EconometricsJournal(2003),volume6,pp.146–166. Moments of the ARMA–EGARCH model M. KARANASOS1 AND J. KIM UniversityofYork,Heslington,York,UK Received: February2003 Summary This paper considers the moment structure of the general ARMA–EGARCH model.Inparticular,wederivetheautocorrelationfunctionofanypositiveintegerpowerof thesquarederrors.Inaddition,weobtaintheautocorrelationsofthesquaresoftheobserved process and cross correlations between the levels and the squares of the observed process. Finally,thepracticalimplicationsoftheresultsareillustratedempiricallyusingdailydataon fourEastAsiaStockIndices. Keywords: Autocorrelations,ExponentialGARCH,Stockreturns. 1.INTRODUCTION One of the principal empirical tools used to model volatility in asset markets has been the ARCHclassofmodels.FollowingEngle’s(1982)ground-breakingidea,severalformulationsof conditionally heteroscedastic models (e.g. GARCH, Fractional Integrated GARCH, Switching GARCH, Component GARCH) have been introduced in the literature (see, for example, the survey of Bollerslev et al. (1994)). These models form an immense ARCH family. Many of the proposed GARCH models include a term that can capture correlation between returns and conditional variance. Models with this feature are often termed asymmetric or leverage volatility models.2 One of the earliest asymmetric GARCH models is the EGARCH (exponential generalized ARCH) model of Nelson (1991). In contrast to the conventional GARCHspecification,whichrequiresnon-negativecoefficients,theEGARCHmodeldoesnot impose non-negativity constraints on the parameter space since it models the logarithm of the conditionalvariance. Although the literature on the GARCH/EGARCH models is quite extensive, relatively few papers have examined the moment structure of models where the conditional volatility is time dependent. Karanasos (1999) and He and Tera¨svirta (1999) derived the autocorrelations of the squarederrorsfortheGARCH(p,q)model,whileKaranasos(2001)obtainedtheautocorrelation functionoftheobservedprocessfortheARMA-GARCH-in-meanmodel.Demos(2002)studied the autocorrelation structure of a model that nests both the EGARCH and stochastic volatility specifications.Heetal.(2002)consideredthemomentstructureoftheEGARCH(1,1)model. 1Correspondingaddress:DepartmentofEconomicsandRelatedStudies,UniversityofYork,York,YO105DD,UK. E-mail:mk16@york.ac.uk 2Theasymmetricresponseofvolatilitytopositiveandnegativeshocksiswellknowninthefinanceliteratureasthe leverageeffectofstockmarketreturns(Black,1976).Researchershavefoundthatvolatilitytendstoriseinresponse to‘badnews’(excessreturnslowerthanexpected)andtofallinresponseto‘goodnews’(excessreturnshigherthan expected). (cid:13)c RoyalEconomicSociety2003.PublishedbyBlackwellPublishingLtd,9600GarsingtonRoad,OxfordOX42DQ,UKand350Main Street,Malden,MA,02148,USA. MomentsoftheARMA–EGARCHmodel 147 This paper focuses solely on the moment structure of the general