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Master Essay
Department of Statistics
An Empirical Study on Stock Exchange Linkages between Chinese and Western Markets
Heng Wang
Supervisor: Anders Ågren
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An Empirical Study on Stock Exchange Linkages between
Chinese and Western Markets
Heng Wang
Department of Statistics, Uppsala University
May, 2012
Abstract
This paper examines the linkages between Chinese stock exchange
markets and western developed markets. We use SHCOMP and SIASA
representing Chinese exchanges, FTSE100 the European exchange, and
S&P500 the U.S. exchange. We find evidence of bidirectional returns and
volatility spillovers from western markets to Chinese markets by four bivariate
GARCH-BEKK models. The volatility impulse response functions show the
impact from U.S is normally stronger than that from Europe and SIASA is more
sensitive to the spillovers than SHCOMP.
Keywords: Stock market linkages, volatility spillovers, multivariate GARCH,
BEKK, volatility impulse response functions
E-mail: [email protected]
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Dedication
I dedicate this paper to my wonderful family. Particularly to my brave
father, Junguo Cao, who has taught me to be strong and persistent, I wish he
could conquer the disease and get healthy soon. I must also thank my loving
mother, Xiufang Wang, who always supports me to move on, and my
understanding sister, Ling Cao, who has helped me to keep patient for study
and given her fullest encouragement. Finally, I dedicate this work to my
girlfriend, Jiaman Yuan, who accompanied me in spirit during the tough days.
I’m grateful for all of them, who always stand behind me and believe I will
succeed.
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Acknowledgement
I am grateful for the guidance and support of my supervisor Anders Ågren. I
thank my thesis opponent, Yumin Xiao, and many professors in Uppsala
University, especially Lars Forsberg, for helpful suggestions on improving the
paper. The encouragements from families and friends are heartily
acknowledged.
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1. Introduction
This paper models the stock markets in Shanghai and Shenzhen, representatives of
Chinese stock markets, focusing on the structure with global influences and studies
whether these two Chinese markets are linked to the western developed markets in Britain
or U.S. Under the development of economic globalization, there are more and more mutual
trades and investment between China and the rest of the world. The European Union and
the U.S are the two largest trading partners of China. Europe has been the largest exporting
region of China since 2004. At the end of 2011, China also became the largest exporting
country of Europe. Besides, the Chinese oversea investment in E.U. exceeded that in U.S. for
the first time, making Europe the No. 1 position in Chinese international trading and
economics. In the meanwhile, China is known as the country which owns most American
national debts and dollar reserves. Therefore U.S. plays a very important role in the
Chinese economy. As one component of the economy, the stock exchange reflects the status
of the domestic economy. The relationship between Chinese and western economies is
believed to produce the reasonable linkage between stock exchange markets. NYSE and
NASDAQ, both involved with American and European companies, take the leading two
positions of market capitalization in the world. And for individual European countries,
British and German market capitalizations get the rank of 41 and 10. The overall
capitalizations of Chinese markets, including Shanghai exchange (rank 5) and Shenzhen
exchange (rank 12), would exceed Japan (rank 3) to generate the third largest market.
Through the economic globalization, the large stock markets should not be isolated. The
co-movements are frequently observed when big economic events strike the world
economic system. A simple example is during the financial crisis in U.S around 2008, when
most markets in different regions crashed with the disaster from the American market.
Our study is based on the assumption that Chinese stock exchanges are associated with
western markets.
As far as Chinese markets are concerned, the study of the regional interaction of the
Chinese stock markets begins very early. Brooks and Ragunathan (2003) and Wang et al.
1 Data from “List of stock exchange” in Wikipedia: http://en.wikipedia.org/wiki/List_of_stock_exchanges
http://en.wikipedia.org/wiki/List_of_stock_exchanges
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(2004) use a GARCH model to investigate the interactions between Chinese A and B shares
traded on the Shanghai and the Shenzhen stock exchanges. In the further studies, a
conjecture of global interaction between the Chinese stock markets and the western stock
markets has drawn a lot of attentions.. Chow (2003) studies the relationship between
Shanghai and New York stock price indices by simple correlation and multiple regression
methods. However the capital markets are found not to be integrated in his paper. Li (2007)
examines the linkage between stock exchanges in China (including Shanghai, Shenzhen and
Hong Kong) and in U.S. through a multivariate GARCH model. The return linkages between
the stock exchange in mainland China and in U.S. are indirectly depending on the return
linkage between the stock exchange in Hong Kong and the U.S. market. Hong Kong has
acted as a go-between in the information flow. Diekmann (2011) analyzes the stock market
integration of mainland China indices, Hong Kong indices and Dow Jones Industrial index
from 1998 to 2006, proving an evidence of global and regional integration. In the newest
research, Zhou et al. (2011) measure the dynamic volatility spillovers between Chinese
stock indices and western indices by variance decompositions in a generalized VAR
framework. Their study shows that from 1996 to 2005 the Chinese market was slightly
affected by western markets and from 2005 to 2009, the Chinese stock market had a
significant volatility spillover effect on western markets, indicating that the influence of the
Chinese stock market was greatly enhanced during the years. They also show that the US
market had dominant volatility impacts on the Chinese markets during the subprime
mortgage crisis.
