Research on Dynamic Correlation Between A&B Stock Index of Shanghai and
Shenzhen Exchange Based on DCC-MGARCH Model
Jin-fang Tian, Wen-jing Wang
Department of Statistics, Shandong University of Finance and Economics, Jinan, China
(20038029@sdufe.edu.cn)
Abstract - The paper dynamically investigates the
correlation of A and B-share in Shanghai and Shenzhen
Stock Exchange using their daily trading data based on the
Dynamic Conditional Correlation Multivariate Generalized
Autoregressive Conditional Hetero skedasticity (DCCMGARCH) model. We show that the A and B-share market
have reflected a certain degree of consistency characteristics
since 2001. However, since China’s stock market isn’t
mature, the dynamic correlation between A and B-share
stock index in both Shanghai and Shenzhen is still volatile
and their segmentation feature is still evident.
Keywords - DCC-MGARCH, A&B stock index, dynamic
correlation
I. INTRODUCTION
Correlation analysis in the financial asset or financial
markets is an important foundation of modern finance,
which also has been playing an important role in building
the investment portfolio and risk management in multiasset and multi-market, especially. Therefore, seeking a
reliable estimate of the correlation coefficient has become
the hot issue that many researchers concerned about. In
order to study conveniently, they often set the indicators of
the correlation among assets to a constant. Whereas this
assumption divorced from reality, in which the correlation
between most financial assets are dynamically changing.
Therefore, close attention must be paid to analyze the
dynamic evolution process of the correlation among assets.
The researches into the dynamic correlation between
the financial assets originated in the promotion/
generalization of the univariate GARCH model. Reference
[1]-[4] extends GARCH model to the case of multivariate,
proposing the multivariate GARCH model. Since the
evolution path of all the individual elements of the
covariance matrix of the model have been made a
GARCH-like description, the indicators of the correlation
between the variables obey a dynamic process. The model
quite comprehensively considers the fluctuation process of
the random vector; as well brings problems about
parameters excessive and make estimation difficulty.
Furthermore, if we do not further setting the coefficient
matrix in the model, we cannot guarantee the positive
definiteness of the estimated covariance matrix. To solve
this problem, Reference [1] proposed the BEKK model,
which makes a useful constraint on the coefficient matrix
of the covariance matrix’s evolution path, and makes it
possible to guarantee the positive definiteness of the
covariance matrix. Although the BEKK model solved the
problem of positive definiteness, since all the parameters
need to be put together to estimate, it brings the estimation
problem in the case of multivariable. From the perspective
of estimation, Reference [5]-[8] proposed the assumption
of constant correlation coefficient, namely Constant
Conditional Correlation (CCC) model. In the CCC model,
the correlation coefficient is set to fixed, although this
assumption has made a huge advantage on estimation, it is
always difficult to satisfied in reality. On the basis of
predecessors' achievements, Reference [9]-[10] expanded
the CCC model and proposed the Dynamic Conditional
Correlation (DCC) model.
This paper focuses on the introduction and the
application of the Dynamic Conditional Correlation
multivariate GARCH model (DCC - MGARCH Model)
proposed by Engle. The advantages of the model lie in: (1)
obvious computational advantage, even the large
correlation coefficient matrix can be estimated. The DCC
model limits the long-term variance-covariance matrix for
the sample variance-covariance matrix; this restriction can
reduce the number of parameters to be estimated and is
well supported by the empirical data. (2) The two-step
estimation method adopted by DCC makes the parameters
to be estimated in related processes independent of the
number of related sequences, the method can easily
estimate the large correlation matrix with more variables.
In order to have a more comprehensive understanding
of the dynamic relationship between A and B-share in both
China’s Shanghai and Shenzhen Stock Exchange, this
paper studies the time-varying correlation between them
based on the DCC-MGARCH model.
II. DCC-MGARCH MODEL
The
et  is a white-noise process with iid(independent
identically distributed), which stands for K kind of the rate
of return on assets information with mean 0 and covariance
matrix
H t  ,
that is,
et t 1 ~ N 0, H t  , and t  is
the information set at t time. The basic structure of the
model is defined as:
r t  u t  et
et t 1~ N (0, t)
 DRD
t
R
t
t
t
t
   
*
 Q
t
1
Q Q*t
t
1
Q
M
N
M
N
In order to study the characteristics of the return series


 1   m     Q   m t m t  m    Q
n
n
t
t  n and their volatility, we draw the time plots of the daily
n 1
m 1
n 1
 m1

