Master Essay

Department of Statistics

An Empirical Study on Stock Exchange Linkages between Chinese and Western Markets

Heng Wang

Supervisor: Anders Ågren

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: arsewang@gmail.com

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.

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

2

(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.

8

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

16

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.

19

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

20

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

21

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

22

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.

24

References

1. Ågren, M., 2006. Does oil price uncertainty transmit to stock markets? Working paper in

Department of Economics of Uppsala University, 2006:23.

2. Bollerslev, T., 1986. Generalized Autoregressive Conditional Heteroskedasticity. Journal

of Econometrics, 31: 307-327.

3. Bollerslev, T., Engle, R., Wooldrige, J., 1988. A capital asset pricing model with

time-varying covariances. Journal of Political Economy, 96: 116-131.

4. Brooks, RD., Ragunathan, V., 2003. Returns and volatility on the Chinese stock

markets. Applied Financial Economics, 13: 747–52.

5. Chow, R., Lin, J., Wu, C., 1999. Modeling the Taiwan stock market and international

linkages. Pacific Economic Review, 4: 305-320.

6. Chow, G., Lawler, C., 2003. A time series analysis of the Shanghai and New York stock

price indices. Annals of Economics and Finance, 4:17-35.

7. Diekman, K., 2011. Are there Spillover Effects from Hong Kong and the United States to

Chinese Stock Markets? Working Papers with number 89 in Institute of Empirical

Economic Research, 2011-12-12.

8. Engle, R., 1982. Autoregressive conditional heteroscedasticity with estimates of the

variance of United Kingdom inflation. Econometrica, 50: 987-1007.

9. Engle, R., Kroner, K., 1995. Multivariate simultaneous generalized ARCH. Econometric

Theory, 11: 122-150.

10. Francq, C., Zakoian, JM., 2010. GARCH Models: Structure, Statistical Inference and

Financial Applications. Wiley, New York.

11. Hafner, C., Herwartz, H., 1998. Volatility impulse functions for multivariate GARCH

models. CORE Discussion Papers from Universite catholique de Louvain, Center for

Operations Research and Econometrics, 2001-09-01.

12. Hafner, C., Herwartz, H., 2006. Volatility impulse responses for multivariate GARCH

models: an exchange rate illustration. Journal of International Money and Finance, 25:

719-740.

25

13. Haroutounian, M., Price, S., 2001. Volatility in the transition markets of central Europe.

Applied Financial Economics, 11: 93-105.

14. Kim, Y and Shin, J., 2000. Interactions among China-related stocks. Asia-Pacific

Financial Markets, 7: 97–115.

15. Li, H., 2007. International linkages of the Chinese stock exchanges: a multivariate

GARCH analysis. Applied Financial Economics, 17: 285-297.

16. Li, H., 2011. China’s stock market reforms and its international stock market linkages.

BMRC-QASS conference on Macro and Financial Economics, 2011-05-24.

17. Li, H., Majerowska E., 2007. Testing stock market linkages for Poland and Hungary: A

multivariate GARCH approach. Research in International Business and Finance, 22:

247-266.

18. Pen., LY., Sevi, B., 2009. Volatility transmission and volatility impulse response functions

in European electricity forward markets. Energy Economics, 32: 758-770.

19. Wang, P., Liu, A., Wang, P., 2004. Returns and risk interactions in Chinese stock

markets. Journal of International Financial Markets, Institutions and Money, 14: 367–84.

20. Worthington, A., Higgs, H., 2004. Transmission of equity returns and volatility in Asian

developed and emerging markets: a multivariate GARCH analysis. International Journal

of Finance and Economics, 9: 71-80.

21. Xiang, Z., Weijin, Z., Jie, Z., 2011. Volatility spillovers between the Chinese and world

equity markets. Pacific-Basin Finance Journal, 20: 247-270.

22. Yang, J., Hsiao, C., Li, Q., Wang, Z., 2006. The emerging market crisis and stock market

linkages: future evidence. Journal of Applied Econometrics, 21: 727-744.

23. Zhou, X., Zhang, W., Zhang, J., 2011. Volatility spillovers between the Chinese and world

equity markets. Pacific-Basin Finance Journal, 20: 247-270.