Comparison of GARCH Models based on Different ... - Semantic Scholar
JOURNAL OF COMPUTERS, VOL. 7, NO. 8, AUGUST 2012
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Comparison of GARCH Models based on Different Distributions Yan Gao School of Science, Hebei United University, Xin Hua Street 46, Tangshan, 063009, Hebei, P. R. China Business school of Jilin University, Changchun 130012, Jilin Province, China Email: [email protected]
Chengjun Zhang and Liyan Zhang Market Department, China Mobile Group Hebei Co, Ltd. Tangshan Branch, Xing Yuan road 139, Tangshan, 063004, Hebei, P. R. China; Business school, Hebei University of Economics and business, Xue Fu road 47, Shijiazhuang, 050061, Hebei, P. R. China Email: [email protected] ; [email protected];
Abstract— Since ARCH and GARCH models are presented, more and more authors are interested in the study of volatilities in financial markets with GARCH models. Method for estimating the coefficients of GARCH models is mainly the maximum likelihood estimation. Now we consider another method—MCMC method to substitute for maximum likelihood estimation method. Then we compare three GARCH models based on it. MCMC method developed based on Markov chain, which is one kind of straggling time and state random process with no offspring imitates. It attracts extensive attention because of its applications in many fields. In this article, we will compare GARCH models based on different distributions with MCMC method. At last we have the conclusion that both in uni-variable case and binary variable case, GED-GARCH is the best model to describe the volatility compared to other two models, and we will provide the application of binary GED-GARCH models in forecasting the volatility in China’s stock markets. Index Terms—MCMC, China’s stock markets, Gibbs sampling, GED-GARCH
I. INTRODUCTION In 1907, A. A. Markov began the study of an important new type of chance process. In this process, the outcome of a given experiment can affect the outcome of the next experiment. This type of process is called a Markov chain [1]. Markov chain is one kind of straggling time and state random process with no offspring imitates, which attracts more and more attention because of the extensive application. MCMC method is presented in 1950s, which has their roots in the Metropolis algorithm. Almost at the same time, Monte Carlo method is used to calculate the complicated integration. The main problem in integration is how to sampling from the complicated distribution. The MCMC method can solve the problem by using the stable distribution and the Markov chain.
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Since ARCH and GARCH models are raised, the application of them appears in many research and practice fields. The yield series of the financial markets generally have the characters of leptokurtotic, fat-tail with skew, volatility clustering and long memory. It violated the Normal distribution assumption for stochastic errors, so Engel [5] imposed ARCH model in 1982, and then in 1986, Bollerslev [6] put its generalized form GARCH forward. Literatures suppose that the errors obey to Normal distribution, but for many financial series, it cannot describe the character of the fat tails, so we need consider using other distribution to describing fat-tail distribution, e.g. t-distribution and generalized error distribution. Many authors are studying the applications of MCMC in volatility analysis. Pan Haitao [10] (2010) use Markov chain Monte Carlo (MCMC) method to estimate the parameters of normal-based GARCH (1, 1) model. The results based on MCMC are more reliable and also show results based on MCMC are better than that of ML based by using real financial data. MA Fu1ing[12] studies the Application of MCMC method in the estimation of the binary stochastic volatility model, Shen xia [14] calculates the VAR by MCMC Simulation method and makes an empirical analysis on certain Shanghai Securities Composite Index in order to prove the advantages of MCMC Method over MC. The sampling methods of MCMC conclude the Metropolis-Hastings sampler, the Gibbs sampler, Importance sampler and Slice sampler, in which, the Gibbs sampler is in common use, we will describe it in detail below. LI Wei-Guo [15] use The MCMC method and the maximum likehood estimation method analyze the correlation coefficient stationary series based on the classical ARMA were applied on the real electric net load monthly data of Guangxi. The result show that the MCMC method provides the most precise prediction．
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II. UNIVARIATE N-GARCH, T-GARCH AND GED-GARCH
A. Mutiple Student Distribution
In order to complete the GARCH [8] specification, we require an assumption about the conditional distribution of the error term
t
vt ~ N (0,1)
(1)
ht 0 1ht 1 1 t21
yR
v ( y )
1
( y )
vn 2
(7)
n
yt c t ~ N (c, ht )
(2)
Many positive study show distributions of yield series have fat-tail, so Nelson (1991) and Hamilton (1994) use generalized error distribution (GED) and t-distribution to adjust the deviation of the tail. We will introduce the two distributions separately. For the student’s t-distribution, the distribution density for it is given below:
(( r 1) / 2) [( r 2) ]1/ 2
(1 x 2 /( r 2)) ( r 1) / 2 (3)
C1
v n v n 2 v ( ) v2
2
1
2
r
2 r (3 / r ) 1 / 2 (3 / r ) x 2 f (x , , r) exp . 2 (1 / r ) 3 / 2 (1 / r )
(4)
() is GAMMA function. If a variate x is
2 drawn from this p.d.f., we can denote x ~ G( , , r ) . We can describe the model below:
t I t 1 ht vt vt ~ G (0,1, r )
(5)
v , v 2 v2
(8)
B. Mutiple Generalized Error Distribution Now we propose the definition of multiple generalized error distribution [2] below: n
n 3 2 (1 ) ( ) d nx 2 r dF ( x , , r ) n (1 n ) ( 1 ) r r (9)
3 2 ( r ) exp ( x )T 1 ( x ) . ( 1 ) r If n dimension vector x obey to this distribution, we remark it as x ~ GED( , , r ) , ,, r are parameters, is the mode of the distribution, and also is mean. The relationship between VAR-COV matrix V and can be shown below:
n2 1 )(1 ) r r V 3 n ( )(1 ) r r (
(10)
C. Definition of GARCH model
1ht 1 1 t21
Then we have
yt c t ~ GED(c, ht , r ) .
E (Y ) , cov(Y , Y )
r
r is constant. For the GED-GARCH model, fist we give out the distribution density for it:
(6)
III. MUTIPLE N-GARCH, T-GARCH AND GED-GARCH MODEL We are familiar to multiple Normal distributions, so we just introduce multiple student t distribution and multiple generalized error distribution here.
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In which,
We use Tn (v, , ) to denote multiple student t distribution.