Q T
Q
*
t
 diag (
q
Among these formulas,
T
 
1
t
t 1
11,t
,

q
Rt 
22,t
t
,
q
kk ,t
)
is dynamic correlation
coefficient matrix,
Dt  diag
 h , h
it
pt
it
 i   e
p 1
2
ip it  p
qt
  iqhtq
q 1
It Means every asset return to obey a GARCH process.
 t  Dt1et
error term, and Q is
is the standard
unconditional variance matrix with standard residual error.
The elements in the
are defined as
Rt
 
 ij,t 
qij ,t
return and the squared return series of SHA、SHB(Figure
1) and SZA 、 SZB(Figure 2). In both figures, r1
represents the return series of A-share stock index and r2
represents the return series of B-share stock index. The
time plot of return series of Shanghai and Shenzhen A and
B stock index shows a number of abnormal peaks, and
abnormal volatility appears suddenly and significantly as
well as obvious clustering phenomenon. We also found
that B-share market’s volatility is higher than the A-share
market’s volatility (pay attention to the different scales of
the vertical axis). From the time plots of the squared return
series we can see that abnormal volatility’s burstiness and
volatility clustering phenomenon is more pronounced.
Accordingly, we can preliminarily determine that the
volatility of the return series is conditional
heteroscedasticity.
*
qii,t q jj ,t
, and Qt is diagonal matrix. In
Qt , the elements contain qij,t 、 qii,t and q jj ,t . Besides,
m
and
n
are known as the coefficients of DCC
models(m and n are the lagging numbers), and
m
reflect
the impact of the standardization residual product of m
lags to dynamic correlation coefficient, and  n reflect the
persistent
feature
of
correlation,
M
N
m 1
n 1
where,
 m  0,  n  0,   m    n  1 .
Figure 1 Time plots of Shanghai A and B-share stock index daily return
and squared return
Estimation of DCC-MGARCH models can be divided
two steps. First, to estimate the univariate GARCH process
of every asset, then we use the conditional variance hit 
that has gained to divide residual
standardized residuals
 it  .
eit  ,
Next we will use
and get
 it  to
estimate the parameters of the dynamic related structure by
maximum likelihood method. By the way, the DCC
estimator has the consistency and asymptotic normality
through the two steps approach.
III.
DESCRIPTIVE STATISTICS FOR STOCK RETURNS
The data used in this paper are the daily closing price
of Shanghai A-share index (SHA) 、 Shanghai B-share
index (SHB) Shenzhen A-share Index (SZA) and
Shenzhen B-share index (SZB). All of the data series were
from February 19, 2001 to December 15, 2011. There were
2668 observations for the sample. We will use the
following notation in this paper. The daily closing price
process is denoted by Pt . Firstly, calculate the
 
