1. Adaptive Quasi-Maximum Likelihood Estimation of GARCH models
with Student’s t Likelihood 1
Xiaorui Zhu 2, Li Xie3
Abstract This paper proposes an adaptive quasi-maximum likelihood estimation when forecast-
ing the volatility of financial data with the generalized autoregressive conditional heteroscedas-
ticity(GARCH) model. When the distribution of volatility data is unspecified or heavy-tailed,
we worked out adaptive quasi-maximum likelihood estimation based on data by using the scale
parameter ηf to identify the discrepancy between wrongly specified innovation density and the
true innovation density. With only a few assumptions, this adaptive approach is consistent and
asymptotically normal. Moreover, it gains better efficiency under the condition that innovation
error is heavy-tailed. Finally, simulation studies and an application show its advantage.
Keywords quasi likelihood, GARCH Model, adaptive estimator, heavy-tailed error
JEL Classification: C13; C22
1 Introduction
With the development of derivatives, volatility has been a crucial variable in not only
modeling financial data, but also designing trading strategies and implementing risk man-
agement. Among various models of analysis of volatility, GARCH(generalized autoregres-
sive conditional heteroscedasticity) model is a well-known and useful one. It was proposed
by Bollerslev(1986) as follows:



ut = σt|t−1εt
σ2
t|t−1 = ω +
∑p
i=1 αiu2
t−i +
∑q
j=1 βjσ2
t−j
(1)
Primarily, the estimation of ARCH/GARCH model is based on the maximum likeli-
hood estimation(MLE) when the innovation subjects to conditional Gaussian distribution.
However, if the distribution of innovation εt is not normal, as is prevalent in plenty of em-
pirical data, quasi-maximum likelihood estimation would be more suitable. At first, plenty
of literature discussed Gaussian quasi-maximum likelihood estimation when the innovation
distribution is not normal. Weiss’s(1986) [19] research showed that even under special con-
dition that the data is un-normalized and have finite fourth moments, the Gaussian-MLE
is consistent asymptotically normal. After that, Bollerslev(1986) [1], Hsieh(1989) [12],
and Nelson(1991) [14] proposed the issue of parameter estimation by generalized Gaussian
quasi-maximum likelihood estimation(GQMLE) when the innovation distribution is not
normal, and has also derived the consistency and efficiency of this method. Bougerol and
Picard(1992) [2] discussed the necessary stationarity and ergodicity of GARCH models. To
1:We appreciate all the helpful suggestions from the editor and the reviewers, thoughtful comments from
A.P.Gaorong Li and Yuyang Zhang, and financial support from the National Natural Science Foundation
(Grant No.11171011) and the National Social Science Foundation(Grant No.13BGL007).
2:College of Applied Sciences, Beijing University of Technology, Pingleyuan 100, Chaoyang District, Bei-
jing, 100124, China.
E-mail: xiaorui.zhu@emails.bjut.edu.cn
3:College of Applied Sciences, Beijing University of Technology, Pingleyuan 100, Chaoyang District, Bei-
jing, 100124, China.
E-mail:xieli@bjut.edu.cn