Then we have
ht 0
1 2
() is gamma function, then
t I t 1 ht vt
0r ,
. There are three assumptions
commonly employed when working with GARCH models [9]: normal (Gaussian) distribution, student’s t-distribution, and the generalized error distribution (GED) [7], we discuss them in detail below. For Normal (Gaussian) distribution, we have
f ( x, r )
•f Y ( y ) C1
y1t Suppose we have n assets, Yt is a vector with y nt n elements, the model is given below:
Yt X t t
(11)
Where t is vector, if: 1) t I t 1 ~ N (0, H t )
(12)
JOURNAL OF COMPUTERS, VOL. 7, NO. 8, AUGUST 2012
2)
t I t 1 ~Tn (v, , )
3) t
1969
(13)
I t 1 ~ GED(X t , H t , r )
(14)
Scalar parameters a sliu and bsliu is the u-th element of a sli and bsli separately. We will use this method to analysis the volatilities. IV. THE MCMC METHOD
I t 1 denotes all the information before t moment. The VAR-COV matrix is
H t C C BH t 1B A t 1 ( A t 1 )
(15)
In which, C, B, A are n n matrix, H t is conditional VAR-COV matrix. We call the model constructed by equations (11), (12) and (15) multiple N-GARCH; the model constructed by equations (11), (13) and (15) multiple T-GARCH; the model constructed by equations (11), (14) and (15) multiple GED-GARCH. D. BEEKmodel BEEK model is presented by Engel and Kroner in 1995 [4].This model denotes the conditional covariance matrix H t by the form of quadratic terms, which guarantees its Positive definiteness, so we needn’t put any limits on unknown parameters. The definition of conditional covariance matrix H t is shown below:
H t C 0 C 0
k11
q
p
il
b
h( x)dx
a
E p ( x ) [ f ( x)]
t i Bil
i 1 l 1
C0 is a lower triangular matrix of m ranks, which m(m 1) has unknown parameters, Ail and Bil are both 2 m dimension unknown parameters square matrix. The element at (i, j ) location of H t , which denotes hijt can
hijt cij
(a sli Yt s )( a slj Yt s )
i 1 l 1
p
k12
b
(17)
sli H t i Bslj
i, j 1, , m
i 1 l 1
The unknown parameter c ij is the (i, j ) element for C C0C0 , a sli is the ith row for matrix Asl , bsli is the ith row for matrix. The equation above can be express as scalar form:
hijt cij
q k11
m
a
p
m
b
sliu bsljv huvt s
s 1 l 1 u ,v 1
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(18)
i, j 1,, m
i
i 1
f ( y x) p( x) ,
which is
approximate by n
f (y x ) i
(20)
i 1
B. Definite of Markov Chains Stochastic process is the quantify description of a series of random events. It is an important tool for many studies, e.g. natural science, engineering science and society science. And it has extensive applications. We often need the theory of stochastic process to establish mathematical models. Markov process is a classic stochastic process. Suppose X (t ) is a stochastic process, when the state of the process at t 0 is known, and the state at t1 is unconcerned to the state before t0 , this character is called
sliu asljv yut s yvt s
s 1 l 1 u ,v 1 k12
f (x )
Consider the integral I ( y )
1 Iˆ( y) n
k11
1 n
(19)
n
The formula is called Monte Carlo integration, which can be used to approximate posterior (or marginal posterior) distributions required for a Bayesian analysis.
be given: q
b
f ( x) p( x)dx
(16)
k12
B H
from the density p(x) , then
a
Ail Yt i Yti Ail
i 1 l 1
A. Monte Carlo Method If the problem requested to be solved is the probability of some event, or is the expected value of some random variable, we can get the frequency of the event or the mean value of the variable, and take it as the solution of the problem, which is the basic thought of the Monte Carlo method. It takes a probability model as base, according to the process described as the model, then get the result of the simulated experiment as the approximate solution of the problem. We can summarize the process of solving the Monte Carlo problem into three steps: Constructing or describing the probability process; Drawing samples from given probability distribution; Establishing different estimators. Now we give out the definition of Monte Carlo integration. If we get random variables x1 , x2 ,, xn
no aftereffect. The stochastic process which has no aftereffect is called Markov process. The time and state for Markov process can be continuous and discrete. We call the Markov process with discrete time and discrete state Markov chains. In another word, we call the
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JOURNAL OF COMPUTERS, VOL. 7, NO. 8, AUGUST 2012
( x, y ) as below: First, we get a random number u from U (0,1) , if u ( x, y ) , set X n1 y ;or set X n1 x . If we take x which makes ( x) 0 as the starting pot
stochastic process is a Markov process if the transition probabilities between different values in the state space depend only on the random variable’s last state, i.e.
Pr( X t 1 ) s j X t si ,, X 0 sk ) (21)
Pr( X t 1 s j X t si )
of the Markov chain, then for equation (23), the denominator of the ratio will not equal to zero.
A Markov chain refers to a sequence of random variables X 0 ,, X n generated by a Markov process. A particular chain is defined by its transition probabilities,
P(i, j) P(i j) Pr( X t 1 s j X t si )
The Gibbs sampler method [3] didn’t have impact on the field of statistics until rather recently. Now we consider the principle. Consider a binary variable ( x, y ) , if we want to get the marginal distribution f (x) or f ( y) , instead of using f ( x, y) , we use the integration of conditional distribution to get the marginal distribution. First, we choose a starting value
(22)
Let the transition probabilities be the elements, we can get the transition matrix C. Sampler Methods of MCMC
Metropolis-Hastings Algorithm Metropolis-Hastings
Algorithm
The Gibbs sampler method
for [11]
is
a
y,
then we draw a random variable
x0
by f ( x y y0 ) , then we get a new y
powerful Markov chain method to simulate
by f ( y x x0 ) , the sampling process is below:
multivariate distributions. The thought of M-H algorithm is to construct a Markov chain whose
xi ~ f ( x y yi1 )
(24)
target distribution is an invariant distribution.
yi ~ f ( y x xi )
(25)
In order to achieve the target, M-H Algorithm recur to a assistant probability density function q ( x, y ) , which usually need to satisfy three
Repeat the process k times, we will get the Gibbs series, subset ( x j , y j ),1 j m k is
conditions(s denotes state space): a). for a settled x , q ( x, ) is a probability density
take by the stimulation sample from the joint distribution [13], we can get m by: 1). Eliminate the affected by starting value by enough simulated Annealing Algorithm, 2). Make all the samples use simulated Annealing Algorithm, then we will get a stable distribution which is irrelevant to the starting values. We use this algorithm to analyze the univariate case and the binary case.
