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Journal of Business & Economic Statistics

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Score Tests for Hyperbolic GARCH Models

Muyi Li, Guodong Li & Wai Keung Li

To cite this article: Muyi Li, Guodong Li & Wai Keung Li (2011) Score Tests for Hyperbolic

GARCH Models, Journal of Business & Economic Statistics, 29:4, 579-586, DOI: 10.1198/ jbes.2011.10024

To link to this article: http://dx.doi.org/10.1198/jbes.2011.10024

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Supplementary materials for this article are available online. Please click the JBES link athttp://pubs.amstat.org.

Score Tests for Hyperbolic GARCH Models

Muyi L

School of Economics and Wang Yanan Institute for Studies in Economics, Xiamen Unviersity, Xiamen, China (limuyi1981@gmail.com)

Guodong L

and Wai Keung L

Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong (gdli@hku.hk;hrntlwk@hku.hk)

Davidson(2004) recently proposed the hyperbolic GARCH model to capture the phenomenon of long-range dependence in volatility, with the extent of such dependence measured by the geometric or hy-perbolic decay of the coefficients in an ARCH(∞) model. In this article, we reinterpret the hyperbolic GARCH model by building a link with the common GARCH model, and construct a simplified score test to check the presence of the hyperbolic decay. We derive the asymptotic of the test statistic under the null hypothesis and the local alternatives. We conduct Monte Carlo simulation experiments to study the perfor-mance of this test, and report an illustration on two log return sequences. This article has supplementary material online.

KEY WORDS: Hyperbolic decay; Long-range dependence. 1. INTRODUCTION

Many asset prices exhibit the phenomena of long-range de-pendence in their squared or absolute returns (seeTsay 2005; Li and Li 2008a; etc.), and many models have been proposed to capture such behavior (seeDing, Granger, and Engle 1993; Breidt, Crato, and Lima 1998;Giraitis, Leipus, and Surgailis 2009and references therein). Among these, parametric models within the framework of ARCH(_∞)(Robinson 1991) models seem most attractive because of the simplicity of ARCH-type models (Engle 1982; Bollerslev 1986) in modeling conditional heteroscedasticity. Baillie, Bollerslev, and Mikkelsen (1996) first proposed a fractionally integrated GARCH (FIGARCH) model to explain the long-range dependence in volatility. This has some limitations, however; for example, the second mo-ment of the sequence is infinite. To measure such dependence, long memory in volatility is usually defined by the nonsumma-bility of the autocovariances for the squared sequence (see Giraitis, Leipus, and Surgailis 2009). However, the autocovari-ances of ARCH(_∞)processes with finite fourth moment are always summable (see Giraitis, Kokoszka, and Leipus 2000; Zaffaroni 2004).

Davidson(2004) considered the geometric or hyperbolic de-cay of the coefficients for ARCH(_∞)processes to measure the short- and long-range dependence in volatility. Note that coeffi-cients in the ARCH(_∞)representations of both the FIGARCH model and the fractional GARCH model (Robinson and Zaf-faroni 2006) have the property of hyperbolic decay.Davidson (2004) extended the FIGARCH model to a more general hyper-bolic GARCH (HYGARCH) model,

et=εth1/2t ,

(1.1) ht=

γ β(1)+

1− δ(B)

β(B)[1−a+a(1−B)

]

e2_t,

where {εt} is an iid sequence with mean 0 and variance 1,

a_≥0, 0<d<1, γ >0, B is the backshift operator,δ(B)₌ 1−δ1B− · · · −δqBq, and β(B)=1−β1B− · · · −βpBp.

Model (1.1) includes the FIGARCH model as a special case

whena₌1 and reduces to the common GARCH model when a₌0 ord₌1. Similar to autoregressive fractional integrated moving average (ARFIMA) models (Beran 1994), both HY-GARCH and FIHY-GARCH models mimic the hyperbolic decay by the structure of(1−B)d₌1−∞j=1πjBj, where

πj=

dŴ(j₋d)

Ŵ(1−d)Ŵ(j₊1)=O(j −1−d₎

as 0<d<1. Kwan, Li, and Li (2010) discussed estimation and diagnostic checking of ARFIMA–HYGARCH models based on the Stu-dentt likelihood. Other types of hyperbolic ARCH(_∞) mod-els have been addressed byZaffaroni(2004) andRobinson and Zaffaroni(2006).

Hyperbolic models for the conditional variance encounter more complicated situations than long-memory models for the conditional mean. Many probabilistic properties, includ-ing the admissible parameter space for strict stationarity, re-main unknown. In contrast, the properties of its counterpart, the common GARCH model, are well developed (Nelson 1991; Bougerol and Picard 1992; Francq and Zakoian 2004). It seems worthwhile to discuss the relationship between the hyperbolic GARCH model and the common GARCH model, which allows us to borrow these well-developed results for the HYGARCH model and other hyperbolic models.

Note thatetis unobservable, but we observeyt=μ+etfor

some unknownμ. Because we concentrate on the structure of the second moment, it is reasonable and simple to assume that μ₌0 throughout, that is,yt=et. Let{et}be a strictly stationary

and ergodic time series generated by common GARCH models, et=εth1/2t , ht=γ+

i=1

αie2t−i+ P

j=1

βjht−j, (1.2)

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580 Journal of Business & Economic Statistics, October 2011

where{εt}is defined as that in model (1.1),γ >0,αi≥0 for

i₌1, . . . ,Qandβj≥0 forj=1, . . . ,P(seeBollerslev 1986).

Denote the polynomialδ∗(z)₌1−(α1+β1)z− · · · −(αM+

βM)zM byM=max{P,Q}, whereαi=0 fori>Qor βj=0

forj>P. Note thatαi’s andβj’s are all nonnegative. The

poly-nomialδ∗(z) is a decreasing function on R+ withδ∗(0)₌1, δ∗(1)₌1−Qi=1αi−

j=1βj andδ∗(z)→ −∞asz→ ∞.

This implies that the equationδ∗(z)₌0 has a unique root on R+, say 1/a, and the polynomial δ∗(z) can be rewritten as δ(z)(1−az). Note that such a decomposition is unique under the restriction ofa_≥0. By some algebra, the conditional vari-ance in (1.2) can be rewritten as

ht=

γ β(1)+

1−δ(B)

β(B)[1−a+a(1−B)]

e2_t. (1.3)

Similar toBaillie, Bollerslev, and Mikkelsen(1996) and David-son (2004), we can arrive at the HYGARCH model (1.1) with ordersp₌Pandq₌M₋1 by replacing(1−B)by(1−B)d. It is natural to restrict model (1.1) bya>0 and 0<d_≤1, and the case wherea<1 corresponds to finite variance in model (1.2). The foregoing observation builds a bridge between the geomet-ric model (1.2) and the hyperbolic model (1.1), and thus pro-vides some new insights for the HYGARCH models. In partic-ular, we may interpretaas a mixing parameter for a geometric GARCH component and a hyperbolic FIGARCH component.