2. 2
get an estimation when the innovation distribution is unknown, Elie and Jeantheau(1995)
proposed the Gaussian quasi-maximum likelihood estimator that is consistent and asymp-
totically normal. Besides, there are also crucial achievements of Gaussian QMLE in recent
years. Berkes, Horvath and Kokoszka (2003) [11] studied the structure of a GARCH(p,q)
and proved the consistency and asymptotic normality of the QMLE under mild conditions.
Strong consistency and asymptotic normality of QMLE were also proved in the study of
Francq and Zakoian (2004) [9].
Other researches that involve improving Gaussian QMLE include: Engle and Gonzalez-
Rivera(1991) [5] published a procedure that can improve the efficiency of GQMLE. Drost
and Klaassen (1997) [4] put forward the adaptive estimation in ARCH model. Sun and
Stengos (2006) [18] proposed adaptive two-step semi-parametric procedures on the condi-
tion of symmetric error or asymmetric error separately. A Self-weighted and local QMLE
for ARMA-GARCH models has been discussed in the study of Ling (2007) [13].
With the developing of quasi-maximum likelihood estimation, some non-gaussian QM-
LEs were proposed to improve the estimator when the innovations are heavy-tailed or
skewed. Xiu(2010) [21] discussed quasi-maximum likelihood estimation of stochastic volatil-
ity model with high frequency data. Ossand´on and Bahamonde(2011) [16] proposed
a novel estimation for GARCH models based on the Extended Kalman Filter(EKF).
Zhu(2012) [22] put forward a mixed portmanteau test for ARMA-GARCH model by
quasi-maximum likelihood estimator. Francq et al.(2011) [8] has developed a two-stage
non-Gaussian quasi-maximum likelihood estimation to rectify the value of parameter es-
timation. This procedure allows the use of generalized Gaussian likelihood and proposes
a test that can determine whether the more efficient Quasi-MLE is required with a non-
Gaussian density. Other notable achievements lay in that the Student’s t likelihood func-
tion has been taken into consideration, which is called three-step non-Gaussian Quasi-MLE
approach in the study of Fan et al.(2013) [6]. Because the Pearson’s Type IV(PIV) distri-
bution can capture a large range of the asymmetry and leptokurtosis of innovation error,
Zhu and Li (2014) [23] have proposed a novel Pearson-type QMLE of GARCH(p,q) models
to capture not only the heavy-tailed but also the skewed innovations.
This article focuses on adaptive analyzing procedure which will increase efficiency of
the estimator of the GARCH model, and proposes an adaptive procedure of QMLE aiming
at minimizing the discrepancy between true and specified distribution of innovation. The
scale parameter ηf is built in the sense of Kullback-Leibler Information Criterion (KLIC).
With the Adaptive-QMLE procedure can not only find the approximate degree of freedom
of innovation distribution, but also get the optimized quasi-maximum likelihood estimator
after several iterations. This general estimation doesn’t rely on models, so it may be
used in other general models. And the idea of Adaptive-QMLE can also be used in
other methods such as GQMLE, NGQMLE and PQMLE. The simulation studies confirm
that convergence rate of this adaptive QMLE procedure is very high, especially when the
innovation is heavy-tailed. It performes well with high frequency data when the empirical
distribution of innovations is often heavy-tailed.
This paper is organized as follows. In Section 2, we introduce the GARCH model
and quasi-maximum likelihood estimation. In Section 3, we describe the assumptions,
propositions of our new adaptive quasi-maximum likelihood estimation, where we also
explain the details and proofs of this procedure. Some simulation studies are provieded in
Section 4. Real data analyses are shown in Section 5. The Section 6 concludes this paper.

3. 3
2 Quasi-MLE in GARCH model
2.1 The GARCH Model
Common type of GARCH(p,q) model has been shown in the Section 1. Let θ =
(ω, ⃗α′, ⃗β′)′ be the unknown parameters of GARCH(p,q), where ⃗α = (α1, · · · , αp)′, ⃗β =
(β1, · · · , βq)′ are the heteroscedastic parameters. {εt, −∞ < t < ∞} are innovation of
model. Θ ∈ R1+p+q
o is the parameter space and Ro = [0, ∞). The following assumptions
for GARCH model are necessary:
Assumption 1.
(i)The GARCH process {ut} is strictly stationary and ergodic.
(ii)For each θ ∈ Θ, α(z) and β(z) have no common root, α(1) ̸= 0, αp + βq ̸= 0 and
∑q
j=1 βj < 1, where α(z) =
∑p
i=1 αizi and β(z) = 1 −
∑q
j=1 βjzj.
(iii)εt is a nondegenerate and i.i.d random variable with Eεt = 0, Eε2
t = 1 and un-
known density g(·).
The stationarity and ergodicity of GARCH models in Assumption 1(i) can be found in
Bougerol and Picard(1992) [2]. The identifiability conditions for GARCH(p,q) are given in
Berkes, Horvath and Kokoszka (2003) [11]. For a general GARCH model the conditional
variance σ2 cannot be expressed in terms of a finite number of the past observations ut−1,
ut−2.... In many researches, conditional variance ˜σ2
t is [7]:
˜σ2
t =
ω
1 − Σq
j=1αj
+
p∑
i=1
αiu2
t−i +
p∑
i=1
αj
∞∑
k=1
q∑
j1=1
· · ·
q∑
jk=1
βj1 · · · βjk
u2
t−i−j1−···−jk
(2)
By doing so, {˜σ2
t } in (2) is a function of sample u∗
t−1 = {us, −∞ < s ≤ t − 1}.
2.2 Quasi-Maximum Likelihood Estimation
Fortunately, it is easy to derive the likelihood function of a GARCH model with normal
error. Under the Eu2
t < ∞ assumption, the log-likelihood function of the GARCH model
is as follow:
L(ω, ⃗α, ⃗β) = −
n
2
log(2π) −
1
2
n∑
t=1
{log(σ2
t|t−1) +
u2
t
σ2
t|t−1
} (3)
Under mild conditions that εt isn’t specified as standard normal, Berkes, Horvath and
Kokoszka (2003) [11] proved the consistency and asymptotic normality of quasi-MLE.
Apart from normal distribution, student’s t-distributions and generalized Gaussian dis-
tributions are considered frequently. The deduction of the generalized quasi-maximum
likelihood estimator of GARCH model is as follows:
ˆθ = argmaxθ
1
2
n∑
t=1
{− log(σt|t−1) + log g(
ut
σt
)} (4)
In this paper, we consider innovation is standardized t-distribution, so the true prob-
ability density function of {εt, −∞ < t < ∞} which is the g(·) in the above formula
is:
g(x) =
Γ((ν + 1)/2)
(πν)1/2Γ(ν/2)
(1 +
x2
ν
)−
(ν+1)
2 (5)
where:ν > 0 may be treated as continuous parameter.