function; b). for x, y S , q ( x, y ) can be calculate; c). for for a settled
x
, random
number can be produced from q ( x, y ) . When the three conditions above be satisfied, the choice of q ( x, y ) can be arbitrary. Now we give out the procedure of the algorithm, X n is supposed to be equal to x , then 1) Produce a new state y from q ( x, y ) ; 2) Calculate the acceptance probability;
( x, y ) min( 1,
( y )q ( y, x) ) ( x ) q ( x, y )
(23)
3) Set X n1 y by probability of ( x, y ) ,
X n1 x 1 ( x, y) .
set In
actual
by
calculation,
probability the
choice
of of
q ( x, y ) have big influence to the efficiency of the algorithm. Generally we think that the more q ( x, y ) is near to the target distribution, the effect of simulation is better. In step 3, if we set X n1 y , we call it “accept proposal”, or we call “reject proposal”. In actual calculation, we can achieve accepting y by probability of © 2012 ACADEMY PUBLISHER
V. POSITIVE STUDY AND CONCLUSION A.
Data and Data Processing We get data from Shanghai composite index and Shenzhen Stock Exchange Component Index, example period is from Jan forth, 2000 to April 29th, 2011, data come from Yahoo financial website. We take SS to denote Shanghai composite index, SZ to denote Shenzhen Stock Exchange Component Index, and we use y1 , y 2 to denote the take log difference series for SS and SZ, which closely to the daily yield series .the graphs is shown below.
JOURNAL OF COMPUTERS, VOL. 7, NO. 8, AUGUST 2012
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clustering and long memory, so we choose to calculate the kurtosis and auto-correlation function as the standard for comparing the GED-GARCH and the N-GARCH.
.12 .08
B. Positive Analysis For Univariable Case
.04
TABLE II.
.00
The estimation result for SS
-.04 Model
-.08 -.12
0
1
1
r (v)
2.79e-06
0.927037
0.06366
(3.61e-07)
(0.003047)
(0.003291)
7.32e-07
0.837227
0.142771
(1.77e-06)
(0.016511)
(0.017977)
N-GARCH
-.16 00 01 02 03 04 05 06 07 08 09 10 Figure1.
The Yield Series of Shenzhen Stock Exchange Component Y2 Index
2
GED-GARCH
0.978988
3.7e-06
0.911674
0.090173
(1.12e-06)
(0.011059)
(0.014605)
t-GARCH
.12 .08
3.508736
We use the Eviews 5.0 software to analyze, we simulate for 20000 times, in which 4000 times are used to eliminate the affect by starting value, then we have the estimating result.
.04 .00
TABLE III.
-.04
The estimation result for SZ Model
-.08
N-GARCH
-.12 GED-GARCH
00 01 02 03 04 05 06 07 08 09 10 Figure2.
t-GARCH
The Yield Series of Shanghai Composite Index Y1
From figure1 and figure 2, we can see the volatility clustering and long memory. We use the deviation series to describe the model fitting,
zit yit
1 T
T
y
it , i
1,2, t 1,2,, T .
(26)
0
1
3.38e-06 (4.83e-07) 1.39e-05 (3.10e-06) 4.40e-06 (1.30e-06)
0.926615 (0.003654) 0.805567 (0.020427) 0.914290 (0.010894)
2 0.96643 3.79696
The kurtosis coefficients and coefficientsof the Std. dev Model
Kurtosis SS
TABLE I.
r (v)
TABLE IV.
t 1
T denotes the number of observations. The basic statistics for the deviation series of y1 and y2 is below:
1 0.063861 (0.003806) 0.169296 (0.023543) 0.082782 (0.012991)
Std. dev SZ
SS
SZ
N-GARCH
Cannot be estimated
Cannot be estimated
1.001073
1.000468
t-GARCH
6.438044
5.873222
0.210226
0.235555
GED-GARCH
8.288008
7.669474
0.121165
0.112245
Basic statistics for the deviation series of y1 and y2
Mean
Std. dev
Skewness
Kurtosis
z1
4.14e-07
0.01636
-0.16794
7.90059
z2
2.55e-07
-0.0179
-0.20821
7.19304
JarqueBera (Probability ) 2968.827 (0.00000) 2184.601 (0.00000)
From Table I, we can see, the yield series have the characters of leptokurtotic, fat-tail with skew, volatility © 2012 ACADEMY PUBLISHER
Table IV describes the kurtosis coefficients and the std. dev between the auto correlation given by model and the real auto correlation. We can conclude that the GED-GARCH model is better than t-GARCH, and t-GARCH is better than N-GARCH. The result declare that the distribution for yield series don’t obey to Normal distribution, we should consider use the GED-GARCH model to dispose the leptokurtotic, fat-tail in financial market, and we also can consider the application in dimension case in the further research.
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JOURNAL OF COMPUTERS, VOL. 7, NO. 8, AUGUST 2012
C. Positive Analysis For Binary Variable Case Now we still use the data simulated above to compare the binary N-GARCH(1,1), T-GARCH and GED(0,r,1)-GARCH(1,1) models in studying the volatility between Shanghai stock market and Shenzhen Stock market, then according to the rule of evaluation, we will use the better model to analyze the volatility. We use the beek method, now we can see the estimation results below. The estimation results For N-GARCH is shown in Table V, the coefficients of 3 and 3 are not significant, which means that there is no Spillover Effect from shanghai stock market to Shenzhen stock market. It indicates that the volatility in Shanghai stock market will not affect the Shenzhen stock market. The estimation results For T-GARCH is shown in Table VI. For T-GARCH model, the coefficients of are also not significant, which is the same 3 and 3 as N-GARCH, T- GARCH mode is better than N-GARCH model.
2
-0.00976
-1.79854
0.0900
3
0.00875
0.84334
0.3753
4
0.86986
132.88765
0.0000
ESTIMATION RESULT FOR GED-GARCH(R=1.25) Model
Parameter
Coefficient
Z-statistics
P-value
1
0.001604
28.34937
0.0000
2
-0.000355
-9.666305
0.0000
3
0.000867
423.6704
0.0000
1
0.55392
(29.996)
0.0000
2
0.016203
8.095982
0.0000
3
0.144000
8.963469
0.0000
Coefficient
Z-statistics
P-value
1
0.001673
12.07842
0.0000
2
0.001924
11.89724
0.0000
4
0.039155
13.61430
0.0000
3
0.000356
4.569905
0.0000
1
0.970622
576.7073
0.0000
1
0.214558
20.24739
0.0000
2
-0.076553
-144.5016
0.0000
2
0.022326
6.719294
0.0000
3
0.921634
157.0822
0.0000
3
-0.0227
-1.5709
0.1162
4
0.008790
14.58418
0.0000
4
0.290876
27.31948
0.0000
1
9.75E-01
162.0012
0.0000
2
-0.0087
-1.75061
0.0800
3
0.005734
0.751661
0.4523
4
0.953381
147.8597
0.0000
ESTIMATION RESULT FOR T-GARCH(V=3.4)
T-GARCH
0.0000
Parameter
TABLE VI.
