremandSlutsky’sTheorems,from(41)followsthat # # 1 (cid:12) U pn n (cid:0) 0 d (cid:0) 0 J (cid:0) 1K (cid:15)2;(cid:27)2 : (43) b (cid:28) n ! (cid:0)! (cid:0) (cid:18) 1 (cid:0) (cid:11) 0(cid:0) (cid:12) 0(cid:19) 0U (cid:15) 1 4 = ; 2 (cid:27)4 1 @ A b Thepowerof(43)isthattheleft-hand-sidehasaproperlimitingdistribution,andtherateofconvergenceis known. Moreover, given (42), it is evident that the left-hand-side of (43) has a proper limiting distribution under Cases 3–6. In fact, the left-hand-side of (43) has a proper limiting distribution under all of the cases consideredinSection5. InCases1and2,theresultistrivial,since(cid:28) hasadegeneratelimit,withthestann dardpnrateofconvergence. InCase3,(cid:28) maintainsadegeneratelimit;however,therateofconvergence n b tothatdegeneratelimitisnowunknown. Fortunately,therateofconvergencetoanon-degeneratelimitfor b 20

# # is also unknown and happens to depend upon the same latent factor in such a precise way that n 0 (cid:0) (cid:16)theeffecto(cid:17)fthis(common)latentfactorcancelsout. InCases4–6,thelimitof(cid:28) becomesnon-degenerate, b n but the rate of convergence to that non-degenerate limit remains precisely aligned with the rate of converb # # gencein # n (cid:0) # 0 insuchawaythatthenormalizedstatistic n (cid:28) (cid:0) n 0 convergesatthestandardpnrate. (cid:16) (cid:17) # # (cid:16)b (cid:17) Owingtot b hisresult, n (cid:28) (cid:0) 0 canbebootstrappedusingthere-sam b plingschemedescribedinHallandYao n (2003, Section 3.2), w(cid:16)h b ich th(cid:17)en approximates the limiting result in (43), as demonstrated by Hall and Yao b (2003,Theorem3.2). 7 Monte Carlo Experiments The sequence (cid:15) is drawn from the skewed student’s-t density of Hansen (1994). This density has two t f g parameters, (cid:21) and (cid:17), with the former governing skewness, the latter governing the tails, and up to the (cid:17)th momentofthedistributionbeingwelldefined. Valuesfortheseparametersare (cid:21) = 0:00; 0:40; 0:80; 0:99 ; (cid:17) = 8:5; 4:5; 4:0; 3:5 : 0 0 (cid:16) (cid:17) (cid:16) (cid:17) As (cid:21) increases, so, too, does skewness, while as (cid:17) decreases, tail thickness increases. (cid:17) = 8:5 is a (rel- 0 atively) thin-tailed case, while the remaining values for (cid:17) correspond with heavy-tailed cases. When 0 (cid:17) = 4:5, QMLE is asymptotically normal (AN). When (cid:17) = 4:0, AN of QMLE is preserved, but with 0 0 aconvergencerateslowerthanpn,whilewhen(cid:17) = 3:5,QMLEisnolongerAN,insteadconvergingtoa 0 limitthatappearsqualitativelysimilartotheonediscoveredforVTNGQMLE. Non-zero skewness levels are considered for two reasons. First, (cid:21) = 0 introduces a density misspeci- 6 fication, since f is symmetric. Second, non-zero skewness is an empirical feature of both SPX log returns and VIX levels, especially, in recent times (see Figure 16), where the former tends to be negative and the latterstronglypositive.13 Unreportedresultsindicatenomaterialdifferencesbetweenpositiveandnegative valuesfor(cid:21) ;consequently,onlyresultsforpositivevaluesarereported.14 0 Across the different parameterizations of the innovation density, the different GARCH(1;1) model parametersaregiveninTable1. TheestimatorsunderstudyareQMLE,NGQMLE,VTNGQMLE,FAN,and LSE.NGQLMEistheStudent’s-testimatorofBollerslev(1987),whileFANandLSEaretheestimatorsof Fan et al. (2014) and Preminger and Storti (2017), respectively. Away from the case (cid:21) = 0, NGQMLE is not robust, while FAN, LSE, and QMLE are all robust estimators. Samples sizes for the simulations range 13ThepositiveskewnessinVIXlevelsis"natural,"inthesensethatVIX>0becauseitisavolatility. 14Simulationresultsusingnegativevaluesof(cid:21) areavailableuponrequest. 0 21

from 500 100;000, with all simulations conducted over 10;000 trials.15 Summary statistics for the sim- (cid:0) ulations include mean bias and inter-decile range. Also reported are ratios of the root-mean-squared- and mean-absolute-error(eachmeasuredwithrespecttothetrueparametervalue)dividedbythecorresponding measure for QMLE. Termed "efficiency ratios," values less than one indicate improved efficiency of the givenestimatoroverQMLE.Figures17–24depictresultsforSpecificationIII. 7.1 Bias For (cid:11) , as (cid:21) increases, NGQMLE displays a growing bias that can be quite severe, particularly in large n 0 samples (see Figure 17). Bias in VTNGQMLE (relative to NGQMLE), in contrast, behaves quite differb ently, tending to decrease (rather sharply) as the sample size increases. At very large samples (> 10;000), VTNGQMLE appears to retain a small amount of bias. Large sample results in Section 5 are based on a first-order approximation to the score of the likelihood function. This residual bias, then, is consistent withhigher-ordereffects. Sincethisresidualbiasis(1)orders-of-magnitudesmallerthanthebiasaffecting NGQMLE and (2) additionally materially smaller than the bias displayed by VTQMLE, the latter being a consistent estimator (see Francq et al., 2011, Theorem 1.1), any retained bias in VTNGQMLE, and the higher-ordertermscausingit,appearstobeonlyofsecondaryimportance. Consequently,simulationresults confirmVTNGQMLEtobeaconsistentestimator,comparabletoFAN,LSE,andQMLE. For(cid:12) ,biasinNGQMLEdecreasessharplywiththesamplesize,indicatingNGQMLEtobeaconsistent n estimatorfor(cid:12) (seeFigure18). Infact,for(cid:12) ,NGQMLEtendstodisplaytheleastbiasofalltheestimators b 0 n beingstudiedandunderallthesimulationdesignsconsidered. Thisresultconfirmsthetheoreticalprediction b inFanetal. (2014)thatbiasinNGQMLEsourcestounder-identificationofscale. (cid:12) isunaffectedbyscale, n in which case, NGQMLE is a robust estimator for (cid:12) . VTNGQMLE of (cid:12) tends to be close to NGQMLE 0 n b in terms of bias and, consequently, tends to display among the least bias of the estimators being studied, b exceptunderverylargesamplesizes. Parallelto(cid:11) ,anyretainedbiasin(cid:12) fromVTNGQMLEsourcesto n n higher-ordereffects,which,owingtoresultsthatfollow,areof(decidedly)second-orderimportance. b b 7.2 Dispersion For (cid:11) , except in the largest samples, NGQMLE and VTNGQMLE tend to be noticeably less disperse n than the other estimators. Under all simulation designs considered, the rate of convergence for FAN and b 15Thefirst200observationswithineachtrialaredroppedinordertoavoidinitializationeffects.Verylargesamplesareconsidered because of the slow convergence rates identified under Cases 4 and 6 (see, also, Hall and Yao, 2003, Theorem 2.1(b)–(c), for QMLE). 22

NGQMLE should be pn. The rate of convergence for VTNGQMLE, however, should always be less than pn and should be the slowest in the case where (cid:17) = 3:5. Consistent with these predictions, the rate of 0 reductionindispersionappearsmutedforVTNGQMLEcomparedtobothFANandNGQMLE(seeFigure 19). Moreover, the difference between rates of reduction in dispersion appears most apparent in the case where(cid:17) = 3:5. 0 For (cid:12) , NGQMLE and VTNGQMLE are consistently the least disperse estimators, followed by FAN n and LSE (see Figure 20). QMLE and VTQMLE are the bottom-two, in terms of dispersion, and appear b (effectively)indistinguishable. 7.3 Efficiency For(cid:11) ,as(cid:21) increases,thebiasinNGQMLEgrowsinimportanceandeventuallydominatesbothefficiency n 0 ratios,causingNGQMLEtobecometheleastefficientestimator(seeFigures21and23). Thisdominance, b however, takes a surprisingly long while to set in, only severely and adversely impacting the very largest samplesizes. Forempirically-relevantsamplesizes(i.e.,T 500; 2;500 ),NGQMLEbeatsallother 2 estimators except VTNGQMLE in terms of RMSE (see Figurhe 21), with a simiilar result holding for MAE (see Figure 23). The source of this outperformance appears to be (despite the materially higher bias) the materialreductionindispersionthattheStudent’s-tlikelihoodaffords(specifically,estimationofadegreesof-freedomparameter)relativetothecompetingestimators. Moreover,exceptinthelargestsamplesconsidered( 50;000),NGQMLEsizablyoutperformsQMLE,intermsofbothRMSEandMAE.Consequently, (cid:21) as a practical matter, in heavy-tailed cases, and despite the presence of material bias, NGQMLE appears preferabletoQMLE. For(cid:11) and(cid:21) > 0,VTNGQMLEconsistentlybeatsNGQMLE(seeFigures21and23). Moreover,and n 0 surprisingly, in these same cases, VTNGQMLE consistently beats both FAN and LSE in samples as large b asT = 2;500,generally. Intheheaviest-tailedcaseof(cid:17) = 3:5,specifically,VTNGQMLEbeatsFANand 0 LSEinsamplesaslargeasT = 10;000. Consequently,gainsinVTNGQMLEoverFANandLSEappearto befinite-samplephenomena;however,inheavy-tailedcases,thesegains(1)arerathersizableand(2)extend intofinitesamplesthatare(very)commontoempiricalapplications. Stillfor(cid:11) ,inthe(relatively)thin-tailedcaseof(cid:17) = 8:5,VTNGQMLEisnevermoreefficient(under n 0 either efficiency ratio) than QMLE in the largest sample size, making gains in VTNGQMLE over QMLE b also a finite-sample phenomena; albeit, one that similarly persists into surprisingly large samples. In cases where (cid:17) 4:5, however, material efficiency gains in VTNGQMLE over QMLE begin appearing even in 0 (cid:20) 23

the largest sample size, when (cid:21) > 0. When (cid:17) = 4:0, VTNGQMLE is more efficient than QMLE across 0 0 allsamplesizesconsidered,withthistendencypreservedintheheaviest-tailedcaseof(cid:17) = 3:5. FromHall 0 andYao(2003,Theorem2.1(b)–(c)),thelargesamplepropertiesofQMLEchangewhen(cid:17) 4:0. Inthese 0 (cid:20) samecases,therelativelargesamplepropertiesbetweenVTNGQMLEandQMLEappeartochangeaswell. For (cid:11) using RMSE, Francq et al. (2011, p. 630) reports that when the true ARCH (1) innovations n are heavy-tailed, VTQMLE "performs remarkably well and even outperforms QMLE." The sample size b upon which this result is based is T = 500. Results in Figures 21 and 23 show that this outperformance of VTQMLE over QMLE (1) extends to the GARCH (1;1) case and (2) covers sample sizes much larger than T = 500. For instance, when (cid:17) 4:0, VTQMLE bests QMLE (in terms of either RMSE or MAE) 0 (cid:20) in samples as large as T = 10;000. However, across all samples considered, VTNGQMLE always bests VTQMLE (and by large amounts), and in the heaviest-tailed cases of (cid:17) 4:0, VTQMLE tends not to 0 (cid:20) outperformQMLEinthelargestsamplesize,whileVTNGQMLEdoes. For (cid:12) overall, VTNGQMLE and NGQMLE tend to be the most efficient estimators, with NGQMLE n performing the best across all specifications considered (see Figures 22 and 24). Only at samples larger b thanT = 10;000doesthereappearanyappreciabledifferencebetweenNGQMLEandVTNGQMLE,with thatdifferencefavoringNGQMLE.Inthesesame(very)largesamplecases( 50;000),bothFANandLSE (cid:21) outperformVTNGQMLEbutneitheroutperformsNGQMLE.Consequently,notonlyis(cid:12) fromNGQMLE n robust to density misspecification (as further explored in the next section), NGQMLE is the best estimator b for(cid:12) outofalltheestimatorsconsidered. 0 Consequently, in the family of robust GARCH estimators, VTNGQMLE appears tough to beat. Compared to both FAN and LSE, VTNGQMLE is also the simplest to implement, requiring the fewest computationalsteps. 8 Explaining the Results Let (cid:27)2 (cid:27)2 = t : (44) t ! 0 Conditionalon(44),themodelof(3)and(4)canbere-castas Y = p! (cid:27) (cid:15) ; (45) t 0 t t 24

(cid:11) (cid:27)2 = 1+ 0 Y2 +(cid:12) (cid:27)2 : (46) t ! t 1 0 t 1 (cid:18) 0(cid:19) (cid:0) (cid:0) From (45), the constant parameter ! can be seen as the scale of the model’s innovations. From (46), 0 "reactivity"oftheconditionalvariancetothepreviousperiod’sinnovationisseentodependonscale.16 Also from(46),theportionofthepreviousperiod’sconditionalvarianceaffectingthecurrentperiod’sconditional variance is seen to be invariant to scale. Consequently, difficulties in estimating ! have the potential 0 to adversely impact the estimation of (cid:11) , while such difficulties should not impact the estimation of (cid:12) . 0 0 Conversely, improvements in the estimation of ! have the potential to benefit the estimation of (cid:11) , as 0 0 shown;e.g.,byFanetal. (2014). Consider,next,thefollowinggeneralizationtothemodelof(3), Y = (cid:17) (cid:27) (cid:15) ; (47) t f;0 t t = (cid:27)(cid:1)(cid:1) (cid:15) t t where (cid:15) (cid:17) = argmaxE log(cid:17) +logf ; (48) f;0 (cid:0) f (cid:17) (cid:17) f >0 " f!# withtheexpectationistakenunderg,and 2 2 (cid:27)(cid:1)(cid:1) = (cid:17)2 ! + (cid:17)2 (cid:11) Y2 +(cid:12) (cid:27)(cid:1)(cid:1) (49) t f;0 0 f;0 0 t 1 0 t 1 (cid:0) (cid:0) 2 = (cid:0)! +(cid:11)(cid:1)Y2(cid:0)+(cid:12) (cid:27)(cid:1)(cid:1) (cid:1) : 0 0 t 1 0 t 1 (cid:0) (cid:0) (47) and (48) herald from Fan et al. (2014, eq. 6), where (cid:17) acts as a scale adjustment parameter.17 f;0 The model of (47) and (48) compliments the finding from Newey and Steigerwald (1997) that GARCHstyle models require additional parameters for correcting discrepancies between f and g, so as to ensure identificationofNGQMLE.18 Wheneitherf = g orf / e(cid:0)2 x2 ,(cid:17) f;0 = 1(seeFanetal., 2014, Proposition 1), in which case, the baseline model of (3) and (4) applies. As a result, no adjustment factor is necessary x2 forthescaleestimatefromQMLE.However,whenf = g andf e(cid:0)2 doesnothold(asisthecasehere), 6 / (cid:17) = 1. Inthiscase,owingto(49),incorrectlyassumingthat(cid:17) = 1resultsinabiasedestimateofscale. f;0 6 f;0 Moreover,thesamebiasimpactingscalewillalso(andequally)impact"reactivity." Consider estimation of (47) and (49) ignoring the presence of (cid:17) and, therefore, implicitly assuming f;0 16Thatis,GARCH"reactivity"istheARCHparameternormalizedbythe(unconditional)scaleofthemodel’sinnovations. 17Specifically,(cid:17) measuresthe"distance"betweenf andg. f;0 18Thosediscrepanciesrelatetolocationandscale. 25