ARMA(r,s)– EGARCH(p,q) model. It would be useful to know the properties of the autocorrelation func- tionofpower-transformedobservationswhencomparingtheEGARCHmodelwiththestandard GARCHmodel.Inparticular,possibledifferencesinthemomentstructureofthesemodelsmay shedlightonthesuccessoftheEGARCHmodelinapplications. We contribute to the aforementioned literature by deriving: (i) the autocorrelation function ofanypositiveintegerpowerofthesquarederrors,(ii)thecrosscorrelationsbetweenthelevels andthesquaresofboththeobservedprocessandthesquarederrors,and(iii)theautocorrelations ofthesquaredobservations.Toobtainthetheoreticalresultsandtocarryouttheestimation,we assumethattheinnovationsaredrawnfromeitherthenormal,doubleexponential,orgeneralized errordistributions.Tofacilitatemodelidentification,theresultsfortheautocorrelationfunction of the power-transformed errors can be applied so that properties of the observed data can be comparedwiththetheoreticalpropertiesofthemodels. The derivation of the autocorrelations of the fitted power-transformed values and their comparison with the corresponding sample equivalents will help the investigator (a) to decide whichisthemostappropriatemethodofestimation(e.g.maximumlikelihoodestimation(MLE), minimum distant estimator (MDE)) for a specific model, (b) to choose, for a given estimation technique,themodel(e.g.asymmetricpowerARCH(APARCH),EGARCH)thatbestreplicates certainstylizedfactsofthedataand,(c)inconjunctionwiththevariousmodelselectioncriteria, toidentifytheoptimalorderofthechosenspecification. This paper is organized as follows. Section 2.1 presents the ARMA(r,s)–EGARCH(p,q) process.Section2.2investigatestheautocorrelationfunctionofanypositiveintegerpowerofthe squared errors for the EGARCH model. Section 2.3 derives the cross correlations between the levelsandthesquaresoftheARMA–EGARCHprocess.Section2.4providestheautocorrelation function of the squared observations. Section 3 discusses the data and presents the empirical results.Intheconclusionswesuggestfuturedevelopments.Proofsarefoundintheappendices. 2.ARMA–EGARCHMODEL 2.1.ARMA(r,s)–EGARCH(p,q)process OfthemanydifferentasymmetricGARCHspecificationstheEGARCHmodelhasbecomeone of the most common. Here we examine the general ARMA(r,s)–EGARCH(p,q) model. The stochasticprocess{y }isassumedtobeacausalARMA(r,s)processsatisfying t 8(L)y =b+2(L)ε , (2.1a) t t where r Y 8(L)≡ (1−φ L), (2.1b) l l=1 s X 2(L)≡1+ θ Ll. (2.1c) l l=1 Further,let{ε }beareal-valuedtimestochasticprocessgeneratedby t 1 ε =e h2, (2.2) t t t (cid:13)c RoyalEconomicSociety2003 148 M.KaranasosandJ.Kim where {e } is a sequence of independent, identically distributed (i.i.d.) random variables with t mean zero and variance 1. h is positive with probability one and is a measurable function of t 6t−1, which in turn is the sigma-algebra generated by {εt−1,εt−2,...