The multivariate, mostly bivariate, BEKK models are widely used to investigate the
interrelation between markets. Chou et al. (1999) use a bivariate BEKK model to examine
the volatility linkage between the Taiwanese and the U.S. stock exchanges and prove that
volatility spillovers from the developed market in the U.S. to the emerging market in
Taiwan exist. Haroutounian and Price (2001) also find a volatility spillover from Poland to
Hungary in a bivariate BEKK model. Worthington and Higgs (2004) use a nine-variable
BEKK model to examine the transmission of equity returns and volatility among nine Asian
developed and emerging markets. They find evidence of return spillovers from the
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developed to emerging markets. Li and Majerowska (2007) test the linkages between
emerging markets in Poland and Hungary and the established markets in Germany and the
U.S under a multivariate BEKK approach. Through a four-variable BEKK model, Li (2011)
investigates stock market linkages among China, Korea, Japan and the U.S. with particular
attention to the impact of Chinese stock market reforms. In this paper, we will concentrate
not only on the stock linkages between Chinese markets and western markets in the recent
unstudied years, but also compare the spillovers from different regions, Europe and U.S, to
different components of Chinese markets, the market in Shanghai and Shenzhen.
We study the differences among linkages based on the impulse response function,
proposed by Sims (1980) for the analysis of VAR models. The impulse response function
has been generalized to nonlinear models and higher conditional moments. Hafner and
Herwartz (1998) apply the impulse response function to the volatility of time series,
defining the volatility impulse response function (VIRF). Hafner and Herwartz (2006)
improve the VIRF to an intuitionistic one, which is employed by many researchers through
years. In this paper, we use Hafner and Herwartz’s VIRF methodology to demonstrate the
shocks in BEKK models. The linkages will be easily distinguished from the illustration of
VIRF.
2. Data and preliminary analysis
There are two main stock markets in China, Shanghai and Shenzhen. In this paper, we
use the Shanghai Stock Exchange Composite Index (SHCOMP) to represent the Shanghai
market and the Shenzhen Stock Exchange Constituent A-Share Index (SIASA) the Shenzhen
market. The stock markets in Britain (FTSE100) and the U.S. (S&P500) are considered to
serve well as proxies for the western developed markets and their economic development.
They are considered to have an influential impact on the Asian financial economies,
especially on the Chinese economy.
Following the previous literature, we use daily observations of the stock indexes of the
markets in Shanghai (SHCOMP), Shenzhen (SIASA), London (FTSE100) and U.S. (S&P500)
in this study. The stock indexes are computed from the closed stock prices in local markets.
The Shanghai stock index, SHCOMP is an overall market index, computed from all the listed
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companies in this market. The Shenzhen stock index, SIASA is a selected index that consists
of the 40 top companies’ A-shares on the Shenzhen Stock Exchange. The reason I choose
different kinds of indexes in these two markets comes from the behavior habit of Chinese
market investors. SCHCOMP represents most large listed companies and SIASA represents
medium and small size companies. The two indexes cover a general situation of Chinese
listed companies, making a quality summary for Chinese stock markets. The London Stock
Exchange is the fourth-largest stock exchange in the world and the largest in Europe, so the
index FTSE100, a share index of the stocks of the top 100 companies listed on the London
Stock Exchange having the highest market capitalization, typically reflects the financial
situation in Europe. The S&P500, known as the combination of the two largest stock
exchanges in U.S. as well as in the world, is the most accurate reflection of the U.S. stock
market. The Chinese data in the paper are downloaded from the stock trading software
Great Wisdom2. The data of FTSE100 and S&P500 are from Yahoo Finance.
Figure 1. Stock indices from June 2006 to March 2012
2 Official website: http://www.gw.com.cn/.
http://www.gw.com.cn/
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The period of the data is from 6 December 2005 to 1 March 2012. Our chosen period
includes the big events in recent years, the bull market increase in China (2005.12 ~
2008.11), American financial crisis (2008.08 ~ 2008.10) and European sovereign debates
crisis (2009.11 ~ now). These events have huge influences on the regional and global
markets, determining the trend of stock exchange. Based on the different stock trends
under the events, we divide the whole period into three separate intervals: Increasing
Period I (2005/12/6 ~ 2007/11/12), Decreasing Period II (2007/11/13 ~ 2008/11/10)
and Depression Period III (2008/11/11 ~ 2012/3/1). Comparison of the results from
different markets in different periods can show if there is any market linkage in the
economic integration age and how it varies if existing.
Figure 1 presents the time plots of the series. The vertical dash lines split out the three
periods. It is impressive that the two Chinese markets follow a similar movement while
FTSE100 and S&P500 have a similar trend. The Chinese indices grew rapidly in the first
period due to the domestic product development. But in the following period, the global
financial crisis made the indices fall down quite a lot to the past level two years ago. Even
with the domestic economic encouragement in the recent three years, 2009, 2010 and
2011, the stock markets couldn’t recover. Moreover, the European sovereign debates crisis
led to a new wave of depressed market. On the contrast, the British and American markets
are more defensive to the economic turnovers. When FTSE100 and S&P500 got to the
lowest after the crisis in 2009, they reacted quickly to get saved, regained the markets
confidence and rebounded to the acceptable level.
Figure 2 presents the returns of stock exchanges within the three periods. We make an
adjustment to extend the natural logarithm a 1000 times to increase returns to an easily
visual level. The Chinese markets have quite high volatility from 2007 to 2009 while the
western markets only have shocks during 2009. Compared to Britain and U.S., as Harvey
(1995) pointed out, emerging markets in Asia have high-expected returns and high
volatility. The cluster phenomenon is also observed in Figure 2. Small volatility is more
likely followed by small volatility and large volatility followed by large volatility, which
implies a regular system in the structure.