In the literature, it is of interest to check whether or not the hyperbolic decay really exists in the fitted hyperbolic models; however, statistical tools for this purpose are lacking. The lack of articles on this topic (Breidt, Crato, and Lima 1998) is partly due to the lack of a well-established theoretical foundation. Our foregoing observation makes it possible to borrow some estab-lished results from the common GARCH models when deriving a score test for checking for the presence of hyperbolic decay in the fitted HYGARCH models.

The article is organized as follows. Section 2 introduces a score test, and Section3presents a simplified version based on the transformed parameter space. Section4studies the asymp-totic behavior of this test under local alternatives. Section5 re-ports the results of a sequence of simulations to offer some in-sight into the finite-sample performance of the proposed score test. As examples, Section6studies the daily log return series for the Hang Seng Index and the Nasdaq Index. Section7 con-cludes. Proofs of both theorems are available in an online sup-plement.

2. SCORE TEST FOR HYPERBOLIC DECAY As pointed out by Davidson (2004), when d ₌ 1, the HYGARCH model (1.1) has the geometric decay for the co-efficients in its ARCH(_∞)representation. In this section we consider the score test for the null hypothesis ofd₌1.

Let λ₌(γ ,a, δ1, . . . , δq, β1, . . . , βp)′. Then ν=(d,λ′)′∈

(0,1] ×Rp+q+2 is the parameter vector of model (1.1). De-note byht(ν), the conditional variance ofet, when the true

pa-rameters in model (1.1) are replaced by the corresponding un-known parameters. By temporarily assuming normality forεt,

we can obtain the quasi–log-likelihood function up to a constant asLn(ν)= −0.5nt₌1lt(ν), where

lt(ν)=

e2_t ht(ν)+

loght(ν).

Then the score function is Sn(λ)=

1 √

t=1

∂lt(λ)

∂d

d=1

=√1

t=1

1− e

ht(1,λ)

1 ht(1,λ)

∂ht(1,λ)

∂d , where the derivative sequence∂ht(1,λ)/∂dsatisfies

∂ht(1,λ)

∂d =aδ(B) e

t−1− ∞

k=2

e2_t₋_k k(k₋1)

j=1

βj

∂ht−j(1,λ)

∂d . For a realized time series, only n values are available; how-ever, the quasi-log-likelihood function and the score function in the foregoing are dependent on past observations infinitely far away. Thus some initial values are needed; we use e2_s ₌ n−1n_t₌₁e2_t fors_≤0. As shown byLing and Li(1997), these initial values can be seen to have a negligible effect. The func-tions evaluated at these initial values are denoted byht(ν),

∂ht(ν)/∂d,lt(ν), andSn(λ).

Note that in the setting of our test, the vector λ _{serves as}

a nuisance parameter, and we need to find a surrogate to replace it. Suppose that_{is a compact set of}_Rp+q+2_{, and that model}

(1.3) [or model (1.1)] is well defined over the set_(or_{₁_}×_).

Consider

λ_n₌_{arg min}

λ_∈

t=1

lt(1,λ).

Denote Dn=

1 n

t=1 1 h2_t(1,λ_n)

∂ht(1,λn)

∂d 2

In=

1 n

t=1 1 h2_t(1,λ_n)

∂ht(1,λn)

∂d

∂ht(1,λn)

∂λ

Jn=

1 n

t=1 1 h2_t(1,λ_n₎

∂ht(1,λn)

∂λ

∂ht(1,λn)

∂λ′

andκε=n−1nt=1[e2t/ht(1,λn)−1]2. The score test statistic

is equal to

Ts=

S2_n(λ_n) κε(Dn−I′nJ−

n In)

and will asymptotically follow a chi-squared distribution with 1 degree of freedom under some regularity conditions.

The key to the foregoing testTsis the existence of a suitable

compact subset_of_Rp+q+2_._Conrad_{(2010) discussed}

restric-tions on the parameter vectorλ _{that ensure the nonnegativity}

of the conditional variance in HYGARCH models. However, some basic problems, such as the existence of a stationary so-lution, remain open, and the available range for the vectorλ_is

unknown.

Note that_{is a feasible set of the vector} λ_{under the null}

hypothesis of geometric decay [i.e., under the form of (1.3)], and, as noted in the previous section, models (1.2) and (1.3) are the same based on a transformation of the parameters. Thus we may derive the parameter space_{through that of the common}

GARCH models.

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3. SIMPLIFIED TEST UNDER A TRANSFORMED PARAMETER SPACE

Denote by θ ₌_{(γ , α}₁_{, . . . , α}_Q_{, β}₁_{, . . . , β}_P₎′ _{the parameter} vector of the common GARCH model (1.2). Note thatλ_{is the}

parameter vector of model (1.3). We can formalize the observa-tion in Secobserva-tion1with the following mapping:

θ₌(γ , α1, . . . , αQ, β1, . . . , βP)′

→ λ₌(γ ,a, δ1, . . . , δq, β1, . . . , βp)′

withp₌Pand q₌max{P,Q_{} −}1, and can denote these by

λ_{(θ )}_, a(θ ), and δ(θ ) respectively. For example, we consider a GARCH(1,2) process with parameter vector θ ₌(γ , α1,

α2, β1)′, which we can transform to a HYGARCH model with d ₌1 and other parameters λ ₌(γ ,a, δ1, β1)′, where

a₌(α1+β1+

(α1+β1)2+4α2)/2 and δ1=(α1+β1−

(α1+β1)2+4α2)/2. Of note, we can derive some conditions onλ_{to ensure that a certain probabilistic property, such as the}

weak stationarity condition, holds for the HYGARCH models even though they are usually very complicated.

Denote by_⊂_RP+Q+1_{the parameter space of the common}

GARCH model (1.2), which has been well studied in the litera-ture. For the restrictions on_{, see}_{Bougerol and Picard}₍₁₉₉₂₎

for the strict stationarity and ergodicity conditions and Ling (1999) for the moment conditions. Let= {λ(θ),θ∈}. Obviously, it (or the set_{1_{} ×}) is a suitable parameter space for model (1.3) [or model (1.1)]. It is of interest to study the score test Ts in the previous section based on the parameter

space.

Let h∗_t(θ)₌ht(1,λ(θ)), l∗t(θ)=lt(1,λ(θ)), ∂h∗t(θ)/∂d =

∂ht(1,λ(θ))/∂d, and

S∗_n(θ)₌Sn(λ(θ))=

1 √

t=1

1− e

h∗_t(θ)

1 h∗_t(θ)

∂h∗_t(θ) ∂d . Then it holds that∂h∗_t(θ)/∂θ₌(∂λ(θ)/∂θ)(∂ht(1,λ)/∂λ)and

∂h∗_t(θ) ∂d =a

δθ(B) e2_t₋₁₋

∞

k=2

e2_t₋_k k(k₋1)

j=1

βj

∂h∗_t₋_j(θ)

∂d , (3.1) where aθ ₌a(θ) and δθ ₌ δ(θ). Denote by h∗_t(θ),l∗_t(θ), ∂h∗_t(θ)/∂d, andS∗_n(θ)the corresponding functions evaluated at the initial values specified in the previous section.

