used to test nested hypotheses. One of the testing hypotheses is that stock prices exhibit GARCH phenomena and leverage effects under whatever conditional
distribution is assumed. Another hypothesis is that normality is not an appropriate description for conditional distributions of stock prices.
The contribution of this study to the literature is the new GARCH models we use in this study, in particular the Poisson – normal – GARCH model which is modified
from Jorion 1988 and other extended models, and the mixed – normal – GARCH model which has never been found in previous literature. For practical purposes,
the estimation also provides information for understanding the stock behavior of the Taiwan stock market. This may have important implications for stock warrants,
and stock index futures markets in pricing, hedging, as well as trading.
The rest of the paper is organized as follows: Section 2 describes the models and methodology used in this paper. Section 3 describes the data and the statistical
characteristics of the sample stock and stock portfolio returns. Section 4 analyzes the empirical results of the MLE and various hypothesis tests applied to the data
for each stock and portfolio. Section 5 is the conclusion.
2. The methodology
Let R
t
denote the return of a stock at time t. It follows the following process: R
t
= m
t
+ o
t
1 where m
t
= ER
t
c
t − 1
is the mean, and h
t
= VR
t
c
t − 1
is the variance of stock returns at time t, conditioned on the information available at time t − 1. o
t
= R
t
− m
t
is the residual of the model, which follows a conditional distribution with a mean of zero and a variance of h
t
.
2
.
1
. Normal and normal – GARCH The standard assumption for stock price S dynamics is a diffusion process
called geometric Brownian motion, that is dSt
St =
a · dt + s · dZt 2
where a is the drift rate, s is the volatility of the stock price, and Zt follows a standard Wiener process. Under this assumption the discretized stock return
continuous compound return process is
R
t
= ln
S
t
S
t − 1
= m + s
· Z
t
3 where m = a − s
2
2, and Z
t
is an independent and identical standard Gaussian process with a zero mean and unit variance. Under this process, the stock return Rt
is normally distributed with a mean of m and variance of s
2
, and is independent across time. To estimate the model, the MLE is used. Thus we have the stock
return process and the log-likelihood function for a random sample with sample size T, lR,u as follows:
2
.
1
.
1
. Model
1
R
t
= m
t
+ h
t
Z
t
; m
t
= m
; h
t
= s
2
ln lR;u = − T
2 ln 2p +
T t = 1
ln 1
h
t
exp −
o
t 2
2h
t
n
4 where u
m,s denotes the parameter set to be estimated.
As specified in the previous section, the observed leptokurtosis of stock return distributions could be explained by a diffusion normal process with conditional
heteroscedasticity. Without losing generality, the simplest form of first-order GJR – GARCH 1,1 model, with an ARMA 1,1 model in the mean to correct serial
autocorrelation, is used. Therefore the stock return process becomes:
2
.
1
.
2
. Model
2
R
t
= m
t
+ h
t
Z
t
; m
t
= m + a
R
t − 1
+ bo
t − 1
; h
t
= s
2
+ f
h
t − 1
+ co
t − 1 2
+ d
·max0, − o
t − 1 2
5 The log-likelihood function for Section 2.1.2 has the same form as that for
Section 2.1.1, except that the parameter set to be estimated becomes u
m, s, a, b, f
, c, d. In addition to the normal distribution, we also propose several GARCH models
in which the conditional distributions are not normal. Alternative explanations for the observed fat tails in the empirical distribution of stock returns involve model
specifications in which the true underlying generating process is a mixture of normal distributions. As noted by Kon 1984, there are two types of mixture, the
continuous mixture of normal distributions, and the discrete mixture of normal distributions.
2
.
2
. Student-t and Student-t – GARCH The first mixture model is a continuous mixture of normal distributions. Assum-
ing that the variance parameter of a normal distribution is drawn from an inverted g
distribution, Blattberg and Gonedes 1974 obtained the resulting posterior distribution, the Student’s t. The Student-t distribution approaches the normal as
the degrees-of-freedom d.f. parameter approaches infinite. As d.f. fall in the range
from 2 to 10, the distribution exhibits leptokurtosis, which is able to explain a part of the observed high kurtosis relative to the stationary normal distribution. Let T
t
be an independent and identical standard Student-t distribution, the stock return process and the log-likelihood function are:
2
.
2
.