Model
187.9876
TABLE VII.
Estimation result for N-GARCH
N-GARCH
0.98653
If we suppose the significance level to be 10%, the but the significance for the two coefficients is stronger than the N-GARCH model, which means the Model of coefficients of 3 can be thought to be significant, which indicate there are mutual Spillover Effect between the two stock markets.
TABLE V.
Model
1
Parameter
Coefficient
Z-statistics
P-value
1
0.002673
14.76384
0.0000
2
0.003924
15.68434
0.0000
3
0.000556
5.78964
0.0000
1
0.314558
23.67543
0.0000
2
0.032326
6.87698
0.0000
3
0.0127
-1.09873
0.09162
4
0.490876
2767493
0.0000
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GED-GARCH
The estimation results For GED-GARCH is shown in Table VII, for GED-GARCH model, the coefficients of all parameters are significant which indicates that there is mutual Spillover Effect between the two stock markets; the statement is consistent with the fact. It has been nearly twenty years since the establishing of the two stock markets, as the developing and improving of the two stock markets, their relationship is more and more closely. Now the two markets show great correlations in both price and volatility. The MCMC give me a method in comparing the different GARCH models.
VI. CONCLUSION We use MCMC method to compare three GARCH models: N-GARCH, T-GARCH and GED-GARCH in univariate case and multiple case separately. In univariate case, we compare the kurtosis coefficients and the std. dev between the auto correlation given by model and the real auto correlation. Then we can conclude that the GED-GARCH model is better than t-GARCH, and t-GARCH is better than N-GARCH.
JOURNAL OF COMPUTERS, VOL. 7, NO. 8, AUGUST 2012
In multiple case, we compare them by Adaptive mean absolute deviation and adaptive root of mean square error criterion [16]. Then will find GED-GARCH is better than T-GARCH, and T-GARCH is better than N-GARCH, which is the same result as in univariate case. Since GED-GARCH is better GARCH based on other distribution, we can change the expanded form of N-GARCH in the future. ACKNOWLEDGMENT The authors thank Professor Xinghuo Wan and Processor Shoudong Chen for their help in completing this article. This work was supported by a grant from Natural Science Foundation of Hebei United University (No. z201114). REFERENCES [1] Tiemey, L. Markov chains for exploring posterior distributions (with discussion [J]. The Analysis of Statistics, 22, 1701-1762． [2] A Generalized Error Distribution, Graham L. Giller. Giller Investments Research Note, August 16, 2005. [3] Geman, S. and D. Geman. Stochastic relaxation, Gibbs distribution and Bayesian restoration of images. IEE Transactions on Pattern Analysis and Machine Intelligence, 1984. [4] Engel, R. E. and K. F. Kroner: "Multivariate Simultaneous Generalized ARCH", Econometric Theory, 1995, 11, pp .12 2-150. [5] Engle Robert F. Autoregressive Conditional Heteroscedasticity with Estimate of the Variance of United Kingdom Inflation [J]. Econometrica, 1982, 50. [6] Bollerslev T. Generalized Autoregressive Conditional Heteroscedasticity [J]. Journal of Econometrics, 1986, 31. [7] CUI Chang. An Analysis of the Stock Price Volatility in China Stock Market Based on the GED Distribution [J]. Journal of Ningbo University (Liberal Arts Edition), Vol. 22 No.4, pp. 94-98July 2009. [8] Pan Hongyu. Time Series Analysis. The Press of University of International Business Economics [M], Jan, 2006. [9] EViews 5 User’s Guide. web: www.eviews.com, April 15, 2004
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[10] Pan Haitao. The application of MCMC arithmetic in time series [J]. The eighth annual meeting of uncertain system in china. 2010, pp. 12-16. [11] Siddhartha Chib and Edward Greenberg. Understanding the Metropolic Hastings Algorithm [J]. The American Statistician. November 1995, Vol.49, No.4 P327-335. [12] MA Fu1ing, LIANG Manfa, Yi Ke. The Application of MCMC Method in the Estimation of the Binary Stochastic Volatility Model [J]. Journal of Hefei University (Natural Sciences Feb)． 2010 Vo1．20 No．1. pp. 27-29. [13] ZHU Huiming, ZHOU Shuaiwei, LI Sufang, ZENG Zhaofa. Bayesian Analysis for Dynamic Panel Data Models Using Gibbs Sampling Algorithm. JOURNAL OF QUANTITATIVE ECONOMICS. Vo1. 28, No. 1, pp. 52-60, M ar. 2011. [14] Shen Xia, Zhao Guoyin and Wang Lifen. MCMC Simulation Method of VaR Calculation [J]. JOURNAL OF TAIYUAN NORMAL UNIVERSITY (Natural Science Edition). V0l.9, No. 3, pp. 13-16, Sept. 2010. [15] LI Weiguo, Xiong Bingzhong. Correlation Coefficient Stationary Series Model Based on MCMC and Its Application. Journal of System Simulation [J]. Vol.20 No.14. pp. 3648-3655, Ju1. 2008. [16] Fan Jianqing, Fan Yingying, Lv Jinchi, Aggregation of non parametric estimators for volatility matrix [J] , Journal of Financial Econometrics, 2007, 5( 3) , 321~ 357. Yan Gao was born in Jixi of Heilongjiang province, on December 18th, 1981. She got Master Degree of Econometrics in JiLin University in July, 2007, which is in Changchun province, in china. Now she is a ph. d candidate of Econometrics there. Her major field of study is financial econometric analysis. She has been working in School of Science, Hebei United University since September, 2007, which located in Xin Hua Street 46, Tangshan, Hebei, P. R. China. Now her Current interests are the application of the multiple GARCH models. Chengjun Zhangwas born in Tangshan of Hebei province, on August 24th, 1980. He got Master Degree of Econometrics in JiLin University. His major field of study is Macro-economy . Now he is working in Market Department, China Mobile Group Hebei Co, Ltd. Tangshan Branch, Xing Yuan road 139, Tangshan, Hebei, P. R. China. Liyan Zhang was born in Tangshan of Hebei province, on December 10th, 1988. She is a student in Business school, Hebei University of Economics and business, Xue Fu road 47, Shi Jiazhuang, Hebei, P. R. China, her major is International economy and trade.
