Measuring financial market risk contagion using dynamic MRS-Copula models: The case of Chinese and other international stock markets

Economic Modelling 51 (2015) 657–671

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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod

Measuring financial market risk contagion using dynamic MRS-Copula models: The case of Chinese and other international stock markets Luo Changqing a,b, Xie Chi a,⁎, Yu Cong a, Xu Yan a a b

College of Business Administration, Hunan University, Lushan Nan Road, No. 2, Changsha, 410082 Hunan Province, PR China Finance School, Hunan University of Commerce, Yuelu Avenue, No. 569, Changsha, 410205 Hunan Province, PR China

a r t i c l e

i n f o

Article history: Accepted 13 September 2015 Available online xxxx Editor: Paresh Narayan Keywords: Risk contagion Lower tail dependency Dynamic Markov Regime Switching Copula Dynamic correlation

a b s t r a c t Considering the asymmetric dependency structure and regime switching process, we construct the dynamic Markov Regime Switching Copula (MRS-Copula) models to measure the financial risk contagion. The dynamic MRS-Copula models consist of the marginal model and dynamic MRS Rotated-Gumbel function, and they are examined by the goodness-of-fit test method. Using the dynamic MRS-Copula models, we calculate the daily lower tail dependency by adopting the international stock market index data from January 1997 to June 2015, and provide evidence of financial risk contagion effects between Chinese stock market and other international stock markets after the reform of the RMB exchange rate system in China, and this is particular the case after in the U.S. subprime mortgage crisis and the European debt crisis. When conducting robust tests with weekly and monthly data, the empirical result basically holds. As for the financial risk contagion channels in Chinese stock market, the fundamental economic linkages play a more important role than liquidity, information and other factors. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The modern international stock markets are highly interdependent with each other. Despite the integration of global financial markets facilitating portfolio management, the financial risk or crisis could spread from one market to another in a very short time. So it is important to measure the risk contagion degree of the financial market for both academicians and practitioners. Scholars have used different definitions, such as comovement, interdependence, volatility spillover and risk contagion to study the risk transmission of the financial markets. King and Wadhwani (1990), Baig and Goldfajn (1999), Barberis et al. (2005), Huyghebaert and Wang (2010), Hwang et al. (2013), and Aloui and Hkiri (2014) focus on the co-movement of the financial markets. Awokuse et al. (2009), Zhu et al. (2014), and He et al. (2015) study the financial market interdependence, and stress on whether a country's financial market is influenced by other international financial markets. While Mensi et al. (2013) and Alotaibi and Mishra (2015) detect the volatility spillover effects to study the financial risk contagion. Dornbusch et al. (2000), Forbes and Rigobon (2002), and Morales and Andreosso-O'Callaghan (2012) propose that contagion is a significant increase in cross-market linkages after a shock to an individual country (or group of countries). Durante and Foscolo (2013) consider contagion as a significant increase in co-movements of prices and quantities across markets, conditional ⁎ Corresponding author. E-mail addresses: [email protected] (L. Changqing), [email protected] (X. Chi), [email protected] (Y. Cong), [email protected] (X. Yan).

http://dx.doi.org/10.1016/j.econmod.2015.09.021 0264-9993/© 2015 Elsevier B.V. All rights reserved.