1
. Model
3
R
t
= m
t
+ h
t
T
t
; m
t
= m
; h
t
= s
2
; ln l R;u
= T ln
G n + 12
p · Gn2 ·
n−2
n
− 1
2 ·
T t = 1
ln h
t
− n +
1 2
·
T t = 1
ln 1 +
o
t 2
h
t
n − 2
n
6 where u
m, s, n, and n is the d.f. parameter for the Student-t distribution.
Combining the Student-t distribution and the GARCH process, the resulting model is
2
.
2
.
2
. Model
4
R
t
= m
t
+ h
t
T
t
; m
t
= m + a
R
t − 1
+ bo
t − 1
; h
t
= s
2
+ f
h
t − 1
+ co
t − 1 2
+ d
·max0, − o
t − 1 2
7 The log-likelihood function for the model has the same form as that for Section
2.2.1, while the parameter set to be estimated is u
m, s, n, a, b, f, c, d.
2
.
3
. Poisson – normal and Poisson – normal – GARCH The second classification of mixture models is a discrete mixture of normal
distributions. An alternative process for stock prices is a mixture of a diffusion process with constant volatility and a Poisson jump process, that is
dSt St
= a
· dt + s · dZt + J · dNt 8
where J is the magnitude of a jump in the stock price, which is assumed to follow a normal distribution with a mean equal to m
J
and variance equal to s
J 2
. Nt is an independent
Poisson process
with the
intensity parameter
l \ 0.
The resulting distribution of the stock returns is an infinite weighted sum of discrete
normal distributions. Under the jump-diffusion process, the discretized stock return process is
R
t
= m + s
Z
t
+
N
t
i = 1
J
i
= m
t
+ h
t
D
t
9 where N
t
is the number of jumps which happen during the period between t − 1 and t. D
t
is the independent and identical standardized mixed Poisson – normal process. According to Akgiray and Booth 1986, the distribution of stock returns is
leptokurtic for l \ 0, and is skewed for m
J
0. The stock return process and the log-likelihood function are:
2
.
3
.
1
. Model
5
R
t
= m + s
Z
t
+
N
t
i = 1
J
i
= m
t
+ h
t
D
t
; m
t
n = ER
t
N
t
= n = m + nm
J
; h
t
n = VR
t
N
t
= n = s
2
+ ns
J 2
; ln lR;u = −
T 2
ln 2p +
T t = 1
ln
n = 0
l
n
e
− l
n ·
1 h
t
n exp
− R
t
− m
t
n
2
2h
t
n
n
10 where m
t
n = ER
t
N
t
= n and h
t
n = VR
t
N
t
= n are the conditional mean and
variance respectively, under the condition that the number of jumps n is known. And the parameter set to be estimated is u
m, s, m
J
, s
J
, l. In order to explore whether the GARCH model or the jump-diffusion process
could explain the observed leptokurtosis, a generalized process which is a combina- tion of a jump-diffusion process and an GJR – GARCH process is used. That is,
conditioned on the information available at time t − 1, the stock return process becomes:
2
.
3
.
2
. Model
6
R
t
= m
t
+ s
t
Z
t
+
N
t
i = 1
J
i
= m
t
+ h
t
D
t
; m
t
n = ER
t
N
t
= n = m + nm
J
+ a
R
t − 1
+ bo
t − 1
; h
t
n = VR
t
N
t
= n = s
2
+ ns
J 2
+ f
h
t − 1
+ co
t − 1 2
+ d
·max0, − o
t − 1 2
; m
t
= m + lm
J
+ a
R
t − 1
+ bo
t − 1
; h
t
= s
2
+ l
m
J 2
+ s
J 2
+ fh
t − 1
+ co
t − 1 2
+ d
·max0, − o
t − 1 2
; o
t
= R
t
− m
t
11 The log-likelihood function for the model has the same form as that for Section
2.3.1. And the parameter set to be estimated is u
m, s, m
J
, s
J
, l, a, b, f, c, d. The above model derived by this study is consistent with the building of the
GARCH model and is different from the Jorion 1988 model, in which the
conditional heteroscedasticity is specified as an ARCH 1 model, and is only allowed in the diffusion part
2
. Vlaar and Palm 1993, Brorsen and Yang 1994, and Drost et al. 1998 also followed the Jorion 1988 model, allowing the
GARCH 1,1 model only in the diffusion component. Although their model is simple and convenient in model estimation, it is not consistent with the structure of
the GARCH model in which the volatility clustering appears in the observed return data, rather than only in the diffusion part of the observations.