}. That is, ht denotes the conditional variance of the errors {εt}, (εt|6t−1) ∼ (0,ht). As regards ht, we assume that it followsanEGARCH(p,q)process B(L)ln(h )=ω+C(L)z , (2.3a) t t ε (cid:20)(cid:12) ε (cid:12) (cid:12) ε (cid:12)(cid:21) zt−l ≡d√t−l +γ (cid:12)(cid:12)√t−l (cid:12)(cid:12)−E(cid:12)(cid:12)√t−l (cid:12)(cid:12) , (2.3b) ht−l (cid:12) ht−l(cid:12) (cid:12) ht−l(cid:12) where q X C(L)≡ c Ll, (2.3c) l l=1 p Y B(L)≡ (1−β L). (2.3d) l l=1 VariouscasesoftheEGARCH(p,q)modelhavebeenappliedbyresearchers.Morespecifically, DonaldsonandKamstra(1997)foundthattheoptimalEGARCHspecificationfortheNIKKEI stockindexwasaflexible3,2.Huetal.(1997)foundthatinthepre-EMSperiod,themajorityof theEuropeancurrenciesfollowedanAR(5)–EGARCH(4,4)model. 2.2.Higher-ordermomentsofthesquarederrors Although the EGARCH model was introduced over a decade ago and has been widely used in empirical applications, its statistical properties have only recently been examined. Engle and Ng (1993) artificially nested the GARCH and EGARCH models, estimated this nested specification,andthenappliedlikelihoodratio(LR)tests(seealsoHuetal.(1997)).Hentschel (1995)developedafamilyofasymmetricGARCHmodelsthatnestsboththeAPARCHmodel andtheEGARCHmodel. InthissectionwefocussolelyonthemomentstructureofthegeneralEGARCH(p,q)model. Assumption1. All the roots of the autoregressive polynomial B(L) lie outside the unit circle (covariance–stationaritycondition). Assumption2. The polynomials C(L) and B(L) in (2.3c) and (2.3d) respectively, have no commonleftfactorsotherthanunimodularones(irreducibilitycondition).Moreover,β ,c 6=0. p q Inwhatfollowsweexamineonlythecasewherealltherootsoftheautoregressivepolynomial B(L) in (2.3d) are distinct. The following proposition establishes the lag-m autocorrelation of {ε2k},thekthpowerofthesquarederrors:ρ(ε2k,ε2k )≡Corr(ε2k,ε2k ). t t t−m t t−m Proposition1. LetAssumptions1and2hold.SupposefurtherthatE(e4k)<∞,E[exp(2kz )]< t t ∞andE[e2kexp(kz )]<∞∀t,foranyfinitepositivescalark.Thentheautocorrelationofthe t t (cid:13)c RoyalEconomicSociety2003 MomentsoftheARMA–EGARCHmodel 149 kthpowerofthesquarederror{ε2k},atlagm (m ∈N),hastheform t (m−2 ρ(εt2k,εt2−km)=E(et2k) Y (cid:2)E(cid:0)exp(cid:0)ξ0,k,izt−i−1(cid:1)(cid:1)(cid:3)E(cid:2)et2−kmexp(cid:0)ξ0,k,m−1zt−m(cid:1)(cid:3) 2 2 i=0  ∞ "∞ #2 ×Y[E(exp(ξm,k,izt−m−i−1))]−E(et2k) Y(cid:2)E(cid:0)exp(cid:0)ξ0,k,izt−i−1(cid:1)(cid:1)(cid:3)  2  i=0 i=0 ( ∞ Y × E(et4k) [E(exp(ξ0,k,izt−i−1))]−[E(et2k)]2 i=0 "∞ #2−1 × Y(cid:2)E(cid:0)exp(cid:0)ξ0,k,izt−i−1(cid:1)(cid:1)(cid:3)  , 2  i=0 (2.4a) where p X ξm,k,i ≡k ζf(λf,m+i+1+λf,i+1), (2.4b) f=1 with  i−1 λfi ≡nP=0ci−nβnf, if i ≤q, (2.4c) λ βi−q, if i >q, fq f βp−1 ζ ≡ f . (2.4d) f Qp (β −β ) n6=f f n n=1 Notethat,whenm =1,thefirstproducttermintheright-handsideof(2.4a)isreplacedby1. Proof. SeeAppendixA. 2 Heetal.