4. 4
3 Adaptive Quasi Maximum likelihood Estimation
If the true innovation distribution cannot be specified, gaussian quasi-maximum likeli-
hood estimation(GQMLE) is always inconsistent as shown in Newey and Steigerwald(1997)
[15]. Weiss(1984), Lee and Hansen(1994) study the asymptotic distributions of GQMLE.
Francq et al.(2011) [8] proposes a two-stage GQMLE that can improve efficiency of esti-
mator. Based on a three step quasi-maximum likelihood estimation, Fan et al.(2013) [6]
derived the asymptotic theory of non-GQMLE when heavy-tailed student’s t distribution
is taken into consideration. These methods need to specify the degree of freedom ν(t
distribution) or the parameter r of the Generalized Error Distribution(GED(r)). And
they adjust estimator one time when the innovation distribution is heavy-tailed. However,
they couldn’t totaly capture the heavy-tail characteristic. Therefore we put forward the
adaptive QMLE procedure by choosing the optimized quasi likelihood function at mini-
mal divergence between quasi innovation density f and the true innovation distribution
density g. This iterative procedure will gain better efficiency than other methods when
the distribution of innovation is heavy-tailed or unknown.
3.1 KLIC and Scale Parameter
In order to measure the divergence between true innovation density g and specified
likelihood function f, Kullback-Leibler divergence is necessary:
I(g; f) =
∫
[log g(u)]g(u)du −
∫
[log f(u)]g(u)du (6)
The scale parameter ηf we use was proposed by White(1982) [20] and Fan et al. [6] as:
ηf = arg maxη>0E{− log η + log f(
ε
η
)}, (7)
where ηf can be computed by using maximum likelihood estimation.
Let W(η) = E{− log η + log f(ε
η )}. In order to derive the consistency property of ˆθ,
another assumption is needed as follows:
Assumption 2. The quasi likelihood is chose from t-distribution family in (5) such
that W(η) has a unique maximmizer ηf > 0.
This assumption helps our find the optimal likelihood within the t-distribution family
that best captured the heavy-tailed characteristic of innovation than others. In other
words, this assumption and the proposition 1 below will determine the best degree of
freedom in adaptive-QMLE, and the best likelihood function will make adaptive-QMLE
have better efficiency. And the scale parameter ηf had a crucial proposition:
Proposition 1. If f ∝ exp(−x2/2) or f = g, then ηf = 1.
Proof. [6] Define the likelihood ratio function G (η) = E
(
log
(
f(ε/η)
η·f(ε)
))
. Suppose G(η)
has no local extremal values. And since log(x) ≤ 2(
√
x − 1),
E
(
log
(
f (ε/η)
η · f (ε)
))
≤ 2E
(√
f (ε/η)
η · f (ε)
− 1
)
= 2
∫ +∞
−∞
√
1
η
f
(
x
η
)
f (x) dx − 2
≤ −
∫ +∞
−∞
(√
1
η
f
(
x
η
)
−
√
f(x)
)2
dx
≤ 0
3.2 Adaptive QMLE