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Figure 2. Stock returns from June 2006 to March 2012
Table 1 shows the summary results of returns. During Period I and Period III, the
larger means and standard errors of the Chinese series prove our statement in the last
paragraph. But in Period II, when the Chinese markets face the strong strike of financial
crisis, there are smaller negative returns and larger volatilities in China than in the
developed markets. These facts prove the higher risk in Chinese markets, especially in the
decreasing time. During Period I, Chinese markets have negative skewness, smaller than
the skewness of FTSE100 and S&P500. It means the Chinese stock returns are more likely
positive and the possibilities of negative returns are larger than those in Britain and U.S. In
Period II and Period II, the Chinese skewness keeps different sign from the mean. For the
western markets, FTSE100 gets a negative skewness but S&P500 is positive-skewed. In
Period III, it is the opposite. In an overall view, all the series are slightly skewed in Period II
due to a small absolute value of skewness. Except SHCOMP in Period I, the kurtosises of the
Chinese stock exchanges are smaller than 3, i.e. in most cases they have significant thinner
tails and shorter peaks, which is not common in the world. On the contrast, FSTE100 and
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S&P500 have skewness larger than 3 in Period II&III, with fatter tails and higher peaks.
Furthermore in Table 1, the Shapiro-Wilk test3 rejects all the normally distributed
hypothesis of the four series except SIASA in Period II. For all the above, GARCH type
models are capable of modeling data with such features.
Table 1 Summary statistics of returns during December 2005 and March 2012
SHCOMP SIASA FTSE100 S&P500
Mean P1 4.0781 4.5881 0.3741 0.3394
P2 -3.8464 -3.2481 -1.3668 -1.8473
P3 0.6798 1.0465 0.4331 0.5387
S.D P1 17.8163 19.8520 8.9983 7.7514
P2 27.6310 29.5603 21.8485 22.1755
P3 15.4937 18.0416 13.6861 15.9025
Skewness P1 -1.1552 -0.8120 -0.4343 -0.5945
P2 0.3175 0.0634 -0.1323 0.0736
P3 -0.4776 -0.4446 0.1485 -0.3551
Kurtosis P1 3.8517 2.8669 2.4370 2.8407
P2 1.0698 0.4154 3.5722 5.9377
P3 2.0277 1.8958 4.1056 4.2110
Shapiro P1 0.9273
(p=0.0000)
0.9551
(p=0.0000)
0.9673
(p=0.0000)
0.9502
(p=0.0000)
P2 0.9813
(p=0.0017)
0.9941
(p=0.4105)
0.9374
(p=0.0000)
0.9034
(p=0.0000)
P3 0.9706
(p=0.0000)
0.9746
(p=0.0000)
0.9567
(p=0.0000)
0.9327
(p=0.0000)
Note: P1, P2 and P3 stand for Period I, Period II and Period III, mentioned above.
3. Methodology
3.1 The AR-GARCH process
As the first principal aim of this paper is to find out the behavior of Chinese stock
returns by exploring the short-run volatility either in a separate scenario or a cross-market
setting, the univariate GARCH model is an appropriate approach for a first trial. Also in the
previous literature, the GARCH model is widely used for financial returns.
3 Shapiro-Wilk test, by Samuel Shapiro and Martin Wilk, tests the null hypothesis that a sample came from a normally
distributed population. We may reject the null hypothesis if the statistic is too small.
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Engle (1982) introduced the ARCH (Autoregressive Conditional Heteroskedastic)
process instead of the classical assumption on time series and econometric models, which
assumes the variance to be constant. The volatility of econometric series usually shows
some random characteristics on its innovation, indicating a possibility assuming a white
noise process. Besides, the clustering of high level volatility also implies autoregressive
possibilities. The GARCH model, developed from the ARCH process by Bollerslev (1986),
combines the lagged noises and conditional variances together to achieve a general
approach.
Let 𝜖𝑡 denote a discrete-time stochastic process, and 𝐼𝑡 the information set at time t
through the past. The GARCH(p,q) process is given as:
ϵt|𝐼𝑡−1~𝑁(0, ℎ𝑡)
ht = 𝜔 +∑𝛼𝑖𝜖𝑡−𝑖2
𝑞
𝑖=1
+∑𝛽𝑖ℎ𝑡−𝑖
𝑝
𝑖=1
where
p > 0, q > 0
ω > 0, 𝛼𝑖 ≥ 0, 𝑖 = 1,… , 𝑞,
βi ≥ 0, 𝑖 = 1,… , 𝑝
In this equation, a conditional normal distribution of ϵt with zero mean and variance
of ht is considered. In a special case, when p=0, the process is an ARCH(q) process
without its own lag effect. And when p=q=0, ϵt is a white noise with variance α0. If the
white noise could be generated from a linear equation, then the new GARCH (p,q) model
has an additional structure with a mean equation, which is presented as.
ϵt = 𝑦𝑡 − 𝑥𝑡′𝑏,
where yt and xt are observed variables and b is a vector of unknown parameters.
For the long-term autoregressive series, an AR process is somehow a better approach to
describe the first order trend. Thus, we adjust the mean equation as
ϵt = 𝑟𝑡 −∑𝑏𝑖𝑟𝑡−𝑖
𝑠
𝑖=1
− 𝜇
where b = (b1, … , bs) is the vector of autoregressive coefficients.
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In independent univariate GARCH models, the conditional variance of each series
could be estimated so that we can capture the trend and volatility of the series. The moving
average model is also a proper approach to illustrate short-term volatility, making it clear
to distinguish the models.