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Comparison of GARCH Models based on Different Distributions Yan Gao School of Science, Hebei United University, Xin Hua Street 46, Tangshan, 063009, Hebei, P. R. China Business school of Jilin University, Changchun 130012, Jilin Province, China Email: [email protected]
Chengjun Zhang and Liyan Zhang Market Department, China Mobile Group Hebei Co, Ltd. Tangshan Branch, Xing Yuan road 139, Tangshan, 063004, Hebei, P. R. China; Business school, Hebei University of Economics and business, Xue Fu road 47, Shijiazhuang, 050061, Hebei, P. R. China Email: [email protected] ; [email protected];
Abstract— Since ARCH and GARCH models are presented, more and more authors are interested in the study of volatilities in financial markets with GARCH models. Method for estimating the coefficients of GARCH models is mainly the maximum likelihood estimation. Now we consider another method—MCMC method to substitute for maximum likelihood estimation method. Then we compare three GARCH models based on it. MCMC method developed based on Markov chain, which is one kind of straggling time and state random process with no offspring imitates. It attracts extensive attention because of its applications in many fields. In this article, we will compare GARCH models based on different distributions with MCMC method. At last we have the conclusion that both in uni-variable case and binary variable case, GED-GARCH is the best model to describe the volatility compared to other two models, and we will provide the application of binary GED-GARCH models in forecasting the volatility in China’s stock markets. Index Terms—MCMC, China’s stock markets, Gibbs sampling, GED-GARCH
I. INTRODUCTION In 1907, A. A. Markov began the study of an important new type of chance process. In this process, the outcome of a given experiment can affect the outcome of the next experiment. This type of process is called a Markov chain [1]. Markov chain is one kind of straggling time and state random process with no offspring imitates, which attracts more and more attention because of the extensive application. MCMC method is presented in 1950s, which has their roots in the Metropolis algorithm. Almost at the same time, Monte Carlo method is used to calculate the complicated integration. The main problem in integration is how to sampling from the complicated distribution. The MCMC method can solve the problem by using the stable distribution and the Markov chain.
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Since ARCH and GARCH models are raised, the application of them appears in many research and practice fields. The yield series of the financial markets generally have the characters of leptokurtotic, fat-tail with skew, volatility clustering and long memory. It violated the Normal distribution assumption for stochastic errors, so Engel [5] imposed ARCH model in 1982, and then in 1986, Bollerslev [6] put its generalized form GARCH forward. Literatures suppose that the errors obey to Normal distribution, but for many financial series, it cannot describe the character of the fat tails, so we need consider using other distribution to describing fat-tail distribution, e.g. t-distribution and generalized error distribution. Many authors are studying the applications of MCMC in volatility analysis. Pan Haitao [10] (2010) use Markov chain Monte Carlo (MCMC) method to estimate the parameters of normal-based GARCH (1, 1) model. The results based on MCMC are more reliable and also show results based on MCMC are better than that of ML based by using real financial data. MA Fu1ing[12] studies the Application of MCMC method in the estimation of the binary stochastic volatility model, Shen xia [14] calculates the VAR by MCMC Simulation method and makes an empirical analysis on certain Shanghai Securities Composite Index in order to prove the advantages of MCMC Method over MC. The sampling methods of MCMC conclude the Metropolis-Hastings sampler, the Gibbs sampler, Importance sampler and Slice sampler, in which, the Gibbs sampler is in common use, we will describe it in detail below. LI Wei-Guo [15] use The MCMC method and the maximum likehood estimation method analyze the correlation coefficient stationary series based on the classical ARMA were applied on the real electric net load monthly data of Guangxi. The result show that the MCMC method provides the most precise prediction．
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II. UNIVARIATE N-GARCH, T-GARCH AND GED-GARCH
A. Mutiple Student Distribution
In order to complete the GARCH [8] specification, we require an assumption about the conditional distribution of the error term
t
vt ~ N (0,1)
(1)
ht 0 1ht 1 1 t21
yR
v ( y )
1
( y )
vn 2
(7)
n
yt c t ~ N (c, ht )
(2)
Many positive study show distributions of yield series have fat-tail, so Nelson (1991) and Hamilton (1994) use generalized error distribution (GED) and t-distribution to adjust the deviation of the tail. We will introduce the two distributions separately. For the student’s t-distribution, the distribution density for it is given below:
(( r 1) / 2) [( r 2) ]1/ 2
(1 x 2 /( r 2)) ( r 1) / 2 (3)
C1
v n v n 2 v ( ) v2
2
1
2
r
2 r (3 / r ) 1 / 2 (3 / r ) x 2 f (x , , r) exp . 2 (1 / r ) 3 / 2 (1 / r )
(4)
() is GAMMA function. If a variate x is
2 drawn from this p.d.f., we can denote x ~ G( , , r ) . We can describe the model below:
t I t 1 ht vt vt ~ G (0,1, r )
(5)
v , v 2 v2
(8)
B. Mutiple Generalized Error Distribution Now we propose the definition of multiple generalized error distribution [2] below: n
n 3 2 (1 ) ( ) d nx 2 r dF ( x , , r ) n (1 n ) ( 1 ) r r (9)
3 2 ( r ) exp ( x )T 1 ( x ) . ( 1 ) r If n dimension vector x obey to this distribution, we remark it as x ~ GED( , , r ) , ,, r are parameters, is the mode of the distribution, and also is mean. The relationship between VAR-COV matrix V and can be shown below:
n2 1 )(1 ) r r V 3 n ( )(1 ) r r (
(10)
C. Definition of GARCH model
1ht 1 1 t21
Then we have
yt c t ~ GED(c, ht , r ) .
E (Y ) , cov(Y , Y )
r
r is constant. For the GED-GARCH model, fist we give out the distribution density for it:
(6)
III. MUTIPLE N-GARCH, T-GARCH AND GED-GARCH MODEL We are familiar to multiple Normal distributions, so we just introduce multiple student t distribution and multiple generalized error distribution here.
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In which,
We use Tn (v, , ) to denote multiple student t distribution.