corresponding daily logarithmic yield from the raw data.
Figure 2 Time plots of Shenzhen A and B-share stock index daily return
and squared return
Furthermore, Table 1 provides some descriptive
statistics of daily returns for selected stock indexes. From
the table, we make the following observations. (a)Daily
returns of selected stock indexes tend to have high excess
kurtosis. B-share stock index has higher excess kurtosis
than A-share stock index, for both Shanghai Stock
Exchange and Shenzhen Stock Exchange. (b)For the entire
selected stock index, the skewnesses are less than zero;
imply the distribution of the return series is not normal.
The JB statistics also shows the normality assumption is
rejected at the 1% significance level. (c)The mean of all
daily return series is close to zero, but B-share stock index
has higher standard deviations than A-share stock index for
both Exchange.(d)The Q(m) statistics of the return series
give high Q(15) value with a p value less than 0.05,
confirming serial correlations exist in the four stock
indexes.(e) The Q(m) statistics of the squared return series
indicate an obvious volatility clustering phenomenon,
which is consistent with the Intuitive judgment from the
time plots.
confirming that the mean equations don’t have
autocorrelation at the 5% significance level . Namely the
four fitted mean equations are adequate. However, the
Ljung-Box statistics for the squared residuals of the four
conditional mean models give Q(20) equal to 412、480、
379、434, respectively, with a p value less than 2.2E-16,
which confirming that the residuals series exist obvious
ARCH effect. We need to establish the corresponding
GARCH model for further analysis. For convenience, we
use the GARCH (1, 1) model to fit the volatility of the
return series, the estimated results are shown in Table 2.
TABLE 1 Descriptive Statistics for Daily Returns of Selected Stock
Indexes
M
STD
skewness
kurtosis
Q(15)
Q^2(15)
JB
SHA
0.0048
1.7
-0.137
4.74
26.7(0.031)
350(0)
2511(0)
SHB
0.0365
2.21
-0.037
7.18
43.5(00)
505(0)
5748(0)
SZA
0.015
1.85
-0.54
4.81
30.1(0.01)
361(0)
2710(0)
SZB
0.055
2.08
-0.03
5.67
68.3(0)
617(0)
3579(0)
We need to test the stationary of the return series
before molding. In this paper, we choose the ADF method.
The outcome shows the ADF test statistics of SHA、SHB
、SZA and SZB is -15、-14、-14 and -13, respectively.
The corresponding p values are all less than 0.01,
confirming the return series don’t have unit root. In other
words, all the return series are stationary time series.
Therefore, we can directly molding.
TABLE 2 Conditional Variance Parameter Using GARCH(1,1)
omega
alpha1
beta1
SHA 0.061(4.29***) 0.076(6.25***) 0.904(60.97***)
SHB 0.201(6.46***) 0.204(9.6***)
0.782(43.15***)
SZA 0.063(4.41***) 0.091(6.8***)
0.893(61.68***)
SZB 0.282(6.18***) 0.216(9.2***)
0.742(31.98***)
Note: the value of the corresponding t -statistics for each
parameter are shown in parentheses. “***” denote the
parameter is not 0 significantly at the 1% significance
level.
From table 2, we can see that all the estimates in the
volatility equation are statistically significant at the 1%
significant level, and    are all less than 1, which
satisfy the assumption of the model. Model checking,
using the residual, indicates that the fitted volatility model
is adequate. Noted that the value of    is very close
to 1, indicating that the volatility of the Shanghai and
Shenzhen return series has a significant persistent
B. Establishment and estimation of DCC-MGARCH
model
We firstly get the standardized residuals from the
A. Modeling and estimating of univariate GARCH model
residual(which can gained from the univariate GARCH
Since the return series of SHA、SHB、SZA and SZB
model) divided by conditional variance, then we use the
have autocorrelation, we use the ARMA model to estimate
standardized residuals to estimate the parameters of the
the conditional mean equations. After several attempts and
dynamic correlation structure by Maximum Likelihood
careful compared by using R software, the fitted models
method. The DCC estimators that we obtained in this way
we ultimately choose are
are consistent as well as asymptotic normality.
rSHA,t  0.0047  0.0538 * rSHA,t  4  0.0391 * rSHA,t  6  aSHA,t
Now we will use the DCC- MGARCH model to
analyze the correlation of A and B-share stock returns for
（0.14） （2.79）
（-2.03）
Shanghai and Shenzhen stock market. Setting the
rSHB,t  0.0364  0.074 * rSHB,t 1  0.04 * rSHB,t  4  0.043 * rSHB,t 6  aSHB,t both
GARCH model as GARCH (1,1) while the degree of the
（0.68） （3.81） （2.09） （2.21）
DCC model is also set as 1, using R software to estimate
parameters of the model, the results are shown in Table
rSZA,t  0.0153  0.0472 * rSZA,t 1  0.0464 * rSZA,t  4  aSZA,the
t
3.
IV.
（0.397）
EMPIRICAL ANALYSIS
（2.433）
（2.392）
rSZB,t  0.0669  0.7631 * rSZB,t 3  aSZB,t  0.7187 * aSZB,t 3
（0.927） （7.112）
（-6.498）
Where, the value of the corresponding t -statistics for each
parameter are shown in parentheses. The Ljung-Box
statistics of the residuals of the four conditional mean
models give Q(20) equal to 23 、 12 、 16 、 12, with
a p value of 0.29 、 0.92 、 0.73 、 0.92, respectively,
TABLE 3 the Results of Parameter Estimates for DCC Model
alpha
beta
Shanghai(A and B) 0.112(112)
0.817(204)
Shenzhen(A and B) 0.073(73)
0.883(110)
From Table 3, we can know that the parameters of the
DCC model are significantly not 0.  is significantly not 0
means that the first-lagged standardization residual product
has an impact on the dynamic conditional correlation;  is
not only significantly not 0, but also very close to 1, which
means the correlation of the A and B-share stock returns
has strong persistent characteristics for both Shanghai and