5. 5
Now, we propose adaptive quasi-maximum likelihood estimation which can be used to
estimate parameters of GARCH model. This method can approximate degree of freedom
of quasi likelihood function based on the Proposition 1. So quasi-maximum likelihood
estimation with the approximative degree will gain efficency significantly.
(a) First, we estimate ˜θ(0)(number in the upper right corner is serial number of esti-
mator) with GQMLE under the assumption of normality by
˜θ(0)
= argmaxθ
1
2
n∑
t=1
{− log(σt|t−1) −
u2
t
σ2
t|t−1
} (8)
.
(b) The {εt} we need to caculate ηf can be replaced by computing from ˜ε
(0)
t =
ut/(˜σ(˜θ(0))) with gaussian maximum likelihood estimator ˜θ(0) in the step (a). Then ˜ηf is
obtained in
˜η
(1)
f = arg maxη>0E{− log η + log f(
˜ε
(0)
t
η
)} (9)
by changing the degree of freedom (df1) of student’s t denstity function (f(·)) above until
|˜η
(1)
f − 1| ≤ δ (δ can be set).
In other words, this step finds the approximate f(df1)(·) of ˜ε
(0)
t under the Proposition
1. Therefore, we can obtain ˜θ(1) under the new density function f(df1) which is more
reliable for true innovation:
˜θ(1)
= arg maxθ
T∑
t=1
(− log(σt|t−1) + log f(df1)(
ut
σt|t−1
)) (10)
(c) Apply the similar procedure in (b) and we will get {˜ε
(1)
t = ut/(˜σt|t−1)} , ˜η
(2)
f in
formula (9), and f(df2)(·) under the condition |˜η
(2)
f − 1| ≤ δ. Then, estimate ˜θ(2) with the
likelihood function f(df2) , and so on.
(d) Finally, obtain a Adaptive estimator ˆθ = ˜θ(n) until |˜θ(n) − ˜θ(n−1)| < λ.
Usually, we set δ at about 0.1 and λ less than 0.5. The smaller the δ and λ, the more
times of iteration are needed.
The adaptive-QMLE is consistent as follow:
THEOREM 3.1. [6] Suppose that Assumption 1(i,ii,iii) and 2 hold. Then ˆθ
p
→ θ0.
Remark 1. The Adaptive-QMLE needs a finite fourth moment for the innovation
because we simply adopt GQMLE in the first step. The finite fourth moment condition
is essential to obtain asymptotic normality. In the future studies, We will use other
alternative estimators in the first step to remove this condition. And the condition Eε2
t = 1
assure the Proposition 1 can be used in this procedure. This paper discusses the case that
Eε2
t < ∞, when Eε2
t = ∞, lots of estimators in this context were studied. Such as Chen
and Zhu(2014) [3], Hill(2014) [10], Peng and Yao(2003) [17] and so on.
This procedure is helpful when the family of distribution of innovation is known. With
the help of KLIC, adaptive-QMLE will increase the efficiency of the estimator and decrease
the discrepancy between true density of innovation and specified density of innovation. So
it will be a better method to process varies situations and a wide variety of data.
4 Simulation studies
We show variation of ηf in Figure 1. Each line represents the variation of the ηf which