(cid:17) = 1.19 Thefollowingtwocasesareconsidered: E Y4 < (thethin-tailedcase);E Y4 = (the f;0 t 1 t 1 VT heavy-tailed case). Let ! denote the NGQMLE estima(cid:0)te of(cid:1)! and ! the VTNGQMLE(cid:0)est(cid:1)imate, with 0 paralleldefinitionsholdingforotherparametersin(49). b b ASSUMPTION8.1 Underboththethin-andheavy-tailedcases, plim ! = ! ; plim (cid:11) = (cid:11) ; plim (cid:12) = (cid:12) : (50) n 0 n 0 n 0 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) b b b UnderAssumption8.1,neither! nor(cid:11) areidentified,owingtothedistortingpresenceof(cid:17) . What 0 0 f;0 areidentified,however,are! and(cid:11) ,whichcanbeinterpretedasreduced-formparameters. Consequently, 0 0 NGQMLE (minus any scale correction) consistently estimates the reduced-form conditional variance in x2 (49). Themistake, then, istreating! n and(cid:11) n asstructuralestimates. Whenf = g andf e(cid:0)2 doesnot 6 / hold,NGQMLEunder-identifiesthe(structural)GARCHmodel. b b Monte Carlo results support Assumption 8.1. In (50), bias in ! and (cid:11) as estimates of ! and (cid:11) , n n 0 0 respectively, is precisely the same, as it stems from the same distorting property introduced by (cid:17) . This b b f;0 prediction is confirmed by comparing Figures 17 and 25. In large samples, the size and sign of the bias in NGQMLE estimates for ! and (cid:11) , respectively, are identical across all simulation designs considered for 0 0 whichf = g. Inaddition,no(asymptotic)biasisdetectedfortheNGQMLEestimatesfor(cid:12) . 0 6 VT Underboththethin-andheavy-tailedcases,! isgivenby n b ! VT = (cid:17)2 ! = (cid:17)2 (cid:27)2 1 (cid:11) VT (cid:12) ; n f;n n f;n n n n (cid:0) (cid:0) n (cid:16) (cid:17)o b b b b b b b in which case, the scale of the GARCH model innovations is, in turn, a scaled version of the unconditional variance of Y , where the scaling coefficients are the parameters governing short-term, conditional t VT variancedynamics. Underthethin-tailedcase,theprobabilitylimitof! is n b plim ! VT = (cid:17)2 (cid:27)2(1 (cid:11) (cid:12) ) (51) n f;0 0 0 0 (cid:0) (cid:0) (cid:16) (cid:17) = (cid:17)2 !(cid:8) (cid:11) (cid:27)2(cid:17)2 (cid:17)2(cid:9) 1 b f;0 0 0 0 f;0 f;0 (cid:0) (cid:0) = plim ! (cid:11) (cid:27)2(cid:0)(cid:17)2 (cid:17)2 (cid:1) 1 ; n 0 0 f;0 f;0 (cid:0) (cid:0) (cid:16) (cid:17) (cid:0) (cid:1) b wherethesecondequalityfollowsfromplim (cid:17)2 = (cid:17)2 byFanetal. (2014,Theorem1)andplim (cid:11) VT = f;n f;0 n (cid:16) (cid:17) 19Fanetal.(2014),incontrast,accountsfortheprese (cid:0) n b ceof (cid:1) (cid:17) f;0 byestimating(48)inapreliminarystep,using f (cid:15) g fromQbMLE. UnlikeinNeweyandSteigerwald(1997),itisnotpossibletojointlyestimate(cid:17) alongwiththeotherGARCHparameters,since f;0 (cid:17) f;0 isnotseparatelyidentified. b 26

VT (cid:11) bythediscussionunderCase2. Inthiscase,! isalsoabiasedestimatorofscale,buttheformofthe 0 n biasdiffersfromthataffecting! . n b VT Undertheheavy-tailedcase,theprobabilitylimitof! becomes b n b plim ! VT = (cid:17)2 (cid:27)2(1 (cid:11) (cid:12) ) (52) n f;0 0 0 0 (cid:0) (cid:0) (cid:16) (cid:17) = (cid:17)2 !(cid:8) (cid:9) b f;0 0 = plim ! ; n (cid:16) (cid:17) b VT VT where the first equality follows from plim (cid:11) = (cid:11) by Theorem 4 or 7. In this case, ! remains a n 0 n biasedestimatorofscale,buttheformofth(cid:16)ebias(cid:17)isnowpreciselythesameasthataffecting! . n b b VT MonteCarloresultssupportthepredictionin(52). Biasesin! and! tendtoberightontopofeach n n b other,incaseswheref = g (seeFigure25). Differencesinthesebiasestendtobeminorand,consequently, 6 b b VT sourcetoplim (cid:11) = (cid:11) beingtruetoafirst-orderapproximation. n 0 Section5di(cid:16)scove(cid:17)rsthatVTNGQMLEisabiasedestimatorfor(cid:11) inthethin-tailedcase(includingthe b 0 borderlinecaseofAssumption3.3)butaconsistentestimatorintheheavy-tailedcase. (51)and(52)reveal thatinneithercaseisVTNGQMLEaconsistentestimatorofscale. Fanetal. (2014)showsthataccounting forthescalecorrectionparameterin(47)and, therefore, solvingtheidentificationproblemevidentin(49), resultsinamoreefficientestimatorthanQMLE,consistentwiththeargumentputforthinSection1. Thatis, in heavy-tailed cases, a heavy-tailed likelihood can distinguish heavy-tailed effects that are static in nature from heavy-tailed effects that arise due to short-run fluctuations in the conditional variance. A Gaussian likelihood, in contrast, cannot make this distinction. Holding heavy-tailed likelihoods back, however, is their inability to identify (and consistently estimate) scale, when those likelihoods depart from the truth. Fanetal. (2014)removesthisimpedimentandshowsthattheresultingefficiencygains(relativetoQMLE) can be substantial. Section 7 shows that VTNGQMLE can be a materially more efficient estimator for (cid:11) 0 than FAN, LSE, and QMLE. But VTNGQMLE does not solve the identification problem associated with scale,wheneverf = g. Sowhatisgoingon? ResultspresentedinSection7hintatananswer. 6 FromSection7,NGQMLEisshown(consistentwithpopularbelief)tobeabiasedestimatorawayfrom the true innovation density. Despite being a biased estimator for (cid:11) , NGQMLE is also shown (contrary to 0 popularbelief)toperformsurprisinglywellagainstrobustalternativesinfinitesamplesofsurprisinglylarge sizes. The reason behind this surprisingly strong performance is the reduction in dispersion afforded by NGQMLE (see Figure 17). Analogous reductions in dispersion tend also to be afforded to the NGQMLE estimates of ! (see Figure 26), where these reductions (relative to robust alternatives) tend to grow as (cid:17) 0 0 27

shrinks. Moreover, thesereductionsindispersiontendtobesogreatastooverwhelmthepresenceofbias, resulting in estimates of ! that tend to be more efficient (in terms of either RMSE or MAE) than robust 0 alternatives in finite samples as large as T = 2;500 (see Figures 27 and 28). From (52) and confirmed in Figure 25, NGQMLE and VTNGQMLE for! are closely linked in terms ofbias. It turns out, NGQMLE 0 andVTNGQMLEfor! arealsocloselylinkedintermsofdispersion(seeFigure26)aswellasintermsof 0 efficiency(seeFigures27and28). Consequently,VTNGQMLEtendstoproduceamoreefficientestimate ofscale,inheavy-tailedcases,thandoeseitherFAN,LSE,orQMLEinsamplesizesaslargeasT = 2;500. In these same cases, the outperformance of the VTNGQMLE estimates of (cid:11) over those from FAN, LSE, 0 orQMLEisatitshighest(compareFigures21and27aswellasFigures23and28). Asaresult,efficiency gainsinVTNGQMLEoverFAN,LSE,andQMLEcanbeattributedtothesamefactoridentifiedinFanet al. (2014): improvementsintheestimateofscalefortheGARCHmodelinnovations. IncontrasttoFanet al. (2014),however,inthecaseofVTNGQMLE,thisimprovementisbeingaffordedbyabiasedestimator. FurtherdepartingfromFanetal. (2014),improvementsintheestimationof! donotappeartobethe 0 only factor contributing to the outperformance of VTNGQMLE over FAN, LSE, and QMLE: the very act ofvariancetargetingitselfappearstobeasecondcontributingfactor. Why? VTQMLEestimatesof! are 0 never more efficient than the QMLE alternatives in any of the simulation designs considered (see Figures 27 and 28). Despite this fact, VTNGQMLE estimates of (cid:11) nonetheless deliver sizable efficiency gains 0 over QMLE alternatives in heavy-tailed cases (as also reported in Francq et al., 2011), where these gains tend to increase as the tails grow thicker. Explaining the difference in gains earned using VTNGQMLE over VTQMLE links to the former’s improvements in estimating scale over the latter’s. Both estimators, however, also appear to enjoy a boost afforded from the very act of VT. Supporting the existence of this shared boost is the fact that efficiency gains in the VTNGQMLE estimates of (cid:11) tend to persist into larger 0 sample sizes even after the efficiency gains in the VTNGQMLE estimates of ! have disappeared (again, 0 compareFigures21and27aswellasFigures23and28). 9 S&P 500 Index-Related Volatility Estimation and Forecasting The S&P 500 Index is a unique financial instrument in that the following three quantities are each directly observed daily: (i) the return on the index; (ii) option-implied volatility on the index (VIX); (iii) optionimplied volatility on VIX, or option-implied volatility of volatility (VVIX). Using a historical time series of (i), it is standard to apply the model of (3) and (4) for the purpose of forecasting return variance and comparing out-of-sample results against the realized (return) variance (see; e.g., Andersen and Bollerslev, 28

1998). Followingthisconvention,Section9.0.1compares# fromQMLE,VTNGQMLE,andFAN,using n the results for each, respective, estimator to generate out-of-sample volatility forecasts 1-, 5-, 10-, and 21b days-aheadtodeterminewhichestimatorproducesthebestforecastsateachhorizon. Aidingtheseforecast comparisons are the loss functions RMSE and QLIKE, since both are "robust" in the sense discussed by Patton(2011). Applying the model of (3) and (4) to the historical time series of (ii), though less standard, produces estimates of the historical volatility of VIX. Given that (iii) is directly observable, it is feasible to ask whether estimates of the historical volatility of VIX are useful in forecasting VVIX. In this sense, VVIX acts analogously as the realized (return) variance in that both (potentially) can be used to compare the efficacyofcompetingvolatilityforecasts,despitethefactthatVVIXisnotanunbiasedmeasure(orproxy) of the realized volatility of VIX.20 Section 9.1 proposes a forecasting model for VVIX that takes GARCH volatilityofVIXforecastsasinputs.21 1-day-aheadVVIXforecastsarethenconstructedusing1-day-ahead volatilityforecastsofVIXfromQMLE,VTNGQMLE,andFAN,andtheperformanceofthesecompeting VVIXforecastsarecomparedusingtheRMSEandQLIKElossfunctions. 9.0.1 Returns Figures 1 and 2 depict rolling window estimates of (cid:11) from daily S&P 500 returns, first over a lengthy n periodbeginning12/27/1999,andthenoverashortenedperiodimmediatelyfollowingtheworst(financially b speaking)oftheCOVIDcrisis. IntheGARCH(1;1)model,theparameter(cid:11) measuresthe"reactivity"of 0 return variance to the previous period’s return shock. Evident in Figure 1, return variance has become an increasinglyreactiveprocessthroughtime,andinastatisticallysignificantway. EvidentinFigure2,when reactivity is at its highest, (cid:11) from QMLE is the largest, followed by FAN and then by VTNGQMLE. Oh n and Patton (2024) document a tendency for QMLE-based GARCH volatility forecasts to "overshoot" their b target(therealizedreturnvariance)followingalargereturnshock. Figure1suggeststhistendencytobethe most acutein recent times. VTNGQMLEis the leastimpacted by thistendency (compared toboth QMLE and FAN), however, making VTNGQMLE (in some sense) comparable to the local maximum likelihood estimatorofOhandPatton(2024). In the GARCH (1;1) model, (cid:30) = (cid:11) +(cid:12) measures persistence in the variance process. Evident in 0 0 0 20TherealizedvolatilityofVIXisdeterminedunderthehistoricalmeasure, whileVVIXisdeterminedundertherisk-neutral measure. Consequently,thelattercontainsavariance-of-the-varianceriskpremiumnotpresentintheformer(see;e.g.,Huanget al.,2019). 21Thismodelprovidesreduced-formscalecorrectionsforthevariance-of-varianceriskpremium,andsocanbeinterpretedas internalizingthebiasinVVIXasaproxyforthe(latent)realizedvolatilityofVIX. 29

Figure 3, variance persistence has been on the decline in recent years and in a statistically significant way. ThisdeclineisthemostacuteunderQMLE.EvidentinFigure4,VTNGQMLEandFAN,incontrast,both indicate more modest declines in variance persistence. A variance process that is more reactive and less persistentishardertoforecast. ThefactthatbothVTNGQMLEandFANdampenthesetrends,(potentially) foreshadowtheirtendencytoproducemorestableand,thus,morereliablevarianceforecasts. Figure5depicts2 (cid:19) estimatesfromtheHill(1975)estimator,togetherwithone-sided95%confidence (cid:2) 0 bands, constructed using the standard error estimator in Hill (2010).22 In the middle of the sample, these estimatesprovidenoevidenceinfavorof H : (cid:19) < 2; (53) 0 0 thusindicatingQMLEtobepnasymptoticallynormal. UnderCases4–6,incontrast,VTNGQMLEhasa non-Gaussian limit, to which convergence is slower than pn. Collectively, these results imply that, in the middle of the sample (when GARCH volatility was relatively less reactive and relatively more persistent), QMLEperformedbetterthanVTNGQMLE.Towardstheendofthesample,however,thereisnowevidence favoring (53); in fact, evident in Figure 5, (cid:19) < 2. From Hall and Yao (2003, Theorem 2.1), when (cid:19) < 2, n 0 QMLEalsohasanon-Gaussianlimit,withaslowerrateofconvergencecomparedtopn. Attheendofthe b sample,therefore,itislessapparentthatQMLEshouldoutperformVTNGQMLE. Table2summarizesout-of-samplecomparisonsoftheGARCHvolatilityforecastsproducedbyQMLE, VTNGQMLE, and FAN, respectively, using the RMSE and QLIKE loss functions and the standard "RV5" proxyforthelatentvariance. Comparisonsareconductedovertwoforecastevaluationsamples,onebeginningon5/1/2020andoneon1/3/2022(theapproximatedatewherethedifferencebetween(cid:11) fromQMLE n and VTNGQMLE gaps out and remains wide through to the end of the sample; see, Figure 2). Over these b samples, k-period-ahead forecasts are generated each day, where k 1; 5; 10; 21 . By RMSE, 2 QMLEisthebest;although,VTNGQMLEisfairlyclosebehind. FAN,in(cid:16)terestingly,tendsto(cid:17)noticeablylag bothQMLEandVTNGQMLE.ByQLIKE,however,adifferentstoryemerges. Inthiscase,VTNGQMLE istheconsistentwinner,whileFANcontinuestolagbehind. 9.1 VIX It is standard convention to model SPX log returns, since the underlying index levels appear (at least) to be well approximated as an I(1) process. VIX levels, on the other hand (precisely because they measure volatilities), should be both strictly stationary and ergodic (see; e.g., Nelson, 1990, and Lumsdaine, 1996). 22Specifically,depictedinFigure5aretailindexestimatesfor (cid:15) T asdeterminedusingVTNGQMLE. fj tjgt=1 b 30

Consequently,itshouldbenotonlyfeasible,butalsopreferable,toextracttheconditionalvarianceofVIX directlyfromVIXlevels,ratherthanfromVIXlogreturns,withoneimportantcaveat. Whenmodelingdaily SPX log returns, it is also standard to ignore the conditional mean, since it is small and doing so exercises (very)littleimpacton# . TheVIXseries,however(again,becauseitisaseriesofvolatilities),clusters,and n thedegreeofthisclusteringindicatesthatconditionalmeandynamicsareimportant. Andersenetal. (2003) b studytherealizedreturnvarianceseriesandfindittodisplaylong-memoryproperties. ThemodelofCorsi (2009)useslower-frequencycovariatesasproxiesforlong-memoryproperties. Motivatedbytheseresults, considerthefollowingextensionofthemodelin(3)and(4). (cid:26)(L)(1 L)d 0Y t = (cid:18)(L)(cid:15) t ; (54) (cid:0) (cid:15) = (cid:27) (cid:17) ; (cid:17) i:i:d:D(0; 1); (55) t t t t (cid:24) (cid:27)2 = ! +(cid:11) (cid:15)2 +(cid:12) (cid:27)2 ; (56) t 0 0 t 1 0 t 1 (cid:0) (cid:0) where (cid:26)(L) = 1 (cid:26) L ; (cid:18)(L) = 1 (cid:18) L , and L is the lag operator. (54) is an ARFIMA (cid:0) Y;0 (cid:0) (cid:15);0 1; d; 1 m(cid:16)odel for Y(cid:17) , where(cid:0)d 0;(cid:1)0:50 governs long-memory dynamics. The estif t gt Z 2 2 m(cid:16)atorfor(54)(cid:17)isfullmaximumlikelihood(seeS(cid:16)owell,1992)(cid:17). Usingthisestimator, (cid:15) T isobtainedand f t gt=1 fromwhich(55)and(56)areestimatedinasecondstep. b Table 3 summarizes estimation results of (54) on a lengthy VIX sample (see the Notes to Table 3 for additionaldetails). Theestimatedisinsideof, butnear, it’supperbound, indicatingtheVIXseriestobea covariancestationaryand(strongly)long-memoryprocess. Asabenchmark,parameterestimatesincluding b the constraint d = 0 are also summarized in Table 3, where this constraint forces the conditional mean of VIXtodisplayonlyshort-memoryproperties. Noticethat(cid:26) issignificantlydifferentinthetwocases,with Y (cid:26) being much closer to 1 in the case where d = 0, compared to the case where d is (jointly) estimated. Y b When d = 0, (cid:26) is forced to perform "double-duty," controlling for both short- and long-run dynamics. Y b When d is freely estimated, on the other hand, (cid:26) only governs short-run dynamics, while d determines Y b long-run dynamics. In the case of VIX, at least, allowing for long-run dynamics results in less persistent b b short-rundynamics. Figures 7 and 8 depict rolling window estimates of (cid:11) from VIX (see the Notes to Figures7–10 for n additional details). Analogous to the case for return variance in Figures 1 and 2, VIX variance "reactivity" b has been increasing through time. The level of VIX variance "reactivity," however, is higher than that of return variance "reactivity," and in a statistically significant way (compare Figure 8 against Figure 2). 31