on a crisis occurring in one market or group of markets. Although the definitions of co-movement, interdependence and volatility spillover are suitable to investigate the financial market interaction, it is difficult to identify the financial crisis period and the time when the contagion occurs. However, the definition based on correlation coefficients can solve this problem. By judging the decline and increase of correlation coefficients, the risk contagion period can be clearly identified. Thus, in our paper, we take the correlation coefficients among different markets as the basic indicators to detect the financial risk contagion effects, and the financial risk contagion occurs when the correlation between stock market significantly increases. Previous works have applied multivariate GARCH models, conditional probability test and co-integration test methods to obtain the linear correlation or dependency structure of different markets (Bae et al., 2003; Bekaert and Wu, 2000; Hong et al., 2004). Their empirical results show that the linear correlation coefficient in crisis period is greater than that of normal period. Longin and Solnik (1995) study the dependency level changes among international equity markets during 1960– 1990, and find the empirical evidence of crisis contagion among international stock markets. Hong et al. (2004) analyzes the extreme risk spillover effects, reporting that while the risk spillover effects are significant it existed among Chinese A shares market and other stock markets. Sibel's (2012) findings support the evidence of contagion during U.S. subprime crisis for most of the developed and emerging countries. In the research of Wang (2013), the evidence is found that Chinese stock market brings more contagion risk to the Vietnamese market when compared to the U.S. stock market by using a bivariate EGARCH model of dynamic conditional correlation coefficients.

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With the development of the statistic methods, scholars gradually find the drawbacks of the linear correlation or dependency structure calculated by the traditional models. Firstly, most of the linear correlations are not time-varying. Secondly, it is difficult to investigate the asymmetry dependency of different markets by using the linear correlation. Thirdly, the linear correlation has poor ability to capture the extreme downside risk spillover. To deal with the above limitations, scholars develop the Copula methods to describe the dynamic and asymmetric dependency structures. As Copula isolates the dependence pattern from the marginal distributions, it is easy to acquire the different dependency structure and contagion effects (Ang and Chen, 2002; Chollete et al., 2009; Das and Uppal, 2004; Longin and Solnik, 2001; Patton, 2006a, 2006b; Peng and Ng, 2012; Wang and Liu, 2011; Wen et al., 2012; Wu and Zhang, 2010). By applying Copula methods, previous works figure out the asymmetric dependency of the international financial markets, which indicates that dependency between the stock markets in the crisis period is higher than that of the normal period. Thus, in our paper, we apply the tail dependency coefficient instead of linear correlation to study the financial risk contagion between Chinese stock market and other international stock markets. Meanwhile, some scholars (Dueker, 1997; Hamilton and Susmel, 1994; Hess, 2003; Lamoureux and Lastrapes, 1990) recognize the unstable state of the stock market, and argue that the Markov Regime Switching (MRS) model is more precise in modeling time variation of stock market return distribution. More recently, Guo et al. (2011) investigate the contagion among the stock market and other financial markets. Miao et al. (2013) insist that correlation model with regimeswitching method can explicitly point out structure changes for time series variables. Lin et al. (2014) find that the Markov Regime Switching models can explain the dynamics of S&P 500 stock index. Considering the asymmetric dependency, tail dependency and regime switching process, we construct dynamic Markov Regime Switching Copula models to measure the financial risk contagion. There are several advantages to adopt this approach. First, the crisis and calm period of the financial market is objectively identified by judging the smooth probability. Second, by using this approach, it is more efficient to capture the dynamic and asymmetric dependency structures among different markets. Third, instead of the normal distribution, skewed-t and generalized error distribution are applied to model the fat-tailed marginal distribution. The above advantages of the dynamic MRS-Copula models allow us to investigate the financial risk contagion in a more precise way. Taking the Chinese stock market and other international stock markets as the empirical samples, our results show that the dynamic MRS-Copula models can clearly distinguish the different states of the risk contagion, and the lower tail dependency gradually increases after the reform of RMB exchange rate system, and dramatically rises after U.S. sub-prime mortgage crisis in 2007. Later, the lower tail dependence stays at a high level until 2013, and this high level period also covers the European debt crisis. Therefore, our empirical study provides evidence of financial risk contagion between Chinese stock market and other international stock markets. The rest of the paper is organized as follows. In Section 2, we build the dynamic MRS-Copula models, and we specify the marginal model, the Copula function, model test method and the indicator of financial risk contagion. In Section 3, we report the empirical results. In Section 4, we conduct robustness test by using monthly and weekly return data. Finally, conclusions are stated in Section 5. 2. Dynamic MRS-Copula model 2.1. Marginal model specification Traditional GARCH models are estimated under the conditional normality assumption to measure the correlation of time series data. However, this assumption is rejected by the existing literature (Longin

