Discussion Paper No. 545

A SURVEY OF
RECENT THEORETICAL RESULTS
FOR TIME SERIES MODELS
WITH GARCH ERRORS*

W. K. Li
Shiqing Ling
andMichael McAleer

June 2001

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan* To appear in Journal of Economic Surveys.

A Survey of Recent Theoretical Results for Time
Series Models with GARCH Errors∗
2

W.K. Li1 , Shiqing Ling and Michael McAleer3
1

Department of Statistics and Actuarial Science,
University of Hong Kong
2
Department of Mathematics,
Hong Kong University of Science and Technology
3
Department of Economics, University of Western Australia
Revised: June 2001

Abstract
This paper provides a review of some recent theoretical results for time series models with GARCH errors, and is directed towards practitioners. Starting with the simple ARCH model and proceeding to the GARCH model,
some results for stationary and nonstationary ARMA-GARCH are summarized. Various new ARCH-type models, including double threshold ARCH
and GARCH, ARFIMA-GARCH, CHARMA and vector ARMA-GARCH, arealso reviewed.

∗

The authors are most grateful for the helpful comments and suggestions of a referee andseminar participants at the Institute for Monetary and Economic Studies, Bank of Japan, Bocconi University, Milan, Chinese University of Hong Kong, Curtin University of Technology, Edith
Cowan University, Erasmus University Rotterdam, Griffith University, Hiroshima University, Hitotsubashi University, International Foundation for Artificial Intelligence, Tokyo, Kyoto University,
Monash University, Murdoch University, Centre for Financial Engineering, National University ofSingapore, Osaka School of International Public Policy, Osaka University, Queensland Universityof Technology, Yokohama National University, and University of Waikato. For financial support,
the first author would like to thank the Hong Kong Research Grants Council, the second author
wishes to acknowledge the Australian Research Council, and the third author is grateful to theAustralian Research Council and the Institute of Social and Economic Research, Osaka University.

1

Introduction

A primary feature of the autoregressive conditional heteroscedasticity (ARCH) model,
as developed by Engle (1982), is that the conditional variances change over time.
Following the seminal idea, numerous models incorporating this feature have been
proposed. Among these models, Bollerslev’s (1986) generalized ARCH (GARCH)
model is certainly the most popular and successful because it is easy to estimate andinterpret by analogy with the autoregressive moving average (ARMA) time seriesmodel. Analyzing financial and economic time series data with ARCH and GARCH
models has become very common in empirical research, with a huge literature having
been established. Several excellent surveys on ARCH/GARCH models are available,
such as Bollerslev, Chou and Kroner (1992), Bollerslev, Engle and Nelson (1994),
and Bera and Higgins (1993). More recently, the Stochastic Volatility model of Taylor (1986) offers an alternative to GARCH. Stochastic Volatility models will not be
discussed in this paper and interested readers are referred to the review by Shephard (1996). In a series of papers, Nelson has made important contributions to thefiltering theory of ARCH processes. His work has been nicely summarized by Ross
(1996), and hence will not be the focus of attention in this paper. Gourieroux (1997)
provides a summary of some earlier results on GARCH models.
The aim of this paper is to provide a review of some recent theoretical results
for time series models with ARCH/GARCH errors, and is directed towards practitioners. The plan of the paper is as follows. We begin with the simple ARCH
model in Section 2 and proceed to the GARCH model in Section 3. The stationary
ARMA-GARCH model is considered in Section 4, and its nonstationary counterpart
in Section 5. Finally, we review some results for other ARCH-type models, including double threshold ARCH, ARFIMA-GARCH, CHARMA, and vector ARMAGARCH, in Section 6. Concluding marks are given in Section 7.

1

2

ARCH Models

Engle’s (1982) ARCH (r) model can be defined as follows:1/2

εt = ηt ht , ht = α0 + α1 ε2t−1 + · · · + αr ε2t−r ,(2.1)

where α0 > 0, αi ≥ 0 (i = 1, · · · , r) and the ηt are a sequence of independently andidentically distributed (i.i.d.) random variables with zero mean and unit variance.
Denote by Ft the σ−field generated by {ηt , ηt−1 , · · ·}. Then E(ε2t |Ft−1 ) = ht , that is,the conditional variance of the process εt varies over time instead of being constant,
as in traditional time series analysis.

2.1

Basic Properties

When a new time series model is proposed, a basic question concerns the conditions
under which the model will be stationary. Engle (1982) showed that εt is secondorder stationary (i.e. Eε2t < ∞) if and only if all the roots of
zr −

r
X

αi z r−i = 0

(2.2)

i=1

are outside the unit circle. To prove this result, Engle (1982) assumed that εtstarts infinitely far in the past with finite variance, which is impossible to verify inpractice. Using a different method, Milhøj (1985) avoided Engle’s (1982) assumption
and showed that εt is second-order stationary if and only if
α1 + · · · + αr < 1.

(2.3)

In particular, Milhøj (1985) showed that (2.3) is also a sufficient condition for strict
stationarity and ergodicity of εt . Since αi is nonnegative for i = 1, . . . , r, conditions(2.2) and (2.3) are equivalent by Lemma 2.1 in Ling (1999b).
For the first-order ARCH model, Engle (1982) showed that, if ηt is normal, the2mth moment of εt exists if and only if
α1m

m
Y

(2j − 1) < 1,

j=1

2

(2.4)

under the assumption that εt starts infinitely far in the past with finite 2mth moment. Without this assumption, Milhøj (1985) obtained the necessary and sufficientcondition for the existence of the 2mth moment of εt . When ηt is normal and r = 1,
Milhøj’s condition is the same as (2.4). A unique drawback is that Milhøj’s (1985)
condition cannot be given an explicit form when r > 1 and m > 2.
It should be noted that (2.3) is not necessary for the strict stationarity ofmodel (2.1). The necessary and sufficient condition for the strict stationarity ofmodel (2.1) was established by Bougerol and Picard (1992) in terms of the top Lyapunov exponent (see §3.1). The regions of strict stationarity are, in general, muchlarger than those of second-order stationarity. As an illustration, for the first-order
ARCH model, ARCH(1), the various conditions under normality are summarized asfollows:

VariableεtStrict stationarity

2nd

Coefficient

α1(0, 3.56214)(0, 1)

Moments
4th(0, 0.57735)

8th(0, 0.31239)

Non-normality reduces the permissible range of the ARCH(1) parameter for the 4th
and higher moments. It seems difficult to obtain a closed form expression of strict
stationarity in terms of the ARCH(r) parameters for any r > 1.