Differences in (cid:11) between QMLE, VTNGQMLE, and FAN also appear accentuated in the VIX variance n case,comparedtothereturnvariancecase. Specifically,evidentinFigure8,(cid:11) fromVTNGQMLEappears n b materially muted compared to either QMLE or FAN. Since heightened values of (cid:11) tend to be associated n b GARCH volatility forecast "overshoot," VTNGQMLE appears (far) less prone to this difficulty than either b QMLEorFAN. Figures9and10trackpersistenceinVIXvariancethroughtime. Incontrasttothecaseforreturnvariance(seeFigures3and4),whereallthreeestimatorstendtoindicateadecliningtrendinpersistence,inthe case of VIX variance, only VTNGQMLE signals a declining trend; QMLE and FAN both imply increasing trends, occurring at the end of the sample. What’s more, towards the end of the sample, (cid:30) > 1 for n bothQMLEandFAN,indicatingthevarianceofVIXtobeeither"integrated"orevenexplosive,whilefor b VTNGQMLE,(cid:30) remains(comfortably)insideoftheunitboundary. VVIX(owingtoitbeingobservable) n appears to be, not only mean stationary, but also covariance stationary.23 It seems counterintuitive, then, b for implied vol-of-vol to appear covariance stationary, while historical vol-of-VIX appears (under QMLE and FAN, at least) either "integrated" or explosive. Regardless, for QMLE, VTNGQMLE, and FAN, material differences between in-sample estimates foreshadow accentuated differences between out-of-sample volatilityforecasts,comparedtothereturnvariancecase. Figures11and12depictrolling2 (cid:19) estimatesforGARCH(1;1)modelinnovationstoVIX(seethe (cid:2) 0 Notes to Figures 11 and 12 for additional details). Consider the one-side null of (cid:19) 2. The full sample 0 (cid:21) offers (very) little support for this null (see Figure 11), and emerging from the COVID crisis, there is no supportforthisnull(seeFigure12). MonteCarloresultsunderCase6evidencematerialefficiencygainsof VTNGQMLE over both QMLE and FAN. Figures 11 and 12 support Case 6 as being empirically relevant for the variance of VIX. Additionally, notice that if (cid:19) < 2 (as is strongly supported by Figures 11 and 0 12),thenQMLEhasanon-Gaussianlimitandaconvergencerateslowerthanpn(seeHallandYao,2003, Theorem2.1),comparabletothefindingsforVTNGQMLEinSection5. Let(cid:27) denotetheout-of-sampleGARCHvolatilityforecastforVIXfrom(56). Figure13compares t t 1 j (cid:0) (cid:27) from VTNGQMLE to VVIX on date t. Visually, out-of-sample GARCH volatility forecasts for t t 1 j (cid:0) b VIX display similar dynamics compared to VVIX. These visual similarities are confirmed by a correlation b coefficientof0:53between (cid:27) T and VVIX T forthefullforecastevaluationsample(seethe t j t (cid:0) 1 t=1 f t gt=1 NotestoFigures13–15fora(cid:8)dditiona(cid:9)ldetails). AlsovisuallyapparentinFigure13isthatthetwoseriesare b not on the same scale. This visual dissimilarity should not be that surprising, since VVIX is anticipated to 23Estimating (54) on VVIX produces d < 0:50 and (cid:26) < 1. These results are not reported herein but are available upon Y;0 request. b b 32

contain a vol-of-vol risk premium that should not be present in the historical volatility of VIX (see; e.g., T Huang et al., 2019). This scale difference needs to be addressed, however, if (cid:27) is to serve as a t t 1 t=1 j (cid:0) forecastinginstrumentfor VVIX T . Towardsthatend,considerthefollo(cid:8)wingmo(cid:9)delforadjustingthe f t gt=1 b scaleof(cid:27) forthepurposeofforecastingatargetvariableU . Let t t 1 t j (cid:0) U V = t; (57) t (cid:27) t where(cid:27) isgivenin(56). t V = (cid:16) +(cid:26) V +(cid:18) (cid:23) +(cid:23) ; (58) t 0 V;0 t 1 (cid:29);0 t 1 t (cid:0) (cid:0) where f (cid:23) t gt 2 Z(cid:3) arei.i.d. innovationsandZ 2 Z(cid:3) ,sothat V = (cid:16) +(cid:26) V +(cid:18) (cid:23) t t 1 0 V;0 t 1 (cid:29);0 t 1 j (cid:0) (cid:0) (cid:0) and U = V (cid:27) : (59) t t 1 t t 1 t t 1 j (cid:0) j (cid:0) (cid:2) j (cid:0) Inthecurrentapplication,U = VVIX . Dynamicsin(58)arelimitedtobeingshort-memory. Rolling t t estimatesof(cid:26) (notreportedhere,butavailableuponrequest)areallcomfortablyinsideoftheunitbound- V;0 ary,indicatingthatshort-rundynamics(atleastasaproxy),arenotabadfit;especially,sinceonlyshort-run forecastsarebeingmade. (57)–(58)controlforthevol-of-volriskpremiuminVVIX,allowingthatriskpremiumtoexerciseboth constantandtime-varyingeffectsonscale. Thistime-varyingscalefactoristhenforecastout-of-sample,and the resulting out-of-sample forecast is combined with an out-of-sample GARCH volatility of VIX forecast to produce the forecast of VVIX in (59). The complete model of (54)–(59), then, produces a forecast of VVIXthatusestheGARCHvolatilityofVIXasitsprincipleinput. The dynamic scale factor model of (57)–(59) additionally, however, has a more general interpretation. Consider U as an observable proxy for the true (and latent) volatility that (cid:27) is intended to forecast. t t t 1 j (cid:0) Forillustrativepurposes,suppose(cid:27) istheGARCHreturnvolatilityfromtheprevioussection,sothat t t 1 j (cid:0) a good candidate for U is the realized return volatility.24 In this case, (cid:16) = 1 and (cid:26) = 0 in (58), t 0 V;0 since the realized return volatility is an unbiased estimator for the true (and latent) return volatility (see; e.g., Barndorff-Nielsen and Shephard, 2004). Consequently, (cid:27) can be used as an unadjusted forecast t t 1 j (cid:0) 24Inotherwords,U = RV5 . t t p 33

for the realized return volatility, consistent with standard practice. With this illustrative example in mind, consider VVIX, not as the target variable of interest directly, but rather as an observable proxy for the true (andlatent)volatilityofVIX.Owingtothepresenceofavol-of-volriskpremium,VVIXcanbeanticipated tobeabiasedproxyforthevolatilityofVIX.Themodelof(57)–(59),then,canbeseenascorrectingforthis bias, thus allowing VVIX to be used as the predicted variable in an evaluation of the GARCH volatility of VIXforecasts,wherethatevaluationlookstoexaminetheefficacyoftheGARCHvolatilityofVIXforecasts aspredictiveinstrumentsforthetrue(andlatent)volatilityofVIX. Figure14showstheresultsofapplyingthemodelin(57)–(59)toadjust(orcorrect)theGARCHvolatilityofVIXforecastsinFigure13. Asisevident,thepredictedvariable(VVIX)andtheout-of-sampleforecasts are now on the same scale. Moreover, the full-sample correlation between the adjusted forecasts and VVIX is, essentially, the same (0:51 versus 0:53), indicating that adjusting the forecasts does (practically) nothing to alter the predictive power of the GARCH volatility of VIX forecasts. Consequently, Figure 14 evidences that GARCH volatility of VIX is, in fact, useful at forecasting implied volatility-of-volatility (VVIX). Also evidenced in Figure 14 is a tendency for the forecasts to "overshoot" their target. Perhaps this tendencyshouldn’tbetoosurprising,giventheheightenedlevelsofGARCHvariance"reactivity"observed acrossdifferentestimators(seeFigure7). Tohelpmitigatethistendency,thefollowingstrategy(motivated by the "averaged-forecasting" approach used in De Nard et al., 2021) is adopted.25 For any date t, it is possible to generate two forecasts, (cid:27) and (cid:27) . The single point forecast for date t is then given t t 1 t t 2 j (cid:0) j (cid:0) (cid:27) +(cid:27) by t t 1 t t 2, and this average forecast is substituted for (cid:27) in (59). The result of performing j (cid:0) 2 j (cid:0) t t 1 b b j (cid:0) thisbsubstitubtion is evidenced in Figure 15. The effect is a fairly apparent reduction in forecast variability, b generally,and,moreimportantly,forecastextremes,specifically. Interestingly,thecorrelationbetweenthese adjusted average forecasts and VVIX increases to 0:65 (from 0:51). Forecast evaluations performed using (cid:27) +(cid:27) theQMLE,VTNGQMLE,andFANestimatorsallsubstitute t t 1 t t 2 for(cid:27) in(59). j (cid:0) 2 j (cid:0) t t 1 j (cid:0) Table 2 also summarizes out-of-sample comparisons of the GARCH volatility forecasts produced by QMLE, VTNGQMLE, and FAN, respectively, using the RMSE and QLIKE loss functions and VVIX as the predicted variable.26 Comparisons are conducted over the same two forecast evaluation samples used H=21 25Considerthesetofdailyforecasts (cid:27) t+h j t H h= = 1 21.The"averagedforecast"fromthissetisgivenbyH(cid:0) 1 h=1 (cid:27) t+h j t .This "averagedforecast"isaproxyforthem(cid:8)onthlyv(cid:9)olatilityforecast. Byanalogy,thedesiredforecasthereisadaPilyforecast. That dailyforecastisproxiedbyan"averagedforecast"takenoveranearneighborhoodbehindthedesiredforecastdate. Thatis,the H=2 "averagedforecast"isH(cid:0) 1 (cid:27) t t h . h=1 j (cid:0) 26FollowingthediscussionPabove,VVIXcanbeinterpretedeitherasthetargetvariablebeingforecastedofa(biased)proxyfor thetrue(andlatent)volatilityofVIX. 34

in evaluating the SPX return volatility estimates. Over these samples, only 1-day-ahead forecasts are considered. By RMSE, VTNGQMLE is now the clear winner, with FAN second and QMLE a close third. By QLIKE, the rankings remain unaltered. These results are consistent with the conclusions drawn from the parameter estimates depicted in Figures 7–10. These results further bolster the strong performance of VTNGQMLEintheMonteCarloexperiments. Table 4 compares the "average forecasting" method for generating 1-day-ahead VVIX forecasts to the standardmethod(bothofwhicharedescribedabove). Forcomparisonpurposes,out-of-sampleresultsfrom the "average forecasting" method are depicted in Figure 15, while results from the standard method are depicted in Figure 14. Consistent with these figures, the "average forecasting" method beats the standard methodintermsofRSME.Somewhatsurprising, the"averageforecasting"methodalsobeatsthestandard methodintermsofQLIKE. 10 Conclusion Motivated by the NGQMLE of Bollerlsev (1987), this paper considers the VTNGQMLE, determining its limiting properties, studying its finite-sample properties, and applying it in a series of empirical investigationsintovolatilityforecasting. Inheavy-tailedcases,VTNGQMLEisshowntobearobustestimator,like QMLE,FAN,andLSE.Inthesesamecases,whenthelikelihoodfunctionismisspecified,VTNGQMLEis showntoperform(surprisingly)well,bothinsimulationandempirically. Infact,VTNGQMLEisshownto beveryhardtobeat, bothbythepopularQMLEandbyalternative(robust)estimatorsaimedatimproving theQMLEresult. ExplainingthepopularityofQMLEisitbeingrobustandpnasymptoticallynormal,underfairlygeneral conditions. Previous works demonstrate that QMLE loses its Gaussian limit when the model errors become (very) heavy-tailed (see; e.g., Hall and Yao, 2003). In earlier years, this case, while theoretically interesting,didnotappearempiricallyrelevant. Inrecenttimes,however,thiscasehasbecomeempirically relevant. Moreover, in this case, both QMLE and multi-step estimators aimed at producing more efficient (relative to QMLE) estimates perform (relatively) poorly. VTNGQMLE, in contrast, performs markedly better; in part, because of its reliance upon a heavy-tailed (though misspecified) likelihood function that removes some emphasis from the ARCH parameter as the single model parameter responsible for capturingheavy-tailedfeaturesintheunconditionaldistributionoftherandomvariablebeingmodeled. Inrecent times, therefore, VTNGQMLE appears to deserve serious consideration over QMLE and competing estimators because VTNGQMLE (i) is comparable in complexity relative to QMLE but (ii) delivers sizably 35

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[38] Vaynman,I.&B.K.Beare(2014)Stablelimittheoryforthevariancetargetingestimator,inY.Chang, T.B.Fomby&J.Y.Park(eds), EssaysinHonorofPeterC.B.Phillips, vol.33ofAdvancesinEconometrics: EmeraldGroupPublishingLimited,chapter24,639-672. 11 Appendix A (Proofs) ProofofTheorem3. Let I = I (cid:27)2 E (cid:27)2 > a ; J = 1 I = I (cid:27)2 E (cid:27)2 a : tn t n tn tn t n (cid:0) (cid:0) (cid:0) (cid:20) (cid:0)(cid:12) (cid:0) (cid:1)(cid:12) (cid:1) (cid:0)(cid:12) (cid:0) (cid:1)(cid:12) (cid:1) (cid:12) (cid:12) (cid:12) (cid:12) Then n n n a 1 W = a 1 W I +a 1 W J (cid:0)n t (cid:0)n t tn (cid:0)n t tn (cid:2) (cid:2) t=1 t=1 t=1 X X X = I(b)+II(b) n I(b) = a 1 (cid:15)2 1 (cid:27)2 I (cid:15)2 1 E (cid:27)2 I + (cid:15)2 1 E (cid:27)2 I (cid:0)n t t tn t tn t tn (cid:0) (cid:2) (cid:2) (cid:0) (cid:0) (cid:2) (cid:2) (cid:0) (cid:2) (cid:2) t=1 X(cid:8)(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:9) n n = a 1 (cid:15)2 1 (cid:27)2 E (cid:27)2 I +a 1 (cid:15)2 1 E (cid:27)2 I (cid:0)n t t tn (cid:0)n t tn (cid:0) (cid:2) (cid:0) (cid:2) (cid:0) (cid:2) (cid:2) t=1 t=1 X(cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) X(cid:0) (cid:1) (cid:0) (cid:1) = I(c)+II(c) ByMarkov’sInequality, n P ( II(c) > C) C 1E a 1 (cid:15)2 1 E (cid:27)2 I (cid:0) (cid:0)n t tn j j (cid:20) (cid:12) (cid:0) (cid:2) (cid:2) (cid:12)! C (cid:0) 1na (cid:0)n 1 (cid:12) (cid:12) (cid:12)E X t (cid:15) = 2 t 1 (cid:0) 1 (cid:1) E (cid:27)2 (cid:0) (cid:1) E I (cid:12) (cid:12) (cid:12)(cid:27)2 t E (cid:27)2 > a n (cid:20) (cid:12) (cid:0) (cid:2) (cid:2) (cid:12) (cid:0) Ca 1nP (cid:27)(cid:0)2(cid:12) E (cid:12)(cid:27)(cid:1)2 >(cid:0)a (cid:1) (cid:0) (cid:0)(cid:12) (cid:0) (cid:1)(cid:12) (cid:1)(cid:1) (cid:0)n t(cid:12) (cid:12) n (cid:12) (cid:12) (cid:20) (cid:0) Ca 1 a(cid:0) 2 n (cid:12) (cid:12) P (cid:27)2 t (cid:0) (cid:0)E(cid:1)(cid:12) (cid:12) (cid:27)2 >(cid:1)a n (cid:20) (cid:0)n H(a ) (cid:0)(cid:12) (cid:0)n (cid:1)(cid:12) (cid:1)! (cid:12) (cid:12) 0 (cid:0)! asn ,wherethefourthinequalityfollowsfrom(11),andthe(weak)convergenceresultfollows ! 1 39

from(12). Next,andalsobyMarkov’sInequality, n P ( I(c) > C) C 1E a 1 (cid:15)2 1 (cid:27)2 E (cid:27)2 I (cid:0) (cid:0)n t t tn j j (cid:20) (cid:12) (cid:0) (cid:2) (cid:0) (cid:2) (cid:12)! C (cid:0) 1na (cid:0)n 1 (cid:12) (cid:12) (cid:12)E X t= (cid:15) 1 2 t (cid:0) 1 (cid:1) (cid:27)2 t (cid:0) E (cid:27)2 (cid:0) (cid:1)(cid:1) I tn (cid:12) (cid:12) (cid:12) (cid:20) (cid:12) (cid:0) (cid:2) (cid:0) (cid:2) (cid:12) Ca 1nE (cid:27)(cid:0)2(cid:12)(cid:0) E (cid:27)(cid:1)2 (cid:0) I (cid:27)2(cid:0) E(cid:1)(cid:1)(cid:27)2 >(cid:12)(cid:1)a (cid:0)n t(cid:12) t (cid:12) n (cid:20) (cid:0) (cid:2) (cid:0) a E (cid:27)2 E (cid:27)2 I (cid:27)2 E (cid:27)2 > a n (cid:0)t(cid:12) (cid:0) (cid:1)(cid:12) (cid:0)t(cid:12) (cid:0) (cid:1)(cid:12) n (cid:1)(cid:1) C (cid:12)(cid:0) (cid:2)(cid:12) (cid:12)(cid:0) (cid:12) (cid:20) H(a ) (cid:0)(cid:12) (cid:0) (cid:1)(cid:12) (cid:0)n(cid:12) (cid:0) (cid:1)(cid:12) (cid:1)(cid:1) (cid:12) (cid:12) (cid:12) (cid:12) 0 (cid:0)! asn ,wherethefourthinequalityfollowsfrom(11),andthe(weak)convergenceresultfollows ! 1 from(13). Consequently, n n a 1 W = a 1 W J +o (1); (cid:0)n t (cid:0)n t tn p (cid:2) t=1 t=1 X X and n Var a 1 W J = na 2E (cid:15)2 1 2 (cid:27)4 I (cid:27)2 E (cid:27)2 a (60) (cid:0)n t tn (cid:0)n t t t n (cid:2) (cid:0) (cid:2) (cid:2) (cid:0) (cid:20) ! X t=1 (cid:16) (cid:0) (cid:1) (cid:0)(cid:12) (cid:0) (cid:1)(cid:12) (cid:1) (cid:17) = Ca 2nE (cid:27)4 I (cid:27)2 E (cid:27)(cid:12)2 a (cid:12) (cid:0)n t t n (cid:2) (cid:0) (cid:20) E(cid:0)(cid:27)4 t I (cid:0)(cid:12)(cid:27)2 t E(cid:0)(cid:27)2 (cid:1)(cid:12) a n(cid:1)(cid:1) C (cid:2) (cid:12) (cid:0) (cid:12)(cid:20) ; (cid:20) (cid:2) E (cid:27)4 I (cid:27)2 a (cid:0) (cid:0)t(cid:12) t(cid:0) (cid:1)n(cid:12) (cid:1)(cid:1)! (cid:12)(cid:2) (cid:20) (cid:12) (cid:0) (cid:0) (cid:1)(cid:1) wheretheinequalityfollowsfrom(11). Forsufficientlylargen, I (cid:27)2 E (cid:27)2 a I (cid:27)2 a (61) t n t n (cid:0) (cid:20) (cid:21) (cid:20) (cid:0)(cid:12) (cid:0) (cid:1)(cid:12) (cid:1) (cid:0) (cid:1) (cid:12) (cid:12) Given(61),becauseE (cid:27)2 doesnotdependonn, aC suchthat 9 (cid:0) (cid:1) I (cid:27)2 E (cid:27)2 a = I (cid:27)2 a +C (62) t n t n (cid:0) (cid:20) (cid:20) (cid:0)(cid:12) (cid:0) (cid:1)(cid:12) (cid:1) (cid:0) (cid:1) (cid:12) (cid:12) 40