2
.
4
. Mixed – normal and mixed – normal – GARCH A difficulty with a Poisson function is that it contains an infinite sum. This sum
has to be truncated for the process to become estimable. Ball and Torous 1983 derived a simplified jump process involving a two normal distributions model from
a Bernoulli jump process to describe informational arrivals. When the information arrival follows a Poisson process, the resulting process is the jump-diffusion process
described above. Christie 1982 has formulated a discrete mixture of two normal distributions model where returns drawn from the distribution with the higher
variance represent information events while the other distribution generates non-in- formation random variables. Kon 1984 proposed a model, which is a mixture of
normal distributions with different means and variances to describe the stock return distribution. Without a priori knowledge of the identification of returns with
specific probability distributions, each return observation is viewed as a drawing from one of a finite number of normal distributions with the mean m
j
and variance s
j 2
with some probability l
j
. When l
j
\ 0 for some j, then the stock return
distribution is leptokurtic. And if m
i
m
j
for some i and j and i j, then the stock return distribution is skewed. Assume that M
t
is the probability density function for the standardized generalized discrete mixture of normal distributions with L
regimes, the model is
2
.
4
.
1
. Model
7
R
t
= m
t
+ h
t
M
t
; m
t
j = ER
t
Qt = j = m
j
; h
t
j = VR
t
Qt = j = s
j 2
; ln lR,u = −
T 2
ln 2p +
T t = 1
ln
L j = 1
l
j
h
t
j exp
− R
t
− m
t
j
2
2h
t
j
n
;
2
In our terminology, Jorion 1988 assumed h
t
n = s
2
+ b
R
t − 1
− m
J 2
+ ns
J 2
. The model consistent with the structure of an ARCH model is h
t
n = s
2
+ b
R
t − 1
− m −
nm
J 2
+ ns
J 2
, since in the jump-diffu- sion process, under the condition that n is known, the mean of the stock returns is m + nm
J
, rather than m
.
L j = 1
l
j
= 1
12 where Qt denotes the regime status which the observation belongs to. And the
parameter set is u
m
j
, s
j
, l
j
; j = 1,2,…, L. Combining the mixed – normal distribution and the GARCH process, the result-
ing model is
2
.
4
.
2
. Model
8
R
t
= m
t
+ h
t
M
t
; m
t
j = ER
t
Qt = j = m
j
+ a
R
t − 1
+ bo
t − 1
; h
t
j = VR
t
Qt = j = s
j 2
+ f
h
t − 1
+ co
t − 1 2
+ d
·max0, − o
t − 1 2
; m
t
=
L j = 1
l
j
m
j
+ a
R
t − 1
+ bo
t − 1
; h
t
=
L j = 1
l
j
m
j 2
+ s
j 2
−
L j = 1
l
j
m
j 2
+ f
h
t − 1
+ co
t − 1 2
+ d
max
0, − o
t − 1 2
; o
t
= R
t
− m
t
13 The log-likelihood function for the model has the same form as that for Section
2.4.1. While the parameter set to be estimated is u
m
j
, s
j
, l
j
, a, b, f, c, d; j = 1,2,…, L.
2
.
5
. Model estimation and testing In the above GJR – GARCH models, the non-negativity condition is satisfied
provided that f ] 0 and c + d ] 0. And if the leverage effect holds, we expect to find d \ 0 The parameters are obtained by a numerical maximization of the
log-likelihood function. The process is recursive until the maximum value of the log likelihood function, lR,u is reached. To test hypotheses, likelihood ratio tests are
used. Assume that u
is the restricted parameter set under the null hypothesis, and u
1
is the unrestricted parameter set under the alternative hypothesis. The statistics −
2[lR,u − lR,u
1
] have a Chi-square x
2
distribution with d.f. equal to the difference in the number of parameters between the two models.
To test which model is most likely, i.e. with highest posterior probability, we use the Schwarz 1978 criterion to choose the model for which
SC = ln lR,u − 1
2 p · ln T
14 is largest. The value p is the number of estimated parameters. The Schwartz
criterion does not depend on a particular prior distribution, hence it can be applied to all estimations for the models used in this study.
3. The data