2.2

Sample ACVF and ACF

In time series analysis, the autocovariance function (ACVF) and autocorrelation
function (ACF) are important because they usually provide meaningful information
about the series. Define the sample ACVF and sample ACF, respectively, by
γnε (k) =n
1 Xεt εt−k ,n t=k+1

ρnε (k) =γn,ε (k),γn,ε (0)3where n is the sample size and k ≥ 0. Correspondingly, the true values are givenby:γε (k) = E(ε0 εk ) ,γε (k).
γε (0)

ρε (k) =As the ARCH process εt is an uncorrelated white noise sequence, γε (k) = ρε (k) = 0
if k > 0. Under the fourth moment condition, Milhøj (1985) showed that γnε (k) and√ρnε (k) are consistent estimators of γε (k) and ρε (k), respectively, and n[γnε (k) −√γε (k)] and n[ρnε (k) − ρε (k)] are asymptotically normal.
It is natural to ask if Milhøj’s results still hold if the fourth moment condition is
not satisfied. This is a difficult problem because ARCH processes exhibit a strong
heavy-tailed feature when Eε4t = ∞. Using the point process technique, Davis andMikosch (1998) showed that, if Eε2t < ∞ but Eε4t = ∞, thenn1−2/q L(n)−2 γnε (k) →d Vq (k) ,n1−2/q L(n)−2 ρnε (k) →dVq (k),
Eε2t

where q ∈ (2, 4) is the unique solution to E(α1 ηt2 )q/2 = 1, Vq (k) is q/2−stable in R,
and L(n) is some slowly-varying function. From the above results, γnε (k) and ρnε (k)are consistent estimators of γε (k) and ρε (k), respectively, but the convergence rateis slower than the usual n1/2 . This result is different from those for linear processes
with i.i.d. regularly varying noise. Davis and Resnick (1985, 1986) showed that the
sample ACF is still asymptotically normal with scaling n1/2 if the i.i.d. noise has
finite variance but infinite fourth moment.
Furthermore, Davis and Mikosch (1998) showed that, if E|ε|p < ∞ for 0 < p < 2
but Eε2t = ∞, thenn1−2/q L(n)−2 γnε (k) →d Vq (k) ,ρnε (k) →dVq (k),Vq (0)

4

where q ∈ (0, 2). In this case, the estimator of the ACF is inconsistent. This result
is quite different from that for linear processes with i.i.d. regularly varying noise, inwhich the sample ACF converges to the true ACF with a convergence rate greaterthan n1/2 (see Davis and Resnick 1985, 1986).
The sample ACVF and ACF of ε2t have also been investigated by Davis andMikosch (1998). Although they considered only the first-order ARCH model, theirresults can be extended to higher-order ARCH models. de Vries (1991) demonstrated
that, under certain conditions, GARCH processes can generate realizations that havea stable distribution unconditionally.

2.3

Parameter Estimation

The parameters of model (2.1) can be estimated by several methods. The simplest
method is the least squares estimator (LSE). First, write model (2.1) as
ε2t = α0 + α1 ε2t−1 + · · · + αr ε2t−r + ξt ,(2.5)

where ξt = ε2t − ht and ξt can now be considered as a martingale difference. Let
δ = (α0 , α1 , · · · , αr )0 and ε̃t = (1, ε2t , · · · , ε2t−r+1 )0 . Then the LSE of δ isδ̂ = (nXnX

ε̃t−1 ε̃0t−1 )−1 (t=2

ε̃t−1 ε̃t ).

t=2

Weiss (1986) and Pantula (1989) showed that δ̂ is consistent and asymptoticallynormal. However, their results assume that the 8th moment of εt exists, which is astrong condition.
In general, maximum likelihood estimation (MLE) is used to estimate the parameter δ. Given observations εt , t = 1, · · · , n, the conditional log-likelihood can be
written as
n
X

1
1 ε2t
L(δ) =
lt , lt = − ln ht −
,
2
2 ht
t=1

(2.6)

where ht is treated as a function of εt . Assume that δ ∈ Θ, a compact subset of
Rr+1 , and that the true value of δ is δ0 . Defineδ̂ = argmaxδ∈Θ L(δ).

5

(2.7)

Since the conditional error ηt is not assumed to be normal, δ̂ is called the quasimaximum likelihood estimator (QMLE). Under the fourth moment condition, Weiss(1986) and Pantula (1989) showed that the QMLE λ̂ is consistent and asymptoticallynormal. Ling and McAleer (1999b) proved that the QMLE of δ is consistent andasymptotically normal under only the second moment condition. It is expected that,when εt is strictly stationary but Eε2t = ∞, the QMLE will still be consistent andasymptotically normal. The BHHH algorithm is often used to determine δ̂. However,
Mak, Wong and Li (1997) suggested that the BHHH algorithm has a convergenceproblem if the starting values are not sufficiently close to the solutions and that afull Newton-Raphson procedure should instead be used.
When ηt is not normal, the QMLE is not efficient, that is, its asymptotic covariance matrix is not minimal in the class of asymptotically normal estimators. In
order to obtain an efficient estimator, one needs to know or estimate the density
function of ηt and use an adaptive estimation procedure. This was considered by
Linton (1993) and Drost, Klaassen and Werker (1995), who proved that the ARCH
model belongs to the locally asymptotically normal (LAN) family. After suitablere-parameterisation, they also constructed adaptive estimators for the parametersof interest.

3

GARCH Models

Bollerslev (1986) extended the ARCH model to the generalized autoregressive conditional heteroscedasticity (GARCH (r, s)) model:qεt = ηt ht ,ht = α0 +

(3.1)

r
X

αi ε2t−i +

i=1

s
X

βi ht−i ,i=1where α0 > 0, αi ≥ 0, βi ≥ 0, and ηt is defined as in (2.1).