Given(62), a +C n (cid:27)4f (cid:27)2 d(cid:27)2 E (cid:27)4 I (cid:27)2 E (cid:27)2 a t t n 0 (cid:2) (cid:0) (cid:20) = (63) E (cid:27)4 I (cid:27)2 a aR n (cid:0) (cid:1) (cid:0) (cid:0)t(cid:12) (cid:12)(cid:2) t(cid:0) (cid:20) (cid:1)n(cid:12) (cid:12) (cid:1)(cid:1) (cid:27)4f((cid:27)2)d(cid:27)2 (cid:0) (cid:0) (cid:1)(cid:1) 0 R a +C n (cid:27)4f (cid:27)2 d(cid:27)2 a = 1+ n aR (cid:0) (cid:1) n (cid:27)4f((cid:27)2)d(cid:27)2 0 R Given(63), n lim Var a 1 W J = C < : (64) (cid:0)n t tn n (cid:2) ! 1 !1 t=1 X n Given (64), in turn, it is possible to apply a CLT to a 1 W following analogous arguments given (cid:0)n t t=1 byHallandYao(2003,p. 306-307). LettheresultofthisPapplicationbe n a 1 W d N 0; V : (65) (cid:0)n t (cid:0)! W t=1 X (cid:0) (cid:1) (30)thenfollowsfromSlutsky’sTheorem.(cid:4) Remark9 An alternative way of establishing that (60) is bounded is to note that, given Assumption 3.3, E((cid:27)4 I( (cid:27)2 E((cid:27)2) a )) t(cid:2) j t(cid:0) j(cid:20) n isaratioofslowlyvaryingfunctionsthat,assuch,hasafinitelimit. E((cid:27)4 I((cid:27)2 a )) (cid:18) t(cid:2) t(cid:20) n (cid:19) ProofofTheorem4. Fori = 1;2,let (cid:17) (cid:17) ifi = 1 n 0 X(i) = (cid:0) : (66) 8 9 Z ifi = 2 < n = b e : ; UnderCase2,itisestablishedthatpnX(i)convergestoastablelimit,solongas(cid:19) > 1. Consider 0 41

then na 1X(i) = n1=2a 1 n1=2X(i) (cid:0)n (cid:0)n (cid:2) (cid:16) (cid:17) (cid:16) 1 (cid:17) = n1=2 Cn1=(cid:20) 0 (cid:0) n1=2X(i) (cid:2) (cid:18) (cid:19) (cid:16) (cid:17) (cid:16) (cid:17) = C n (cid:20) 2 0 (cid:20) (cid:0) 0 2 n1=2X(i) (cid:2) (cid:2) (cid:18) (cid:19) (cid:16) (cid:17) = C o (1) O (1) p p (cid:2) (cid:2) = o (1): p where the third equality follows, since n1=2X(i) converges to a stable limit, and n (cid:20) 2 0 (cid:20) (cid:0) 0 2 0 as (cid:0)! n ! 1 ,since(cid:20) 0 < 2.(cid:4) Remark10 n1=2 isincreasingatafasterratethann (cid:20) 2 0 (cid:20) (cid:0) 0 2 isdecreasing. Consequently,n1=2X(i)reaches (cid:20)0(cid:0) 2 itsstablelimitfirstandthenisdriventowardszerobyn 2(cid:20)0 . ProofofTheorem5. Startingfrom(29),fora" > 0, n n n a 1 W = a 1 W I (cid:27)2 > a " +a 1 W I (cid:27)2 a " (67) (cid:0)n t (cid:0)n t t n (cid:0)n t t n (cid:2) (cid:2) (cid:20) t=1 t=1 t=1 X X (cid:0) (cid:1) X (cid:0) (cid:1) = I(d)+II(d) Var(II(d)) = na 2Var (cid:15)2 1 (cid:27)2 I (cid:27)2 a " (68) (cid:0)n t t t n (cid:0) (cid:2) (cid:2) (cid:20) = na 2E (cid:0)(cid:15)2(cid:0) 1 2(cid:1) (cid:27)4 I (cid:0)(cid:27)2 a " (cid:1)(cid:1) (cid:0)n t t t n (cid:0) (cid:2) (cid:2) (cid:20) (cid:16) (cid:17) = Cna 2E(cid:0)(cid:27)4 I(cid:1) (cid:27)2 a "(cid:0) ; (cid:1) (cid:0)n t t n (cid:2) (cid:20) (cid:0) (cid:0) (cid:1)(cid:1) wherenecessaryforthethirdequalityisE (cid:15)2 1 2 < , whichisestablishedbyE (cid:15)4 < . t t (cid:0) 1 1 Since given Theorem 1, (cid:27)2 is regularly va(cid:16)ry(cid:0)ing wit(cid:1)h(cid:17)tail index (cid:20) , for a function L tha(cid:0)t is(cid:1)slowly t 0 42

varyingat , 1 a " n E (cid:27)4 I (cid:27)2 a " = (cid:27)2 2 f (cid:27)2 d(cid:27)2 (69) t t n (cid:2) (cid:20) Z (cid:0) (cid:0) (cid:1)(cid:1) 0 (cid:0) (cid:1) (cid:0)(cid:0) (cid:1)(cid:1) a " n C( (cid:20) ) (cid:27)2 2 (cid:0) (cid:20) 0(cid:0) 1 L (cid:27)2 d(cid:27)2 0 (cid:24) (cid:0) Z 0 (cid:0) (cid:1) (cid:0) (cid:1) (cid:24) C( (cid:0) (cid:20) 0 )(2 (cid:0) (cid:20) 0 ) (cid:0) 1 (cid:27)2 2 (cid:0) (cid:20) 0L (cid:27)2 j a 0 n " C( (cid:20) 0 )(2 (cid:20) 0 ) (cid:0) 1n (a(cid:0) n ")(cid:1)2(a n ") (cid:0) (cid:0)(cid:20) 0L(cid:1)(a n " o ) (cid:24) (cid:0) (cid:0) Ca2"2P (cid:27)2 > a " n n (cid:24) (cid:0) (cid:1) where the first follows from Mikosch (1999, Theorem 1.2.9), the second from Mikosch (1999, (cid:24) (cid:24) Theorem1.2.6),andthelast fromLemma1. Putting(69)and(68)together, (cid:24) Var(II(d)) C"2nP (cid:27)2 > a " ; n (cid:24) (cid:0) (cid:1) inwhichcase, lim Var(II(d)) C "2 (cid:0) (cid:20) 0 n (cid:24) (cid:2) !1 byDefinition1,andfurther lim limVar(II(d)) 0; n " 0 (cid:24) !1 ! since(cid:20) (1;2). Asaresult, 0 2 n n a 1 W = a 1 W I (cid:27)2 > a " +o (1): (70) (cid:0)n t (cid:0)n t t n p (cid:2) t=1 t=1 X X (cid:0) (cid:1) Let n U = a 1 W I (cid:27)2 > a " ; (71) n;" (cid:0)n t t n (cid:2) t=1 X (cid:0) (cid:1) 2 anddefinethefunctionT " : M P R+ nf 0 g (cid:0)! Ras (cid:16) (cid:17) 1 1 T (cid:14) = X X I X > " ; " X i;2 i;1 i;1 i (cid:0) (cid:2) ! i=1 i=1 X X(cid:0) (cid:1) (cid:0) (cid:1) d such that, given (14), U = T (N ). From Lemma 2, N N as n , in which case, n;" " n n (cid:0)! ! 1 43

d T (N ) T (N)asn bythecontinuousmappingtheorem. Lastly, " n " (cid:0)! ! 1 d T (N) U ; " 0; (72) " (cid:27)2 (cid:0)! ! by Davis and Hsing (1995, Theorem 3.1(ii)), where U is a (cid:20) -stable random variable that can be (cid:27)2 0 expressedintermsoftheP ’sandQ ’sinLemma2. Given(29),(72)thenresultsin i ij 1 (cid:12) U d (cid:0) 0 U : n (cid:0)! 1 (cid:11) (cid:12) (cid:27)2 (cid:18) (cid:0) 0(cid:0) 0(cid:19) (cid:4) Remark11 (71) is a special case of Vaynman and Beare (2014, eq. 35). Consequently, the result in (72) also follows from the proof of Vaynman and Beare (2014, Theorem 4), starting from equaton 35 and proceedingtotheend. ProofofTheorem6. Given(67), n II(d) = a 1 (cid:15)2 1 I (cid:15)2 > b " (cid:27)2 I (cid:27)2 a " (cid:0)n t t n t t n (cid:0) (cid:2) (cid:2) (cid:2) (cid:20) t=1 X(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) n +a 1 (cid:15)2 1 I (cid:15)2 b " (cid:27)2 I (cid:27)2 a " (cid:0)n t t n t t n (cid:0) (cid:2) (cid:20) (cid:2) (cid:2) (cid:20) t=1 X(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) = I(e)+II(e): Var(II(e)) = a 2n Var (cid:15)2 1 I (cid:15)2 b " (cid:27)2 I (cid:27)2 a " (cid:0)n t t n t t n (cid:2) (cid:0) (cid:2) (cid:20) (cid:2) (cid:2) (cid:20) = E (cid:15)2 1 2(cid:0)(cid:0) I (cid:15)2(cid:1) b "(cid:0) a 2(cid:1)n E (cid:27)4 (cid:0) I (cid:27)2 (cid:1)a(cid:1)" t t n (cid:0)n t t n (cid:0) (cid:2) (cid:20) (cid:2) (cid:2) (cid:2) (cid:20) (cid:16) (cid:17) = E(cid:0)(cid:15)4 I(cid:1)(cid:15)2 (cid:0)b " +C(cid:1) a 2n E (cid:0)(cid:27)4 I (cid:0)(cid:27)2 a "(cid:1)(cid:1) t t n (cid:0)n t t n (cid:2) (cid:20) (cid:2) (cid:2) (cid:2) (cid:20) (cid:8)b2n(cid:0) 1+C(cid:0) a 2n(cid:1)(cid:1) E (cid:9)(cid:27)4 I (cid:27)2 a(cid:0) " (cid:0) (cid:1)(cid:1) n (cid:0) (cid:0)n t t n (cid:20) (cid:2) (cid:2) (cid:2) (cid:20) (cid:8)C a 2n (cid:9)E (cid:27)4 I (cid:27)2(cid:0) a " (cid:0) (cid:1)(cid:1) (cid:0)n t t n (cid:20) (cid:2) (cid:2) (cid:2) (cid:20) 0; (cid:0) (cid:0) (cid:1)(cid:1) (cid:0)! asn and" 0,wherebothinequalitiesfollowfromAssumption5.1,andconvergencesources ! 1 ! 44

to(69)andtheresultsthatfollow. Var(I(e)) = E (cid:15)2 1 2 I (cid:15)2 > b " a 2n E (cid:27)4 I (cid:27)2 a " t t n (cid:0)n t t n (cid:0) (cid:2) (cid:2) (cid:2) (cid:2) (cid:20) (cid:16) (cid:17) = E(cid:0)(cid:15)4 I(cid:1)(cid:15)2 >(cid:0)b " +C(cid:1) a 2n E (cid:0)(cid:27)4 I (cid:0)(cid:27)2 a "(cid:1)(cid:1) t t n (cid:0)n t t n (cid:2) (cid:2) (cid:2) (cid:2) (cid:20) = (cid:8)E (cid:15)(cid:0)4 I (cid:15)(cid:0)2 > b " (cid:1)(cid:1) a 2(cid:9)n E (cid:27)4 I(cid:0)(cid:27)2 a(cid:0)" (cid:1)(cid:1) t t n (cid:0)n t t n (cid:2) (cid:2) (cid:2) (cid:2) (cid:20) +C(cid:0) a 2(cid:0)n E (cid:27)4(cid:1)(cid:1) I (cid:27)2 a "(cid:0) (cid:0) (cid:1)(cid:1) (cid:0)n t t n (cid:2) (cid:2) (cid:2) (cid:20) = b2n 1 b 2n(cid:0) E (cid:15)4 (cid:0) I (cid:15)2 > b(cid:1)(cid:1)" a 2n E (cid:27)4 I (cid:27)2 a " n (cid:0) (cid:0)n t t n (cid:0)n t t n (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:20) (cid:0)+C a(cid:1) 2n(cid:0) E (cid:1)(cid:27)4 (cid:0) I (cid:27)2(cid:0) a " (cid:1)(cid:1) (cid:0) (cid:0) (cid:1)(cid:1) (cid:0)n t t n (cid:2) (cid:2) (cid:2) (cid:20) b2n 1 E (cid:15)(cid:0) 4 t (cid:2) I (cid:15)(cid:0) 2 t > b n " (cid:1)(cid:1) a 2n E (cid:27)4 I (cid:27)2 a " (cid:20) n (cid:0) (cid:2) H(b ) (cid:2) (cid:0)n (cid:2) t (cid:2) t (cid:20) n (cid:0) (cid:0) n (cid:1)(cid:1)! (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) +C a 2n E (cid:27)4 I (cid:27)2 a " (cid:0)n t t n (cid:2) (cid:2) (cid:2) (cid:20) b n 1 b n E(cid:0) (cid:15)4 t (cid:2) I(cid:0)(cid:15)2 t > b n "(cid:1)(cid:1) a 2n E (cid:27)4 I (cid:27)2 a " (cid:20) n (cid:0) (cid:2) H(b ) (cid:2) (cid:0)n (cid:2) t (cid:2) t (cid:20) n (cid:0) (cid:0)n (cid:1)(cid:1)! (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) +C a 2n E (cid:27)4 I (cid:27)2 a " ; (cid:0)n t t n (cid:2) (cid:2) (cid:2) (cid:20) (cid:0) (cid:0) (cid:1)(cid:1) followingfrombothAssumption5.1and(13)adaptedfor(cid:15)2. Notingthatb n 1 = o(1), n (cid:0) lim Var(I(e)) o(1) "2 (cid:0) (cid:20) 0 +C "2 (cid:0) (cid:20) 0 n (cid:20) (cid:2) (cid:2) !1 0; (cid:0)! as " 0, (see, again, (69) and the results that follow). Consequently, (70) continues to hold and, ! fromwhich,theresultin(33)follows(seetheproofofTheorem5),sincethelimitingrandomvariable continuestobedeterminedby(only)theextremesof(cid:27)2 t ,despiteE (cid:15)2 t = .(cid:4) 1 (cid:0) (cid:1) ProofofTheorem7. Given(66), na 1b 1X(i) = n1=2a 1 b 1 n1=2X(i) (cid:0)n (cid:0)n (cid:0)n (cid:0)n (cid:2) (cid:2) = (cid:16) C n (cid:20) 2 0 (cid:17) (cid:20) (cid:0) 0 2 (cid:0) (cid:1) n(cid:0)(cid:19)0 1(cid:16) n1=2 (cid:17) X(i) (cid:2) (cid:2) (cid:2) (cid:18) (cid:19) (cid:18) (cid:19) (cid:16) (cid:17) = C o(1) o(1) O (1) p (cid:2) (cid:2) (cid:2) = o (1); p wherethethirdequalityfollowssince(cid:20) <2,andn1=2X(i)convergestoastablelimit(seetheproof 0 45