Letθ_n₌_{arg min}_θ

∈

n t=1l∗t(θ),

D∗_n₌1 n

t=1 1 [h∗t(θn)]2

_∂_h_∗

t(θn)

∂d 2

I∗_n₌1 n

t=1 1 [h∗_t(θ_n)_]2

∂h∗

t(θn)

∂d

∂h∗

t(θn)

∂θ

J∗_n₌1 n

t=1 1 [h∗_t(θ_n)_]2

∂h∗_t(θ_n)

∂θ

∂h∗_t(θ_n)

∂θ′

and κ_ε∗ ₌n−1n_t₌_1[e2_t/h_t∗(θ_n)₋1]2. Of note, λ_n ₌λ(θ_n), an=a(θn),δn=δ(θn), Dn∗ =Dn, I∗n=(∂λ(θ)/∂θ)In, J∗n =

(∂λ(θ)/∂θ)Jn(∂λ(θ)/∂θ′), andκε∗=κε. Then the score test statisticTscan be rewritten as

Ts= [

S_n∗(θ_n₎_]2

κ∗

ε(D∗n−In∗′J∗−n 1I∗n)

Theorem 1. Under the null hypothesis of d₌1, if the se-ries{et}is strictly stationary and ergodic withE(e4t) <∞, then

Ts→dχ₁2, where χ₁2 is the standard chi-squared distribution

with 1 df.

Francq and Zakoian (2004) derived the asymptotic nor-mality of the quasi-maximum likelihood estimatorθ_n

with-out moment conditions, which are the sharpest results so far. Note that the quantity ∂h∗_t(θ₀)/∂d is a linear combination of e2_t

−1,e 2

t−2, . . . with hyperbolically decaying coefficients, and the conditional variance ht has a geometrically decaying

co-efficients. The technique ofFrancq and Zakoian(2004) cannot be used here, and the finite fourth moment ofet is required for

D₌E_[h−_t 2(∂h_t∗(θ₀)/∂d)2_]<_∞.

The iterative Equation (3.1) can be used to calculate the se-quence{∂h∗_t(θ_n)/∂d_}in the test statisticTs. Note that

h∗_t(θ)₌γ₊

i=1

αie2t−i+ P

j=1

βjh∗t−j(θ),

l∗_t(θ₎₌ e

h∗_t(θ)+logh ∗

t(θ),

andθ_n_{is the Gaussian quasi-maximum likelihood estimator of}

the common GARCH model (1.2). Thus, all other quantities in the test statistic Ts are all based on the common GARCH

models only.

4. ASYMPTOTIC POWERS

To investigate the asymptotic power of the test proposed in the previous sections, we consider the asymptotic behavior of Tsunder the following alternatives:

H1n:dn=1+

√_n, whereμ <0 is a fixed value.

Denote by ν₀₌(1,λ′

0)′,λ0 the true parameter value of λ, and suppose that there exists a strictly stationary and ergodic time series{et}satisfying model (1.1) with the parameter

vec-torν₀₊n−1/2vn, wherevnis a bounded sequence inRp+q+3.

LetFZ_{be the Borel}σ-field onRZ_with_Z_{= {}₀,_±1,_±2, . . ._}, let P be a probability measure on (FZ,RZ), and let Pnν be the restriction ofP onF_n_{, where}F_n _{is the} σ-field generated by {en, . . . ,e1,E0} and E0= {e0,e−1, . . .} is the initial vec-tor. Suppose that the errors{ε1(ν), . . . , εn(ν)}underPnν are iid with density function f(_·)and are independent of E0, where

εt(ν)=et/√ht(ν)andht(ν)is as defined in Section2. Assume

that the initial vectorE0has the same distribution underPnνfor differentν_{, and then the log-likelihood ratio}λ_n(ν₁,ν₂)ofPnν₂ toPnν₁ is

λ_n(ν₁,ν₂)₌

t=1

[loggt(ν2)−loggt(ν1)],

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582 Journal of Business & Economic Statistics, October 2011

wheregt(ν)=f(εt(ν))/√ht(ν).

Let

D₌E

1 h2_t

∂ht(ν0)

∂d 2

I₌E

1 h2t

∂ht(ν0)

∂d

∂ht(ν0)

∂λ

J₌E

1 h2_t

∂ht(ν0)

∂λ

∂ht(ν0)

∂λ′

andξ(x)₌1+xf′(x)/f(x), where the expectations are taken underPnν₀ andf′(·)is the derivative function off(·).

Assumption 1. The density f ofεt is positive onRand

ab-solutely continuous with derivative f′ almost everywhere and 0<ξ2(x)f(x)dx<_∞.

By a method similar to theorem 2.1 and remark 2.1 ofLing and McAleer (2003), if Assumption 1 and the conditions in Theorem1are satisfied, we can show that

λ_n

ν₀,ν₀₊_√vn

= − v

′

2√n

t=1

ξ(εt)

∂ht(ν0)

∂ν +

ξ0 8v

′

nvn+op(1),

underPnν₀, andPnν₀ andPn_ν

0+vn/√nare contiguous, whereξ0= E_[ξ2(εt)]and

₌

D I′

I J

;

see also theorem 3.1 ofLi and Li(2008b). Without confusion, we rewrite the score function Sn(λ) as Sn(1,λ). Expanding

Sn(1,λn)around the true parameter vector(dn,λ′₀)′, we have

that

Sn(1,λn)=Sn(dn,λ0)+ 1 √

∂Sn(ν∗n)

∂d ·(−μ)

+√1_n∂Sn(ν ∗

∂λ ·

√

n(λ_n₋λ₀₎

=(1,₋I′J−1)_·_√1 n

t=1

∂lt(dn,λ0)

∂ν

−μ(D₋I′J−1I)₊op(1),

whereν∗_n_{is a sequence between}(1,λ′_n)′and(dn,λ′0)′. Note that underH1n,

(1,₋I′J−1)_·_√1 n

t=1

∂lt(dn,λ0)

∂ν →dN(0, κε[D−I

′_J−1_I ]), whereκε=E(ε4t)−1. We then have the following theorem.

Theorem 2. UnderH1n, if Assumption1and the conditions

in Theorem1hold, then

(a) Sn(1,λn)→dN(−μ[D−I′J−1I], κε[D−I′J−1I]),

(b) Ts→dχ12(),

where₌μ2κ_ε−1(D₋I′J−1I), andχ₁2()is the chi-squared distribution with 1 df and noncentral parameter.

The foregoing theorem shows that the score testTshas

non-trivial asymptotic power under local alternativesH1n.