3.2 The VAR-GARCH-BEKK process
Our second task is to detect the relationship between Chinese stock markets and
western markets, requiring a new model to combine separated series together. The cross
market impact will be in two forms. One is an instant impact which means that a change
from one market effects another immediately and lasts no more. The other one is a
continuous impact which will exist during a period. In a univariate GARCH model, the
mean equation gives an instant impact to the returns and the variance equation results in a
continuous indirect impact in volatility of returns. Both impacts should be analyzed in a
cross-market setting, with cross effects both from series and volatilities. So the series will
be dependent series in a multivariate case. Then, dynamic covariance or correlation
becomes important. Since cross-effects are not considered in a univariate model, a
higher-dimension model is needed for examining cross-market effects.
The BEKK model, proposed by Engle and Kroner (1995), is a multivariate GARCH
model. Specifically, the following model presents the BEKK process, a joint process
covering the multivariate financial return series to be studied.
Yt = 𝛼 + Γ𝑌𝑡−1 + 𝜖𝑡
ϵt|𝐼𝑡−1~𝑁(0,𝐻𝑡),
where Yt is a N × 1 vector of multivariate series at time t, α is a N × 1 vector of
intercepts and Γ is a N × N coefficient matrix associated with lagged own effects. The
diagonal elements in Γ measure the autoregressive effects and the off-diagonal elements
describe spillovers across the markets. The N × 1 vector of noise, or the random errors,
ϵt is the innovation for every market at time t. In this multivariate model, the conditional
distribution of ϵt based on the former information It−1, is in a multivariate normal form
with a conditional variance-covariance matrix, Ht.
Bollerslev et al. (1988) assume that Ht is a linear function of cross products of errors
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and related to its lagged value, Ht−1. So one kind of the BEKK model is given as:
vech(Ht) = c +∑𝐴𝑖𝑣𝑒𝑐ℎ(𝜖𝑡−1𝜖𝑡−1′ )
𝑞
𝑖=1
+∑𝐺𝑖𝑣𝑒𝑐ℎ(𝐻𝑡−𝑖)
𝑝
𝑖=1
,
where vech is an operator transforming the lower/upper triangular part of a symmetric
matrix into a vector. Since the number of parameters to be estimated is large, results
through the formula, which is also the conditional covariance matrix, may not be
guaranteed to be positive definite, making it unreasonable. Years later, Engle and Kroner
(1995) modified the model to a general decomposition one, the well-known BEKK model,
explained better in Ht than its vech form:
Ht = 𝐶′𝐶 + 𝐴′𝜖𝑡−1
′ 𝜖𝑡−1𝐴 + 𝐺′𝐻𝑡−1𝐺
In the BEKK model, 𝐶 is a 𝑁 × 𝑁 lower triangular matrix of constants. 𝐴 and 𝐺
are 𝑁 × 𝑁 parameter matrixes. The diagonal elements in 𝐴 and 𝐺 represent the lagged
effects from single series and its volatility on the conditional variance while off-diagonal
elements measure the spillovers’ effect on the conditional variance as well. Compared to
the univariate GARCH model, the BEKK model has considered all cross effects from a single
series and its volatility. If every matrix in a BEKK model is a diagonal matrix, then the
BEKK model becomes a set of univariate models. It is convenient to model multi series
with a BEKK model, since fixing different elements in the parameter matrices means
different assumption of the effects. At the same time, the quadratic forms ensure the
positive definiteness of the equation. The parameter matrices correspond to specific parts
in the 𝑣𝑒𝑐ℎ formula, but in a very complicated relation.
In our paper, to study the interaction among regional markets and distinguish the
differences, we choose the BEKK (1,1) model with one Chinese series and two western
series. One key point is to reduce the computational processes. On the other hand, to
remove the interference within highly related Chinese markets is also important. Thus, we
get two BEKK models, one with SHCOMP, FTSE100 and S&P500, another one with SIASA,
FTSE100 and S&P500.
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3.3 Volatility Impulse Response Functions
In Section 3.2, we know that for every BEKK model there is a unique vec model
specifying the same structure. And for the BEKK(1,1) model, the equivalent vec
specification is
vech(Ht) = c + 𝐴1𝑣𝑒𝑐ℎ(𝜖𝑡−1𝜖𝑡−1′ ) + 𝐺1𝑣𝑒𝑐ℎ(𝐻𝑡−𝑖).
When we get the long-term expectation of the model, the unconditional covariance matrix
Σ could be found from the formula:
vech(Σ) = (I − A1 − G1)−1𝑐.
If Ht reaches to Σ, in the stationary condition, the volatility will remain the same
even when the time goes on. But in this case, any change in Ht will break the current
balance. If the system is stationary, it will recover from the interruption. If it is not, the
strike from interruption will last forever. As Σ is a multivariate term, associated with
dependent series, any change in one series will influence others. Assume that the volatility
is stable at time t = 0, Σ0 = Σ. We introduce a shock ξ0, a vector related to each series.
One component of ξ0 is set to be 1 and the others are set to be 0, which means the series
with shock 1 will have effects on the system. Here the Volatility Impulse Response Function,
VIRF, is defined as the expectation of the volatility conditional on the initial shock ξ0 and
the initial volatility Σ0. The formula of VIRF is given by
Vt(𝜉0) = 𝐸[𝑣𝑒𝑐ℎ(Σ𝑡)|𝜉0, Σ0 = Σ].