and Solnik, 2001; Hong et al., 2004; Patton, 2006b; Peng and Ng, 2012, etc.). By contrast, the use of Copula function allows us to consider the marginal distributions and the dependence structure both separately and simultaneously, thus the non-normality of the joint distribution and the dependency structure can be modeled more appropriately. For modeling the marginal distribution, it is generally assumed that financial asset volatility is time-varying and clustered, and TGARCH model can well describe these volatility characteristics of financial assets. Therefore, we apply the AR(p)–TGARCH(p, q) as the marginal models of asset returns to construct the Copula models. According to Nelson (1991), we suppose that the standard residual complies with the skewed-t distribution or generalized error distribution (GED). The model is given by:

rt ¼ μ þ

p X

ρi r t−i þ ε t

ð1Þ

i¼1

ht ¼ c þ

p X

φi ε2t−i þ

i¼1

q X

δ j ht− j þ γI ðε t b0Þε 2t :

ð2Þ

j¼1

pffiffiffiffiffi Here, zt ¼ εt = ht  skewed−tðυ; ηÞ or zt ~ GED(ς), εt is the residual of the return, and ht is the conditional variance. zt refers to independent and identically distributed standard residual that complies with skewed-t distribution or generalized error distribution. kt − 1 = 1 when εt − 1 is negative, otherwise kt − 1 = 0. The density function of the skewed-t distribution is: 8  2 !−υþ1=2 > > 1 bz þ a a > > ; zb− > < bc 1 þ υ−2 1−η b skewed−t ðzjυ; ηÞ ¼ :  2 !−υþ1=2 > > bz þ a a > > bc 1 þ 1 ; z ≥ − > : υ−2 1−η b

ð3Þ

The values of a, b, and c are defined as: a ¼ 4η

υ−2 Γ ðυ þ 1=2Þ ; b ¼ 1 þ 3η2 −a2 ; c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi υ−1 πðυ−2ÞΓ ðυ=2Þ

where υ is the freedom degree parameter and η is the asymmetry parameter. These two parameters are restricted to 4 b υ b 30 and −1 b η b 1. A negative η indicates that the distribution is left skewed, which suggests that there is a greater probability of negative returns. Thus, the asymmetric and fat-tailed features of the asset returns can be well defined by skewed-t distribution. If the distribution of the asset returns shows a significant asymmetry characteristic, then the assumption that the standardized residuals obey the skewed-t distribution is appropriate, otherwise, the generalized error distribution is applied in the process of modeling. The generalized error distribution is advantageous on adjusting the parameter ς to fit different distribution forms. The density function of GED can be expressed as:  # 1  ς ς  exp − z=λj 2   f ðz; ς Þ ¼ ς þ1 1 λ  2½ ς   Γ "

ð4Þ

ς

ς

where λ = [2−2/ ⋅ Γ(1/ς)/Γ(3/ς)]0.5, Г(·) is the gamma function and ς is a scale parameter, or degrees of freedom to be estimated endogenously. For ς = 2, the GED yields the normal distribution, while for yields the Laplace or, double exponential distribution.

ς = 1 it

L. Changqing et al. / Economic Modelling 51 (2015) 657–671

2.2. The dynamic MRS Gumbel–Clayton Copula function A Copula function describes a flexible dependence structure between two random variables. According to Sklar's theorem (Sklar, 1959), for a joint distribution function, the marginal distributions and the dependence structure represented by a Copula function can be separated. Let r1,t and r2,t, two random variables, denote two different asset returns at period t. Then we assume that the conditional cumulative distribution functions (CDF) of z1,t and z2,t are F1(z1,t|Ωt − 1) and F2(z2,t|Ωt − 1), where Ωt − 1 denotes the information set at period t − 1. The conditional Copula function is obtained as follows:   Φt z1;t ;z2;tjΩt−1 ¼ C t ðut ; vt jΩt−1 Þ  ¼ C t F 1 z1;t jΩt−1 ; F 2 z2;t jΩt−1 Ωt−1