5. SIMULATION EXPERIMENTS

Here we report three Monte Carlo simulation experiments conducted to investigate the finite-sample performance of the proposed score testTs. The data-generating process in the first

experiment is et=εth

1/2

t ,

ht=0.05+

1−(1+0.2B)_[1−a₊a(1−B)d_]e2_t, where{εt}are iid random variables with the standard normal

distribution, d₌1 corresponds to the size, and d<1 corre-sponds to the power. Four types of data-generating processes are considered with the probabilistic properties under the null hypothesis of ARCH(2) models: (1) finite fourth moment, (2) fi-nite second moment and infifi-nite fourth moment, (3) an inte-grated GARCH process, and (4) strictly stationary with infinite variance. Note that our test can handle only the first type in the-ory, because the finite fourth moment is required. The value of ais set to 0.5, 0.8, 1, and 1.15, respectively, for the four types, and all selected values of(d,a)satisfy the constraint of non-negativeness ofht (Conrad 2010). The sample sizes is set to

n₌500, 1000, or 2000, and there are 1000 replications. For each replication, we generaten₊10,000 values, and then dis-card the first 10,000 values to mitigate the effect of the initial values. Two significance levels, 0.05 and 0.10, are considered, and the rejection rates are presented in Table1. The empirical sizes are all close to the nominal rates, and the test is more pow-erful as the value ofa grows larger. The HYGARCH process can be explained as the weighted summation of a geometric GARCH process and a hyperbolic FIGARCH process, and the parameterais simply the weight of the hyperbolic component. As pointed out by an anonymous referee, the test will tend to have zero power when the value ofaapproaches 0.

In the second experiment, we consider a data-generating pro-cess with the null hypothesis of a GARCH(1,1)model,

et=εth1/2t ,

(5.1) ht=

0.05 1−β1+

1− 1

1−β1B[

1−a₊a(1−B)d_]

e2_t,

where all settings are the same as those in the first experiment, and(β1,a)=(0.2,0.7),(0.3,0.95),(0.35,1), and(0.35,1.25), respectively, correspond to the four types of models. The em-pirical sizes and powers are given in Table 2. It can be ob-served that (1) the empirical sizes are all close to the nomi-nal rates except that the test is a little sensitive for the fourth model, that is,(β1,a)=(0.35,1.25); (2) the empirical powers are lower compared with the results in the first experiment; and (3) the powers exhibit the property of nonmonotonicity; that is, the rejection rates first increase and then decrease when the departure from the null hypothesis increases. Note that the geo-metric decaying pattern of the GARCH(1,1)model may com-pensate for the hyperbolic decaying pattern of the generated HYGARCH process to some extent when estimating a real time series with finite sample size; some evidence of this in terms of

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Table 1. Rejection rates of the testTswith the null hypothesis of ARCH(2)models

d₌1.0 d₌0.9 d₌0.7 d₌0.5

n 0.05 0.10 0.05 0.10 0.05 0.10 0.05 0.10

a₌0.5

500 0.048 0.104 0.079 0.135 0.200 0.281 0.262 0.341 1000 0.048 0.115 0.083 0.147 0.328 0.440 0.493 0.592 2000 0.049 0.097 0.150 0.232 0.570 0.672 0.756 0.823

a₌0.8

500 0.043 0.087 0.262 0.340 0.607 0.695 0.714 0.795 1000 0.054 0.094 0.430 0.514 0.897 0.932 0.945 0.973 2000 0.052 0.099 0.676 0.740 0.996 1.000 1.000 1.000

a₌1.0

500 0.043 0.089 0.592 0.671 0.917 0.940 0.934 0.953 1000 0.056 0.111 0.869 0.912 0.999 1.000 0.999 1.000 2000 0.060 0.103 0.973 0.986 1.000 1.000 1.000 1.000

a₌1.15

500 0.047 0.092 0.934 0.956 0.996 0.999 0.999 0.999 1000 0.054 0.098 0.987 0.993 1.000 1.000 0.999 1.000 2000 0.062 0.105 1.000 1.000 1.000 1.000 1.000 1.000

β1is presented in Table3. This may partially explain the sec-ond observation. For the third observation,Vogelsang(1999) andGiraitis et al.(2003) encountered a similar problem, and argued that it was due to the behavior of the sample variance. Agiakloglou and Newbold(1994) also observed a similar phe-nomenon when testing an ARFIMA(1,0,1)process against an ARFIMA(1,d,1)process. We may experience the same prob-lem asAgiakloglou and Newbold(1994), and we conducted the following experiment to gain more insight into our findings.

In this (the third) experiment, we considered the data-generating process (5.1) with(β1,a)=(0.2,0.7); that is, we fo-cused on the first model of the second experiment. Note that our test is a likelihood-based test, and the likelihood function ac-tually is a joint conditional density. Denote byf(H)₍_e

1, . . . ,en)

andf(G)(e1, . . . ,en)the joint densities of a HYGARCH process

and a GARCH process with initial values specified in Section2. The Kullback–Leibler information can be used to measure the distance between these two processes,

KL=

· · ·

f(H)(e1, . . . ,en)log

f(H)(e1, . . . ,en)

f(G)₍_e

1, . . . ,en)

de1· · ·den

=Elogf(H)(E1, . . . ,En)−logf(G)(E1, . . . ,En),

where_{E1, . . . ,En}is a segment from the HYGARCH process.

This quantity can be approximated by averaging over samples (realizations). Note that the densitiesf(H) andf(G) depend on the values of the corresponding parameter vectors. Then the first step (Step A) is to find a GARCH(1,1) process that is clos-est to the predetermined HYGARCH process. The natural way to do so is by maximizing logf(G)(E1, . . . ,En)with respect to

Table 2. Rejection rates of the testTswith the null hypothesis of GARCH(1,1)models

d₌1.0 d₌0.9 d₌0.7 d₌0.5

n 0.05 0.10 0.05 0.10 0.05 0.10 0.05 0.10

(β1,a)=(0.2,0.7)

500 0.041 0.086 0.051 0.099 0.071 0.124 0.062 0.118 1000 0.043 0.087 0.092 0.149 0.099 0.167 0.069 0.128 2000 0.046 0.097 0.147 0.217 0.208 0.285 0.087 0.153

(β₁,a)₌(0.3,0.95)

500 0.050 0.099 0.167 0.250 0.194 0.280 0.109 0.185 1000 0.053 0.107 0.301 0.369 0.350 0.447 0.139 0.221 2000 0.058 0.110 0.498 0.581 0.652 0.748 0.178 0.259

(β1,a)=(0.35,1.0)

500 0.046 0.104 0.235 0.300 0.221 0.308 0.164 0.232 1000 0.059 0.101 0.386 0.489 0.446 0.562 0.166 0.272 2000 0.052 0.106 0.627 0.696 0.720 0.803 0.257 0.347

(β1,a)=(0.35,1.25)

500 0.096 0.149 0.953 0.973 0.964 0.979 0.793 0.835 1000 0.098 0.159 0.996 0.998 0.998 0.998 0.976 0.978 2000 0.098 0.152 1.000 1.000 1.000 1.000 1.000 1.000

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584 Journal of Business & Economic Statistics, October 2011