6

(3.2)

3.1

Basic Properties

Bollerslev (1986) showed that the necessary and sufficient condition for the secondorder stationarity of models (3.1)-(3.2) is:
r
X

αi +

s
X

i=1

βi < 1.

(3.3)

i=1

For the GARCH(1,1) model, Nelson (1990) obtained the necessary and sufficientcondition for strict stationarity and ergodicity as follows:E( ln(α1 ηt2 + β1 )) < 0.

(3.4)

Condition (3.4) allows α1 + β1 to be 1, or slightly larger than 1, in which caseEε2t = ∞. For the general model (3.1)-(3.2), the necessary and sufficient condition
for strict stationarity and ergodicity was established by Bougerol and Picard (1992)
and Nelson (1990). Ling and Li (1997c) proved that, under (3.3), there exists a
unique Ft -measurable and second-order stationary solution to model (3.1)-(3.2),
and that the solution is strictly stationary and ergodic, with the following causal
representation:
ht = α0 +

∞
X

0

c (

j=1

j
Y

At−i ) ξt−j a.s.,

(3.5)

i=1

where ξt = (α0 ηt , 0, · · · , 0, α0 , 0, · · · , 0)(r+s)×1 , with the first component α0 ηt and(r + 1)-th component α0 , c = (α1 , · · · , αr , β1 , · · · , βs )0 , and


α1 ηt



At = 

 α1

···

αr ηt

β1 ηt

I(r−1)×(r−1) O(r−1)×1
···

···

βs ηt

O(r−1)×sαrβ1

O(s−1)×r

···

βs




.



(3.6)

I(s−1)×(s−1) O(s−1)×1

Bollerslev (1986) provided the necessary and sufficient condition for the existenceof the 2mth moment of the GARCH(1,1) model, and the necessary and sufficientcondition for the fourth-order moments of the GARCH(1,2) and GARCH(2,1) models. Using a similar method as in Bollerslev (1986), He and Teräsvirta (1999a)
provided the moment conditions of a family of GARCH(1,1) models. Ling and7

McAleer (1999d) derived the sufficient condition for the existence of the stationary solution of this family of GARCH(1,1) models, showed that He and Terävirta’s(1999a) condition is necessary but not sufficient, and provided the sufficient moment
condition. He and Teräsvirta (1999b) and Karanasos (1999) examined the fourth
moment structure of the GARCH(p, q) process. From the proof in Karanasos (1999),
it can be seen that the condition is necessary but not sufficient. He and Teräsvirta(1999b) stated that their condition is necessary and sufficient. Ling and McAleer(1999c) showed that the necessary condition for the existence of the fourth moment
is incomplete, that the condition is not sufficient for the existence of the fourth moment, and also derived the necessary and sufficient conditions for the existence of
all the moments.
Based on Theorem 2.1 in Ling and Li (1997c) and Theorem 2 in Tweedie (1988),
Ling (1999b) showed that a sufficient condition for the existence of the 2mth momentof model (3.1)-(3.2) isρ[E(A⊗mt )] < 1,(3.7)

where ρ(A) = max{eigenvalues of a matrix A}. Ling’s result does not need toassume that the GARCH(r, s) process starts infinitely far in the past with finite
2mth moment, as is required in Bollerslev (1986) and He and Teräsvirta (1999a, b),
and has a far simpler form as compared with that of Milhøj (1985). Ling and McAleer(1999c) further showed that condition (3.7) is also necessary for the existence of the2mth moment. Thus, the moment structure of the GARCH(r, s) model in (3.1)(3.2) has now been established completely. Bera, Higgins and Lee (1996) considereda random coefficient formulation of GARCH processes. An asymptotic theory forthe sample autocorrelations and extremes of a GARCH(1,1) process is provided in
Mikosch and Stărică (2000). As an extension of the GARCH(r, s) process, Ling and
McAleer (1999c) also derived the necessary and sufficient moment conditions of theasymmetric power GARCH(r, s) model of Ding et al. (1993).

8

3.2

Quasi-Maximum Likelihood Estimation

The GARCH model is usually estimated by the quasi-maximum likelihood method.
However, the properties of the QMLE are not completely clear. Consider the simple
but important GARCH(1,1) model. In this case, the likelihood can be written as
L(δ) =nX

1
1 ε2
lt , lt = − ln ht − t ,
2
2 ht
t=1

(3.8)

where ht is treated as a function of εt , and the parameter δ = (α0 , α1 , β1 )0 and ht
are calculated through the following recursion:ht = α0 + α1 ε2t−1 + β1 ht−1 , h0 = a positive constant.

(3.9)

Lee and Hansen (1994) and Lumsdaine (1996) proved that the local QMLE is consistent and asymptotically normal, assuming that E(ln(α1 ηt2 + β1 )) < 0, which is thenecessary and sufficient condition for strict stationarity. However, Lee and Hansen(1994) required that all the conditional expectations of ηt2+κ < ∞ uniformly with
κ > 0, while Lumsdaine (1996) required that Eηt32 < ∞. In addition, Lee andHansen (1994) showed that the global QMLE is consistent if εt is second-order stationary. Lee and Hansen (1994) and Lumsdaine (1996) stated that their methods
are valid only for the simple GARCH(1,1) model and cannot be extended to moregeneral cases.
For the general order GARCH(r, s) model, Ling and Li (1997b) proved that the
local QMLE is consistent and asymptotically normal if Eε4t < ∞. Based on uniform
convergence as a modification of a theorem in Amemiya (1985, page 116), Ling and
McAleer (1999b) proved the consistency of the global QMLE under only the secondorder moment condition. They also derived the asymptotic normality of the global
QMLE under the 6th moment condition.
When ηt is not normal, the QMLE is inefficient. Drost and Klaassen (1997)
investigated adaptive estimation of the GARCH(1,1) model. This method was extended to nonstationary ARMA models with higher-order GARCH(r, s) errors by

9

Ling and McAleer (1999a). Francq and Zakoı̈an (2000) consider the estimation of
weak GARCH representations (Drost and Nijman, 1993) characterized by an ARMA
structure for the squared error terms.