ð5Þ

where ut = F1(z1,t|Ωt − 1) and vt = F2(z2,t|Ωt − 1) follow (0,1) uniform distribution. Assuming all CDFs are differentiable, the bivariate conditional density function of z1,t and z2,t is given by:         φt z1;t ; z2;t jΩt−1 ¼ ct F 1 z1;t jΩt−1 ; F 2 z2;t jΩt−1 jΩt−1  f 1 z1;t jΩt−1  f 2 z2;t jΩt−1 ;

ð6Þ

where ct(ut,vt|Ωt − 1) = ∂2Ct(ut,vt|Ωt − 1)/∂ut ∂vt is the conditional Copula density function. f1(z1,t|Ωt − 1) is the conditional density function of z1,t, and f2(z2,t|Ωt − 1) is the conditional density function of z2,t. Let the parameters in the conditional density functions for ct(·), f1(·) and f2(·) be denoted as θc, θ1 and θ2, respectively. At time t, by taking the logarithm of Eq. (6), there comes the log-likelihood function:     logφt z1;t ; z2;t jΩt−1 ¼ logct ðut ; vt jΩt−1 Þ þ logf 1 z1;t jΩt−1 þ logf 2 z2;t jΩt−1 :

ð7Þ

For a simpler representation, Eq. (7) can be written as: LðθÞ ¼ Lc ðθc Þ þ L1 ðθ1 Þ þ L2 ðθ2 Þ:

ð8Þ

Parameter sets θ = (θc, θ1, θ2) and Lk(·) represent the log-likelihood function of Copula (k = c), asset 1 (k = 1) and asset 2 (k = 2) densities. It could be difficult to realize the optimization when the dimension of the estimated model is high, in this backdrop, Joe and Xu (1996) propose a two-stage estimation procedure which is called the inference functions for the margins (IFMs). This approach allows us to estimate the marginal densities and the Copula density separately. Under the IFM framework, the estimation process can be divided into two stages. In the first stage, the parameters of the marginal densities are estimated by: T X   ^θ1 ≡ arg max logf 1 z1;t jΩt−1 ; θ1

ð9Þ

t¼1 T X   ^θ2 ≡ arg max logf 2 z2;t jΩt−1 ; θ2 :

ð10Þ

t¼1

In the second stage, given the estimation of Eqs. (9) and (10), the Copula parameters are estimated by:

the lower and upper tails have been taken into account. Thus, we focus on the lower tail dependency since it can better describe the risk contagion. In our paper, two types of Copula are employed to construct the dynamic Markov Regime Switching Rotated Gumbel–Clayton Copula function. The first Copula is the Rotated Gumbel Copula, and it is also known as the Gumbel survival Copula. The relationship between Copula function and Copula survival function is: ^ ðu; vÞ ¼ u þ v−1 þ C ð1−u; 1−vÞ C

In the first stage, we can generate the estimated sequences of values, ut and vt, by probability integral transformation according to the parameters of the AR(p)–TGARCH(p, q) models. In the second stage, these sequences of values are successively employed to estimate the Copula dependence structure between z1,t and z2,t. Chen et al. (2014) argue that the estimated dependence structure is an unreliable measurement of the financial risk contagion when both

ð12Þ

where Ĉ(·,·) is the Copula survival function. Therefore, the Rotated Gumbel Copula can be expressed as G ð13Þ C RG t ðut ; vt jκ t Þ ¼ ut þ vt −1 þ C t ð1−ut ; 1−vt Þ ¼ ut þ vt −1 n