Table 3. Results for the third simulation experiments

d₌0.9 d₌0.8 d₌0.7 d₌0.6 d₌0.5

Step A

γ 0.0462 (0.0048) 0.0416 (0.0054) 0.0361 (0.0060) 0.0297 (0.0060) 0.0231 (0.0058)

β1 0.2958 (0.0439) 0.3948 (0.0493) 0.4964 (0.0559) 0.5975 (0.0557) 0.6930 (0.0550)

a 0.7213 (0.0392) 0.7462 (0.0392) 0.7760 (0.0422) 0.8092 (0.0411) 0.8430 (0.0409) TLLF ₋1216.7 ₋1271.0 ₋1312.8 ₋1319.6 ₋1286.1 MLLF ₋1215.6 ₋1270.5 ₋1312.6 ₋1319.0 ₋1284.6 Step B

KL 6.3036 7.2665 7.8372 7.3488 6.9010 Power 0.1955 0.3120 0.2720 0.1455 0.1155

NOTE: The values inside the brackets correspond to the sample standard deviations.

replication. Table3 gives the sample means and standard de-viations of the values of these estimated parameters. The sam-ple standard deviations are small, and the samsam-ple mean can be treated as the parameter vector of the GARCH model that is closest to the corresponding HYGARCH process. Table3also presents the average of the true log-likelihood (TLLF) for the generated HYGARCH process and the average of the maxi-mized log-likelihood (MLLF) under the GARCH model. All of the values of MLLF exceed those of TLLF, and the sam-ple mean for the persistent parameter β1 increases as the de-pendence parameter further departs from the null hypothesis of d₌1.

rates at the 0.05 significance level are reported in Table3. The values of the Kullback–Leibler information have a similar pat-tern as the empirical powers. The difference between the hy-perbolic GARCH models and the geometric GARCH models in terms of the Kullback–Leibler information is also nonmono-tonic whendmoves away from 1. This may partially explain the third observation in the second experiment. However, as sug-gested by a referee, the nonmonotonicity phenomenon does not exist in the asymptotic theory, and thus this phenomenon may be due to initial values in generating a realization. Note that we need more initial values when d is smaller, and the effect of these initial values is not negligible for some asymptotic in-ferences whend<0.5 (seeRobinson and Zaffaroni 2006). We also checked with the other three models in the second experi-ment and found similar results.

6. REAL EXAMPLES

In this section we consider two log return (in percentage) se-quences: the daily closing prices of the Hang Seng Index from January 2, 2002, to January 7, 2010, with 2002 observations, and the daily closing prices of the Nasdaq Index from Jan-uary 4, 1999, to November 30, 2009 with 2745 observations. Some descriptive statistics were summarized in Table 4, and

Figures1and2present their time plots, sample autocorrelation functions (ACFs), and sample ACFs of the squared sequences. The sample ACFs are all insignificant, and the sample ACFs of the squared sequences are significant even at lag 200, although the amplitudes are small. Thus these two sequences are usually identified to have short-range dependence in mean and long-range dependence in volatility.

Given that the focus of this article is on volatility, we first considered using the ARFIMA(0,d,0) model to remove com-ponents of possible long-range dependence in the conditional mean. The fitted models for the Hang Seng Index and the Nas-daq Index are

(1₋B)−0.0208(yt−0.0337)=et,

(1−B)−0.0357(yt+0.0011)=et,

check the adequacy of the models. The estimating results of the HYGARCH model and the p-values of the portmanteau tests are given in Table5. In terms of autocorrelations of stan-dardized residuals [Qρ(M)] and squared standardized residuals [QR(M)], the fitted models explain these two datasets well. The

estimated value ofdis significantly different from that for the Nasdaq Index; however, it is insignificant for the Hang Seng Index. It is interesting to check whether the fitted HYGARCH model for the Hang Seng Index really provides a better fit than the common GARCH model. Note that the estimated value of the parameterδ1is also insignificant for the Nasdaq Index, and we keep it here for the sake of comparison.

Table 4. Descriptive statistics of two log return sequences Hang Seng Index Nasdaq Index Sample period 01/02/2002–01/07/2010 01/04/1999–11/30/2009 Sample size 2002 2745

Mean 0.0377 ₋0.0011 Std. dev. 1.6660 1.9172 Skewness 0.1027 0.0758 Kurtosis 12.5261 6.8248

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Figure 1. Log returns (in percentage) of daily closing prices for the Hang Seng Index from January 2, 2002, to January 7, 2010.

Finally, we applied the proposed score testTs to these two

datasets, confirming geometric decay for the Hang Seng Index but not for the Nasdaq Index (see Table5). Note that the “dot-com bubble” or “IT bubble” occurred roughly between 1998 and 2000, with a climax on March 10, 2000, when the Nasdaq Index peaked at 5132.52, and our dataset covers this period. Investors may have the crisis event in their mind for a longer period, resulting in the phenomenon of hyperbolic decay of volatility in the Nasdaq Index.

7. CONCLUSION

In this article we have reinterpreted Davidson’s HYGARCH models by building a link with common GARCH models. It is natural to consider a score test for the presence of the hyper-bolic decay in the HYGARCH model. Our test requires the fi-nite fourth moment of the HYGARCH process when deriving the asymptotic null distributions. This constraint is not easily relaxed in theory, and our simulation experiments show that the

Figure 2. Log returns (in percentage) of daily closing prices for the Nasdaq Index from January 4, 1999, to November 30, 2009.

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586 Journal of Business & Economic Statistics, October 2011

Table 5. Estimating and testing results for two log return sequences Hang Seng Index Nasdaq Index Estimating results

γ 0.0157 (0.0648_×10−3) 0.0294 (0.1436_×10−3)

δ₁ ₋0.0260 (4.4422_×10−3) 0.0039 (3.2050_×10−3)

β1 0.9068 (1.2054×10−3) 0.8072 (3.7785×10−3)

a 0.9940 (0.0368_×10−3) 0.9947 (0.0806_×10−3)

d 0.9905 (10.2175_×10−3) 0.7873 (12.3010_×10−3) Portmanteau tests for adequacy

Qρ(25) 0.4275 0.3990

QR(25) 0.2827 0.9139

Score test for hyperbolic decay

Ts 0.0272 27.8789

pvalue 0.8690 1.2752_×10−7

NOTE: The values inside the brackets correspond to the standard errors.

test still works well even if the moment condition is broken. It can be shown that in theory, the extent of the hyperbolic de-cay in HYGARCH processes will increase as the parameterd decreases from 1 to 0. However, our simulation experiments showed that this may not be true in finite-sample cases. Finally, the HYGARCH is not the unique hyperbolic ARCH(_∞)model, and our method can be used to design tests for other hyperbolic ARCH(_∞) models, such as the fractional GARCH model of Robinson and Zaffaroni(2006).

SUPPLEMENTARY MATERIALS

Appendix: Files providing some regularity conditions for common GARCH models and the detailed proofs of The-orems1and2. (ScoreTest_JBES_supplement.pdf)

ACKNOWLEDGMENTS

The authors thank Professor J. Wright, an Associate Editor, and two anonymous referees for valuable comments that led to the substantial improvement of this manuscript. This work was partially supported by Hong Kong Research Grants Coun-cil Grant HKU 702908P.