4

Stationary ARMA-GARCH Models

The ARCH process is a non-independent white noise sequence, which first appearedin the regression model of Engle (1982). Engle’s original motivation seems to havebeen that an ARCH structure provides improved statistical inference for the meanof the regression model, such as confidence intervals and forecasting. Over the last
decade, there has been a tendency to employ the ARCH/GARCH model to analyze
the volatilities of financial and economic data, while ignoring the specification andestimation of the conditional mean. However, if the conditional mean is not specified
adequately, then it may not be possible to construct consistent estimates of the true
ARCH process, for which statistical inference and empirical analysis regarding the
ARCH component might be misleading. Thus, even though the primary interest
might be on the volatilities in the data, the specification and estimation of theconditional mean are still important.
The conditional mean is typically given as an AR or ARMA model. However,since the conditional variances of the white noise are not constant, the generating
mechanism of the AR or ARMA model is quite different from the traditional AR or
ARMA model with i.i.d. errors, or martingale differences with a constant conditionalvariance. As a number of statistical properties of the traditional AR or ARMA
model cannot be extended to the present case, it is necessary to have a thoroughinvestigation of these types of models.
We define the ARMA-GARCH model by the following equations:
yt =

p
X
i=1

ϕi yt−i +qqX

ψi εt−i + εt ,

(4.1)

i=1

εt = ηt ht , ht = α0 +

r
X
i=1

10

αi ε2t−i +

s
X
i=1

βi ht−i .

(4.2)

There is no paper which is especially devoted to the ARMA-GARCH model, although it is a special case of Ling and Li (1997c, 1998) and Ling and McAleer(1999b). When s = 0, the ARMA-GARCH model reduces to the ARMA-ARCH
model, which is a special case of the ARMA-ARCH model of Weiss (1986). Whenq = 0, s = 0 and r = 1, the AR-ARCH(1) model was investigated by Pantula (1988).
The properties of the ARMA-GARCH model appear in Ling and Li (1997c). When
all the roots of φ(z) = z p −Pp

i=1 ϕi zp−ilie outside the unit circle, yt is strictly

stationary if εt is strictly stationary, and yt is 2mth order stationary if εt is 2mth
stationary. Thus, in this section, we consider estimation of only the ARMA-GARCH
model.
The parameters in (4.1)-(4.2) consist of two sets: one set includes the parametersof the conditional mean, denoted by m, and another set includes the parameters of
the conditional variance ht , denoted by δ. In practice, m is first estimated andthen the residuals from the estimated conditional mean are calculated. When theresiduals have been obtained, δ can be estimated using the methods in Sections 2-3.
Furthermore, the estimated ht is used to obtain a more efficient estimator of m.
If the density function of ηt is symmetric, the MLE of m and δ can be obtained
through a separate iteration procedure without loss of asymptotic efficiency. The
following section examines the estimation of m when δ is assumed to be known.

4.1

Least Squares Estimation

Denote the true value of m by m0 . Given observations y1 , · · ·, yn , the LSE of m0 ,m̂, is defined as the values in Θ which minimize
Sn =

n
X

ε2t .

(4.3)

t=1

For the ARMA-ARCH model, Weiss (1986) showed that m̂ is consistent for m0 and√n(m̂ − m0 ) −→L N (0, A),

11

(4.4)

with

"

A=E

−1

#

"

#

"

#

∂εt ∂εt2 ∂εt ∂εt−1 ∂εt ∂εt
E
ε
E
.
t
m m0
m m0
m m0 m=m0

Pantula (1989) also obtained the asymptotic distribution of the LSE for the AR
model with ARCH(1) errors, and gave an explicit form for A. The results in Weiss(1986) and Pantula (1989) require that yt has finite fourth moment. As yet, no one
seems to have considered the LSE of m0 for the ARMA-GARCH model. However,
the result in Weiss (1986) for the LSE can be easily extended to the ARMA-GARCH
model. When GARCH reduces to an i.i.d. white noise process, the LSE is equivalent
to the MLE of m0 .
There is presently no asymptotic theory for the LSE of the ARMA-GARCH
model when the fourth moment condition is not satisfied. From the results of Davisand Mikosch (1998), it would be expected that the LSE is inconsistent if the variance√of εt is infinite, but is consistent but with a slower convergence rate than n if εt
has finite variance and infinite fourth moment. In such cases, the results would be
different from those in Davis and Resnick (1985, 1986).

4.2

Quasi-Maximum Likelihood Estimation

Although the LSE is consistent and asymptotically normal if the fourth moment
is finite, it is inefficient for ARMA-ARCH/GARCH models. In such cases, it isstandard to use MLE. The maximum likelihood method was first used by Engle(1982) for both the AR-ARCH model and a fixed design regression with ARCH
errors. First, the log-likelihood function can be written as
L(m) =nX

1
1 ε2
lt , lt = − ln ht − t ,
2
2 ht
t=1

(4.5)

where ht is treated as a function of yt and m, and is calculated through the following
recursion:ht = α0 +rX
i=1

αi ε2t−i +sXβi ht−i , h0 = a positive constant.

i=1

12

(4.6)

Define m̂ = maxm∈Θ L(m). Since ηt is not assumed to be normal, m̂ is referred to as
the QMLE of m. For the ARMA-ARCH model, Weiss (1986) showed that the QMLE
is consistent and asymptotically normal under a finite fourth moment condition.
From Ling and Li (1997c), there exists a locally consistent and asymptotically normal
QMLE for the ARMA-GARCH model if it has finite fourth moment. When ηt is√normal, the asymptotic covariance matrix of n(m̂ − m0 ) is"#−1

1 ∂ εt ∂ εt
1 ∂ ht ∂ht
B=E
+ 2
.
0
ht ∂ m ∂ m
2ht ∂ m ∂ m0 m=m0

(4.7)

Engle (1982) demonstrated that the MLE is more efficient than the LSE througha simple fixed design regression model and a first-order ARCH process. Pantula(1989) also showed that the MLE is more efficient than the LSE for the AR modelwith ARCH(1) errors. In fact, it can be shown that A ≥ B for the general ARMAGARCH case.
Under the existence of the second moment, Ling and McAleer (1999b) showedthat the global QMLE is consistent. However, in order to derive the asymptotic
normality of the global QMLE, the model must satisfy the sixth moment condition.
For the ARMA-GARCH(1, q) model, it is possible to show that the global QMLE
of m0 is consistent and asymptotically normal, even if the fourth moment condition
is not satisfied.