Then we have
ht 0
1 2
() is gamma function, then
t I t 1 ht vt
0r ,
. There are three assumptions
commonly employed when working with GARCH models [9]: normal (Gaussian) distribution, student’s t-distribution, and the generalized error distribution (GED) [7], we discuss them in detail below. For Normal (Gaussian) distribution, we have
f ( x, r )
•f Y ( y ) C1
y1t Suppose we have n assets, Yt is a vector with y nt n elements, the model is given below:
Yt X t t
(11)
Where t is vector, if: 1) t I t 1 ~ N (0, H t )
(12)
JOURNAL OF COMPUTERS, VOL. 7, NO. 8, AUGUST 2012
2)
t I t 1 ~Tn (v, , )
3) t
1969
(13)
I t 1 ~ GED(X t , H t , r )
(14)
Scalar parameters a sliu and bsliu is the u-th element of a sli and bsli separately. We will use this method to analysis the volatilities. IV. THE MCMC METHOD
I t 1 denotes all the information before t moment. The VAR-COV matrix is
H t C C BH t 1B A t 1 ( A t 1 )
(15)
In which, C, B, A are n n matrix, H t is conditional VAR-COV matrix. We call the model constructed by equations (11), (12) and (15) multiple N-GARCH; the model constructed by equations (11), (13) and (15) multiple T-GARCH; the model constructed by equations (11), (14) and (15) multiple GED-GARCH. D. BEEKmodel BEEK model is presented by Engel and Kroner in 1995 [4].This model denotes the conditional covariance matrix H t by the form of quadratic terms, which guarantees its Positive definiteness, so we needn’t put any limits on unknown parameters. The definition of conditional covariance matrix H t is shown below:
H t C 0 C 0
k11
q
p
il
b
h( x)dx
a
E p ( x ) [ f ( x)]
t i Bil
i 1 l 1
C0 is a lower triangular matrix of m ranks, which m(m 1) has unknown parameters, Ail and Bil are both 2 m dimension unknown parameters square matrix. The element at (i, j ) location of H t , which denotes hijt can
hijt cij
(a sli Yt s )( a slj Yt s )
i 1 l 1
p
k12
b
(17)
sli H t i Bslj
i, j 1, , m
i 1 l 1
The unknown parameter c ij is the (i, j ) element for C C0C0 , a sli is the ith row for matrix Asl , bsli is the ith row for matrix. The equation above can be express as scalar form:
hijt cij
q k11
m
a
p
m
b
sliu bsljv huvt s
s 1 l 1 u ,v 1
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(18)
i, j 1,, m
i
i 1
f ( y x) p( x) ,
which is
approximate by n
f (y x ) i
(20)
i 1
B. Definite of Markov Chains Stochastic process is the quantify description of a series of random events. It is an important tool for many studies, e.g. natural science, engineering science and society science. And it has extensive applications. We often need the theory of stochastic process to establish mathematical models. Markov process is a classic stochastic process. Suppose X (t ) is a stochastic process, when the state of the process at t 0 is known, and the state at t1 is unconcerned to the state before t0 , this character is called
sliu asljv yut s yvt s
s 1 l 1 u ,v 1 k12
f (x )
Consider the integral I ( y )
1 Iˆ( y) n
k11
1 n
(19)
n
The formula is called Monte Carlo integration, which can be used to approximate posterior (or marginal posterior) distributions required for a Bayesian analysis.
be given: q
b
f ( x) p( x)dx
(16)
k12
B H
from the density p(x) , then
a
Ail Yt i Yti Ail
i 1 l 1
A. Monte Carlo Method If the problem requested to be solved is the probability of some event, or is the expected value of some random variable, we can get the frequency of the event or the mean value of the variable, and take it as the solution of the problem, which is the basic thought of the Monte Carlo method. It takes a probability model as base, according to the process described as the model, then get the result of the simulated experiment as the approximate solution of the problem. We can summarize the process of solving the Monte Carlo problem into three steps: Constructing or describing the probability process; Drawing samples from given probability distribution; Establishing different estimators. Now we give out the definition of Monte Carlo integration. If we get random variables x1 , x2 ,, xn
no aftereffect. The stochastic process which has no aftereffect is called Markov process. The time and state for Markov process can be continuous and discrete. We call the Markov process with discrete time and discrete state Markov chains. In another word, we call the
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( x, y ) as below: First, we get a random number u from U (0,1) , if u ( x, y ) , set X n1 y ;or set X n1 x . If we take x which makes ( x) 0 as the starting pot
stochastic process is a Markov process if the transition probabilities between different values in the state space depend only on the random variable’s last state, i.e.
Pr( X t 1 ) s j X t si ,, X 0 sk ) (21)
Pr( X t 1 s j X t si )
of the Markov chain, then for equation (23), the denominator of the ratio will not equal to zero.
A Markov chain refers to a sequence of random variables X 0 ,, X n generated by a Markov process. A particular chain is defined by its transition probabilities,
P(i, j) P(i j) Pr( X t 1 s j X t si )
The Gibbs sampler method [3] didn’t have impact on the field of statistics until rather recently. Now we consider the principle. Consider a binary variable ( x, y ) , if we want to get the marginal distribution f (x) or f ( y) , instead of using f ( x, y) , we use the integration of conditional distribution to get the marginal distribution. First, we choose a starting value
(22)
Let the transition probabilities be the elements, we can get the transition matrix C. Sampler Methods of MCMC
Metropolis-Hastings Algorithm Metropolis-Hastings
Algorithm
The Gibbs sampler method
for [11]
is
a
y,
then we draw a random variable
x0
by f ( x y y0 ) , then we get a new y
powerful Markov chain method to simulate
by f ( y x x0 ) , the sampling process is below:
multivariate distributions. The thought of M-H algorithm is to construct a Markov chain whose
xi ~ f ( x y yi1 )
(24)
target distribution is an invariant distribution.
yi ~ f ( y x xi )
(25)
In order to achieve the target, M-H Algorithm recur to a assistant probability density function q ( x, y ) , which usually need to satisfy three
Repeat the process k times, we will get the Gibbs series, subset ( x j , y j ),1 j m k is
conditions(s denotes state space): a). for a settled x , q ( x, ) is a probability density
take by the stimulation sample from the joint distribution [13], we can get m by: 1). Eliminate the affected by starting value by enough simulated Annealing Algorithm, 2). Make all the samples use simulated Annealing Algorithm, then we will get a stable distribution which is irrelevant to the starting values. We use this algorithm to analyze the univariate case and the binary case.