[Received January 2010. Revised February 2011.]

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Kwan, W., Li, W. K., and Li, G. (2010), “On the Estimation and Diagnostic Checking for the ARFIMA–HYGARCH Model,”Computational Statistics & Data Analysis, to appear,doi:10.1016/j.csda.2010.07.010. [579,584] Li, G., and Li, W. K. (2008a), “Least Absolute Deviation Estimation for

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(2008b), “Testing for Threshold Moving Average With Conditional Heteroscedasticity,”Statistica Sinica, 18, 647–665. [582]

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3. SIMPLIFIED TEST UNDER A TRANSFORMED PARAMETER SPACE

Denote by θ ₌_{(γ , α}₁_{, . . . , α}_Q_{, β}₁_{, . . . , β}_P₎′ _{the parameter} vector of the common GARCH model (1.2). Note thatλ_{is the} parameter vector of model (1.3). We can formalize the observa-tion in Secobserva-tion1with the following mapping:

θ₌(γ , α1, . . . , αQ, β1, . . . , βP)′

→ λ₌(γ ,a, δ1, . . . , δq, β1, . . . , βp)′

withp₌Pand q₌max{P,Q_{} −}1, and can denote these by λ_{(θ )}_, a(θ ), and δ(θ ) respectively. For example, we consider a GARCH(1,2) process with parameter vector θ ₌(γ , α1,

α2, β1)′, which we can transform to a HYGARCH model with d ₌1 and other parameters λ ₌(γ ,a, δ1, β1)′, where

a₌(α1+β1+

(α1+β1)2+4α2)/2 and δ1=(α1+β1−

(α1+β1)2+4α2)/2. Of note, we can derive some conditions onλ_{to ensure that a certain probabilistic property, such as the} weak stationarity condition, holds for the HYGARCH models even though they are usually very complicated.

Denote by_⊂_RP+Q+1_{the parameter space of the common} GARCH model (1.2), which has been well studied in the litera-ture. For the restrictions on_{, see}_{Bougerol and Picard}₍₁₉₉₂₎ for the strict stationarity and ergodicity conditions and Ling (1999) for the moment conditions. Let= {λ(θ),θ∈}. Obviously, it (or the set_{1_{} ×}) is a suitable parameter space for model (1.3) [or model (1.1)]. It is of interest to study the score test Ts in the previous section based on the parameter space.

Let h∗_t(θ)₌ht(1,λ(θ)), l∗t(θ)=lt(1,λ(θ)), ∂h∗t(θ)/∂d = ∂ht(1,λ(θ))/∂d, and

S∗_n(θ)₌Sn(λ(θ))₌_√1 n

t=1

1− e

t h∗_t(θ)

1 h∗_t(θ)

∂h∗_t(θ)

∂d .

Then it holds that∂h∗_t(θ)/∂θ₌(∂λ(θ)/∂θ)(∂ht(1,λ)/∂λ)and

∂h∗_t(θ)

∂d =a

δθ(B) e2_t₋₁₋

∞

k=2

e2_t₋_k k(k₋1)

+ p

j=1

βj

∂h∗_t₋_j(θ)

Letθ_n₌_{arg min}_θ

∈

t=1l∗t(θ), D∗_n₌1

n n

t=1 1 [h∗t(θn)]2

_∂_h_∗

t(θn)

∂d

I∗_n₌1

n n

t=1 1 [h∗_t(θ_n)_]2

∂h∗

t(θn)

∂d

∂h∗

t(θn)

∂θ

J∗_n₌1

n n

t=1 1 [h∗_t(θ_n)_]2

∂h∗_t(θ_n)

∂θ

∂h∗_t(θ_n)

∂θ′

and κ_ε∗ ₌n−1n_t₌₁_[e2_t/h_t∗(θ_n)₋1]2. Of note, λ_n ₌λ(θ_n),

an=a(θn),δn=δ(θn), Dn∗ =Dn, I∗n=(∂λ(θ)/∂θ)In, J∗n =

(∂λ(θ)/∂θ)Jn(∂λ(θ)/∂θ′), andκε∗=κε. Then the score test statisticTscan be rewritten as

Ts= [

S_n∗(θ_n₎_]2 κ∗

ε(D∗n−In∗′J∗−n 1I∗n) .

Theorem 1. Under the null hypothesis of d₌1, if the se-ries{et_}is strictly stationary and ergodic withE(e4_t) <_∞, then Ts_→dχ₁2, where χ₁2 is the standard chi-squared distribution with 1 df.

Francq and Zakoian (2004) derived the asymptotic nor-mality of the quasi-maximum likelihood estimatorθ_n with-out moment conditions, which are the sharpest results so far. Note that the quantity ∂h∗_t(θ₀)/∂d is a linear combination of e2_t

−1,e 2

t−2, . . . with hyperbolically decaying coefficients, and

the conditional variance ht has a geometrically decaying co-efficients. The technique ofFrancq and Zakoian(2004) cannot be used here, and the finite fourth moment ofet is required for D₌E_[h−_t 2(∂h_t∗(θ₀)/∂d)2_]<_∞.

The iterative Equation (3.1) can be used to calculate the se-quence{∂h∗_t(θ_n)/∂d_}in the test statisticTs. Note that

h∗_t(θ)₌γ₊ Q

i=1

αie2t−i+

j=1

βjh∗t−j(θ),

l∗_t(θ₎₌ e 2

h∗_t(θ)+logh ∗ t(θ),

andθ_n_{is the Gaussian quasi-maximum likelihood estimator of} the common GARCH model (1.2). Thus, all other quantities in the test statistic Ts are all based on the common GARCH models only.

4. ASYMPTOTIC POWERS

To investigate the asymptotic power of the test proposed in the previous sections, we consider the asymptotic behavior of Tsunder the following alternatives:

H1n:dn=1+ μ √_n,

whereμ <0 is a fixed value. Denote by ν₀₌(1,λ′

0)′,λ0 the true parameter value of λ, and suppose that there exists a strictly stationary and ergodic time series{e_t}satisfying model (1.1) with the parameter vec-torν₀₊n−1/2vn, wherevnis a bounded sequence inRp+q+3. LetFZ_{be the Borel}σ-field onRZ_with_Z_{= {}₀,_±1,_±2, . . ._}, let P be a probability measure on (FZ,RZ), and let Pnν be the restriction ofP onF_n_{, where}F_n _{is the} σ-field generated

by {en, . . . ,e1,E0} and E0= {e0,e−1, . . .} is the initial

vec-tor. Suppose that the errors{ε1(ν), . . . , εn(ν)}underPnν are iid with density function f(_·)and are independent of E0, where

εt(ν)=et/√ht(ν)andht(ν)is as defined in Section2. Assume that the initial vectorE0has the same distribution underPnνfor differentν_{, and then the log-likelihood ratio}λ_n(ν₁,ν₂)ofPnν₂ toPnν₁ is

λ_n(ν₁,ν₂)₌ n

t=1

[loggt(ν2)−loggt(ν1)],

(2)

wheregt(ν)=f(εt(ν))/√ht(ν). Let

D₌E

1 h2_t

∂ht(ν0)

∂d

I₌E

1 h2t

∂ht(ν0)

∂d

∂ht(ν0)

∂λ

J₌E

1 h2_t

∂ht(ν₀)

∂λ

∂ht(ν₀)

∂λ′

andξ(x)₌1+xf′(x)/f(x), where the expectations are taken underPnν₀ andf′(·)is the derivative function off(·).