4.3

Adaptive Estimation

The QMLE of m0 in the stationary ARMA-GARCH model is efficient only if ηt isnormal. When ηt is not normal, adaptive estimation is useful for obtaining efficientestimators. A comprehensive account of the theory and method of adaptive estimation can be found in Bickel (1982) and Bickel, Klaassen, Ritov and Wellner (1993),
with valuable surveys available in Robinson (1988) and Stoker (1991).
In the time series context, Kreiss (1987a) investigated the stationary ARMA
model with i.i.d. errors. He proved the local asymptotic normality (LAN) property

13

of the model and constructed adaptive estimators of m0 . Unlike Bickel (1982),
Kreiss’ adaptive procedure avoids the split sample technique, and hence is quite
useful for practical applications. Jeganathan (1995) and Koul and Schick (1996)
constructed adaptive estimators without splitting the sample for some nonlinear AR
time series with i.i.d. noise. Koul and Schick (1996) also showed through simulationthat the adaptive estimator without splitting the sample is superior to those basedon the split sample technique.
Lee and Tse (1991) and Engle and González-Rivera (1991) are among the firstto have used a semiparametric approach for models (4.1)-(4.2), but they did not
obtain any theoretical results. Koul and Schick (1996) investigated adaptive estimation for a random coefficient AR model, which is an ARCH-type time seriesmodel. Jeganathan (1995) and Drost, Klaassen and Werker (1997) developed general frameworks suitable for stationary ARCH-type times series. The results in Ling
and McAleer (1999a) include the development of the adaptive method for stationary
ARMA-GARCH models and the conditions required for adaptive estimation.

5

Nonstationary ARMA-GARCH Models

Nonstationary time series have now been extensively investigated for the last two
decades. Some important results for nonstationary AR models can be found in Fuller(1976), Dickey and Fuller (1979), Phillips (1987), Chan and Wei (1987, 1988), Tsay
and Tiao (1990) and Jeganathan (1995), among many others. However, research
on nonstationary time series is almost always limited to innovations with constantconditional variances. Under the framework of Phillips and Durlauf (1986) andPhillips (1987), the long-run variance and the innovation variances are equal in thepresence of heteroscedasticity, but it does not include conditional heteroscedastic
processes as defined in (3.1)-(3.2).
The ARMA-GARCH model is called nonstationary if the characteristic polyno-

14mial φ(z) has a root on the unit circle. Consider the simple AR(1) case:yt = φyt−1 + εt

(5.1)

where φ = 1, and εt follows the GARCH(1, 1) process, that is,qεt = ηt ht , ht = α0 + α1 ε2t−1 + β1 ht−1 .

(5.2)

When β1 = 0, in which case εt follows a first-order ARCH process, Pantula (1989)
derived the asymptotic distribution of the LSE of the unit root under the fourth
moment condition. Ling and Li (1997b) obtained the same result under the second
moment condition, namely α1 + β1 < 1. The asymptotic distribution is
R1

B(t)dB(t)n(φ̂LS − 1) −→ 0R 1 2,
0 B (t)dt
L

where φ̂LS = (

Pn

2
−1 Pn
t=1 yt−1 ) ( t=1 yt yt−1 ) and B(t) is a standard Brownian motion.

Thus, the Dickey-Fuller test statistic can still be used. However, Peters and Veloce (1988) and Kim and Schmidt (1993) provided simulation results showing that
Dickey-Fuller tests based on the LSE are generally not robust.
It should be noted that, for stationary ARMA-GARCH models, the QMLE is
more efficient than the LSE. It seems natural to expect this advantage to extend to
nonstationary time series, in which case unit root tests based on the MLE in thepresence of ARCH/GARCH innovations should be useful. According to standard
statistical theory, an efficient estimator will often provide locally most powerful tests[e.g. see Rao(1973, Chapter 7)]. For this reason, unit root tests based on QMLE
would be expected to be more powerful than those based on LSE.
Note that Leybourne, McCabe and Tremayne (1996) observed that heteroscedasticity will be present automatically if φ is actually a random variable fluctuating
about 1. They developed a score test for such a randomized unit root.

5.1

Quasi-Maximum Likelihood Estimation

In this section, we assume that the characteristic polynomial φ(z) has only a unit
root of +1. The general case was investigated in Ling and Li (1998). Since ϕ(z) has

15

a unit root, it can be decomposed as (1 − z)φ(z), where φ(z) = 1 −Pp−1ii=1 φi z .

Let

wt = (1 − B)yt , where B is the backshift operator. Model (4.1) can be rewritten as
yt = γyt−1 + wt , wt =p−1
X

φi wt−i +

i=1

q
X

ψi εt−i ,

(5.3)

i=1

where γ = 1 and εt is defined by (4.2). The parameters in model (5.3) are γ and m =0(φ0 , ψ 0 )0 , where φ = (φ1 , · · · , φp−1 ) and ψ = (ψ1 , · · · , ψq )0 . As in the stationary case,
we assume that the parameters in (4.2) are known or can be estimated consistently.
Given the observations y1 , · · · , yn , with initial values yi = 0, or some constants,
for i ≤ 0, the log-likelihood function can be written as
L(λ) =nX

1
1 ε2
lt , lt = − ln ht − t ,
2
2 ht
t=1

(5.4)

where λ = (γ, m0 )0 , and ht is treated as a function of yt and λ. Ling and Li (1998)
showed that there exists a locally consistent QMLE such that0 0G−1n (λ̂ − λ) −→L (ξM L , N ) ,

(5.5)

where
R

c 1 w1 (t)dw2 (t)ξM L = 0 R 1 2,
F 0 w1 (t)dt(5.6)

c = [1 − φ(1)]−1 , N is a normal random vector independent of ξM L , F is a constantdepending on the GARCH parameters, κ = Eηt4 − 1, and (w1 (t), w2 (t)) is a bivariate
Brownian motion with covariance tΩ. When r = s = 1,
Ã

Ω=

Eht
1
2 P∞
1 E(1/ht ) + κα k=1 β 2(k−1) E(ε2t−k /h2t )and when ηt is normal, κ = 2 and F = E(1/ht ) + 2α2

P∞

k=1 β

!