6. 6
the distribution of {εt} in formula (9) is fixed as noted in the upper left. The horizontal
axis represents degree of freedom of likelihood function. We compares 4 types of Student’s
t distribution and three types of Generalized Gaussian distribution with different shape
parameter. It is shown that if the degree of freedom of quasi likelihood function is larger(or
smaller) than the degree of freedom of {εt}, the ηf > 1(or < 1). When the degree of
freedom of likelihood function is smaller than the true innovation degree freedom the
ηf < 1 . It’s no doubt that ηf approximately equals to 1 when the specified the quasi
likelihood function equals to the true innovation function. For example, we can find that
the bold line in Figure 1, when innovation is t2 and quasi likelihood function is Student’s
t density with degree of freedom equal to 2, the ηf is approximately equal to 1. The same
things occured in each line.
In order to show the advantages of adaptive-QMLE clearly, we took the ordinary
GARCH(1,1) model with true parameters (ω, α1, β1) = (0.02, 0.6, 0.3) into consideration.
With the bound of |˜η
(2)
f −1| ≤ 0.2, adaptive estimator is convergent after several iterations
as showed in Table 1. The innovation distribution ranges from thin-tailed t20, which is
approximately normal distribution, to heavy-tailed t2. At the same time, we compared
different situation with sample size ranging among 500, 1000 or 2000. The “Step” column
display the order of iteration. “df” column is the degree of freedom of student’s t quasi
likelihood function which was found with the bound |˜η
(1)
f −1| ≤ 0.2. “df=Gauss” means the
gaussian assumption of innovation distribution. What can be found when the innovations
are heavy-tailed, such as t2, t3, is that the maximum-likelihood estimator with Gaussian
assumption is intolerable. However, from the results in the table we could see the Adaptive-
Quasi-Maximum likelihood estimators are better than Maximum likelihood estimators.
And, the adaptive estimation procedure approximate the true degree of freedom and true
parameters.
In Table 2, we use the same setting of GARCH(1,1) model in Table 1. The sample size
is various among 250,500 and 1000. And repeat time of simulation is 500 at each sample
size. The innovation was set as Student’s t distribution for various degrees of freedom from
heavy-tailed to thin-tailed in Table 2. There are three things need to be emphasized in
Table 2. The first one is that NGQMLE is the three-step estimation procedure proposed
by Fan et al(2013) [6]. The auxiliary innovation distribution of NG-QMLE is t4. The
second is GQMLE represent maximum-likelihood estimation with the assumption that
distribution of innovations is normal. Third is that MLE is ordinary maximum likelihood
estimation with the true distribution of innovation set at the beginning of simulations.
In Table 2, gaussian quasi-maximum likelihood estimators are always terrible when
the innovations are heavy-tailed such as t2 or t3. Especially, when the innovations are
t2 or t3, a fourth moment doesn’t exist, the RMSE of ω of GQMLE is intolerable. So
the relative RMSE ratio of other estimators against GQMLE is meaningless. We won’t
display them in the Table 2. But we can find that when the innovations are heavy-tailed,
the Adaptive-QMLE is better than the other two estimations, NGQMLE and GQMLE. In
the table, the relative RMSE ratio of A-QML estimator against GQMLE is smaller than 1.
Meanwhile, the relative RMSE ration of adaptive-QMLE shown that the adaptive-QMLE
is better than NGQMLE. It is clearly shown in the Table 2 that most relative RMSE
ratio of A-QMLE against GQMLE is close to relative RMSE ratio of MLE, which means
A-QMLE can be an best alternative estimator when the innovation is unknown. On the
other hand, approximate df of true innovation, showed in “dfB” column, is close to the
exact degree of freedom. These evidences implies adaptive QMLE, no matter whether

7. 7
the tail of innovation is heavy or thin, is an optimized estimator which is close to the
maximum likelihood estimator with true innovation distribution.
5 Application
In this section, first, we summary daily returns of six indexes, namely S&P500, FTSE,
NADQ, CAC, DAX, HSI. The time period we considered is taken from January 2, 2000
to March 31, 2014. In the Table 3, the number of samples, the mean, standard deviation
and excess kurtosis of daily return are showed. The we can find that the excess kurtosis of
all the indexes are bigger than zero, which means that they are all heavy-tailed. So, when
one want to analyze stock returns with GARCH model, the Adaptive-QMLE precedure
is not only necessary but also helpful. Second, we use this adaptive-QMLE procedure
to estimate parameters of GARCH model. Table 4 displays estimators of GARCH(1,1)
model of quasi-maximum likelihood estimation and adaptive quasi-maximum likelihood.
In the table 4, “dfB” represent approximate degree of freedom of student’s t density. From
the results in Table 4 we could find that although the adaptive QML estimator is a little
different from the quasi-maximum likelihood estimator, we get an approximate degree of
freedom. These estimated degree of freedom imply the same heavy-tailed characteristic
of data in Table 3. S&P500, NASDAQ and HSI have bigger kurtosis, so these three
approximate degree of freedom are smaller than the other three indexes, in other words
their tails are heavier than others’.
6 Conclusions
This article focuses on improving efficiency of the estimator of the GARCH model when
innovation distribution is unknown, and proposes the Adaptive-QMLE which gains better
efficiency. By using ηf = 1 in the sense of KLIC which can identify the quasi likelihood
function f, the Adaptive-QMLE is stable no matter whether the tail of innovation is heavy
or not.
The most important thing is that, without specifying the distribution of innovation, the
Adaptive-QMLE is very close to MLE with true innovation distribution. Hence, it is very
helpful and accurate when we don’t know the distribution of innovation, especially when
the innovation is heavy-tailed. So it’s a general quasi-maximum likelihood estimation
which can be used in more situations, such as the financial field or genetics. Possible
extension of the Adaptive-QMLE includes introducing other distributions of innovation
and considering more models.