−1 o κt κ t κ t þ exp − ð− ln ð1−ut ÞÞ þ ð− ln ð1−vt ÞÞ G where CRG t (·) and Ct (·) represent the distribution function of Rotated Gumbel Copula and Gumbel Copula at time t. The parameters κt = (1 − τt)−1, τt ∈ (−1,1), and τt are Kendall's τ. The second Copula considered in our paper is the Clayton Copula. The distribution of the Clayton Copula is defined as:

 −κ −1 t t t þ v−κ −1 C Ct ðut ; vt jκ t Þ ¼ u−κ t t

ð14Þ

where CCt (·) explains the distribution function of Clayton Copula at time t. The dependence parameter κt = 2τt (1 − τt)−1. The Rotated Gumbel Copula and Clayton Copula can be applied to capture the lower tail dependency. For capturing the lower tail dependency in different states, we incorporate a state variable st into the Copula function, assuming that st obeys the Markov state transition process. In this way, the parameters of the Copula function change as state variable st does. The cross-market correlations among different asset returns in a different state have been varying. Consequently, the MRS-Copula function is a dynamic Copula function. In this paper, we assume that the unobserved latent state variable follows a first-order, two-state Markov process with transition probability:  P¼

p00 1−p11

1−p00 p11

 ð15Þ

where the pij explains the probability of transition from state i at time t to state j at time t + 1. The dependence structure is modeled with the following dynamic MRS Rotated Gumbel–Clayton Copula:    RG  C ut ; vt jΩt−1 ; κ RG C t ut ; vt jΨt−1 ; κ RG t t ; κ t ; st ¼ ð1−st ÞC t   þ st C Ct ut ; vt jΩt−1 ; κ Ct ;

ð16Þ

and κCt are the parameters of Rotated Gumbel and where κRG t Clayton Copulas, respectively.1 Refer to research of Patton (2006b), we specify dependence parameters as dynamic evolution equations: 0

κ RG t

T      X   ^θc ≡ arg max logct F 1 z1;t Ωt−1 ; ^θ1 ; F 2 z2;t Ωt−1 ; ^θ2 jΩt−1 ; θc : ð11Þ t¼1

659

12 10   1 X RG   @ ¼ 1 þ ω1 þ α 1 κ t−1 þ β1 u −vt− j A ; 10 j¼1 t− j 0

κ Ct

12 10   1 X C   @ ¼ ω2 þ α 2 κ t−1 þ β2 u −vt− j A : 10 j¼1 t− j

ð17Þ

ð18Þ

1 Unlike traditional MRS Copula model, we assume that dependence parameters change over time.

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L. Changqing et al. / Economic Modelling 51 (2015) 657–671

Eqs. (17) and (18) are used to keep κtRG and κtC in [1,+∞) and [0, +∞) respectively. These two equations reveal that the dependence parameter κt depends on the previous dependence parameter κt − 1 and the lagged 10 absolute difference of the transformed variables ut − j and vt − j. The unknown parameters of the Copula models are θc = (p00, p11, ω1, α1, β1, ω2, α2, β2), which can be estimated by maximizing the following log-likelihood function:

Lðθc Þ ¼

T X

log½ct ðut ; vt jΩt−1 Þ

ð19Þ

2.4. The measurement of financial risk contagion Financial risk contagion is usually caused by the extreme risk events, and hence the traditional correlation is not an ideal indicator to measure the financial risk contagion. However, lower tail dependency can statistically account for the danger of financial market crisis. In the study of risk contagion, lower tail dependency can well reflect the risk contagion effect between markets when negative extreme events happen. Therefore, it is reasonable to use lower tail dependency to measure the financial risk contagion. Lower tail dependency is:

t¼1

τL ¼ lim P ðUbεjVbεÞ ¼ lim ε→0

where the log-likelihood function can be expressed as: ct ðut ; vt jΩt−1 Þ ¼ cRG t ðut ; vt jΩt−1 ; st ¼ 0Þ  P ðst ¼ 0jΩt−1 Þ þ cCt ðut ; vt jΩt−1 ; st ¼ 1Þ  P ðst ¼ 1jΩt−1 Þ