Assumption 1. The density f ofεt is positive onRand ab-solutely continuous with derivative f′ almost everywhere and 0<ξ2(x)f(x)dx<_∞.

By a method similar to theorem 2.1 and remark 2.1 ofLing and McAleer (2003), if Assumption 1 and the conditions in Theorem1are satisfied, we can show that

λ_n

ν₀,ν₀₊_√vn

= − v ′ n 2√n

t=1

ξ(εt)

∂ht(ν0)

∂ν +

ξ0

8v ′

nvn+op(1), underPnν₀, andPnν₀ andPn_ν

0+vn/√nare contiguous, whereξ0=

E_[ξ2(εt)]and

₌

D I′

I J

;

see also theorem 3.1 ofLi and Li(2008b). Without confusion, we rewrite the score function Sn(λ₎ _as _Sn₍₁_,λ₎_{. Expanding} Sn(1,λn)around the true parameter vector(dn,λ′₀)′, we have

that

Sn(1,λn)=Sn(dn,λ0)+ 1 √ n

∂Sn(ν∗_n)

∂d ·(−μ)

+√1_n∂Sn(ν ∗ n)

∂λ ·

√

n(λ_n₋λ₀₎

=(1,₋I′J−1)_·_√1 n

t=1

∂lt(dn,λ0)

∂ν

−μ(D₋I′J−1I)₊op(1),

whereν∗_n_{is a sequence between}(1,λ′_n)′and(dn,λ′0)′. Note that underH1n,

(1,₋I′J−1)_·_√1

n n

t=1

∂lt(dn,λ0)

∂ν →dN(0, κε[D−I

′_J−1_I

]),

whereκε=E(ε4t)−1. We then have the following theorem. Theorem 2. UnderH1n, if Assumption1and the conditions in Theorem1hold, then

(a) Sn(1,λn)→dN(−μ[D−I′J−1I], κε[D−I′J−1I]), (b) Ts→dχ12(),

where₌μ2κ_ε−1(D₋I′J−1I), andχ₁2()is the chi-squared distribution with 1 df and noncentral parameter.

The foregoing theorem shows that the score testTshas non-trivial asymptotic power under local alternativesH1n.

5. SIMULATION EXPERIMENTS

Here we report three Monte Carlo simulation experiments conducted to investigate the finite-sample performance of the proposed score testTs. The data-generating process in the first experiment is

et=εth

1/2

t , ht=0.05+

1−(1+0.2B)_[1−a₊a(1−B)d_]e2_t, where{ε_t}are iid random variables with the standard normal distribution, d₌1 corresponds to the size, and d<1 corre-sponds to the power. Four types of data-generating processes are considered with the probabilistic properties under the null hypothesis of ARCH(2) models: (1) finite fourth moment, (2) fi-nite second moment and infifi-nite fourth moment, (3) an inte-grated GARCH process, and (4) strictly stationary with infinite variance. Note that our test can handle only the first type in the-ory, because the finite fourth moment is required. The value of ais set to 0.5, 0.8, 1, and 1.15, respectively, for the four types, and all selected values of(d,a)satisfy the constraint of non-negativeness ofht (Conrad 2010). The sample sizes is set to n₌500, 1000, or 2000, and there are 1000 replications. For each replication, we generaten₊10,000 values, and then dis-card the first 10,000 values to mitigate the effect of the initial values. Two significance levels, 0.05 and 0.10, are considered, and the rejection rates are presented in Table1. The empirical sizes are all close to the nominal rates, and the test is more pow-erful as the value ofa grows larger. The HYGARCH process can be explained as the weighted summation of a geometric GARCH process and a hyperbolic FIGARCH process, and the parameterais simply the weight of the hyperbolic component. As pointed out by an anonymous referee, the test will tend to have zero power when the value ofaapproaches 0.

In the second experiment, we consider a data-generating pro-cess with the null hypothesis of a GARCH(1,1)model,

et=εth1t/2,

(5.1) ht=

0.05 1−β1+

1− 1

1−β1B[

1−a₊a(1−B)d_]

e2_t, where all settings are the same as those in the first experiment, and(β1,a)=(0.2,0.7),(0.3,0.95),(0.35,1), and(0.35,1.25), respectively, correspond to the four types of models. The em-pirical sizes and powers are given in Table 2. It can be ob-served that (1) the empirical sizes are all close to the nomi-nal rates except that the test is a little sensitive for the fourth model, that is,(β1,a)=(0.35,1.25); (2) the empirical powers are lower compared with the results in the first experiment; and (3) the powers exhibit the property of nonmonotonicity; that is, the rejection rates first increase and then decrease when the departure from the null hypothesis increases. Note that the geo-metric decaying pattern of the GARCH(1,1)model may com-pensate for the hyperbolic decaying pattern of the generated HYGARCH process to some extent when estimating a real time series with finite sample size; some evidence of this in terms of

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Table 1. Rejection rates of the testTswith the null hypothesis of ARCH(2)models

d₌1.0 d₌0.9 d₌0.7 d₌0.5

n 0.05 0.10 0.05 0.10 0.05 0.10 0.05 0.10

a₌0.5

500 0.048 0.104 0.079 0.135 0.200 0.281 0.262 0.341 1000 0.048 0.115 0.083 0.147 0.328 0.440 0.493 0.592 2000 0.049 0.097 0.150 0.232 0.570 0.672 0.756 0.823

a₌0.8

500 0.043 0.087 0.262 0.340 0.607 0.695 0.714 0.795 1000 0.054 0.094 0.430 0.514 0.897 0.932 0.945 0.973 2000 0.052 0.099 0.676 0.740 0.996 1.000 1.000 1.000

a₌1.0

500 0.043 0.089 0.592 0.671 0.917 0.940 0.934 0.953 1000 0.056 0.111 0.869 0.912 0.999 1.000 0.999 1.000 2000 0.060 0.103 0.973 0.986 1.000 1.000 1.000 1.000

a₌1.15

500 0.047 0.092 0.934 0.956 0.996 0.999 0.999 0.999 1000 0.054 0.098 0.987 0.993 1.000 1.000 0.999 1.000 2000 0.062 0.105 1.000 1.000 1.000 1.000 1.000 1.000

1, . . . ,en)

andf(G)(e1, . . . ,en)the joint densities of a HYGARCH process

and a GARCH process with initial values specified in Section2. The Kullback–Leibler information can be used to measure the distance between these two processes,