Start with t = 1,
V1(𝜉0) = 𝑐 + 𝐴1𝑣𝑒𝑐ℎ (Σ12𝜉0𝜉0
′Σ12) + G1𝑣𝑒𝑐ℎ(Σ).
For t ≥ 2,
Vt(𝜉0) = 𝑐 + (𝐴1 + 𝐺1)𝑉𝑡−1(𝜉0).
In the formula above, Vt(𝜉0) is a 3 dimensional vector. For every component in Σ,
there is one corresponding component in Vt(𝜉0), which measures the change of the
covariance of the series due to the shock at time t. In our paper, we focus on the change of
the Chinese indexes, SHCOMP and SIASA, influenced by shocks in western markets.
Therefore we set ξ0 to be (0,1) and pick up the first element in Vt(𝜉0), Vt,1(𝜉0) as the
Chinese response. To make clear how the responses differ, we calculate the differences
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between Vt,1(𝜉0) and V0,1(𝜉0), DVt,1(𝜉0), to study the situation of Chinese series in the
whole system. Generally, in the stationary system, DVt,1(𝜉0) will go to zero after a period
of shocks.
3.4 Estimation
We adopt a maximum likelihood framework to estimate the BEKK models. The
log-likelihood function of the joint distribution is calculated from the conditional
multi-normal distribution. Denote 𝐿𝑡 as the log-likelihood of observation at time 𝑡, 𝑛 as
the number of series and 𝐿 as the sum of log-likelihood of all time. The calculation is given
as:
𝐿 = ∑𝐿𝑡
𝑇
𝑡=1
𝐿𝑡 =𝑛
2ln(2𝜋) −
1
2ln|𝐻𝑡| −
1
2𝜖𝑡′𝐻𝑡
−1𝜖𝑡
Usually, by the BHHH4 method we use first order and second order derivatives to
iterate and get the estimated parameters after achieving a required tolerance. However, in
the BEKK model, the number of parameters is large and all parameters are in a matrix form
so the likelihood function may be non-differentiable or even differentiable derivatives are
hard to be calculated. Another numerical procedure, the Nelder-Mead algorithm5, as a
modification of BHHH, works reasonably well for non-differentiable and complicated
functions by approximating the derivatives using function values. To search for the optimal
value efficiently, a proper initial value is essential. The estimation can be divided into two
steps. The first step is to find out all the significant univariate models. The second step is to
extend the univariate set to a multivariate model through the Nelder-Mead algorithm.
Therefore, the initial parameter matrices we use are all diagonal matrices which come
from univariate models.
In the larger order BEKK model, the extended form of the matrix equation, which
means the transformation from BEKK to 𝑣𝑒𝑐ℎ form, is hard to specify. Thus the
interpretations of coefficients are not easy to understand. In this case, we apply the
4 BHHH, introduced by Berndt, B. Hall, R. Hall and J. Hausman in “Estimation and Inference in Nonlinear Structural Models”
(1974), is a non-linear optimization algorithm similar to the Gauss-Newton algorithm. 5 The Nelder-Mead method was first proposed by John Nelder & Roger Mead (1965) in “A simplex method for function
minimization”. It is a nonlinear technique for minimizing an objective function in a many-dimensional space.
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conditional covariance to measure the linkage between volatilities.
4. Empirical Results
In this section, we use the GARCH and BEKK models to model our stock returns data.
The approaches by the GARCH method to Period I&II are not efficient in this case.
Significant models and reasonable diagnostics cannot be given under the present study.
Therefore we concentrate on the longest Period III (Depression Period) to continue
searching for the existence of time-varying returns and volatilities in each series as well as
market linkages. Then we analyze the fitness of models through comparison between the
moving averages volatility and estimated ones. We try different approaches to the model,
step by step, from univariate to multivariate. In Section 4.1, we report the ARCH effects in
the stock return series and estimate 4 univariate GARCH models to do the preliminary
fitting. Then we improve the univariate model set to a combined multivariate BEKK model
in Section 4.2. In the next Section 4.3, we perform some diagnostic checking on the BEKK
model. Then in Section 4.4, we report the market linkages demonstrated by time-varying
conditional correlations. In Section 4.5, we introduce the VIRF function to distinguish the
strength of linkages.
Table 2. Estimated coefficients for univariate models
SHCOMP SIASA FSTE100 SP500
𝜇 0.870 *
(0.368)
1.043 **
(0.340)
𝜔 5.782*
(2.747)
10.078 *
(4.926)
3.048 **
(1.437)
2.538 **
(0.876)
𝛼 0.050 ***
(0.013)
0.048 ***
(0.012)
0.101 ***
(0.022)
0.123 ***
(0.019)
𝛽 0.922 ***
(0.021)
0.919 ***
(0.024)
0.882 ***
(0.024)
0.866 ***
(0.017)
AIC 8.236 8.569 7.858 7.924
ARCH-LM Stat 11.340
(p=0.000)
5.276
(p=0.021)
9.314
(p=0.002)
76.342
(p=0.000)
Note: Values in parentheses are standard errors. ***, **and * represent significant levels at 0.1%, 1%
and 5%.
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Figure 3. Comparison between 7-days moving average and model estimates
4.1. The evidence of ARCH effects
As often done in previous studies, our first approach is to fit our series by
AR(1)-GARCH(1,1) and get some estimations. The estimates of the autoregressive
coefficients are not significantly different from zero and are therefore removed from the
mean equation. To guarantee the significance of the model, we remove the constant terms
in the mean equation of the Chinese series. Table 2 shows the results of the estimation
without autoregressive effects in the mean equation including diagnostic statistics. From
the table, except for two autoregressive constants in SHCOMP and FSTE100, all other
coefficients are significant, implying a positive model fitting. The ARCH-LM tests for
examining the existence of ARCH effects also prove the reasonable ARCH coefficients.