(2002)derivedtheautocorrelationsofpositivepowersoftheabsolute-valuederrors oftheEGARCH(1,1)model. In the following theorem we provide the autocorrelations of the kth power of the squared errors{ε2k}oftheEGARCH(p,q)model. t Theorem1. Letkbeanyfinitepositiveinteger.Then,whenthedistributionof{e }isgeneralized t error,thesecondmomentandtheautocorrelationfunctionof{ε2k}aregivenby t 2k"ω−γ0(cid:0)0v2(cid:0)(cid:1)1λ(cid:1)2v1 Plq=1cl# E(εt4k)=exp Qp (1v−β ) µ(4gk)B0(g,k), (2.5a)  f=1 f  (cid:13)c RoyalEconomicSociety2003 150 M.KaranasosandJ.Kim " (cid:18) (cid:19)2# µ(g) A(g) D(g) B(g) −µ(g) B(g) 2k m−1,k m−1,k m,k 2k 0,k ρ(ε2k,ε2k )= 2 , (A(g) ≡1), (2.5b) t t−m " (cid:18) (cid:19)2# 0,k µ(g)B(g)− µ(g)B(g) 4k 0,k 2k 0,k 2 where A(g) ≡mY−1(X∞ h2v1λξ iτ (cid:2)(γ +d)τ +(γ −d)τ(cid:3) 0(cid:0)1+vτ(cid:1) ), (2.5c) m,k i=0 τ=0 0,k2,i 20(cid:0)v1(cid:1)0(1+τ) Bm(g,)k ≡Y∞ (X∞ h2v1λξm,k,iiτ [(γ +d)τ +(γ −d)τ]20(cid:0)01(cid:1)(cid:0)01+v(1τ(cid:1)+τ)), (2.5d) i=0 τ=0 v and D(g) ≡22vkλ2kX∞ (cid:16)λ2v1ξ (cid:17)τ [(γ +d)τ +(γ −d)τ] 0(cid:0)τ+2vk+1(cid:1) , (2.5e) m,k τ=0 0,k2,m 20(cid:0)v1(cid:1)0(1+τ) (cid:2)0(cid:0)1(cid:1)(cid:3)k−10(cid:0)2k+1(cid:1) µ(g)≡ v v , (2.5f) 2k (cid:2)0(cid:0)3(cid:1)(cid:3)k v with λ≡(2−v20(cid:18)1(cid:19)(cid:20)0(cid:18)3(cid:19)(cid:21)−1)12 , v v where ξm,k,i is defined in Proposition 1, v are the degrees of freedom of the generalized error distribution and 0(·) is the Gamma function. When v > 1, the summations in (2.5c)–(2.5e) are finite; when v < 1, the three summations are finite if and only if ξ γ +|ξ d| ≤ 0, 0,k,i 0,k,i ξm,k,iγ +|ξm,k,id|≤0andξ0,k,mγ +|ξ0,k,md|≤0,respectively(seeNel2son,(1991)2). 2 2 Proof. SeeAppendixA. 2 Oneofthemostwidelyusedmodelsinfinancialeconomicstodescribeatimeseries,r ,of t thereturnsfromsomeassetisthemartingaleprocess p r ≡ε =e h , t t t t wheree isi.i.d.(0,1)andh isaGARCHtypeprocess. t t Inmanyapplicationsinfinancialeconomics,itisnotreasonabletoassumethenormalityof e ,becauseofthesubstantialexcesskurtosispresentintheconditionaldensityofreturns.Hence t investigators often use MLE assuming some fat-tailed conditional density such as generalized error. Therefore, when it comes to model identification, practitioners in this area may find the resultsinTheorem1quiteuseful. In the following proposition, when the innovations are drawn from the double exponential distribution,weprovidetheautocorrelationsof{ε2k}foranyfinitepositiveintegerk. t (cid:13)c RoyalEconomicSociety2003 MomentsoftheARMA–EGARCHmodel 151 Proposition2. Whenthedistributionof{e }isdoubleexponential,theautocorrelationfunction t ofthekthpowerofthesquarederrorsisgivenby " (cid:18) (cid:19)2# µ(d) A(d) D(d) B(d) −µ(d) B(d) 2k m−1,k m−1,k m,k 2k 0,k ρ(ε2k,ε2k )= 2 , (A(d) ≡1), (2.6a) t t−m " (cid:18) (cid:19)2# 0,k µ(d)B(d)− µ(d)B(d) 4k 0,k 2k 0,k 2 with √ m−1 2− 2ξ γ A(d) ≡ Y √ 0,k2,i , (2.6b) m,k i=0 2− 2ξ0,k,iγ +ξ02,k,i(γ2−d2) √ 2 B(d) ≡Y∞ √ 2− 2ξm,k,iγ , (2.6c) m,k i=02−2 2ξm,k,iγ +ξm2,k,i(γ2−d2) and D(d) ≡2−(k+1)0(2k+1) m,k ( " ξ (γ +d)# " ξ (γ −d)#) × F 2k+1; 0,k2,m√ +F 2k+1; 0,k2,m√ , (2.6d) 2 2 µ(d)≡ 0(2k+1), (2.6e) 2k 2k where F(·) is the hypergeometric function (see Abadir, (1999)) and ξm,k,i is defi√ned in Proposition 1. Expressions (2.6b) and (2.6c) hold if and only if ξ γ +|ξ d| < 2 and √ 0,k2,i 0,k2,i ξm,k,iγ +|ξm,k,id| < √2, respectively; the right-hand side of (2.6d) converges if and only if ξ γ +|ξ d|< 2(seeNelson,(1991)). 