,

(5.7)

2(k−1)

E(ε2t−k /h2t ). For

higher-order GARCH models, the structure of Ω can be found in Ling and Li (1998).
Note also that, unlike the least squares case, the moving average parameters do notappear in (5.6) and (5.7).
The above results were derived under the fourth moment condition in Ling andLi (1998). Furthermore, under the second moment condition, Ling and Li (1997b)

16derived the same result for models (5.1)-(5.2) when c = 1. If the second moment
condition is not satisfied, the asymptotic distribution for the LSE or QMLE of theunit root is as yet unknown. For the unit root process with i.i.d. errors having
infinite variance and in the domain of attraction of an α-stable law, Chan and Tran
(1989) and Chan (1990) showed that n−1/α (φ̂LS − 1) converges to a functional of a
Levy process with α ∈ (0, 2). It is conjectured that there is a similar asymptotic
distribution for the LSE or QMLE of the unit root when the GARCH noise has an
infinite variance.

5.2

Unit Root Tests Based on QMLE

The asymptotic distribution for the QMLE of the unit root can be used to constructa unit root test. For simplicity, we consider only models (5.1)-(5.2). Denote φ̃M L as
the QMLE of φ, and let
1
1
B1 (t) = w1 (t) and B2 (t) = − 2
σ
σ

s

σ2
w1 (t) +σ2K − 1s

σ2
w2 (t),
σ2K − 1

where σ 2 = Eht and K is the (2,2)th element of Ω. Then B1 (t) and B2 (t) are two
independent standard Brownian motions. As shown in Ling and Li (1998),√
R1R
σ 2 K − 1 01 B1 (t)dB2 (t)L
0 B1 (t)dB1 (t)n(φ̃M L − 1) −→ 2 R 1 2
+
.
(5.8)
R1 2
σ2F
σ F 0 B1 (t)dt
0 B1 (t)dt√
R
The second term in (5.8) can be simplified to [ σ 2 K − 1/F σ 2 ] ( 01 B12 (t) dt)−1/2 ξ,Rwhere ξ is a standard normal random variable independent of 01 B12 (t)dt (see Phillips,
1989). Thus,
R1

B1 (t)dB1 (t)n(φ̂M L − 1) −→ 02 R 1 2+
σ F 0 B1 (t)dtL√

σ2K − 1 Z 1 2( B1 (t)dt)−1/2 ξ.
σ2F
0

(5.9)

From (5.8)-(5.9), we see that the asymptotic distribution of φ̂M L can be represented
as a combination of the asymptotic distribution of φ̂LS and a scale mixture of normals. This property is similar to that of the least absolute deviation estimator ofunit roots given in Herce (1996). Ling and Li (1998) showed that the QMLE of φ ismore efficient than the LSE.

17

As the asymptotic distribution in (5.9) includes nuisance parameters, we cannot
use it directly to test for a unit root. There are two methods to overcome thisdifficulty. The first is to combine the LSE and QMLE to construct a unit root test,
as in Ling and Li (1997b). Letn
1 X
Lφ = n(φ̂LS − 1), Lt = ( 2y 2 )1/2 Lφ ,n t=1 t−1

where ȳ = n−1Pn

t=1 yt−1 . Furthermore, defineσ̂ 2 F̂Mφ = q{n(φ̂M L − 1) − (F̂ σ̂ 2 )−1 [n(φ̂LS − 1)]},σ̂ 2 K̂ − 1n
1 X
Mt = ( 2y 2 )1/2 Mφ .
n t=1 t−1
Ling and Li (1997b) showed thatL

Mφ −→ [Z 1
0L

B12 (t)dt]−1/2 ξ and Mt −→ ξ,

where ξ is a standard normal random variable independent ofR1

2
0 B1 (t)dt.

The limiting distributions of Mφ and Mt are the same as those based on theleast absolute deviations estimators of Herce (1996). However, the test statistics
themselves are quite different. Empirical critical values of these distributions werereported in Ling, Li and McAleer (1999), who showed that Mφ and Mt can overcome
the excessive sizes, as reported in Peters and Veloce (1988) and Kim and Schmidt(1993), and have power comparable to that of the Dickey-Fuller test.
Another method of overcoming the presence of nuisance parameters is to construct a unit root test without using the LSE, as used in Seo (1999). First, rewrite(5.9) asRR1
ρ 1 B1 (t)dB1 (t) q
B1 (t)dB2 (t)nc1 (φ̃M L − 1) −→ 0R 1 2+ 1 − ρ2 0 R 1 2,
0 B1 (t)dt
0 B1 (t)dt

L(5.10)

σF
where c1 = √ and ρ2 = 1/(σ 2 K) ∈ (0, 1). The t-statistic is then given by
K
R

R

1
1
n
1 X
B1 (t)dB1 (t) q
B1 (t)dB2 (t)
L ρ
2
2 R0
nc2 ( 2yt−1
)1/2 (φ̃M L − 1) −→ R 10 2
+
1
−
ρ
,
1
−1/2
n t=1
( 0 B1 (t)dt)( 0 B12 (t)dt)−1/2(5.11)

18

where c2 = c1 /σ. Seo (1999) tabulated the limiting distribution in (5.11) for different
values of ρ. The simulation results in Seo (1999) showed that the unit root test basedon (5.11) not only overcomes the size distortion problem, but is also consistently
more powerful than tests based on the LSE. These results confirm the expectationthat more efficient estimates of unit roots yield more powerful unit root tests.
When the conditional errors ηt are not normal, the estimator of the unit root is
not efficient. Ling and McAleer (1999c) investigated adaptive estimation of the nonstationary ARMA model with GARCH errors. They obtained the locally asymptotic
quadratic form of the log-likelihood ratio, and showed that it was neither locallyasymptotic normal nor locally asymptotic mixed normal. A new efficiency criterion
was given for a class of defined M -estimators. When the conditional error density
is known, Ling and McAleer (1999c) showed that efficient estimators can be constructed using the kernel estimator for the score function. It is also shown that theadaptive procedure for the parameters in the conditional mean part uses the fullsample.