ofTheorem4).(cid:4) ProofofTheorem8. Startingfrom(35),theanalogto(67)is n n n a 1b 1 W = a 1b 1 W I (cid:27)2 > a " +a 1b 1 W I (cid:27)2 a " (cid:0)n (cid:0)n t (cid:0)n (cid:0)n t t n (cid:0)n (cid:0)n t t n (cid:2) (cid:2) (cid:20) t=1 t=1 t=1 X X (cid:0) (cid:1) X (cid:0) (cid:1) = I(d)+II(d); preservingthesamenotationfromtheproofofTheorem5. Consider n I(d) = a 1b 1 W I (cid:15)2 > b " I (cid:27)2 > a " (cid:0)n (cid:0)n t t n t n (cid:2) (cid:2) t=1 X (cid:0) (cid:1) (cid:0) (cid:1) n +a 1b 1 W I (cid:15)2 b " I (cid:27)2 > a " (cid:0)n (cid:0)n t t n t n (cid:2) (cid:20) (cid:2) t=1 X (cid:0) (cid:1) (cid:0) (cid:1) = I(di)+I(dii); and n I(d) = a 1b 1 W I (cid:15)2 > b " I (cid:27)2 a " (cid:0)n (cid:0)n t t n t n (cid:2) (cid:2) (cid:20) t=1 X (cid:0) (cid:1) (cid:0) (cid:1) n +a 1b 1 W I (cid:15)2 b " I (cid:27)2 a " (cid:0)n (cid:0)n t t n t n (cid:2) (cid:20) (cid:2) (cid:20) t=1 X (cid:0) (cid:1) (cid:0) (cid:1) = II(di)+II(dii): Var(II(dii)) = a 2b 2nVar (cid:15)2 1 I (cid:15)2 b " (cid:27)2 I (cid:27)2 a " (cid:0)n (cid:0)n t t n t t n (cid:0) (cid:2) (cid:20) (cid:2) (cid:2) (cid:20) = n 1 b (cid:0)2(cid:0)n E(cid:1) (cid:15)2(cid:0) 1 2 I(cid:1) (cid:15)2 b "(cid:0) a(cid:1)(cid:1)2n E (cid:27)4 I (cid:27)2 a " (cid:0) (cid:0)n t t n (cid:0)n t t n (cid:2) (cid:2) (cid:0) (cid:2) (cid:20) (cid:2) (cid:2) (cid:2) (cid:20) n (cid:16) (cid:17)o = (cid:0)n 1(cid:1) II(cid:0)(diii)(cid:1) II(d(cid:0)iv); (cid:1) (cid:0) (cid:1) (cid:8)(cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1)(cid:9) (cid:0) (cid:2) (cid:2) (cid:0) (cid:1) where II(diii) = b 2n E (cid:15)4 I (cid:15)2 b " 2E (cid:15)2 I (cid:15)2 b " +E I (cid:15)2 b " (cid:0)n t t n t t n t n (cid:2) (cid:2) (cid:20) (cid:0) (cid:2) (cid:20) (cid:20) = (cid:0)b 2n(cid:1) (cid:8)E(cid:0)(cid:15)4 I(cid:0)(cid:15)2 b "(cid:1)(cid:1)+C (cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:0) (cid:1)(cid:1)(cid:9) (cid:0)n t t n (cid:2) (cid:2) (cid:20) = (cid:0)b 2n(cid:1) (cid:8)E (cid:15)(cid:0)4 I (cid:15)(cid:0)2 b " (cid:1)(cid:1)+o(1(cid:9)); (cid:0)n t t n (cid:2) (cid:2) (cid:20) (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)(cid:1) 46

since 1 b (cid:0)n 2n = Cn(cid:19) 1 0 (cid:0) n (73) (cid:18) (cid:19) (cid:19)0=2 = C n (cid:19)0 (cid:2) (cid:18) (cid:19) 0 (cid:0)! asn ,given(cid:19) < 2. ! 1 0 b " n E (cid:15)4 I (cid:15)2 b " = (cid:15)2 2 f (cid:15)2 d(cid:15)2 (74) t t n (cid:2) (cid:20) Z (cid:0) (cid:0) (cid:1)(cid:1) 0 (cid:0) (cid:1) (cid:0) (cid:1) b " n C ( (cid:19) ) (cid:15)2 2 (cid:0) (cid:19) 0(cid:0) 1 L (cid:15)2 d(cid:15)2 0 (cid:24) (cid:2) (cid:0) (cid:2) Z 0 (cid:0) (cid:1) (cid:0) (cid:1) (cid:24) C (cid:2) ( (cid:0) (cid:19) 0 ) (cid:2) (2 (cid:0) (cid:19) 0 ) (cid:0) 1 (cid:2) (cid:15)2 2 (cid:0) (cid:19) 0L (cid:15)2 j b 0 n " C (b n ")2(b n ") (cid:0) (cid:19) 0L(b n ") n (cid:0) (cid:1) (cid:0) (cid:1) o (cid:24) (cid:2) C (b ")2 P (cid:15)2 > b " ; n n (cid:24) (cid:2) (cid:2) (cid:0) (cid:1) where the first follows from Mikosch (1999, Theorem 1.2.9), the second from Mikosch (1999, (cid:24) Theorem1.2.6),andthefinalfromLemma1. Consequently, II(diii) = C (")2 nP (cid:15)2 > b " ; n (cid:2) (cid:2) (cid:0) (cid:1) inwhichcase, lim II(diii) = C (")2 (cid:0) (cid:19) 0; n (cid:2) !1 which,inturn,impliesthat lim limII(diii) 0; n " 0 (cid:24) !1 ! since(cid:19) < 2;and 0 lim limII(div) 0; n " 0 (cid:24) !1 ! given(68)andtheargumentsthat(immediately)follow. Asaresult, lim limVar(II(di)) 0: (75) n " 0 (cid:24) !1 ! 47

Next, n II(di) = a 1b 1 (cid:15)2 I (cid:15)2 > b " I (cid:15)2 > b " (cid:27)2 I (cid:27)2 a " (cid:0)n (cid:0)n t t n t n t t n (cid:2) (cid:0) (cid:2) (cid:2) (cid:20) t=1 X(cid:8) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9) (cid:0) (cid:1) n = n (cid:0) 1 2b (cid:0)n 1 (cid:15)2 t I (cid:15)2 t > b n " I (cid:15)2 t > b n " a (cid:0)n 1 n 1 2 (cid:27)2 t I (cid:27)2 t a n " (cid:2) (cid:0) (cid:2) (cid:2) (cid:2) (cid:2) (cid:20) (cid:16) (cid:17)X t=1 (cid:8) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9) n (cid:0) (cid:1) o n C n (cid:0) 1 2b (cid:0)n 1 (cid:15)2 t I (cid:15)2 t > b n " I (cid:15)2 t > b n " (cid:20) (cid:2) (cid:2) (cid:0) (cid:16) (cid:17)X t=1 (cid:8) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9) n n C n (cid:0)2 1 b (cid:0)n 1 (cid:15)2 t I (cid:15)2 t > b n " b (cid:0)n 1 n (cid:0) 1 2 (cid:15)2 t I (cid:15)2 t > b n " (cid:20) (cid:2) (cid:2) (cid:0) (cid:2) ( ( )) (cid:16) (cid:17)X t=1 (cid:0) (cid:1) X t=1 (cid:0) (cid:1) n C n (cid:0)2 1 b (cid:0)n 1 (cid:15)2 t I (cid:15)2 t > b n " o(1) O p (1) (cid:20) (cid:2) (cid:2) (cid:0) (cid:2) ( ) (cid:16) (cid:17)X t=1 (cid:0) (cid:1) n C n (cid:0) 1 2 b (cid:0)n 1 (cid:15)2 t I (cid:15)2 t > b n " +o p (1) (cid:20) (cid:2) (cid:2) (cid:2) ( ) t=1 X (cid:0) (cid:1) where the first inequality follows from (68) and the results that (immediately) follow, and the third inequality follows from a central limit theorem for i.i.d. data. Since (1) (cid:15) is i.i.d., and (2) (8) f t gt Z 2 holdsfor(cid:15)andthenormalizingconstantsb , n n lim limb 1 (cid:15)2 I (cid:15)2 > b " = Z ; (76) (cid:0)n t t n (cid:15)2 n " 0 (cid:2) !1 ! t=1 X (cid:0) (cid:1) where Z follows a (cid:19) -stable law (see; e.g., LePage et al., 1981, Theorem 1). Given (1) and (2), (cid:15)2 0 sufficient for (76) is that the distribution of (cid:15) has a balanced tail (see, Feller, 1971), as defined in DavisandHsing(1995,eq. 1.2)). Given(76), II(di) C o(1) O (1)+o (1) o (1); p p p (cid:20) (cid:2) (cid:2) (cid:20) n in which case, given (75), II(d) o (1). Consequently, the asymptotic limit of a 1b 1 W is p (cid:0)n (cid:0)n t (cid:20) t=1 P 48

determinedbyI(di)andI(dii). n I(dii) = a 1b 1 (cid:15)2 1 I (cid:15)2 b " (cid:27)2 I (cid:27)2 > a " (cid:0)n (cid:0)n t t n t t n (cid:0) (cid:2) (cid:20) (cid:2) (cid:2) t=1 X(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) n = a 1b 1 (cid:15)2 I (cid:15)2 b " I (cid:15)2 b " (cid:27)2 I (cid:27)2 > a " (cid:0)n (cid:0)n t t n t n t t n (cid:2) (cid:20) (cid:0) (cid:20) (cid:2) (cid:2) t=1 X(cid:8) (cid:0) (cid:1) (cid:0) (cid:1)(cid:9) (cid:0) (cid:1) n = a 1b 1 (cid:15)2 I (cid:15)2 b " (cid:27)2 I (cid:27)2 > a " (cid:0)n (cid:0)n t t n t t n (cid:2) (cid:20) (cid:2) (cid:2) t=1 X (cid:0) (cid:1) (cid:0) (cid:1) n a 1b 1 (cid:27)2 I (cid:27)2 > a " I (cid:15)2 b " (cid:0)n (cid:0)n t t n t n (cid:0) (cid:2) (cid:2) (cid:20) t=1 X (cid:0) (cid:1) (cid:0) (cid:1) = I(e)+II(e): n n II(e) a 1 (cid:27)2 I (cid:27)2 > a " b 1 I (cid:15)2 b " (cid:0)n t t n (cid:0)n t n (cid:20) (cid:2) (cid:2) (cid:20) ( ) ( ) t=1 t=1 X (cid:0) (cid:1) X (cid:0) (cid:1) n n a (cid:0)n 1 (cid:27)2 t I (cid:27)2 t > a n " b (cid:0)n 1n 1 2 n (cid:0) 1 2 I (cid:15)2 t b n " (cid:20) (cid:2) (cid:2) (cid:20) ( ) ( ) X t=1 (cid:0) (cid:1) (cid:16) (cid:17) X t=1 (cid:0) (cid:1) n a 1 (cid:27)2 I (cid:27)2 > a " o(1) O (1); (cid:0)n t t n p (cid:20) (cid:2) (cid:2) (cid:2) ( ) t=1 X (cid:0) (cid:1) wherethefinalinequalityfollowsfrom(73)andacentrallimittheoremfori.i.d. data. GivenLemma 1and(8), n lim lima 1 (cid:27)2 I (cid:27)2 > a " = Z ; (77) (cid:0)n t t n (cid:27)2 n " 0 (cid:2) !1 ! t=1 X (cid:0) (cid:1) whereZ followsa(cid:20) -stablelaw(seeDavisandHsing,1995,Theorem3.1.ii). Consequently, (cid:27)2 0 II(e) O (1) o (1) o (1): (78) p p p (cid:20) (cid:2) (cid:20) Next, n I(e) = n (cid:0)2 1 a (cid:0)n 1 b (cid:0)n 1n 1 2 (cid:15)2 t I (cid:15)2 t b n " (cid:27)2 t I (cid:27)2 t > a n " (79) (cid:2) (cid:2) (cid:2) (cid:20) (cid:2) (cid:2) ( ) (cid:16) (cid:17) X t=1n(cid:16) (cid:17) (cid:0) (cid:1) o (cid:8) (cid:0) (cid:1)(cid:9) n C n (cid:0)2 1 a (cid:0)n 1 (cid:27)2 t I (cid:27)2 t > a n " (cid:20) (cid:2) (cid:2) (cid:2) ( ) (cid:16) (cid:17) X t=1 (cid:8) (cid:0) (cid:1)(cid:9) C o(1) O (1); p (cid:20) (cid:2) (cid:2) where the first inequality follows from (74) and the results that (immediately) follow, and the third 49

inequality follows from (77). Combining (78) and (79) implies that I(dii) o (1). What then p (cid:20) remainstoconsideris n I(di) = a 1b 1 (cid:15)2 1 I (cid:15)2 > b " (cid:27)2 I (cid:27)2 > a " (cid:0)n (cid:0)n t t n t t n (cid:0) (cid:2) (cid:2) (cid:2) t=1 X(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) n = a 1b 1 (cid:15)2 I (cid:15)2 > b " (cid:27)2 I (cid:27)2 > a " (cid:0)n (cid:0)n t t n t t n (cid:2) (cid:2) (cid:2) t=1 X (cid:0) (cid:1) (cid:0) (cid:1) n a 1b 1 (cid:27)2 I (cid:27)2 > a " I (cid:15)2 > b " (cid:0)n (cid:0)n t t n t n (cid:0) (cid:2) (cid:2) t=1 X (cid:0) (cid:1) (cid:0) (cid:1) = I(f)+II(f): First, n n II(f) a 1 (cid:27)2 I (cid:27)2 > a " b 1 I (cid:15)2 > b " (cid:0)n t t n (cid:0)n t n (cid:20) (cid:2) (cid:2) ( ) ( ) t=1 t=1 X (cid:0) (cid:1) X (cid:0) (cid:1) n n a (cid:0)n 1 (cid:27)2 t I (cid:27)2 t > a n " n 1 2b (cid:0)n 1 n (cid:0)2 1 I (cid:15)2 t > b n " (cid:20) (cid:2) (cid:2) (cid:2) ( ) ( ) X t=1 (cid:0) (cid:1) (cid:16) (cid:17) X t=1 (cid:0) (cid:1) I(g) o(1) II(g); (cid:20) (cid:2) (cid:2) w d wherethethirdinequalityfollowsfrom(73). Moreover,sinceI(g) Z (see77),andII(g) (cid:27)2 (cid:0)! (cid:0)! N (0;V)givena(standard)centrallimittheoremfori.i.d. data,II(f) o (1). p (cid:20) Finally,consider n I(f) = a 1b 1 (cid:15)2 I (cid:15)2 > b " (cid:27)2 I (cid:27)2 > a " (80) (cid:0)n (cid:0)n t t n t t n (cid:2) (cid:2) (cid:2) t=1 X(cid:8) (cid:0) (cid:1)(cid:9) (cid:8) (cid:0) (cid:1)(cid:9) = U : n;" e LetY =(cid:15)2andZ = (cid:27)2,notingthatY andZ areindependent. Further,letX = Y ; Z ,with t t t t t t t t t the associated polar coordinates X t ; (cid:18) X t , where A 2 B [ 0; e 2(cid:25) (cid:16) )d (cid:0) 1 , the (cid:17) Borel (cid:16) (cid:12) (cid:12) (cid:16) (cid:17) (cid:17) (cid:16) (cid:17) (cid:12) (cid:12) e e (cid:12) (cid:12) 50

subsetsof[ 0; 2(cid:25) )d (cid:0) 1,for1 d 2. Forar > 0,consider (cid:20) (cid:20) P X > ur; (cid:18) X A t t 2 (cid:16) (cid:12) (cid:12) (cid:16) (cid:17) (cid:17) (cid:12) (cid:12)P X > r (cid:12) e (cid:12) t e (cid:16)(cid:12) (cid:12) (cid:17) P Y > u(cid:12)r;e (cid:12)Z > ur; (cid:18)(X ) A = j t j (cid:12) (cid:12)j t j t 2 (cid:16) (cid:17) P Y > r; Z > r t t j j j j P ( Y >(cid:16)ur) P ( Z > u(cid:17)r) = j t j j t j ; P ( Y > r) (cid:2) P ( Z > r) (cid:26) j t j (cid:27) (cid:26) j t j (cid:27) inwhichcase, P X > ur; (cid:18) X A t t u l ! im 1 (cid:16) (cid:12) (cid:12) (cid:12) e (cid:12) (cid:12) (cid:12) P X t > (cid:16) r e (cid:17) 2 (cid:17) = (cid:8) r (cid:0) (cid:19) 0 (cid:9) (cid:2) (cid:8) r (cid:0) (cid:20) 0 (cid:9) < 1 (81) (cid:16)(cid:12) (cid:12) (cid:17) (cid:12) (cid:12) e r, given Assumption 3.2 and Lem(cid:12)ma(cid:12)1. Given (81), in turn, Davis and Mikosch (1988, eq. 2.1) is 8 satisfied for X (see also Resnick, 1986). Moreover, given Carrasco and Chen (2002, Corollary 6), t X isstrongmixing,inwhichcase,DavisandMikosch(1998,eq2.3)isalsosatisfied. Next,note t e tnhattoime-dependencein X drivesentirelyfrom Z ,where e t f t g n o e (cid:27)2 = !+(cid:11)Y2 +(cid:12)(cid:27)2 (82) t t 1 t 1 (cid:0) (cid:0) = !+(cid:27)2 (cid:11)(cid:15)2 +(cid:12) t 1 t 1 (cid:0) (cid:0) = (cid:27)2 A +(cid:0)B (cid:1) t 1 t t (cid:0) t t t = A (cid:27)2+ A B i 0 j i i=1 i=1j=i+1 = I Q (cid:27)2+I P ; Q t;1 0 t;2 where the final equality follows from recursive substitution. From Mikosch and Sta˘rica˘ (2000), (82) isavalidstochasticrecurrenceequation,forwhich a y a y P (cid:27)2 > a y (cid:27)2 > a y P I (cid:27)2 > a y (cid:27)2 > n +P I > n (cid:27)2 > a y t n j 0 n (cid:20) t;1 0 n j 0 2 t;2 2 j 0 n (cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) I(h)+II(h): (cid:20) 51