0,k,m 0,k,m 2 2 Proof. SeeAppendixA. 2 AlsonotethatthecoefficientsoftheWoldrepresentationofthekthpoweroftheconditional variance are needed for the computation of the autocorrelations of the kth power of the squarederrors(seeDemos(2002)).InAppendixA(seeequation(A.1))weprovideexactform solutionswhichexpresstheabovecoefficientsintermsoftheparametersofthemovingaverage polynomialandtherootsoftheautoregressivepolynomial. In the following proposition, when the errors are conditionally normal, we derive the autocorrelationfunctionofthekthpowerofthesquarederrors{ε2k},k ∈N. t Proposition3. Whenthedistributionof{e }isnormal,theautocorrelationsofthekthpowerof t thesquarederrorsaregivenby " (cid:18) (cid:19)2# µ(n) A(n) D(n) B(n) −µ(n) B(n) 2k m−1,k m−1,k m,k 2k 0,k ρ(ε2k,ε2k )= 2 , (A(n) ≡1), (2.7a) t t−m " (cid:18) (cid:19)2# 0,k µ(n)B(n)− µ(n)B(n) 4k 0,k 2k 0,k 2 (cid:13)c RoyalEconomicSociety2003 152 M.KaranasosandJ.Kim with A(mn,)k ≡mYi=−01exp(γ +d2)2ξ02,k2,i12h1+exp(cid:16)−2γdξ02,k2,i(cid:17)i+ (γ +√d2)πξ0,k2,i × F1;3;(γ +d)2ξ02,k2,i+ (γ −√d)ξ0,k2,i ×F1;3;(γ −d)2ξ02,k2,i, 2 2 2π 2 2  (2.7b) B(n) ≡Y∞ (exp (γ +d)2ξm2,k,i!1[1+exp(−2γdξ2 )]+ (γ +√d)ξm,k,i m,k 2 2 m,k,i 2π i=0 × F 1;3;(γ +d)2ξm2,k,i!+ (γ −√d)ξm,k,i ×F 1;3;(γ −d)2ξm2,k,i!), 2 2 2π 2 2 (2.7c) and Dm(n,)k ≡ 12∂hξ0,k,m(∂γ −d)i2k expξ02,k2,m(2γ −d)2"1+8 ξ0,k2,m√(γ2−d)!# 2 + ∂ expξ02,k2,m(γ +d)2"1+8 ξ0,k2,m√(γ +d)!#, ∂hξ0,k,m(γ +d)i2k  2 2  2 (2.7d) k µ(n)≡ Y[2k−(2j −1)], (2.7e) 2k j=1 where∂ denotespartialderivative,8(·)istheerrorfunctionofthestandardnormaldistribution andξm,k,i isgivenby(2.4b). Proof. SeeAppendixA. 2 Several previous articles dealing with financial market data—e.g. Dacorogna et al. (1993), Dingetal.(1993)andMulleretal.(1997)—havecommentedonthebehavioroftheautocorre- lationfunctionofpositivepowersofthesquaredreturns,andthedesirabilityofhavingamodel which comes close to replicating certain stylized facts in the data (abstracted from Baillie and Chung (2001)). In this respect, one can apply the results in this section to check whether the EGARCH model can effectively replicate the observed pattern of autocorrelations of power- transformedreturns. 2.3.Dynamicasymmetry In this section we examine the cross correlations between the levels and the squares of the ARMA–EGARCHprocessin(2.1)–(2.3). (cid:13)c RoyalEconomicSociety2003 MomentsoftheARMA–EGARCHmodel 153 Proposition4. Let the distribution of {e } be generalized error and k a finite positive integer. t SupposefurtherthatE(e4k)<∞,E[e2k−1exp(kz )]<∞andE[exp(2kz )]<∞∀t.Then,if t t t t Assumptions1and2hold,thecrosscorrelationsbetweenthe2kthand(2k−1)thpowersof{ε } t aregivenby