6

Other ARCH-type Models

In this section, some other ARCH-type models are considered, namely double threshold ARCH, ARFIMA-GARCH, CHARMA, and vector ARMA-GARCH.

6.1

Double Threshold ARCH Models

Given the success of Tong’s (1978, 1980) threshold model in nonlinear time series, it
is natural to consider threshold structures for the conditional variance specification.
The use of thresholds to model asymmetries is supported by well known empiricalcharacteristics as to the likely asymmetric behaviour of volatility in the stock market(see, for example, French et al. (1987)).
Li and Li (1996) proposed the double threshold AR conditional heteroskedastic

19

(DTARCH) time series model:(j)yt = φ0 +

pi
X
(j)

φi yt−i + εt , aj−1 < yt−b ≤ aj ,

(6.1)

i=1
1

εt = ηt ht2 ,(k)

ht = α0 +

(6.2)
rk
X

(k)

αi ε2t−i , ck−1 < yt−d ≤ ck ,

(6.3)

i=1

where j = 1, · · · , v1 ; k = 1, · · · , v2 ; and b and d ≥ 1 are the delay parameters. In(6.1)-(6.3), the threshold parameters satisfy −∞ = a0 < a1 < · · · < av1 = ∞ and(j)−∞ < c0 < c1 < · · · < cv2 = ∞, φi(k)and αi(k)

are constants, α0(k)> 0 and αi≥ 0. The model generalizes the threshold AR model of Tong (1978, 1980) to include
a threshold ARCH component. Tong (1990) referred to this type of hybrid model
as a second generation model. Note that other indicator variables may be usedin place of yt−b and yt−d . The threshold variables are typically defined as a linear
combination of the lagged values of the observed process, but van Dijk, Teräsvirta
and Franses (2000) relaxed this definition of threshold variables to include non-linear
combinations of the lags of the observed process as well as of other variables. Li
and Lam (1995) combined the threshold autoregressive model with a fixed ARCH
specification in studying the asymmetry of a stock index. Extension to a doublethreshold GARCH model was considered by Brooks (2001).
Ling (1999b) showed that, ifPp(j)
i=1 maxj |φi | < 1

and

Pr

(k)
< 1, theni=1 maxk αithere exists a strictly stationary solution {yt , εt } satisfying models (6.1)-(6.3), andEπ1 (|yt |) and Eπ2 (ε2t ) are finite, where π1 and π2 are the stationary distributionsof {yt } and {εt }, respectively. However, the uniqueness and ergodicity conditions
are as yet unknown. If the second threshold, ck−1 < yt−d ≤ ck , is replaced byck−1 < εt−d ≤ ck , the strict stationarity and ergodicity condition has been obtainedby Liu , Li and Li (1997).
Under the assumption that yt is strictly stationary and ergodic, and the thresholdparameters ai and ci are known, Li and Li (1996) proved that the MLE is consistentand asymptotically normal. In practice, the threshold parameters ai and ci are20

unknown and can be estimated by the maximum likelihood method. However, theasymptotic distributions of the estimators are as yet unknown. For the threshold AR
model with i.i.d. errors, Chan (1993) showed that the estimator of the thresholdparameter has a convergence rate of n and an asymptotic distribution associatedwith the compound Poisson process. This method could possibly be used for the
DTARCH model.
Pesaran and Potter (1997) considered a floor and ceiling model of US output
which may be interpreted as a double threshold ARCH model. Rabemanjara andZakoı̈an (1993) examined an asymmetric ARCH model which may be regarded as a
special case of the DTARCH model. Fornari and Mele (1997) considered a similar
formulation to handle asymmetry in volatility. Lee and Li (1998) developed a smoothtransition double threshold model. Lundbergh and Teräsvirta (1998a) used a doublesmooth AR-GARCH model to analyse some high-frequency exchange rate data.
Wong and Li (1997) considered tests for the presence of autoregression under ARCH,
while Wong and Li (1999) examined tests for the null of AR-ARCH against the
double threshold ARCH model.
In the spirit of threshold nonlinear models Wong and Li (2000), Wong and Li(2001a, b) considered mixtures of autoregressive models and mixtures of autoregressive models with ARCH. Some interesting features of these types of models are thatsome components of the mixture can be non-stationary while the entire series canbe stationary, the predictive distributions can be multimodal, and it is fairly easy
to derive the conditions for stationarity and expressions for the autocorrelations.

6.2

Fractional ARIMA Models

Let {yt } satisfy
φ(B)(1 − B)d (yt − µ) = θ(B)εt ,εt | Ft−1 ∼ N (0, ht ), ht = α0 +

r
X
i=1

21

(6.4)
αi ε2t−i +

s
X
i=1

βi ht−i ,(6.5)

where (1 − B)d is defined by the binomial series:(1 − B)d =∞X(k + d − 1)!
k=0

k!(d − 1)!

Bk.

(6.6)

The specifications in (6.4)-(6.5) are referred to as the fractional ARIMA-GARCH orequivalently the ARFIMA-GARCH model, which was investigated by Ling and Li(1997c). Baillie, Chung and Tieslau (1995) considered a fractional ARIMA(0, d, 1)GARCH(1,1) model for the CPI series of 10 different countries. Note that exact
maximum likelihood estimation of (6.4) with ht = a constant has been considered
as early as in 1981 in the University of Western Ontario Ph.D. Thesis by W.K. Li.
Sufficient conditions for stationarity, ergodicity and the existence of higher-ordermoments of the fractional ARIMA model were derived by Ling and Li (1997c).
Under some mild conditions, it is shown that the MLE is locally consistent andasymptotically normal. It is well known that, when p = q = 0 so that (1−B)d yt = εt ,the MLE of d converges to N (0, 6/π 2 ) in distribution if εt is i.i.d. (see Li and McLeod,
1986). However, when εt is a GARCH process, Ling and Li (1997c) showed that theasymptotic variance isΩγ = E[1 ∂ εt 2
1 ∂ht 2() + 2() ],ht ∂ d2ht ∂ dwhich is no longer independent of d and is less than 6/π 2 . Ling and Li (1997c) also
examined the large sample distributions of the residual autocorrelations and thesquared-residual autocorrelations, and two portmanteau test statistics. Robinson(1991) considered tests for conditional heteroskedasticity in long memory processes.
More recently, Beran and Feng (1999) considered local polynomial estimation of afractional ARIMA model similar to the above.