Actually in a long-term financial returns case, ARCH effects on volatility exist widely.
Another point in Table 2 is that we could find that returns in a similar economic
environment (SHCOMP&SIASA, FSTE100&SP500) have similar scale of coefficients in the
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variance equation. The larger 𝛽 and smaller 𝛼 in the Chinese cases means the changing
volatility is more possibly caused by cumulative effects and keeps more stable in the
long-term financial development.
Figure 3 shows the comparison of volatilities between moving averages and model
estimates. I choose the lag as 7 in the moving average to describe weekly volatilities, which
could somehow express the characters of conditional volatilities. Through each row in
Figure 3, we find that most of the sharpest peaks from the moving average are removed in
the model estimates while the overall scale of volatilities remains the same. As we
mentioned in Data Section, the volatilities in theChinese markets vary more dramatically
than in the western markets. Almost in the whole period, the Chinese volatilities keep a
heavy fluctuation, especially within 2007 to 2009. Compared to the Chinese markets, the
British and American markets only have obvious changes in the second half year of 2008.
The reason is known as the world-wide financial crisis. The more fluctuation in Chinese
markets may be a result of the domestic multi-macroeconomic regulation and control.
4.2 The evidence of market linkages
The VAR-BEKK models are estimated by the maximum log-likelihood method and the
results are reported in Table 3 and Table 4.
Table 3 reflects models associated with SHCOMP. The elements of the three matrices,
γij, aij and gij, represent the impact from series i to series j of returns and volatilities.
Since we have removed the autoregressive terms, the first two rows of coefficients, 𝛾12
and 𝛾21, capture the instant cross-market effects in the mean equation. γ12 captures the
instant impact from SHCOMP to FTSE100/S&P500 and γ21 captures the instant impact
from FTSE100/S&P500 to SHCOMP. The mutual interaction exists due to the significant
r12 and r21. The positive γ12 and γ21 show the positive synergies between SHCOMP and
western exchange. That is an increase in western indices/SHCOMP also promotes an
increase in SHCOMP/western indexes, and a decrease in western indexes/SHCOMP will
lead a decrease in SHCOMP/western indices. The following 𝑎1𝑖 and 𝑎2𝑖 measure the
ARCH effect in the variance equation. In model 1 and model 2, the significant a1i means
SHCOMP has a reasonable impact on the western stock volatilities. On the opposite, not
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both British and American indexes have a significant effect on SHCOMP’s volatilities. The
impact from FTSE100, measured by a211 , is insignificant but the impact from S&P500,
measured by a212 , is significant. At last, the rest of the 𝐺 matrix, consisting of 𝑔1𝑖 and 𝑔2𝑖,
are all statistically significant, indicating volatility spillovers between FTSE100/S&P500
and SHCOMP. From the two models, we could conclude that the American market has
multiple impacts on SHCOMP.
Table 3. Estimated coefficients for bivariate VAR-BEKK model focus on SHCOMP
Model 1: SHCOMP & FTSE100 Model 2: SHCOMP & SP500
SHCOMP FTSE100 SHCOMP SP500
𝛾1𝑖 0.055*** 0.0844***
𝛾2𝑖 0.044*** 0.088***
𝑎1𝑖 -0.117* 0.188*** 0.054* -0.059***
𝑎2𝑖 0.025 0.166*** -0.009*** 0.368***
𝑔1𝑖 0.990*** 0.018*** 0.996*** 0.007***
𝑔2𝑖 0.013*** 0.954*** 0.011*** 0.926***
LBQ(10) 8.895 8.150 7.497 10.433
Probability 0.542 0.614 0.677 0.403
LBQ(20) 21.137 19.725 21.315 17.408
Probability 0.389 0.475 0.378 0.626
LBQs(10) 18.273 52.529 52.566 18.067
Probability 0.050 0.000 0.000 0.054
LBQs(20) 28.950 60.077 72.236 25.303
Probability 0.088 0.000 0.000 0.190
LLR -10597.49 -10638.09
Note: In the following context, we use an upper index γij1 and γij
2 to distinguish γij in Model 1
and Model 2, as well as aij and gij. LBQ(10) and LBQ(20) represent the Ljung-Box Q-statistic of
standardized residuals with a lag equals to 10 and 20. LBQs(10) and LBQs(20) represent the
Ljung-Box Q-statistic of squared standardized residuals with a lag of 10 and 20. These notes are
also available in Table 4.
The Ljung-Box Q statistics for the 10th and 20th orders in the standardized residuals of
two bivariate models indicate an appropriate specification of the mean equation. However,
when we examine the Ljung-Box Q statistics for the 10th and 20th orders in squared
standardized residuals, we cannot always get independent squared residuals. In the
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SHCOMP&FTSE100 model, the squared residuals of the conditional variance of FTSE100
fail to be random. So is SHCOMP in the SHCOMP&SP500 model. Since we strictly take the
order of variance equation from the univariate model and ensure a right selection for the
univariate model, we cannot deny the model to be a good approach. And we insist to study
the linkages from this BEKK model.