µ(g)A(g) D(g) B(g) ρ(ε2k,ε2k−1)= 2k m−1,k m−1,(k) m,(k) , (m ∈N), (2.8a) t t−m s(cid:20) (cid:21) µ(g)B(g)−(B(g))2 µ(g) B(g) 4k 0,k 0,k 4k−2 0,(4k−1) 2 4 with Dm(g,)(k)≡22kv−1λ2k−1Xτ∞=0(cid:16)λ2v1ϕ0,2k4+1,m(cid:17)τ (cid:2)(γ +d)τ −(γ −d)τ(cid:3)20(cid:0)0v1(cid:0)(cid:1)τ0+v(21k+(cid:1) τ), (2.8b) Bm(g,)(k)≡Y∞ (X∞ h2v1λϕm,k,iiτ [(γ +d)τ +(γ −d)τ]20(cid:0)01(cid:1)(cid:0)01+v(1τ(cid:1)+τ)), (2.8c) i=0 τ=0 v and p X ϕm,k,i ≡ (kλf,m+i+1+(k−0.5)λf,i+1)ζf, f=1 where A(g), B(g) and µ(g) are defined in Theorem 1. Note that, when m is a negative integer, m,k m,k 4k ρ(ε2k,ε2k−1)=0. t t−m TheproofofProposition4issimilartothatofTheorem1. Demos(2002)derivedthecrosscorrelationsbetweenthelevelsandthesquaresofanobser- ved series, under the assumption that the mean parameter is time varying and the conditional variancefollowsaflexibleparameterizationwhichneststheautoregressivestochasticvolatility and the exponential GARCH specifications.3 In the same spirit, the following theorem obtains the cross correlations between the levels and the squares of the ARMA(r,s)–EGARCH(p,q) processin(2.1)–(2.3). Assumption3. Alltherootsoftheautoregressivepolynomial8(L)lieoutsidetheunitcircle. Assumption4. Thepolynomials8(L)and2(L)areleftcoprime. Theorem2. Let Assumptions 1–4 hold. Suppose further that E(e4) < ∞, E[e exp(z )] < ∞ t t t andE[exp(2z )]<∞∀t.Thenthecrosscorrelationsbetweenthesquaresandthelevelsof{y } t t aregivenby (cid:18)η−1(cid:19)12 ρ(yt2,yt−m)= κ −1 (Fm +Hm), (m ∈N), (2.9a) 3Demoscalledthismodeltimevaryingparametergeneralizedstochasticvolatilityinmean(TVP-GSV-M). (cid:13)c RoyalEconomicSociety2003 154 M.KaranasosandJ.Kim where Pj+m−1P∞ δ2δ ρ(cid:0)ε2,ε (cid:1) F ≡ i=0 j=0 i j t t−(j+m−i) , (2.9b) m (cid:0)P∞ δ2(cid:1)2 l=0 l H ≡ 2Pi∞=j+m+1P∞j=0δiδjδj+mρ(cid:0)εt2,εt−(i−j−m)(cid:1). (2.9c) m (cid:0)P∞ δ2(cid:1)2 l=0 l Furthermore (cid:18)η−1(cid:19)12 ρ(y ,y2 )= (K +L ), (m ∈N), (2.10a) t t−m κ −1 m m with P∞ P∞ δ δ2ρ(cid:0)ε2,ε (cid:1) K ≡ i=j+m+1 j=0 i j t t−(i−j−m) , (2.10b) m (cid:0)P∞ δ2(cid:1)2 l=0 l L ≡ 2Pi∞=j+1P∞j=0δiδjδj+mρ(cid:0)εt2,εt−(i−j)(cid:1), (2.10c) m (cid:0)P∞ δ2(cid:1)2 l=0 l whereη andκ denotethekurtosisofε and y respectively;δ istheithcoefficientintheWold t t i representation of the ARMA(p,q) process in (2.1) (see equation (B.2a)). Note that when the distributionof{et}isgeneralizederror,ρ(εt2,εt−m)isgiveninProposition4,andη,κ aregiven inProposition5below. Proof. SeeAppendixB. 2 Also observe that when there is no leverage effect (d = 0), the D(g) term in (2.8b) is m,(k) zero and hence the cross correlations between the levels and the squares of both the errors and the observed process are zero. Demos (2002), for the TVP-GSV-M model, does not need the asymmetric EGARCH effect to obtain dynamic asymmetry even under the assumption of conditionalnormality. 