6.3

CHARMA Models

Tsay (1987) proposed the conditional heteroskedastic autoregressive moving average(CHARMA) model, given by:yt − µ =pX
i=1

ψi (yt−i − µ) +

q
X
i=1

22

θi εt−i + εt ,

(6.7)

εt =

r
X

δit εt−i +

i=1

s
X

wit (yt−i − µ) + w0t (ŷt−1 (1) − µ) + et ,

(6.8)

i=1

where the orders p, q, r and s are finite and non-negative integers; µ, ψi and θi areconstant; δit , wit and et are random variables; and ŷt−1 (1) = E(yt |Ft−1 ), where Ft−1
is the σ−field generated by {et−i , wt−i , δt−i |i = 1, 2, · · ·}, wt = (w0t , w1t , · · · , wst )0 ,and δt = (δ1t , · · · , δrt )0 .
The LSE method can be used to estimate the parameters in (6.7). Tsay (1987)
proved that the LSE is consistent if Eε4t < ∞, and is asymptotically normal if
Eε8t < ∞. Since the model is an extension of the random coefficient AR model, theasymptotic MLE results can be obtained using the method in Nicholls and Quinn(1982). Basic properties such as strict stationarity, ergodicity and the moment
structure are given in Ling (1999a).
The CHARMA model has been extended to the multivariate case. Wong andLi (1997) considered a stationary multivariate CHARMA model, and Li, Ling andWong (1999) investigated a partially nonstationary AR model with conditional heteroscedasticity, as follows:
Yt = Φ1 Yt−1 + · · · + Φp Yt−p + εt

(6.9)εt = α1t εt−1 + · · · + αqt εt−q + et ,(6.10)

andwhere the Φi are constant matrices; det{Φ(z)} = |I − Φ1 z − · · · − Φp z p | = 0 has
d ≤ m unit roots and other roots outside the unit circle; rank[Φ(1)] = m − d;δt = (α1t , · · · , αqt ) is a sequence of i.i.d. matrices with mean zero and nonnegative
covariance E[vec(δt )vec0 (δt )] = Ω; and et is an i.i.d. random vector with mean zeroand positive covariance E(et e0t ) = G.
Under the condition for the finite fourth moment, Li, Ling and Wong (1998)
derived the asymptotic distributions of the LSE, a full rank MLE, and a reduced
rank MLE. When the multivariate ARCH process reduces to the innovation with a23

constant covariance matrix, these asymptotic distributions are the same as in Ahn
and Reinsel (1990). However, in the presence of multivariate ARCH innovations, theasymptotic distributions of the full rank MLE and the reduced rank MLE involvetwo correlated multivariate Brownian motions, which are different from those given
in Ahn and Reinsel (1990). The asymptotic results in Li, Ling and Wong (1998)
can be used to construct cointegration tests based on the MLE.

6.4

Vector ARMA-GARCH Models

Ling and McAleer (1999b) proposed the vector ARMA-GARCH model:
Φ(B)(Yt − µ) = Ψ(B)εt ,1/2

εt = Dt ηt , Ht = W +(6.11)rX
i=1

Ai ε̃t−i +sX

Bi Ht−i ,(6.12)

i=1

where Dt = diag(h1t , · · · , hmt )0 , Ht = (h1t , · · · , hmt )0 , Φ(B) = I − Φ1 B − · · · − Φp B pand Ψ(B) = I + Ψ1 B + · · · + Ψq B q are polynomials in B, ε̃t = (ε21t , · · · , ε2mt )0 ,and ηt = (η1t , · · · , ηmt )0 is a sequence of i.i.d. random vectors with mean zero andcovariance Γ.
Ling and McAleer (1999b) obtained the conditions for strict stationarity andergodicity, and the higher-order moments of the model. The consistency of the
global QMLE is proved under the existence of only the second-order moment. In
order to derive the asymptotic normality of the global QMLE, the results require thesecond moment condition for the vector ARCH model, the fourth moment condition
for the vector ARMA-ARCH model, and the sixth moment condition for the vector
ARMA-GARCH model.

7

Conclusion

Most of the theoretical results for GARCH-type processes require that the fourthor higher-order moments exist. In practice, this condition may not be satisfied.
When the fourth moment of the GARCH process is infinite, it exhibits the feature24

of heavy tails. At present, a theory is lacking for ARMA models derived fromthis type of GARCH specification, even for ARMA models with i.i.d. heavy-tailed
noise (see Resnick (1997)). Since heavy-tailed phenomena are often encountered infinance and economics, an analysis of data exhibiting heavy tails would seem to be
an important direction for future research.
Although there have been many contributions to the ARCH/GARCH literature,
it seems that until recently very little attention has been paid to model selection.
Apart from the diagnostic checking method of Li and Mak (1994) and its extension
by Ling and Li (1997a), there would seem to be few formal tools for checking model
adequacy. Tse and Zuo (1997) provided a simulation study of the Li–Mak test.
More recently, Lundbergh and Teräsvirta (1998b) showed that the Li–Mak test is
equivalent to a Lagrange multiplier test of no residual ARCH. Tse (1999) provides a
recent review of this literature. A generalization of Li and Mak (1994) is obtained by
Horvath and Kokoszka (2001). A robustified version of Li and Mak (1994) against
outliers is developed by Jiang, Shao and Hui (2001). All order selection methods for
ARMA models, such as those in Hannan (1980), Potscher (1983, 1989), Tsay (1984),
and Wei (1992), require that the error processes are i.i.d. or martingale differenceswith supt E(ε2t |Ft−1 ) ≤ a constant. However, ARCH-type models generally do notsatisfy these conditions. It is important to develop a theory for order selection of
ARCH, GARCH and ARMA-GARCH models, with Wong and Li (1996) and An,
Fong and Li (1999) being two useful attempts in this direction.

25

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