Table 4. Estimated coefficients for bivariate VAR-BEKK model focus on SIASA
Model 3: SIASA & FTSE100 Model 4: SIASA & SP500
SIASA FTSE100 SIASA SP500
𝛾1𝑖 0.206*** 0.351
𝛾2𝑖 0.035*** 0.000***
𝑎1𝑖 -0.103* 0.148* 0.003* 0.147***
𝑎2𝑖 0.007*** 0.171*** -0.033*** 0.251***
𝑔1𝑖 0.992*** 0.009 0.995*** -0.005***
𝑔2𝑖 0.022*** 0.959*** 0.029*** 0.941***
LBQ(10) 11.216 7.811 11.216 7.811
Probability 0.340 0.647 0.340 0.647
LBQ(20) 22.560 16.427 22.560 16.427
Probability 0.310 0.689 0.310 0.689
LBQs(10) 21.712 59.346 21.713 59.346
Probability 0.016 0.000 0.016 0.000
LBQs(20) 59.346 70.012 39.504 70.012
Probability 0.000 0.000 0.005 0.000
LLR -10895.70 -10930.87
Table 4 shows the model estimation associated with SIASA and western markets.
Similarly as Table 3, most coefficients are statistically significant. One difference is in
model 4, where γ124 , which means the spillovers from SIASA to S&P500 in the mean
equation, is insignificant. Another point is, in model 3, where g123 is also insignificant,
indicating the absence of an unreasonable volatility spillover from SIASA to FTSE100.
These two insignificant parameters show that SIASA has a weaker external influence than
SHCOMP, but is influenced more from western markets.
The Ljung-Box diagnostic on the residuals of the two bivariate models supply a good
support for the residuals independence. And the Ljung-Box diagnostic on the squared
residuals shows the same dependent results as above.
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4.3 Diagnostic checking on the bivariate VAR-BEKK models
The stationary condition of the BEKK(1,1) model is that all the eigenvalues of the
matrix A1 ⊗𝐴1 + 𝐺1 ⊗𝐺1, where ⊗ is the Kronecker product of two matrixes, are
smaller than one in modulus.6 Table 5 shows the eigenvalues of each model. It is clear that
the modulus of all the eigenvalues is smaller than 1, guaranteeing stationarity of all the
models.
Table 5. Eigenvalues of bivariate BEKK models
Eigenvalues Model 1 Model 2 Model 3 Model 4
𝜆1 0.9972 0.9969 0.9992 0.9969
𝜆2 0.9687 0.9878 0.9721 0.9631
𝜆3 0.9201 0.9458 0.9324 0.9420
𝜆4 0.8954 0.9415 0.9072 0.9104
In the previous section, we reported the results of the bivariate BEKK models and
analyzed the residuals. Now we carry out the likelihood ratio test to examine if the series
are appropriately constructed in the models. We begin by checking the variance equation.
We introduce the restrictions that all the off-diagonal parameters, the coefficients in the A
and G matrices are zeros. These restrictions limit the interdependence of each series. The
bivariate BEKK model with such restrictions, diagonal BEKK model, is equivalent to two
univariate GARCH models. If we reject the validity of the restrictions, the combination of
univariate models is not appropriate. The likelihood ratio test 1 statistics reported in Table
6 are all quite large, rejecting the null hypothesis that the off-diagonal parameters are
zeros.
Recall that the coefficient we estimated, the insignificant r12 stands for the linkage of
returns between Chinese and western markets. To examine whether this linkage exists or
not, we test the log-likelihood ratio between the present model and the model without this
linkage. The likelihood ratio test 2 shows that the null hypothesis for the zero linkage is
6 See proof in section 11.3.1 of “GARCH Models—Structure, Statistical Inference and Financial Applications” by Christian
Francq and Jean-Michel Zakoian.
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rejected statistically. That is, though we get insignificant coefficients, they still make sense
anyway.
The likelihood ratio test 3 is to examine the cross-market spillovers in volatility, both
from lagged volatility and lagged returns. The results prove the conjecture which the
Chinese stock volatility is significantly affected by British and American markets. The cross
spillovers in volatility strengthen the global linkage. When the shock happens in a market,
others will react, too.
A summary test for the cross spillovers is given by likelihood ratio test 4. All the
cross-markets effects on Chinese markets are set to be zero in the null hypothesis. And the
statistics suggest that the cross spillovers should be included in the model, implying a
widely related global market.
Table 6. Restriction tests concerning the bivariate BEKK model
Log-likelihood ratio test
statistics
SHCOMP
&FTSE100
SHCOMP
&SP500
SIASA
&FTSE100
SIASA
&SP500
1.Diagonal BEKK,
H0:𝑎𝑖𝑗 = 𝑔𝑖𝑗 = 0 df = 4
129.56 16.74 39.44 97.44
2.Instant cross market Impact
from FTSE100 and S&P500,
H0: γ12=0 df=1
58.9
63.62
34.68
47.2
3.Cross spillovers in volatility
from FTSE100 and S&P500,
H0: 𝑎12 = 𝑔12 = 0 df=2
50.64
12.06
6.02
12.14
4.Cross spillovers in both
mean and variance equation,
H0: γ12 = 𝑎12 = 𝑔120 df=3
94.88
74.68
51.66
57.26
4.4 Market linkage demonstration
Under our study of the depression period, we find that Chinese markets, both in
Shanghai and Shenzhen, are linked in terms of returns and volatility with western
developed markets in Britain and U.S. Figure 4 is the illustration of the conditional
correlation of the series pairs in our models. The figure supports our evidence of market
linkages nowadays. The fluctuations of dynamic correlations follow a similar trend in the
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whole period, showing a co-movement of global stock markets. And from Figure 4, we can
also observe that the correlation is not so large, mostly between -0.1 to 0.3, except for
some individual intervals. Thus we know the linkages exist but the influences are limited. It
is true in the economic environment since the domestic economy is mainly affected by
domestic factors. The external impacts function on the economy could not threaten the
basement.