8. 8
List of Figures and Tables
Figure 1: Variations of ηf across Student’s-t likelihood QMLE
2 4 6 8 10
0.61.01.4
Degree of freedom
η
t2
t4
t6
t10

9. 9
Table 1: Some samples to show Convergence of Adaptive-QMLE
Innov. Step N = 500 N = 1000 N = 2000
Si ω α β df ω α β df ω α β df
t2 S1 2.571 0.387 0.613 Gauss. 0.386 0.267 0.733 Gauss. 0.984 0.336 0.663 Gauss.
t2 S2 1.748 0.576 0.420 6 0.196 0.463 0.536 20 0.191 0.568 0.432 11
t2 S3 0.022 0.618 0.365 2.5 0.069 0.620 0.375 5 0.042 0.649 0.349 4
t2 S4 0.013 0.627 0.312 2.0 0.033 0.656 0.326 3 0.032 0.679 0.317 3
t2 S5 0.013 0.605 0.305 1.8 0.027 0.643 0.312 2.5 0.032 0.679 0.317 3
t2 S6 0.027 0.643 0.312 2.5
t3 S1 0.000 0.758 0.298 Gauss. 0.029 0.709 0.285 Gauss. 0.371 0.540 0.459 Gauss.
t3 S2 0.015 0.686 0.269 2.8 0.021 0.656 0.309 3.8 0.040 0.644 0.356 20
t3 S3 0.016 0.690 0.274 3 0.020 0.643 0.302 3.5 0.302 0.678 0.321 9
t3 S4 0.016 0.690 0.274 3 0.027 0.692 0.307 7
t3 S5 0.025 0.701 0.298 6
t3 S6 0.024 0.707 0.293 5.5
t5 S1 0.014 0.666 0.249 Gauss. 0.022 0.608 0.220 Gauss. 0.018 0.416 0.356 Gauss.
t5 S2 0.012 0.542 0.328 4 0.019 0.606 0.248 3 0.039 0.619 0.378 20
t5 S3 0.012 0.542 0.328 4 0.019 0.606 0.248 3 0.032 0.632 0.364 15
t5 S4 0.029 0.637 0.357 10
t5 S5 0.028 0.641 0.351 5
t10 S1 0.082 0.689 0.302 Gauss. 0.096 0.639 0.359 Gauss. 0.087 0.755 0.243 Gauss.
t10 S2 0.018 0.579 0.354 20 0.027 0.706 0.272 20 0.023 0.750 0.223 20
t10 S3 0.017 0.566 0.356 15 0.026 0.703 0.267 15 0.023 0.750 0.223 20
t10 S4 0.017 0.566 0.356 15 0.025 0.695 0.261 11
t10 S5 0.024 0.684 0.256 10
t10 S6 0.023 0.675 0.254 9
t20 S1 0.056 0.613 0.382 Gauss. 0.080 0.735 0.261 Gauss. 0.078 0.582 0.291 Gauss.
t20 S2 0.017 0.537 0.365 20 0.024 0.630 0.253 20 0.017 0.623 0.306 20
t20 S3 0.017 0.537 0.365 20 0.024 0.630 0.253 20 0.017 0.623 0.306 20
a True paremeters (ω, α1, β1) = (0.02, 0.6, 0.3).