UsingMarkov’sinequality, P I (cid:27)2 > a n y ; (cid:27)2 > a y t;1 0 2 0 n I(h) = (cid:16) P (cid:27)2 > a y (cid:17) 0 n P I (cid:27)2 I (cid:27)2 > a y > a n y = t;1 0(cid:2) (cid:0) 0 n(cid:1) 2 P (cid:27)2 > a y (cid:0) (cid:0)0 n (cid:1) (cid:1) 2 (a y) 1 E I (cid:27)2 I (cid:27)2 > a y (cid:2) n (cid:0) (cid:0) (cid:2) t;1 (cid:1)0(cid:2) 0 n (cid:20) P (cid:27)2 > a y (cid:0)0 n (cid:0) (cid:1)(cid:1) 2 (a y) 1 E I E (cid:27)2 I (cid:27)2 > a y (cid:2) n (cid:0) (cid:2) (cid:0) t;1 (cid:2) (cid:1) 0(cid:2) 0 n ; (cid:20) P (cid:27)2 > a y (cid:0) (cid:1)0 n(cid:0) (cid:0) (cid:1)(cid:1) (cid:0) (cid:1) where t t E I = E A = E(A ) = E(A)t = bt; t;1 i i (cid:18)i=1 (cid:19) i=1 (cid:0) (cid:1) Q Q b < 1,and 1 E (cid:27)2 I (cid:27)2 > a y = (cid:27)2f (cid:27)2 d(cid:27)2 0 0 n 0 0 0 (cid:2) Z (cid:0) (cid:0) (cid:1)(cid:1) a n y (cid:0) (cid:1) 1 C ( (cid:20) ) (cid:27)2 (cid:0) (cid:20) 0L (cid:27)2 d(cid:27)2 0 0 0 0 (cid:24) (cid:2) (cid:0) Z a n y(cid:0) (cid:1) (cid:0) (cid:1) C ( (cid:20) 0 )( (cid:20) 0 +1) (cid:0) 1(a n y) (cid:0) (cid:20) 0 +1L(a n y) (cid:24) (cid:2) (cid:0) (cid:0) C (a n y) (a n y) (cid:0) (cid:20) 0L(a n y) (cid:24) (cid:2) (cid:2) C (a y) P (cid:27)2 > a y ; n 0 n (cid:24) (cid:2) (cid:2) (cid:0) (cid:1) with the first following from Mikosch (1999, Theorem 1.2.9), and the third following from (cid:24) (cid:24) Mikosch(199,Theorem1.2.6(b)). Asaresult, 2 (a y) 1 bt C (a y) P (cid:27)2 > a y I(h) (cid:2) n (cid:0) (cid:2) (cid:2) (cid:2) n (cid:2) 0 n (83) (cid:20) P (cid:27)2 > a y 0 n (cid:0) (cid:1) C bt: (cid:0) (cid:1) (cid:20) (cid:2) 52

Next,byindependenceandMarkov’sinequality, a y II(h) = P I > n (84) t;2 2 (cid:16) (cid:17) t t a y = P A B > n j i 2 i=1j=i+1 ! P Q a y P 1 1 A B > n (cid:20) j i 2 i=1j=i+1 ! P Q a y 1 n (cid:0) E 1 1 A B (cid:20) 2 (cid:2) j i i=1j=i+1 ! (cid:16) (cid:17) P Q a y 1 n (cid:0) E 1 1 E A B (cid:20) 2 (cid:2) j i i=1j=i+1 (cid:16) (cid:17) C a1; P Q (cid:0) (cid:1) n (cid:20) (cid:2) sinceE(A) < 1. Forasequenceofpositiveintegers r ,wherer and n asn , f n g n ! 1 r n ! 1 ! 1 puttingtogethertheresultsin(83)and(84)produces lim lim supP (cid:27)2 > a y (cid:27)2 > a y t n 0 n k n 0 j 1 !1 !1 k t r (cid:20)_j j(cid:20) n @ r +m A n lim lim sup2(m+1) P (cid:27)2 > a y (cid:27)2 > a y t n 0 n (cid:20) k n j !1 !1 t=k X (cid:0) (cid:1) 1 lim C bt (cid:20) k (cid:2) !1 t=k X 0; (cid:20) thusestablishingthatDavisandMikosch(1998,eq2.10)holdsforX ,which,inturn,establishesthat t (cid:13) inDavisandMikosch(1998,eq2.11)exists. Supposethat(cid:13) = 0. 6 e 2 ConsiderthefunctionT " : M P R+ nf 0 g (cid:0)! Ras (cid:16) (cid:17) e 1 1 T (cid:14) = Z I(Z > ") Y I(Y > ") : " Y Z i i i i i i f (cid:2) gf (cid:2) g ! i=1 i=1 X X e Furtherlet n N = (cid:14) : (85) n t=1 a(cid:0)n 1b(cid:0)n 1Y t Z t P e 53

Given(80), U = T N : n;" " n (cid:16) (cid:17) FromDavisandMikosch(1998,Propositieon3.1,eRemeark3.2),N d N asn ,andthenfrom n (cid:0)! ! 1 d the continuous mapping theorem, T N T N as n . Lastly, from Davis and Hsing " n (cid:0)! " !e 1 e (1995,Theorem3.1(ii)), (cid:16) (cid:17) (cid:16) (cid:17) e e e e d T N U ; " 0; (86) " (cid:15)2;(cid:27)2 (cid:0)! ! (cid:16) (cid:17) whereU isa(cid:20) -stablerandome vareiablethatcanbeexpressedintermsofquantitiesqualitatively (cid:15)2;(cid:27)2 0 similartotheP ’sandQ ’sinLemma2,inwhichcase, i ij d U U : n;" (cid:15)2;(cid:27)2 (cid:0)! e (cid:4) ProofofTheorem9. Aunivariateanalogto(15)is n Si = Yi for i = 2;4: n t t=1 X Giventhisunivariateanalog, na 2b 2(cid:28)2 = a 2b 2 S4 na 2b 2 n 1S2 2 (cid:0)n (cid:0)n n (cid:0)n (cid:0)n n (cid:0)n (cid:0)n (cid:0) n (cid:2) (cid:0) (cid:2) = (cid:0)a 2b 2(cid:1) S4 (cid:0)na 2b 2(cid:1) (cid:0)n 1a b(cid:1) 2 a 1b 1 S2 2 b (cid:0)n (cid:0)n (cid:2) n (cid:0) (cid:0)n (cid:0)n (cid:2) (cid:0) n n (cid:2) (cid:0)n (cid:0)n (cid:2) n = (cid:0)a 2b 2(cid:1) S4 (cid:0)n 1 (cid:1)a (cid:0)1b 1 S(cid:1)2 2 (cid:0)(cid:0) (cid:1) (cid:1) (cid:0)n (cid:0)n n (cid:0) (cid:0)n (cid:0)n n (cid:2) (cid:0) (cid:2) (cid:2) = (cid:0)a 2b 2(cid:1) S4 o(cid:0)(1)(cid:1) O(cid:0)(cid:0)(1) (cid:1) (cid:1) (cid:0)n (cid:0)n n p (cid:2) (cid:0) (cid:2) = (cid:0)a 2b 2(cid:1) S4 +o (1); (cid:0)n (cid:0)n n p (cid:2) (cid:0) (cid:1) wherethefourthequalityfollowsfromtheproofofTheorem(8). Consequently,theasymptoticlimit of(cid:28)2 isdeterminedbyS4. n n Also from the proof of Theorem 8, X is regularly varying with tail index (cid:20) . By Mikosch (1999, b t 0 Proposition1.5.14),X2 = Y2; Z2 isregularlyvaryingwithtailindex(cid:20) =2. t t et 0 n (cid:16) (cid:17) LetN2 = (cid:14) e . GivenN d N asn ,fromtheproofofTheorem8,N2 d N2 n t=1 a(cid:0)n 2b(cid:0)n 2Y t 2Z t 2 n (cid:0)! ! 1 n (cid:0)! P e e e e e 54

asn ,givenRemark2. Since(1)X2 isregularlyvaryingand(2)N2 d N2 asn ,Davis t n ! 1 (cid:0)! ! 1 andHsing(1995,Theorem3.1(i))canbeappliedtoestablish e e e a 2b 2 S4 d U ; (cid:0)n (cid:0)n n (cid:15)4;(cid:27)4 (cid:2) (cid:0)! (cid:0) (cid:1) whereU isa((cid:20) =2)-stablerandomvariablethatcanbeexpressedintermsofthelimitingpoints (cid:15)4;(cid:27)4 0 forY t 2=b2 n andZ t 2=a2 n .(cid:4) ProofofTheorem11. RecallingthedefinitionofU from(80),let n;" n e U2 = a 2b 2 (cid:15)4 I (cid:15)2 > b " (cid:27)4 I (cid:27)2 > a " ; n;" (cid:0)n (cid:0)n t t n t t n (cid:2) (cid:2) (cid:2) t=1 X(cid:8) (cid:0) (cid:1)(cid:9) (cid:8) (cid:0) (cid:1)(cid:9) e anddefinethefunctionT " : M P R 2 + nf 0 g (cid:0)! R 2 as (cid:16) (cid:17) e T 1 (cid:14) = 1 Z I(Z > ") Y I(Y > ") ; 1 Z2 I(Z > ") Y2 I(Y > ") : " Y i Z i f i(cid:2) i gf i(cid:2) i g i (cid:2) i i (cid:2) i ! i=1 i=1 i=1 (cid:18) (cid:19) X P P(cid:8) (cid:9)(cid:8) (cid:9) e FurtherrecallthedefinitionofN from(85). Then n e U ; U2 = T N : n;" n;" " n (cid:16) (cid:17) (cid:16) (cid:17) e e e e Theconditionsrequisitefor d N N; n n (cid:0)! ! 1 areestablishedintheproofofTheoreme8. Thate d T N T N ; n " n " (cid:0)! ! 1 (cid:16) (cid:17) (cid:16) (cid:17) e e e e thenfollowsfromthecontinuousmappingtheorem. Lastly, d T N U ; U ; " 0 " n (cid:0)! (cid:15)2;(cid:27)2 (cid:15)4;(cid:27)4 ! (cid:16) (cid:17) (cid:16) (cid:17) e e results from Davis and Mikosch (1998, Proposition 3.3), where the marginal limits are described at theendoftheproofstoTheorems(8)and(9),respectively.(cid:4) 55

12 Appendix B (Tables) Table1: ParameterConfigurations (cid:18) 0 Specification ! (cid:11) (cid:12) 0 0 0 I 0.05 0.05 0.90 II 0.05 0.10 0.85 III 0.05 0.20 0.75 IV 0.05 0.30 0.65 NotestoTable1.DifferentGARCH(1;1)parametervaluesconsideredintheMonteCarlosimulations. Table2: Out-of-SampleForecastComparisons Eval. Loss ForecastHorizon Asset Sample(Beg.) Function Estimator 1-Step 5-Steps 10-Steps 21-Steps SPX 5/1/2020 RMSE QMLE 6.687 7.152 7.367 7.487 VTNGQMLE 6.820 7.338 7.593 7.713 FAN 7.057 7.718 8.143 8.611 QLIKE QMLE 3.463 3.481 3.492 3.502 VTNGQMLE 3.463 3.480 3.491 3.501 FAN 3.468 3.488 3.503 3.522 1/3/2022 RMSE QMLE 6.476 6.790 7.074 7.213 VTNGQMLE 6.550 6.902 7.206 7.263 FAN 6.873 7.414 7.937 8.449 QLIKE QMLE 3.496 3.510 3.524 3.538 VTNGQMLE 3.494 3.506 3.519 3.532 FAN 3.500 3.516 3.535 3.559 VIX 5/1/2020 RMSE QMLE 34.026 VTNGQMLE 26.301 FAN 31.441 QLIKE QMLE 5.636 VTNGQMLE 5.628 FAN 5.633 1/3/2022 RMSE QMLE 27.446 VTNGQMLE 18.747 FAN 24.307 QLIKE QMLE 5.557 VTNGQMLE 5.548 FAN 5.554 NotestoTable2. DailySPX,VIX,andVVIXlevelssourcetoBloomberg,L.P.TheGARCH(1;1)modelof(3)and(4) isestimatedonSPXlogreturnsusingafixed10-yearlook-backwindowbeginningon4/30/2020androllingthroughtheend ofthesampleon10/29/2024. Fork 1; 5; 10; 21 ,oneachdayofthesample,k-period-aheadGARCHvolatility 2 forecastsareconstructedusingQMLE,VTNGQMLE,andFAN.RMSEandQLIKElossfunctionsevaluatetheefficacyofthese (cid:0) (cid:1) GARCHforecasts,usingthestandard"RV5"proxyforthelatentvariance. TheARFIMA(1; d; 1)modelof(54)isestimated onVIXlevelsusingafixed20-yearlook-backwindowbeginningon4/30/2020androllingthroughtheendofthedatasample on10/29/2024.TheGARCH(1;1)modelof(55)and(56)isthenestimatedonthedailyARFIMA(1; d; 1)modelinnovations usingQMLE,VTNGQMLE,andFANonafixed10-yearlook-backwindowbeginningon4/30/2020androllingthroughtheend ofthedatasampleon10/29/2024,soastoproducetheout-of-sampleforecasts(cid:27) and(cid:27) oneachdatetofthesample. t t 1 t t 2 Foreachestimator,thedynamicscalefactormodelof(57)–(59)isestimatedonajfi(cid:0)xed10-yeajrl(cid:0)ook-backwindowbeginningon 4/30/2020androllingthroughtheendofthedatasampleon10/29/2024,soastobproduceU tbt 1 =V\VIX t t 1 foreachdate tinthesample,where (cid:27)t j t (cid:0) 1+ 2 (cid:27)t j t (cid:0) 2 replaces(cid:27) t j t (cid:0) 1 in(59). V\VIX t j t (cid:0) 1 T t=1b fo j rm (cid:0) edusingthecom j p (cid:0) etingestimators arecomparedusingtheRMSEandQLIKElossfunctions. n o 56

Table3: ARFIMA 1; d; 1 EstimatesforVIX model para. est. stderror 95%C.I. (cid:0) (cid:1) ARFIMA 1; d; 1 (cid:26) 0.830 0.045 0.742 0.918 Y (cid:18) -0.494 0.065 -0.622 -0.366 (cid:0) (cid:1) d 0.497 0.002 0.494 0.500 ARFIMA 1; 0; 1 (cid:26) 0.984 0.005 0.975 0.994 Y (cid:18) -0.138 0.032 -0.202 -0.075 (cid:0) (cid:1) d 0.000 NotestoTable3. DailyVIXlevelssourcetoBloomberg,L.P.TheARFIMA 1; d; 1 modelisfittotheseVIX levelsoveralengthysamplebeginning1/2/1990andrunningthrough10/13/2022. Reportedstandarderrorsarerobustinthe (cid:0) (cid:1) Huber-White"sandwich"estimatorsense. Table4: Out-of-SampleVVIXForecastConstructionComparisons Eval. LossFunction Sample(Beg.) Forecast RMSE QLIKE 5/1/2020 1-Step 43.469 5.634 + Avg. 26.301 5.628 1/3/2022 1-Step 26.801 5.552 Avg. 18.747 5.548 Notes to Table 4. Daily VIX levels source to Bloomberg, L.P. The ARFIMA (1; d; 1) model of (54) is estimated on VIXlevelsusingafixed20-yearlook-backwindowbeginningon4/30/2020androllingthroughtheendofthedatasampleon 10/29/2024. TheGARCH(1;1)modelof(55)and(56)isestimatedonthedailyARFIMA(1; d; 1)modelinnovationsusing VTNGQMLEonafixed10-yearlook-backwindowbeginningon4/30/2020androllingthroughtheendofthedatasampleon 10/29/2024,soastoproducetheout-of-sampleforecasts(cid:27) and(cid:27) oneachdatetofthesample. Thedynamicscale t t 1 t t 2 factormodelof(57)–(59)isestimatedonafixed10-yearloojk-(cid:0)backwindojw(cid:0)beginningon4/30/2020androllingthroughtheend ofthedatasampleon10/29/2024,soastoproduceforeabchdatetinbthesampleU t t 1 = V\VIX t t 1 asdenotedby"1- Step",andU t j t (cid:0)T 1 = V\VIX t j t (cid:0) 1 asdenotedby"Avg.",where,inthiscase, b (cid:27)t j j (cid:0) t (cid:0) 1+ 2 (cid:27)t j t (cid:0) 2 re j pla (cid:0) ces(cid:27) t j t (cid:0) 1 in(59). V\VIX formedas"1-Step"and"Avg."arethencomparedusingtheRMSEandQLIKElossfunctions. t jbt (cid:0) 1 t=1 n o 57