function; b). for x, y S , q ( x, y ) can be calculate; c). for for a settled
x
, random
number can be produced from q ( x, y ) . When the three conditions above be satisfied, the choice of q ( x, y ) can be arbitrary. Now we give out the procedure of the algorithm, X n is supposed to be equal to x , then 1) Produce a new state y from q ( x, y ) ; 2) Calculate the acceptance probability;
( x, y ) min( 1,
( y )q ( y, x) ) ( x ) q ( x, y )
(23)
3) Set X n1 y by probability of ( x, y ) ,
X n1 x 1 ( x, y) .
set In
actual
by
calculation,
probability the
choice
of of
q ( x, y ) have big influence to the efficiency of the algorithm. Generally we think that the more q ( x, y ) is near to the target distribution, the effect of simulation is better. In step 3, if we set X n1 y , we call it “accept proposal”, or we call “reject proposal”. In actual calculation, we can achieve accepting y by probability of © 2012 ACADEMY PUBLISHER
V. POSITIVE STUDY AND CONCLUSION A.
Data and Data Processing We get data from Shanghai composite index and Shenzhen Stock Exchange Component Index, example period is from Jan forth, 2000 to April 29th, 2011, data come from Yahoo financial website. We take SS to denote Shanghai composite index, SZ to denote Shenzhen Stock Exchange Component Index, and we use y1 , y 2 to denote the take log difference series for SS and SZ, which closely to the daily yield series .the graphs is shown below.
JOURNAL OF COMPUTERS, VOL. 7, NO. 8, AUGUST 2012
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clustering and long memory, so we choose to calculate the kurtosis and auto-correlation function as the standard for comparing the GED-GARCH and the N-GARCH.
.12 .08
B. Positive Analysis For Univariable Case
.04
TABLE II.
.00
The estimation result for SS
-.04 Model
-.08 -.12
0
1
1
r (v)
2.79e-06
0.927037
0.06366
(3.61e-07)
(0.003047)
(0.003291)
7.32e-07
0.837227
0.142771
(1.77e-06)
(0.016511)
(0.017977)
N-GARCH
-.16 00 01 02 03 04 05 06 07 08 09 10 Figure1.
The Yield Series of Shenzhen Stock Exchange Component Y2 Index
2
GED-GARCH
0.978988
3.7e-06
0.911674
0.090173
(1.12e-06)
(0.011059)
(0.014605)
t-GARCH
.12 .08
3.508736
We use the Eviews 5.0 software to analyze, we simulate for 20000 times, in which 4000 times are used to eliminate the affect by starting value, then we have the estimating result.
.04 .00
TABLE III.
-.04
The estimation result for SZ Model
-.08
N-GARCH
-.12 GED-GARCH
00 01 02 03 04 05 06 07 08 09 10 Figure2.
t-GARCH
The Yield Series of Shanghai Composite Index Y1
From figure1 and figure 2, we can see the volatility clustering and long memory. We use the deviation series to describe the model fitting,
zit yit
1 T
T
y
it , i
1,2, t 1,2,, T .
(26)
0
1
3.38e-06 (4.83e-07) 1.39e-05 (3.10e-06) 4.40e-06 (1.30e-06)
0.926615 (0.003654) 0.805567 (0.020427) 0.914290 (0.010894)
2 0.96643 3.79696
The kurtosis coefficients and coefficientsof the Std. dev Model
Kurtosis SS
TABLE I.
r (v)
TABLE IV.
t 1
T denotes the number of observations. The basic statistics for the deviation series of y1 and y2 is below:
1 0.063861 (0.003806) 0.169296 (0.023543) 0.082782 (0.012991)
Std. dev SZ
SS
SZ
N-GARCH
Cannot be estimated
Cannot be estimated
1.001073
1.000468
t-GARCH
6.438044
5.873222
0.210226
0.235555
GED-GARCH
8.288008
7.669474
0.121165
0.112245
Basic statistics for the deviation series of y1 and y2
Mean
Std. dev
Skewness
Kurtosis
z1
4.14e-07
0.01636
-0.16794
7.90059
z2
2.55e-07
-0.0179
-0.20821
7.19304
JarqueBera (Probability ) 2968.827 (0.00000) 2184.601 (0.00000)
From Table I, we can see, the yield series have the characters of leptokurtotic, fat-tail with skew, volatility © 2012 ACADEMY PUBLISHER
Table IV describes the kurtosis coefficients and the std. dev between the auto correlation given by model and the real auto correlation. We can conclude that the GED-GARCH model is better than t-GARCH, and t-GARCH is better than N-GARCH. The result declare that the distribution for yield series don’t obey to Normal distribution, we should consider use the GED-GARCH model to dispose the leptokurtotic, fat-tail in financial market, and we also can consider the application in dimension case in the further research.
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C. Positive Analysis For Binary Variable Case Now we still use the data simulated above to compare the binary N-GARCH(1,1), T-GARCH and GED(0,r,1)-GARCH(1,1) models in studying the volatility between Shanghai stock market and Shenzhen Stock market, then according to the rule of evaluation, we will use the better model to analyze the volatility. We use the beek method, now we can see the estimation results below. The estimation results For N-GARCH is shown in Table V, the coefficients of 3 and 3 are not significant, which means that there is no Spillover Effect from shanghai stock market to Shenzhen stock market. It indicates that the volatility in Shanghai stock market will not affect the Shenzhen stock market. The estimation results For T-GARCH is shown in Table VI. For T-GARCH model, the coefficients of are also not significant, which is the same 3 and 3 as N-GARCH, T- GARCH mode is better than N-GARCH model.
2
-0.00976
-1.79854
0.0900
3
0.00875
0.84334
0.3753
4
0.86986
132.88765
0.0000
ESTIMATION RESULT FOR GED-GARCH(R=1.25) Model
Parameter
Coefficient
Z-statistics
P-value
1
0.001604
28.34937
0.0000
2
-0.000355
-9.666305
0.0000
3
0.000867
423.6704
0.0000
1
0.55392
(29.996)
0.0000
2
0.016203
8.095982
0.0000
3
0.144000
8.963469
0.0000
Coefficient
Z-statistics
P-value
1
0.001673
12.07842
0.0000
2
0.001924
11.89724
0.0000
4
0.039155
13.61430
0.0000
3
0.000356
4.569905
0.0000
1
0.970622
576.7073
0.0000
1
0.214558
20.24739
0.0000
2
-0.076553
-144.5016
0.0000
2
0.022326
6.719294
0.0000
3
0.921634
157.0822
0.0000
3
-0.0227
-1.5709
0.1162
4
0.008790
14.58418
0.0000
4
0.290876
27.31948
0.0000
1
9.75E-01
162.0012
0.0000
2
-0.0087
-1.75061
0.0800
3
0.005734
0.751661
0.4523
4
0.953381
147.8597
0.0000
ESTIMATION RESULT FOR T-GARCH(V=3.4)
T-GARCH
0.0000
Parameter
TABLE VI.