10. 10
Table 2: Relative RMSE ratio comparison with Student’s t innovation
Para ε T dfB A-tQMLE NGQMLE GQMLE MLE
ω t5
250 7.807 0.084 0.094 0.157 0.059
500 7.555 0.064 0.055 0.139 0.044
1000 7.825 0.051 0.031 0.123 0.032
ω t10
250 11.27 0.148 0.282 0.060 0.130
500 11.961 0.126 0.107 0.055 0.087
1000 10.752 0.137 0.128 0.044 0.078
ω t20
250 13.292 0.158 0.180 0.055 0.148
500 14.151 0.141 0.679 0.048 0.123
1000 14.545 0.127 0.075 0.044 0.084
α t2
250 2.937 0.478 0.801 0.203 0.639
500 2.699 0.408 0.601 0.218 0490
1000 2.644 0.339 0.495 0.254 0.351
α t3
250 4.544 0.701 0.726 0.160 0.828
500 4.010 0.584 0.667 0.140 0.733
1000 4.223 0.647 0.763 0.125 0.747
α t5
250 7.807 0.787 0.773 0.172 0.784
500 7.555 0.653 0.683 0.173 0.557
1000 7.825 0.596 0.561 0.151 0.502
α t10
250 11.27 0.741 0.800 0.228 0.552
500 11.961 0.665 0.749 0.224 0.394
1000 10.752 0.648 0.740 0.249 0.264
α t20
250 13.292 0.665 0.746 0.235 0.508
500 14.151 0.684 0.796 0.220 0.421
1000 14.545 0.667 0.781 0.226 0.293
β t2
250 2.937 0.253 0.757 0.305 0.213
500 2.699 0.172 0.642 0.317 0.136
1000 2.644 0.098 0.562 0.367 0.087
β t3
250 4.544 0.343 0.701 0.234 0.226
500 4.010 0.188 0.630 0.260 0.195
1000 4.223 0.152 0.679 0.258 0.155
β t5
250 7.807 0.530 0.682 0.184 0.184
500 7.555 0.358 0.528 0.171 0.353
1000 7.825 0.252 0.505 0.170 0.267
β t10
250 11.27 0.669 0.720 0.161 0.659
500 11.961 0.477 0.488 0.148 0.450
1000 10.752 0.288 0.359 0.156 0.270
β t20
250 13.292 0.682 0.732 0.166 0.666
500 14.151 0.521 0.566 0.150 0.497
1000 14.545 0.405 0.418 0.131 0.394
a The RMSE of GQMLE are showed between the two bold lines. The
others are the relative RMSE ratio of other estimators against GQMLE.
Almost all are less than 1 means these three estimators are better than
GQMLE.

11. 11
Table 3: Summary of Six stock indexes
ut of Index n mean sd kurtosis
S&P500 3581 0.0002 0.0132 7.9423
FTSE 3596 0.0001 0.0125 6.1308
NASDAQ 3582 0.0007 0.0302 7.2389
CAC 3632 0.0002 0.0414 4.9996
DAX 3633 0.0019 0.0458 4.4939
HSI 3548 0.0019 0.0471 8.2166
a The data are 6 daily stock market returns,
from January 2, 2000 to March 31, 2014. And
the kurtosis in the table is sample excess kur-
tosis.

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Table 4: QMLE and Adaptive-tQMLE of GARCH(1,1) models.
Index Estimator ω α β dfB
S&P500
GQMLE 0.003 0.085 0.902
15
AtQMLE 0.007 0.083 0.905
FTSE
GQMLE 0.013 0.098 0.892
30
AtQMLE 0.021 0.082 0.891
NASDAQ
GQMLE 0.005 0.715 0.236
13
AtQMLE 0.006 0.139 0.824
CAC
GQMLE 0.0002 0.084 0.906
56
AtQMLE 0.0003 0.085 0.906
DAX
GQMLE 0.0002 0.085 0.905
50
AtQMLE 0.0002 0.088 0.899
HSI
GQMLE 0.0002 0.064 0.912
12
AtQMLE 0.0002 0.054 0.929
a The data are 6 daily stock market returns, from Jan-
uary 2, 2000 to March 31, 2014.

13. 13
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