13 Appendix C (Figures) NotestoFigures1–4. DailySPXindexlevelssourcetoBloomberg,L.P.TheGARCH(1;1)modelof(3)and(4)isestimatedondailylogreturns constructedfromtheseindexlevelsusingafixed10-yearlook-backwindowbeginningon12/23/1999androllingthroughtheendofthe datasampleon10/29/2024.TheGARCHestimatorsbeingcomparedareQMLE,VTNGQMLE,andFAN.Fortheseestimators,thefigures plot(cid:11)and(cid:30)=(cid:11)+(cid:12)assolidlines.Alsoplottedasdashedlinesare2-sided,95%confidencebandsfortheQMLEestimates,constructed usingHuber-White"sandwich"standarderrorestimates. b b b b NotestoFigures5–6. DailySPXindexlevelssourcetoBloomberg,L.P.TheGARCH(1;1)modelof(3)and(4)isestimatedondailylogreturns constructedfromtheseindexlevelsusingVTQMLEandafixed10-yearlook-backwindowbeginningon12/23/1999androllingthrough theendofthedatasampleon10/29/2024.Oneachdaybeginning12/23/1999,GARCH(1;1)modelinnovationsforthefixed10-yearlookbackwindowareestimated,towhichthetailindexestimatorofHill(1975)isappliedusingathresholdof5%ofthelargestinnovationsin absolutevalue.27 One-sided,95%confidencebands(dashedlines)fortheHill(1975)estimates(solidlines)arealsoconstructedusingthe robuststandarderrorestimatordevelopedinHill(2010). NotestoFigures7–10. DailyVIXlevelssourcetoBloomberg,L.P.TheseVIXlevelsaremean-filteredusinganARFIMA(1; d; 1)model,where d (0; 0:5),estimatedusingthefullmaximumlikelihoodestimatorofSowell(1992)onafixed20-yearlook-backwindowbeginningon 2 12/31/2009androllingthroughtheendofthedatasampleon10/29/2024. TheGARCH(1;1)modelof(3)and(4)isthenestimatedon thedailyARFIMA(1; d; 1)modelinnovationsusingafixed10-yearlook-backwindowbeginningon12/31/2009androllingthroughthe endofthedatasampleon10/29/2024.TheGARCHestimatorsbeingcomparedareQMLE,VTNGQMLE,andFAN.Fortheseestimators, thefiguresplot(cid:11)and(cid:30)=(cid:11)+(cid:12)assolidlines. NotestoFigures11–12. b DailbyVIX b levbelssourcetoBloomberg,L.P.TheseVIXlevelsaremean-filteredusinganARFIMA(1; d; 1)model, where d (0; 0:5), estimated using the full maximum likelihood estimator of Sowell (1992) on a fixed 20-year look-back window 2 beginningon12/31/2009androllingthroughtheendofthedatasampleon10/29/2024. TheGARCH(1;1)modelof(55)and(56)is thenestimatedonthedailyARFIMA(1; d; 1)modelinnovationsusingVTNGQMLEandafixed10-yearlook-backwindowbeginning on12/31/2009androllingthroughtheendofthedatasampleon10/29/2024. Oneachdaybeginning12/31/2009,GARCH(1;1)model innovationsforthefixed10-yearlook-backwindowareestimated, towhichthetailindexestimatorofHill(1975)isappliedusinga thresholdof5%ofthelargestinnovationsinabsolutevalue.One-sided,95%confidencebands(dashedlines)fortheHill(1975)estimates (solidlines)arealsoconstructedusingtherobuststandarderrorestimatordevelopedinHill(2010). NotestoFigures13–15. DailyVIXlevelssourcetoBloomberg,L.P.TheseVIXlevelsaremean-filteredusinganARFIMA(1; d; 1)model, where d (0; 0:5), estimated using the full maximum likelihood estimator of Sowell (1992) on a fixed 20-year look-back window 2 beginningon05/01/2020androllingthroughtheendofthedatasampleon10/29/2024. TheGARCH(1;1)modelof(55)and(56)is thenestimatedonthedailyARFIMA(1; d; 1)modelinnovationsusingVTNGQMLEandafixed10-yearlook-backwindowbeginning on05/01/2020androllingthroughtheendofthedatasampleon10/29/2024. Oneachdaybeginning04/30/2020,a1-step-aheadand 2-steps-aheadout-of-sampleGARCHvolatilityforecastismade,sothatforeachdatetintheforecast-evaluationsample,therearetwo out-of-sample,GARCHvolatilityforecasts,(cid:27) and(cid:27) . Figure13compares(cid:27) againsttheactualVVIXvalueondate t t 1 t t 2 t t 1 j (cid:0) j (cid:0) j (cid:0) t. Figure14depictstheout-of-sampleVVIXforecastconstructedusing(cid:27) andV ,comparingthatforecastagainsttheactual t t 1 t t 1 V th V at I f X or v e a c l a u s e ta o g n a d in a s t t e t t h . e F a i c g t u u r a e lV 15 V d IX ep v ic a t l s ue th o e n b o d u a t- te of t - . sa I m n p F l i b e g V ur V es IX 14 fo an re d ca 1b s 5 t , j c M o (cid:0) n A s X tru i c s t t e h d b e f l r j a b o r m g (cid:0) es (cid:27) t t V j V t (cid:0) IX 1+ 2 v (cid:27) al t u j e t e (cid:0) v 2 er a o n b d se V r t vejd t .(cid:0) 1 ,comparing b b b NotestoFigure16. DailySPXindexandVIXlevelssourcetoBloomberg,L.P.ForSPXlogreturns,skewnessestimatesonafixed10-yearlookbackwindowbeginningon05/01/2020androllingthroughtheendofthedatasampleon10/29/2024areshown.ForVIXlevels,skewness estimatesfortheinnovationstoanARFIMA(1; d; 1)modelfittoafixed10-yearlook-backwindowbeginningon05/01/2020androlling throughtheendofthedatasampleon10/29/2024areshown. 27This 5% threshold means that the 126 largest daily innovations (in terms of absolute value) are input into the Hill (1975) estimator. 58