2.4.Autocorrelationsofthesquaredobservations In this section, we establish the autocorrelation properties of the squares of the ARMA– EGARCH process in (2.1)–(2.3). Demos (2002) obtained the autocorrelation function of the squaresoftheobservedseriesfortheTVP-GSV-Mmodel. The result presented in the following proposition, which is a special case of Theorem 2 in PalmaandZevallos(2001),ishighlyrelevantsinceithelpstoidentifythenatureoftheprocess. By analyzing the autocorrelation function of the squared series it is possible to discard those theoreticalmodelswhichareincompatiblewiththedataunderstudy. Proposition5. LetAssumptions1–4hold.SupposefurtherthatE(e4) < ∞,E[exp(2z )] < ∞ t t and E[e2exp(z )] < ∞ ∀ t. Then, when the distribution of {ε } is generalized error, the t t t autocorrelationfunctionof{y2}isgivenby t ρ(y2,y2 )= 2[ρ(yt,yt−m)]2 + (κ −3)0m + η−1[G +21 −31 0 ], (m ∈N), t t−m κ −1 (κ −1) κ −1 m m 0 m (2.11a) (cid:13)c RoyalEconomicSociety2003 MomentsoftheARMA–EGARCHmodel 155 where P∞ δ2δ2 0 ≡ i=0 i i+m, (2.11b) m P∞ δ4 l=0 l P∞ P∞ δ δ δ δ ρ(cid:0)ε2,ε2 (cid:1) 1 ≡ i=0 j=0 i j i+m j+m t t−(i−j) , (2.11c) m (cid:0)P∞ δ2(cid:1)2 l=0 l P∞ P∞ δ2δ2ρ(cid:0)ε2,ε2 (cid:1) G ≡ i=0 j=0 i j t t−(m+j−i) , (2.11d) m (cid:0)P∞ δ2(cid:1)2 l=0 l and 2η(cid:0)P∞ δ4(cid:1) κ ≡3− i=0 i +3(η−1)1 , (2.11e) (cid:0)P∞ δ2(cid:1)2 0 l=0 l E(cid:0)ε4(cid:1) η≡ t , (2.11f) (cid:2)E(cid:0)ε2(cid:1)(cid:3)2 t where ρ(cid:0)ε2,ε2 (cid:1) and E(ε4) are given in Theorem 1; δ is defined in Theorem 2 and t t−(i−j) t i ρ(yt,yt−m)isthelag-m autocorrelationof{yt}. The significance of the above result is that it allows us to establish whether the ARMA– EGARCH model in (2.1)–(2.3) is capable of reproducing key features exhibited by the data. Thesefeaturesinclude,forexample,timeserieswithverylittleautocorrelationbutwithstrongly dependent squares. Another potential motivation for the derivation of the results in Theorem 2 is that the autocorrelations of the squared process in (2.1) can be used to estimate the ARMA andGARCHparametersin(2.1)and(2.3)respectively.TheapproachistousetheMDE,which estimates the parameters by minimizing the mahalanobis generalized distance of a vector of sample autocorrelations from the corresponding population autocorrelations (see Baillie and Chung(2001)). Thefollowingpropositionprovidesthelag-m autocorrelationof{hkt},k ∈R+. Proposition6. LetAssumptions1and2hold.SupposefurtherthatE[exp(2kz )]<∞∀t.Then, t whenthedistributionof{e }isgeneralizederror,theautocorrelationfunctionofthekthpower t oftheconditionalvarianceisgivenby(2.5b)wherenowtheterms D(g) andµ(g) arereplaced (g) (g) by1,and A isreplacedby A . m−1,k m,k Proof. SeeAppendixB. 2 Demos(2002)derivedtheautocorrelationsofthekthpoweroftheconditionalvariancefor theGSVmodelundertheassumptionofconditionalnormality. Next,consideraprocess y governedby t yt =E(yt |6t−1)+εt. Further,supposethattheconditionalmeanof y ,giveninformationthroughtimet −1,is t E(yt |6t−1)=δhkt, (k >0). (cid:13)c RoyalEconomicSociety2003

Moments of the ARMA–EGARCH model

Similar Moments of the ARMA–EGARCH model