Model
187.9876
TABLE VII.
Estimation result for N-GARCH
N-GARCH
0.98653
If we suppose the significance level to be 10%, the but the significance for the two coefficients is stronger than the N-GARCH model, which means the Model of coefficients of 3 can be thought to be significant, which indicate there are mutual Spillover Effect between the two stock markets.
TABLE V.
Model
1
Parameter
Coefficient
Z-statistics
P-value
1
0.002673
14.76384
0.0000
2
0.003924
15.68434
0.0000
3
0.000556
5.78964
0.0000
1
0.314558
23.67543
0.0000
2
0.032326
6.87698
0.0000
3
0.0127
-1.09873
0.09162
4
0.490876
2767493
0.0000
© 2012 ACADEMY PUBLISHER
GED-GARCH
The estimation results For GED-GARCH is shown in Table VII, for GED-GARCH model, the coefficients of all parameters are significant which indicates that there is mutual Spillover Effect between the two stock markets; the statement is consistent with the fact. It has been nearly twenty years since the establishing of the two stock markets, as the developing and improving of the two stock markets, their relationship is more and more closely. Now the two markets show great correlations in both price and volatility. The MCMC give me a method in comparing the different GARCH models.
VI. CONCLUSION We use MCMC method to compare three GARCH models: N-GARCH, T-GARCH and GED-GARCH in univariate case and multiple case separately. In univariate case, we compare the kurtosis coefficients and the std. dev between the auto correlation given by model and the real auto correlation. Then we can conclude that the GED-GARCH model is better than t-GARCH, and t-GARCH is better than N-GARCH.
JOURNAL OF COMPUTERS, VOL. 7, NO. 8, AUGUST 2012
In multiple case, we compare them by Adaptive mean absolute deviation and adaptive root of mean square error criterion [16]. Then will find GED-GARCH is better than T-GARCH, and T-GARCH is better than N-GARCH, which is the same result as in univariate case. Since GED-GARCH is better GARCH based on other distribution, we can change the expanded form of N-GARCH in the future. ACKNOWLEDGMENT The authors thank Professor Xinghuo Wan and Processor Shoudong Chen for their help in completing this article. This work was supported by a grant from Natural Science Foundation of Hebei United University (No. z201114). REFERENCES [1] Tiemey, L. Markov chains for exploring posterior distributions (with discussion [J]. The Analysis of Statistics, 22, 1701-1762． [2] A Generalized Error Distribution, Graham L. Giller. Giller Investments Research Note, August 16, 2005. [3] Geman, S. and D. Geman. Stochastic relaxation, Gibbs distribution and Bayesian restoration of images. IEE Transactions on Pattern Analysis and Machine Intelligence, 1984. [4] Engel, R. E. and K. F. Kroner: "Multivariate Simultaneous Generalized ARCH", Econometric Theory, 1995, 11, pp .12 2-150. [5] Engle Robert F. Autoregressive Conditional Heteroscedasticity with Estimate of the Variance of United Kingdom Inflation [J]. Econometrica, 1982, 50. [6] Bollerslev T. Generalized Autoregressive Conditional Heteroscedasticity [J]. Journal of Econometrics, 1986, 31. [7] CUI Chang. An Analysis of the Stock Price Volatility in China Stock Market Based on the GED Distribution [J]. Journal of Ningbo University (Liberal Arts Edition), Vol. 22 No.4, pp. 94-98July 2009. [8] Pan Hongyu. Time Series Analysis. The Press of University of International Business Economics [M], Jan, 2006. [9] EViews 5 User’s Guide. web: www.eviews.com, April 15, 2004
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[10] Pan Haitao. The application of MCMC arithmetic in time series [J]. The eighth annual meeting of uncertain system in china. 2010, pp. 12-16. [11] Siddhartha Chib and Edward Greenberg. Understanding the Metropolic Hastings Algorithm [J]. The American Statistician. November 1995, Vol.49, No.4 P327-335. [12] MA Fu1ing, LIANG Manfa, Yi Ke. The Application of MCMC Method in the Estimation of the Binary Stochastic Volatility Model [J]. Journal of Hefei University (Natural Sciences Feb)． 2010 Vo1．20 No．1. pp. 27-29. [13] ZHU Huiming, ZHOU Shuaiwei, LI Sufang, ZENG Zhaofa. Bayesian Analysis for Dynamic Panel Data Models Using Gibbs Sampling Algorithm. JOURNAL OF QUANTITATIVE ECONOMICS. Vo1. 28, No. 1, pp. 52-60, M ar. 2011. [14] Shen Xia, Zhao Guoyin and Wang Lifen. MCMC Simulation Method of VaR Calculation [J]. JOURNAL OF TAIYUAN NORMAL UNIVERSITY (Natural Science Edition). V0l.9, No. 3, pp. 13-16, Sept. 2010. [15] LI Weiguo, Xiong Bingzhong. Correlation Coefficient Stationary Series Model Based on MCMC and Its Application. Journal of System Simulation [J]. Vol.20 No.14. pp. 3648-3655, Ju1. 2008. [16] Fan Jianqing, Fan Yingying, Lv Jinchi, Aggregation of non parametric estimators for volatility matrix [J] , Journal of Financial Econometrics, 2007, 5( 3) , 321~ 357. Yan Gao was born in Jixi of Heilongjiang province, on December 18th, 1981. She got Master Degree of Econometrics in JiLin University in July, 2007, which is in Changchun province, in china. Now she is a ph. d candidate of Econometrics there. Her major field of study is financial econometric analysis. She has been working in School of Science, Hebei United University since September, 2007, which located in Xin Hua Street 46, Tangshan, Hebei, P. R. China. Now her Current interests are the application of the multiple GARCH models. Chengjun Zhangwas born in Tangshan of Hebei province, on August 24th, 1980. He got Master Degree of Econometrics in JiLin University. His major field of study is Macro-economy . Now he is working in Market Department, China Mobile Group Hebei Co, Ltd. Tangshan Branch, Xing Yuan road 139, Tangshan, Hebei, P. R. China. Liyan Zhang was born in Tangshan of Hebei province, on December 10th, 1988. She is a student in Business school, Hebei University of Economics and business, Xue Fu road 47, Shi Jiazhuang, Hebei, P. R. China, her major is International economy and trade.