Figure1: SPX(cid:11) (FullSample) n b Figure2: SPX(cid:11) (PostCOVID) n b 59

Figure3: SPX(cid:30) (FullSample) n b Figure4: SPX(cid:30) (PostCOVID) n b 60

Figure5: SPX(cid:19) (FullSample) n b Figure6: SPX(cid:19) (PostCOVID) n b 61

Figure7: VIX(cid:11) (FullSample) n b Figure8: VIX(cid:11) (PostCOVID) n b 62

Figure9: VIX(cid:30) (FullSample) n b Figure10: VIX(cid:30) (PostCOVID) n b 63

Figure11: VIX(cid:19) (FullSample) n b Figure12: VIX(cid:19) (PostCOVID) n b 64

Figure13: VIX1-StepGARCHVolForecasts Figure14: VVIX1-StepGARCHVolForecasts 65

Figure15: VVIXAverageGARCHVolForecasts Figure16: SPXandVIXSkewness 66

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:20)(cid:26)(cid:29)(cid:3)(cid:48)(cid:40)(cid:36)(cid:49)(cid:3)(cid:37)(cid:44)(cid:36)(cid:54)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:36)(cid:47)(cid:51)(cid:43)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 6.50% 6.50% 6.50% 6.50% 5.50% 5.50% 5.50% 5.50% 4.50% 4.50% 4.50% 4.50% 3.50% 3.50% 3.50% 3.50% 2.50% 2.50% 2.50% 2.50% 1.50% 1.50% 1.50% 1.50% 0.50% 0.50% 0.50% 0.50% (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)-0.50% -0.50% -0.50% -0.50% -1.50% -1.50% -1.50% -1.50% -2.50% -2.50% -2.50% -2.50% -3.50% -3.50% -3.50% -3.50% -4.50% -4.50% -4.50% -4.50% -5.50% -5.50% -5.50% -5.50% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 11.50% 11.50% 11.50% 11.50% 10.00% 10.00% 10.00% 10.00% 8.50% 8.50% 8.50% 8.50% 7.00% 7.00% 7.00% 7.00% 5.50% 5.50% 5.50% 5.50% 4.00% 4.00% 4.00% 4.00% 2.50% 2.50% 2.50% 2.50% 1.00% 1.00% 1.00% 1.00% -0.50% -0.50% -0.50% -0.50% -2.00% -2.00% -2.00% -2.00% (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19) -3.50% -3.50% -3.50% -3.50% -5.00% -5.00% -5.00% -5.00% -6.50% -6.50% -6.50% -6.50% -8.00% -8.00% -8.00% -8.00% -9.50% -9.50% -9.50% -9.50% -11.00% -11.00% -11.00% -11.00% -12.50% -12.50% -12.50% -12.50% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 11.00% 11.00% 11.00% 11.00% 9.50% 9.50% 9.50% 9.50% 8.00% 8.00% 8.00% 8.00% 6.50% 6.50% 6.50% 6.50% 5.00% 5.00% 5.00% 5.00% 3.50% 3.50% 3.50% 3.50% 2.00% 2.00% 2.00% 2.00% 0.50% 0.50% 0.50% 0.50% -1.00% -1.00% -1.00% -1.00% -2.50% -2.50% -2.50% -2.50% (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19) -4.00% -4.00% -4.00% -4.00% -5.50% -5.50% -5.50% -5.50% -7.00% -7.00% -7.00% -7.00% -8.50% -8.50% -8.50% -8.50% -10.00% -10.00% -10.00% -10.00% -11.50% -11.50% -11.50% -11.50% -13.00% -13.00% -13.00% -13.00% -14.50% -14.50% -14.50% -14.50% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 11.50% 11.50% 11.50% 11.50% 9.50% 9.50% 9.50% 9.50% 7.50% 7.50% 7.50% 7.50% 5.50% 5.50% 5.50% 5.50% 3.50% 3.50% 3.50% 3.50% 1.50% 1.50% 1.50% 1.50% -0.50% -0.50% -0.50% -0.50% -2.50% -2.50% -2.50% -2.50% -4.50% -4.50% -4.50% -4.50% -6.50% -6.50% -6.50% -6.50% (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)-1 - 0 8. . 5 5 0 0 % % -1 - 0 8. . 5 5 0 0 % % -1 - 0 8. . 5 5 0 0 % % -1 - 0 8. . 5 5 0 0 % % -12.50% -12.50% -12.50% -12.50% -14.50% -14.50% -14.50% -14.50% -16.50% -16.50% -16.50% -16.50% -18.50% -18.50% -18.50% -18.50% -20.50% -20.50% -20.50% -20.50% -22.50% -22.50% -22.50% -22.50% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:20)(cid:27)(cid:29)(cid:3)(cid:48)(cid:40)(cid:36)(cid:49)(cid:3)(cid:37)(cid:44)(cid:36)(cid:54)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:37)(cid:40)(cid:55)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 0.00% 0.00% 0.00% 0.00% -0.50% -0.50% -0.50% -0.50% -1.00% -1.00% -1.00% -1.00% -1.50% -1.50% -1.50% -1.50% -2.00% -2.00% -2.00% -2.00% -2.50% -2.50% -2.50% -2.50% -3.00% -3.00% -3.00% -3.00% (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)-3.50% -3.50% -3.50% -3.50% -4.00% -4.00% -4.00% -4.00% -4.50% -4.50% -4.50% -4.50% -5.00% -5.00% -5.00% -5.00% -5.50% -5.50% -5.50% -5.50% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.00% 0.00% 0.00% 0.00% -0.50% -0.50% -0.50% -0.50% -1.00% -1.00% -1.00% -1.00% -1.50% -1.50% -1.50% -1.50% -2.00% -2.00% -2.00% -2.00% -2.50% -2.50% -2.50% -2.50% -3.00% -3.00% -3.00% -3.00% -3.50% -3.50% -3.50% -3.50% (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19)-4.00% -4.00% -4.00% -4.00% -4.50% -4.50% -4.50% -4.50% -5.00% -5.00% -5.00% -5.00% -5.50% -5.50% -5.50% -5.50% -6.00% -6.00% -6.00% -6.00% -6.50% -6.50% -6.50% -6.50% -7.00% -7.00% -7.00% -7.00% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.00% 0.00% 0.00% 0.00% -0.50% -0.50% -0.50% -0.50% -1.00% -1.00% -1.00% -1.00% -1.50% -1.50% -1.50% -1.50% -2.00% -2.00% -2.00% -2.00% -2.50% -2.50% -2.50% -2.50% -3.00% -3.00% -3.00% -3.00% -3.50% -3.50% -3.50% -3.50% -4.00% -4.00% -4.00% -4.00% -4.50% -4.50% -4.50% -4.50% (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19)-5.00% -5.00% -5.00% -5.00% -5.50% -5.50% -5.50% -5.50% -6.00% -6.00% -6.00% -6.00% -6.50% -6.50% -6.50% -6.50% -7.00% -7.00% -7.00% -7.00% -7.50% -7.50% -7.50% -7.50% -8.00% -8.00% -8.00% -8.00% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.00% 0.00% 0.00% 0.00% -0.50% -0.50% -0.50% -0.50% -1.00% -1.00% -1.00% -1.00% -1.50% -1.50% -1.50% -1.50% -2.00% -2.00% -2.00% -2.00% -2.50% -2.50% -2.50% -2.50% -3.00% -3.00% -3.00% -3.00% -3.50% -3.50% -3.50% -3.50% -4.00% -4.00% -4.00% -4.00% -4.50% -4.50% -4.50% -4.50% (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)-5.00% -5.00% -5.00% -5.00% -5.50% -5.50% -5.50% -5.50% -6.00% -6.00% -6.00% -6.00% -6.50% -6.50% -6.50% -6.50% -7.00% -7.00% -7.00% -7.00% -7.50% -7.50% -7.50% -7.50% -8.00% -8.00% -8.00% -8.00% -8.50% -8.50% -8.50% -8.50% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:20)(cid:28)(cid:29)(cid:3)(cid:39)(cid:44)(cid:54)(cid:51)(cid:40)(cid:53)(cid:54)(cid:44)(cid:50)(cid:49)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:36)(cid:47)(cid:51)(cid:43)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 0.24 0.24 0.24 0.24 0.22 0.22 0.22 0.22 0.20 0.20 0.20 0.20 0.18 0.18 0.18 0.18 0.16 0.16 0.16 0.16 0.14 0.14 0.14 0.14 0.12 0.12 0.12 0.12 (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)0.10 0.10 0.10 0.10 0.08 0.08 0.08 0.08 0.06 0.06 0.06 0.06 0.04 0.04 0.04 0.04 0.02 0.02 0.02 0.02 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.35 0.35 0.35 0.35 0.33 0.33 0.33 0.33 0.30 0.30 0.30 0.30 0.28 0.28 0.28 0.28 0.25 0.25 0.25 0.25 0.23 0.23 0.23 0.23 0.20 0.20 0.20 0.20 0.18 0.18 0.18 0.18 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19)0 0 . . 1 1 3 5 0 0 . . 1 1 3 5 0 0 . . 1 1 3 5 0 0 . . 1 1 3 5 0.10 0.10 0.10 0.10 0.08 0.08 0.08 0.08 0.05 0.05 0.05 0.05 0.03 0.03 0.03 0.03 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.40 0.40 0.40 0.40 0.38 0.38 0.38 0.38 0.35 0.35 0.35 0.35 0.33 0.33 0.33 0.33 0.30 0.30 0.30 0.30 0.28 0.28 0.28 0.28 0.25 0.25 0.25 0.25 0.23 0.23 0.23 0.23 0.20 0.20 0.20 0.20 0.18 0.18 0.18 0.18 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19)0.15 0.15 0.15 0.15 0.13 0.13 0.13 0.13 0.10 0.10 0.10 0.10 0.08 0.08 0.08 0.08 0.05 0.05 0.05 0.05 0.03 0.03 0.03 0.03 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.40 0.40 0.40 0.40 0.38 0.38 0.38 0.38 0.35 0.35 0.35 0.35 0.33 0.33 0.33 0.33 0.30 0.30 0.30 0.30 0.28 0.28 0.28 0.28 0.25 0.25 0.25 0.25 0.23 0.23 0.23 0.23 0.20 0.20 0.20 0.20 (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)0 0 . . 1 1 5 8 0 0 . . 1 1 5 8 0 0 . . 1 1 5 8 0 0 . . 1 1 5 8 0.13 0.13 0.13 0.13 0.10 0.10 0.10 0.10 0.08 0.08 0.08 0.08 0.05 0.05 0.05 0.05 0.03 0.03 0.03 0.03 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:21)(cid:19)(cid:29)(cid:3)(cid:39)(cid:44)(cid:54)(cid:51)(cid:40)(cid:53)(cid:54)(cid:44)(cid:50)(cid:49)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:37)(cid:40)(cid:55)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 0.26 0.26 0.26 0.26 0.24 0.24 0.24 0.24 0.22 0.22 0.22 0.22 0.20 0.20 0.20 0.20 0.18 0.18 0.18 0.18 0.16 0.16 0.16 0.16 0.14 0.14 0.14 0.14 0.12 0.12 0.12 0.12 (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)0.10 0.10 0.10 0.10 0.08 0.08 0.08 0.08 0.06 0.06 0.06 0.06 0.04 0.04 0.04 0.04 0.02 0.02 0.02 0.02 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.35 0.35 0.35 0.35 0.33 0.33 0.33 0.33 0.30 0.30 0.30 0.30 0.28 0.28 0.28 0.28 0.25 0.25 0.25 0.25 0.23 0.23 0.23 0.23 0.20 0.20 0.20 0.20 0.18 0.18 0.18 0.18 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19)0 0 . . 1 1 3 5 0 0 . . 1 1 3 5 0 0 . . 1 1 3 5 0 0 . . 1 1 3 5 0.10 0.10 0.10 0.10 0.08 0.08 0.08 0.08 0.05 0.05 0.05 0.05 0.03 0.03 0.03 0.03 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.40 0.400 0.40 0.40 0.38 0.380 0.38 0.38 0.36 0.360 0.36 0.36 0.34 0.340 0.34 0.34 0.32 0.320 0.32 0.32 0.30 0.300 0.30 0.30 0.28 0.280 0.28 0.28 0.26 0.260 0.26 0.26 0.24 0.240 0.24 0.24 0.22 0.220 0.22 0.22 0.20 0.200 0.20 0.20 0.18 0.180 0.18 0.18 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19)0.16 0.160 0.16 0.16 0.14 0.140 0.14 0.14 0.12 0.120 0.12 0.12 0.10 0.100 0.10 0.10 0.08 0.080 0.08 0.08 0.06 0.060 0.06 0.06 0.04 0.040 0.04 0.04 0.02 0.020 0.02 0.02 0.00 0.000 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.45 0.45 0.45 0.45 0.43 0.43 0.43 0.43 0.40 0.40 0.40 0.40 0.38 0.38 0.38 0.38 0.35 0.35 0.35 0.35 0.33 0.33 0.33 0.33 0.30 0.30 0.30 0.30 0.28 0.28 0.28 0.28 0.25 0.25 0.25 0.25 0.23 0.23 0.23 0.23 0.20 0.20 0.20 0.20 (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)0.18 0.18 0.18 0.18 0.15 0.15 0.15 0.15 0.13 0.13 0.13 0.13 0.10 0.10 0.10 0.10 0.08 0.08 0.08 0.08 0.05 0.05 0.05 0.05 0.03 0.03 0.03 0.03 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:21)(cid:20)(cid:29)(cid:3)(cid:53)(cid:50)(cid:50)(cid:55)(cid:16)(cid:48)(cid:40)(cid:36)(cid:49)(cid:16)(cid:54)(cid:52)(cid:56)(cid:36)(cid:53)(cid:40)(cid:39)(cid:16)(cid:40)(cid:53)(cid:53)(cid:50)(cid:53)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:36)(cid:47)(cid:51)(cid:43)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19)0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19)0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 0.25 0.25 0.25 0.25 0.20 0.20 0.20 0.20 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 0.25 0.25 0.25 0.25 0.20 0.20 0.20 0.20 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:21)(cid:21)(cid:29)(cid:3)(cid:53)(cid:50)(cid:50)(cid:55)(cid:16)(cid:48)(cid:40)(cid:36)(cid:49)(cid:16)(cid:54)(cid:52)(cid:56)(cid:36)(cid:53)(cid:40)(cid:39)(cid:16)(cid:40)(cid:53)(cid:53)(cid:50)(cid:53)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:37)(cid:40)(cid:55)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 1.40 1.40 1.40 1.40 1.35 1.35 1.35 1.35 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19)0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 0.25 0.25 0.25 0.25 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19)0 0 . . 5 6 5 0 0 0 . . 5 6 5 0 0 0 . . 5 6 5 0 0 0 . . 5 6 5 0 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 0.25 0.25 0.25 0.25 0.20 0.20 0.20 0.20 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)0 0 . . 4 5 5 0 0 0 . . 4 5 5 0 0 0 . . 4 5 5 0 0 0 . . 4 5 5 0 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 0.25 0.25 0.25 0.25 0.20 0.20 0.20 0.20 0.15 0.15 0.15 0.15 0.10 0.10 0.10 0.10 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:21)(cid:22)(cid:29)(cid:3)(cid:48)(cid:40)(cid:36)(cid:49)(cid:16)(cid:36)(cid:37)(cid:54)(cid:50)(cid:47)(cid:56)(cid:55)(cid:40)(cid:16)(cid:40)(cid:53)(cid:53)(cid:50)(cid:53)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:36)(cid:47)(cid:51)(cid:43)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 1.45 1.45 1.45 1.45 1.40 1.40 1.40 1.40 1.35 1.35 1.35 1.35 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)0 1 1 1 1 1 . . . . . . 9 0 0 1 1 2 5 0 5 0 5 0 0 0 1 1 1 1 1 . . . . . . . 9 9 0 0 1 1 2 0 5 0 5 0 5 0 0 0 1 1 1 1 1 . . . . . . . 9 9 0 0 1 1 2 0 5 0 5 0 5 0 0 0 1 1 1 1 1 . . . . . . . 9 9 0 0 1 1 2 0 5 0 5 0 5 0 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19)0 0 . . 7 7 0 5 0 0 . . 7 7 0 5 0 0 . . 7 7 0 5 0 0 . . 7 7 0 5 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19)0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:21)(cid:23)(cid:29)(cid:3)(cid:48)(cid:40)(cid:36)(cid:49)(cid:16)(cid:36)(cid:37)(cid:54)(cid:50)(cid:47)(cid:56)(cid:55)(cid:40)(cid:16)(cid:40)(cid:53)(cid:53)(cid:50)(cid:53)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:37)(cid:40)(cid:55)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0 0 0 . . . 6 7 7 5 0 5 0 0 0 . . . 6 7 7 5 0 5 0 0 0 . . . 6 7 7 5 0 5 0 0 . . 7 7 0 5 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19)0.60 0.60 0.60 0.65 0.55 0.55 0.55 0.60 0.50 0.50 0.50 0.55 0.45 0.45 0.45 0.50 0.40 0.40 0.40 0.45 0.35 0.35 0.35 0.40 0.30 0.30 0.30 0.35 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 0.30 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE 500 1,000 2,500 10,000 50,000 100,000 FAN LSE FAN LSE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19)0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 0.25 0.25 0.25 0.25 0.20 0.20 0.20 0.20 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)0 0 . . 4 5 5 0 0 0 . . 4 5 5 0 0 0 . . 4 5 5 0 0 0 . . 4 5 5 0 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 0.25 0.25 0.25 0.25 0.20 0.20 0.20 0.20 0.15 0.15 0.15 0.15 0.10 0.10 0.10 0.10 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:21)(cid:24)(cid:29)(cid:3)(cid:48)(cid:40)(cid:36)(cid:49)(cid:3)(cid:37)(cid:44)(cid:36)(cid:54)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:50)(cid:48)(cid:40)(cid:42)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 34.00% 34.00% 34.00% 34.00% 31.50% 31.50% 31.50% 31.50% 29.00% 29.00% 29.00% 29.00% 26.50% 26.50% 26.50% 26.50% 24.00% 24.00% 24.00% 24.00% 21.50% 21.50% 21.50% 21.50% 19.00% 19.00% 19.00% 19.00% 16.50% 16.50% 16.50% 16.50% 14.00% 14.00% 14.00% 14.00% (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)1 9 1 . . 0 50 0% % 1 9 1 . . 0 50 0% % 1 9 1 . . 0 50 0% % 1 9 1 . . 0 50 0% % 6.50% 6.50% 6.50% 6.50% 4.00% 4.00% 4.00% 4.00% 1.50% 1.50% 1.50% 1.50% -1.00% -1.00% -1.00% -1.00% -3.50% -3.50% -3.50% -3.50% -6.00% -6.00% -6.00% -6.00% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 45.00% 45.00% 45.00% 45.00% 40.00% 40.00% 40.00% 40.00% 35.00% 35.00% 35.00% 35.00% 30.00% 30.00% 30.00% 30.00% 25.00% 25.00% 25.00% 25.00% 20.00% 20.00% 20.00% 20.00% 15.00% 15.00% 15.00% 15.00% (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19) 10.00% 10.00% 10.00% 10.00% 5.00% 5.00% 5.00% 5.00% 0.00% 0.00% 0.00% 0.00% -5.00% -5.00% -5.00% -5.00% -10.00% -10.00% -10.00% -10.00% -15.00% -15.00% -15.00% -15.00% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 50.00% 50.00% 50.00% 50.00% 45.00% 45.00% 45.00% 45.00% 40.00% 40.00% 40.00% 40.00% 35.00% 35.00% 35.00% 35.00% 30.00% 30.00% 30.00% 30.00% 25.00% 25.00% 25.00% 25.00% 20.00% 20.00% 20.00% 20.00% 15.00% 15.00% 15.00% 15.00% (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19) 1 5 0 . . 0 00 0% % 1 5 0 . . 0 00 0% % 1 5 0 . . 0 00 0% % 1 5 0 . . 0 00 0% % 0.00% 0.00% 0.00% 0.00% -5.00% -5.00% -5.00% -5.00% -10.00% -10.00% -10.00% -10.00% -15.00% -15.00% -15.00% -15.00% -20.00% -20.00% -20.00% -20.00% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 55.00% 50.00% 55.00% 55.00% 50.00% 45.00% 50.00% 50.00% 45.00% 40.00% 45.00% 45.00% 40.00% 35.00% 40.00% 40.00% 35.00% 30.00% 35.00% 35.00% 30.00% 25.00% 30.00% 30.00% (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19) 1 1 2 2 5 0 5 0 5 . . . . . 0 0 0 0 0 0 0 0 0 0% % % % % 1 1 2 5 0 5 0 . . . . 0 0 0 0 0 0 0 0% % % % 1 1 2 2 5 0 5 0 5 . . . . . 0 0 0 0 0 0 0 0 0 0% % % % % 1 1 2 2 5 0 5 0 5 . . . . . 0 0 0 0 0 0 0 0 0 0% % % % % 0.00% 0.00% 0.00% 0.00% -5.00% -5.00% -5.00% -5.00% -10.00% -10.00% -10.00% -10.00% -15.00% -15.00% -15.00% -15.00% -20.00% -20.00% -20.00% -20.00% -25.00% -25.00% -25.00% -25.00% 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:21)(cid:25)(cid:29)(cid:3)(cid:39)(cid:44)(cid:54)(cid:51)(cid:40)(cid:53)(cid:54)(cid:44)(cid:50)(cid:49)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:50)(cid:48)(cid:40)(cid:42)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 0.09 0.09 0.09 0.09 0.08 0.08 0.08 0.08 0.07 0.07 0.07 0.07 0.06 0.06 0.06 0.06 0.05 0.05 0.05 0.05 (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.12 0.12 0.12 0.12 0.11 0.11 0.11 0.11 0.10 0.10 0.10 0.10 0.09 0.09 0.09 0.09 0.08 0.08 0.08 0.08 0.07 0.07 0.07 0.07 0.06 0.06 0.06 0.06 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19)0.05 0.05 0.05 0.05 0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.13 0.13 0.13 0.13 0.12 0.12 0.12 0.12 0.11 0.11 0.11 0.11 0.10 0.10 0.10 0.10 0.09 0.09 0.09 0.09 0.08 0.08 0.08 0.08 0.07 0.07 0.07 0.07 0.06 0.06 0.06 0.06 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19)0.05 0.05 0.05 0.05 0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE 0.14 0.14 0.14 0.14 0.13 0.13 0.13 0.13 0.12 0.12 0.12 0.12 0.11 0.11 0.11 0.11 0.10 0.10 0.10 0.10 0.09 0.09 0.09 0.09 0.08 0.08 0.08 0.08 0.07 0.07 0.07 0.07 (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)0 0 . . 0 0 5 6 0 0 . . 0 0 5 6 0 0 . . 0 0 5 6 0 0 . . 0 0 5 6 0.04 0.04 0.04 0.04 0.03 0.03 0.03 0.03 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.00 0.00 0.00 0.00 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE QMLE NGQMLE VTQMLE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE VTNGQMLE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:21)(cid:26)(cid:29)(cid:3)(cid:53)(cid:50)(cid:50)(cid:55)(cid:16)(cid:48)(cid:40)(cid:36)(cid:49)(cid:16)(cid:54)(cid:52)(cid:56)(cid:36)(cid:53)(cid:40)(cid:39)(cid:16)(cid:40)(cid:53)(cid:53)(cid:50)(cid:53)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:50)(cid:48)(cid:40)(cid:42)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 1.45 1.45 1.45 1.45 1.40 1.40 1.40 1.40 1.35 1.35 1.35 1.35 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)0 0 . . 8 9 5 0 0 0 . . 8 9 5 0 0 0 . . 8 9 5 0 0 0 . . 8 9 5 0 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19)0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19)0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 0.25 0.25 0.25 0.25 0.20 0.20 0.20 0.20 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0 0 . . 6 6 0 5 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 0.25 0.25 0.25 0.25 0.20 0.20 0.20 0.20 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE

(cid:41)(cid:44)(cid:42)(cid:56)(cid:53)(cid:40)(cid:3)(cid:21)(cid:27)(cid:29)(cid:3)(cid:48)(cid:40)(cid:36)(cid:49)(cid:16)(cid:36)(cid:37)(cid:54)(cid:50)(cid:47)(cid:56)(cid:55)(cid:40)(cid:16)(cid:40)(cid:53)(cid:53)(cid:50)(cid:53)(cid:3)(cid:38)(cid:50)(cid:48)(cid:51)(cid:54)(cid:3)(cid:11)(cid:50)(cid:48)(cid:40)(cid:42)(cid:36)(cid:12) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:19)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:27)(cid:19) (cid:540)(cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:28)(cid:28) 1.45 1.45 1.45 1.45 1.40 1.40 1.40 1.40 1.35 1.35 1.35 1.35 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 (cid:536)(cid:3)(cid:32)(cid:3)(cid:27)(cid:17)(cid:24)(cid:19)0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 0.80 0.80 0.80 0.80 0.75 0.75 0.75 0.75 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.45 1.45 1.45 1.45 1.40 1.40 1.40 1.40 1.35 1.35 1.35 1.35 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:24)(cid:19)0 0 . . 7 8 5 0 0 0 . . 7 8 5 0 0 0 . . 7 8 5 0 0 0 . . 7 8 5 0 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.45 1.45 1.45 1.45 1.40 1.40 1.40 1.40 1.35 1.35 1.35 1.35 1.30 1.30 1.30 1.30 1.25 1.25 1.25 1.25 1.20 1.20 1.20 1.20 1.15 1.15 1.15 1.15 1.10 1.10 1.10 1.10 1.05 1.05 1.05 1.05 1.00 1.00 1.00 1.00 0.95 0.95 0.95 0.95 0.90 0.90 0.90 0.90 0.85 0.85 0.85 0.85 (cid:536)(cid:3)(cid:32)(cid:3)(cid:23)(cid:17)(cid:19)(cid:19)0 0 . . 7 8 5 0 0 0 . . 7 8 5 0 0 0 . . 7 8 5 0 0 0 . . 7 8 5 0 0.70 0.70 0.70 0.70 0.65 0.65 0.65 0.65 0.60 0.60 0.60 0.60 0.55 0.55 0.55 0.55 0.50 0.50 0.50 0.50 0.45 0.45 0.45 0.45 0.40 0.40 0.40 0.40 0.35 0.35 0.35 0.35 0.30 0.30 0.30 0.30 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE FAN LSE FAN LSE 1.45 1.45 1.45 1.45 1.40 1.40 1.40 1.40 (cid:536)(cid:3)(cid:32)(cid:3)(cid:22)(cid:17)(cid:24)(cid:19)0 0 0 0 0 1 1 1 1 1 1 1 1 . . . . . . . . . . . . . 7 8 8 9 9 0 0 1 1 2 2 3 3 5 0 5 0 5 0 5 0 5 0 5 0 5 0 0 0 0 1 1 1 1 1 1 1 1 . . . . . . . . . . . . 8 8 9 9 0 0 1 1 2 2 3 3 0 5 0 5 0 5 0 5 0 5 0 5 0 0 0 0 1 1 1 1 1 1 1 1 . . . . . . . . . . . . 8 8 9 9 0 0 1 1 2 2 3 3 0 5 0 5 0 5 0 5 0 5 0 5 0 0 0 0 0 1 1 1 1 1 1 1 1 . . . . . . . . . . . . . 7 8 8 9 9 0 0 1 1 2 2 3 3 5 0 5 0 5 0 5 0 5 0 5 0 5 0.70 0.75 0.75 0.70 0 0 0 0 0 0 0 0 . . . . . . . . 3 3 4 4 5 5 6 6 0 5 0 5 0 5 0 5 500 1,000 2,500 10,000 50,000 100,000 0 0 0 0 0 0 0 0 0 . . . . . . . . . 3 3 4 4 5 5 6 6 7 0 5 0 5 0 5 0 5 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . 3 3 4 4 5 5 6 6 7 0 5 0 5 0 5 0 5 0 0 0 0 0 0 0 0 0 . . . . . . . . 3 3 4 4 5 5 6 6 0 5 0 5 0 5 0 5 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE 500 1,000 2,500 10,000 50,000 100,000 500 1,000 2,500 10,000 50,000 100,000 NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE NGQMLE VTQMLE VTNGQMLE FAN LSE FAN LSE