Figure 4. Estimated conditional correlations during November 2008 and March 2012
4.5 VIRF for the estimated BEKK models
In the following, we refer to the VIRF plots given in Figure 5 for the estimation results
of BEKK models. In order to visualize the function, we produce a shock in the bivariate
model. The shock in this case is designed as ξ0,1, which means there’s one innovation in
the second series but the first series keeps the same. In the stationary situation, the shock
will be consumed by the long-term progress in the model. Thus, the shock in the volatility
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will disappear after some periods. We manage this change in the volatility, describing it as
the cross-series interferences.
Figure 5. VIRF on Chinese stock after an external shock
In Figure 5, the red dashed line represents for the shock from FTSE100 to SHCOMP,
and the red solid line is the shock from S&P500 to SHCOMP. The blue dashed line and blue
solid line are the shock from FTSE100 to SIASA and the shock from S&P500 to SIASA
respectively. It is obvious that when we introduce ξ0,1 to the model, the volatility of first
series will jump to a high level immediately. It may continue increasing for a period or just
fall down to regain balance.
The first point we find in Figure 5 is that the shock comes from S&P500 is stronger
than that from FTSE100, i.e. solid line is always beyond dashed line. The peak of the
reaction of SHCOMP from S&P500 is around 330 but the reaction of SHCOMP from
FTSE100 is as small as 5.65. It is the same in the case of SIASA. The peak of the reaction of
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SIASA from FTSE100 is around 65.51 on the contrast of a peak about 524.54 from S&P500.
This is an evidence that the U.S. stock market has more influence in Chinese markets,
compared to the British stock market.
The second point is that SHCOMP keeps more stability with the western shocks than
SIASA. It is clear in the figure that the blue solid line is beyond the red solid line and the
blue dashed line is also beyond the red dashed line. In the meanwhile, tails of red lines are
on the left of blue lines, implying that it takes a longer time to regain balance in SIASA than
in SHCOMP. The phenomenon may come from the fact that small companies do not have
enough anti-risk ability. When the external depression strikes the domestic economics, the
small companies are more likely to face the problems of losing orders and get trouble with
capital turnover. However the large companies would have their business distributed to
multiple operations in case of centralized risk. Even if the shock happens, large companies
could recover soon due to its leading market position and rich capital reserve. Therefore
these advantages help large companies to keep their system more stable and the
disadvantages of small size of the listed companies would undertake more risk in
economic globalization.
The last point which cannot be ignored is that the sensitivities of SHCOMP and SIASA
to the shock from FTSE100 or S&P500 are different even SHCOMP takes more stability to
resist the shocks. In Figure 5, at 𝑡 = 91 and 𝑡 = 132, the red lines reach to their peaks,
which means it takes 91 and 132 days to result in the biggest shocks from FTSE100 and
S&P500 to SCHOMP. Therefore the approximate average shock rates are 5.65/91 = 0.06
and 330/132 = 2.5 which the latter one is nearly 42 times of the former one. While for
shocks from FTSE100 and S&P500 to SIASA, it takes 145 and 150 days to reach to the top.
The average shock rates of SIASA are calculated as 65.51/145 = 0.45 and 524.54/150 =
3.5, which the rate from S&P500 is only 8 times of that from FTSE100. From the high
level shock rates of SIASA, it is obvious that the SIASA is more sensitive to the external
spillovers. On the other hand, spillovers from S&P500 are keeping high level impacts both
in SIASA and in SHCOMP while spillovers from FTSE100 show different influences in the
two Chinese markets. In a time of no longer than 150 days, the impact would reach to its
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peak. In case of the bankruptcy due to the heaviest strike, all the companies should try to
guarantee a safe cash flow during this period.
5. Conclusion
This study investigates the linkages between Chinese stock exchange markets and
global developed markets. Under the BEKK approach to daily stock returns from November
2008 to March 2012, we find the evidence supporting the existence of linkages. The
spillovers of returns and volatilities, from western markets to Chinese markets, are found
in the models. On the opposite, the stock spillovers from China to western countries are
also significant, which indicates that Chinese stock markets are developing from the
emerging markets and Chinese financial economics is an important part of the world
financial system.
We illustrate the dynamic linkages through dynamic conditional correlations. From
our illustration linkages are observed neither too weak nor too strong. Thus the volatility
impulse response functions are used to distinguish the different linkages under systematic
shocks. The VIRF explains the co-movement of global stock interaction. The stock exchange
in Shanghai, representing the large listed companies in China, is less sensible to the
western shocks compared to the stock exchange in Shenzhen, which represents mostly the
medium and small companies. The shocks from U.S. markets are also proved to be stronger
than those from European markets. Though Europe is the largest trading partner, in the
financial aspect, U.S. still takes the leading influence on China.
Our study suggests an anti-risk method for Chinese companies. Through our VIRF
analysis, since the power of spillovers from U.S and Europe are different, larger companies
should care more risks from U.S. while small companies should distribute their business
associated with Europe and U.S. in appropriate proportions to minimize the shock from
both regions. In the meanwhile, they should prepare for the continuous shocks within a
specific period.
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