KL=

· · ·

f(H)(e1, . . . ,en)log

f(H)(e1, . . . ,en)

f(G)₍_e

1, . . . ,en)

de1· · ·den

=Elogf(H)(E1, . . . ,En)−logf(G)(E1, . . . ,En),

where_{E1, . . . ,En}is a segment from the HYGARCH process. This quantity can be approximated by averaging over samples (realizations). Note that the densitiesf(H) andf(G) depend on the values of the corresponding parameter vectors. Then the first step (Step A) is to find a GARCH(1,1) process that is clos-est to the predetermined HYGARCH process. The natural way to do so is by maximizing logf(G)(E1, . . . ,En)with respect to

Table 2. Rejection rates of the testTswith the null hypothesis of GARCH(1,1)models

d₌1.0 d₌0.9 d₌0.7 d₌0.5

n 0.05 0.10 0.05 0.10 0.05 0.10 0.05 0.10

(β1,a)=(0.2,0.7)

500 0.041 0.086 0.051 0.099 0.071 0.124 0.062 0.118 1000 0.043 0.087 0.092 0.149 0.099 0.167 0.069 0.128 2000 0.046 0.097 0.147 0.217 0.208 0.285 0.087 0.153

(β₁,a)₌(0.3,0.95)

500 0.050 0.099 0.167 0.250 0.194 0.280 0.109 0.185 1000 0.053 0.107 0.301 0.369 0.350 0.447 0.139 0.221 2000 0.058 0.110 0.498 0.581 0.652 0.748 0.178 0.259

(β1,a)=(0.35,1.0)

500 0.046 0.104 0.235 0.300 0.221 0.308 0.164 0.232 1000 0.059 0.101 0.386 0.489 0.446 0.562 0.166 0.272 2000 0.052 0.106 0.627 0.696 0.720 0.803 0.257 0.347

(β1,a)=(0.35,1.25)

500 0.096 0.149 0.953 0.973 0.964 0.979 0.793 0.835 1000 0.098 0.159 0.996 0.998 0.998 0.998 0.976 0.978 2000 0.098 0.152 1.000 1.000 1.000 1.000 1.000 1.000

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Table 3. Results for the third simulation experiments

d₌0.9 d₌0.8 d₌0.7 d₌0.6 d₌0.5

Step A

γ 0.0462 (0.0048) 0.0416 (0.0054) 0.0361 (0.0060) 0.0297 (0.0060) 0.0231 (0.0058)

β1 0.2958 (0.0439) 0.3948 (0.0493) 0.4964 (0.0559) 0.5975 (0.0557) 0.6930 (0.0550)

a 0.7213 (0.0392) 0.7462 (0.0392) 0.7760 (0.0422) 0.8092 (0.0411) 0.8430 (0.0409) TLLF ₋1216.7 ₋1271.0 ₋1312.8 ₋1319.6 ₋1286.1 MLLF ₋1215.6 ₋1270.5 ₋1312.6 ₋1319.0 ₋1284.6 Step B

KL 6.3036 7.2665 7.8372 7.3488 6.9010

Power 0.1955 0.3120 0.2720 0.1455 0.1155

NOTE: The values inside the brackets correspond to the sample standard deviations.

the parameter vectorθ₌(γ , β1,a). We consider five different HYGARCH processes withd₌0.9, 0.8, 0.7, 0.6, and 0.5, and set the sample size to 3000. There are 2000 replications, and we can obtain a fitted value for the parameter vectorθ_{for each} replication. Table3 gives the sample means and standard de-viations of the values of these estimated parameters. The sam-ple standard deviations are small, and the samsam-ple mean can be treated as the parameter vector of the GARCH model that is closest to the corresponding HYGARCH process. Table3also presents the average of the true log-likelihood (TLLF) for the generated HYGARCH process and the average of the maxi-mized log-likelihood (MLLF) under the GARCH model. All of the values of MLLF exceed those of TLLF, and the sam-ple mean for the persistent parameter β1 increases as the de-pendence parameter further departs from the null hypothesis of d₌1.

In the second step (Step B) of the third experiment, we ap-proximate the Kullback–Leibler information by sample averag-ing and set the number of the generated samples to 2000. We also perform the test Ts for each generated sample. Rejection rates at the 0.05 significance level are reported in Table3. The values of the Kullback–Leibler information have a similar pat-tern as the empirical powers. The difference between the hy-perbolic GARCH models and the geometric GARCH models in terms of the Kullback–Leibler information is also nonmono-tonic whendmoves away from 1. This may partially explain the third observation in the second experiment. However, as sug-gested by a referee, the nonmonotonicity phenomenon does not exist in the asymptotic theory, and thus this phenomenon may be due to initial values in generating a realization. Note that we need more initial values when d is smaller, and the effect of these initial values is not negligible for some asymptotic in-ferences whend<0.5 (seeRobinson and Zaffaroni 2006). We also checked with the other three models in the second experi-ment and found similar results.

6. REAL EXAMPLES

(1₋B)−0.0208(yt₋0.0337)₌et,

(1−B)−0.0357(yt+0.0011)=et,

where short-range dependence is indicated by both models, which is consistent with our observation on the sample ACFs in Figures1and2. We next considered HYGARCH models for both residual sequences, and performed the portmanteau tests Qρ(M)andQR(M)(Kwan, Li, and Li 2010) withM=25 to check the adequacy of the models. The estimating results of the HYGARCH model and the p-values of the portmanteau tests are given in Table5. In terms of autocorrelations of stan-dardized residuals [Qρ(M)] and squared standardized residuals [QR(M)], the fitted models explain these two datasets well. The estimated value ofdis significantly different from that for the Nasdaq Index; however, it is insignificant for the Hang Seng Index. It is interesting to check whether the fitted HYGARCH model for the Hang Seng Index really provides a better fit than the common GARCH model. Note that the estimated value of the parameterδ1is also insignificant for the Nasdaq Index, and we keep it here for the sake of comparison.

Table 4. Descriptive statistics of two log return sequences Hang Seng Index Nasdaq Index Sample period 01/02/2002–01/07/2010 01/04/1999–11/30/2009

Sample size 2002 2745

Mean 0.0377 ₋0.0011

Std. dev. 1.6660 1.9172

Skewness 0.1027 0.0758

Kurtosis 12.5261 6.8248

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Figure 1. Log returns (in percentage) of daily closing prices for the Hang Seng Index from January 2, 2002, to January 7, 2010.

Finally, we applied the proposed score testTs to these two datasets, confirming geometric decay for the Hang Seng Index but not for the Nasdaq Index (see Table5). Note that the “dot-com bubble” or “IT bubble” occurred roughly between 1998 and 2000, with a climax on March 10, 2000, when the Nasdaq Index peaked at 5132.52, and our dataset covers this period. Investors may have the crisis event in their mind for a longer period, resulting in the phenomenon of hyperbolic decay of volatility in the Nasdaq Index.