Moments of the ARMA–EGARCH model

21 Pages·2002·0.15 MB·English

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:[email protected]
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
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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
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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 
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" (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
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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
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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).
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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).
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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)
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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).
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