Shenzhen stock market.
We provide the time plots of the estimation results of
the dynamic conditional correlation in Figure 3 and 4, so
as to facilitate our visual observation of the changes in
dynamic conditional correlation between A and B-share
stock index for Shanghai and Shenzhen Stock Exchange.
Figure 3 DCC of SH
Figure 4 DCC of SZ
Through the time plots of the dynamic conditional
correlation of the yields for both Shanghai and Shenzhen
Stock Exchange, we can come to the following
conclusions: firstly, during the sample interval （ from
February 19, 2001 to December 15, 2011 ）, the timevarying characteristics of the dynamic conditional
correlation are fairly obvious, which means there is a
significant time-varying feature. Secondly, seen from the
time change paths, since the B-share market opened to
domestic investors on February 19, 2001, the dynamic
conditional correlations of A and B-share stock index for
both Shanghai and Shenzhen are very similar and appear
significant positive related. The two dynamic conditional
correlations are both close to even more than 0.9. While in
contrast, the volatility of Shanghai Stock Exchange is
slightly higher than that of Shenzhen Stock Exchange.
Thirdly, the dynamic conditional correlations are
significantly low during 2005-2006 and 2008-2009.
Particllarly, it’s even lower than 0.2 for Shanghai Stock
Exchange. Fourthly, the dynamic conditional correlations
have been in decline and appeared more frequent
fluctuations since 2011 for both Shanghai and Shenzhen
Stock Exchange.
The dynamic conditional correlation is an important
indicator of the degree of movement convergence of
financial assets or financial market. The higher the
dynamic conditional correlation, the larger degree of
convergence of the market share price trend, and the higher
degree of market integration. Conversely, the lower the
dynamic conditional correlation means the two markets
appear lager deviation in price trend, which means obvious
market segmentation. Since each market stock price
movement contains its own driving factors, we can
investigate the dominant factor behind them by analyzing
the relationship between A and B-share stock price
movements and the reflected dynamic conditional
correlation. The dynamic conditional correlations of A and
B-share stock index in Shanghai and Shenzhen were
always in the high level during 2001-2005, which is
closely related with the event that B-share market opened
to domestic investors on February 19, 2001, making the
investors structure of the B-share market presenting
features of convergent to the A-share market. The dynamic
conditional correlations are significantly low during 20052006 and 2008-2009, which are related to the introduction
of share splitting in 2005 and financial crisis in 2008. The
strong impact on China’s securities market by the two
major events exacerbated A and B-share market’s
volatility and instability. According to the Figure 3 and 4,
The dynamic conditional correlations of the two stock
markets are always in a state of decline with more
frequent fluctuations since 2011. Although China's
securities market is growing and increasingly mature and
the merger of A and B-share market is an irresistable
trend, China's securities market is still not mature enough
and the segmentation feature of A and B-share market is
still relatively obvious , which means it will take a
certain time for the two market to embarked on the merger.
V. CONCLUSION
This paper takes daily trading data of A and B-share
index in Shanghai and Shenzhen Stock Exchange as
sample, dynamically investigate and characterize the
correlation between China's Shanghai (and Shenzhen ) A
and the B-share market through the establishment of the
Dynamic Conditional Correlation multivariate GARCH
(DCC-MGARCH) model. The results show that: the
dynamic conditional correlation between A and B-share
market in Shanghai and Shenzhen Stock Exchange is
positive in general, whereas appear significant timevarying characteristics in the time path. Phases of view, the
dynamic conditional correlation between A and B-share
market in Shanghai and Shenzhen Stock Exchange was
relatively large and volatility was relatively weak from
February 19, 2001 to 2005, suggesting that the stock price
in these market showed a certain degree of consistency.
However, the correlation showed a significant downward
trend after 2005. By contrast with the Shenzhen stock
market, the Shanghai stock market’s downward trend was
more obvious. The correlation got a certain degree of
recovery during 2006 to 2008. By influence of the
financial crisis and other factors in 2008, the correlation
between A and B-share market fall again and suffered with
more frequent fluctuations. In addition, the correlation has
been in a state of decline since the second half of 2010.
The dynamic conditional correlation is an important
indicator of the degree of movement convergence of
financial assets or financial market. The securities industry
generally considered that the B-share market merge to the
A-share market is the trend. Through the analysis of the
dynamic conditional correlation, this paper shows that A
and B-share market reflects a certain degree of consistency
characteristics since 2001. However, because the China’s
securities market is not yet mature, the dynamic correlation
between A and B-share market is still very large, and the
characteristics of segmentation between A and B-share
stock index is still very obvious.
ACKNOWLEDGMENT
This research was supported in part by National Funds
of Social Science(10CTJ003, 09BTJ011) and Shandong
Science Foundation (Y2007A25).
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