ð20Þ

where P(st = 0|Ωt − 1) is the probability of being in regime 0 at time t. Let P(st = 0|Ωt − 1) = π0,t, and define P(st = 0|Ωt − 1) as: "

# π 0;t−1 cRG t−1   þ ð1−p11 Þ P ðst ¼ 0jΩt−1 Þ ¼ p00 π cRG þ 1−π0;t−1 cCt−1 " 0;t−1 t−1 #  C 1−π 0;t−1 ct−1   C ;  π0;t−1 cRG t−1 þ 1−π 0;t−1 ct−1

ð23Þ

If τ L exists and is in the interval (0, 1], the random variables U and V are lower tail dependent. If the larger the value is, the greater the lower tail dependence between U and V is. If it equals to zero, the random variables U and V are independent. In this paper, the lower tail dependence coefficient provides the measurement of financial risk contagion level, where the larger the lower tail dependence coefficient τ L is, the greater the extent of financial contagion will be. As to Rotated-Gumbel Copula function and Clayton Copula function, the lower tail dependency coefficients at time t are respectively defined

ð21Þ 3. Empirical study

2.3. Goodness-of-fit test method Along with a goodness-of-fit test, we examine the preciseness of the dependency structure of return sets on assets in dynamic MRS-Copula model. Assume the distributions of random variables U and V are G1(·) and G2(·), and the corresponding Copula functions are C(·,·), then the conditional distribution of random variable V is shown as follows: ∂ C ðu; vÞ ∂u

C ðε; εÞ : ε

as: τLRG ðtÞ ¼ 2−21=κ 1;t , and τLC ðtÞ ¼ 2−1=κ 2;t .

C where cRG t − 1 and ct − 1 represent the conditional density of Rotated Gumbel Copula and Clayton Copula at time t − 1.

C u ðvÞ ¼ C ðV ≤vjU ¼ uÞ ¼

ε→0

ð22Þ

where u = G1(U), v = G1(V), and Cu(v) obey the (0,1) uniform distribution. The Kolmogorov–Smirnov (KS) statistics can be applied to test the goodness-of-fit of the Copula model. When the first-order partial derivatives of Copula function obey the (0, 1) uniform distribution, we can accept that the Copula function is able to capture the dependence structure of the market returns.

3.1. Data sources and sample selection In this paper, we focus on financial risk contagion between Chinese stock market and international stock markets. Referring to P.K. Narayan et al. (2014), S. Narayan et al. (2014), K.S. Narayan et al. (2015), P.K. Narayan et al. (2015) and other existing literature, we select 42 international stock markets from Europe, North America, South America, Africa, Asia and Oceania as the original samples (as shown in Table 1). These 42 international stock markets include major mature and emerging markets, thus, the samples can represent the global markets to some degree. In December 2001, China joined the World Trade Organization. Since then, China's economy gradually increases the degree of openness. Our sample period starts from January 1, 1997 which is 5 years prior to China's accession of WTO, and ends on June 30, 2015, thus the time window covers a long horizon and different stages of China's economic integration to the global economy. The stock market index data is from DataStream and WIND database. The daily market indices are displayed in Fig. 1. Judging by Fig. 1 we can find similar fluctuation trace between Chinese stock market and other international stock markets. Meanwhile, the major market crash of each index incurs almost at the

Table 1 42 countries (regions) and respective symbols. Country

Symbol

Country

Symbol

Country

Symbol

Country

Symbol

Country

Symbol

Argentina Australia Belgium Brazil Canada Chile China Demark Spain

ARG AUS BEL BRA CAN CHI CHN DNK ESP

Finland France United Kingdom Germany Greece Hong Kong Hungary India Indonesia

FIN FRA GBR GER GRE HKG HUN IND INA

Ireland Israel Japan Korea Saudi Arabia Lebanon Malaysia Mexico Netherlands

IRE ISR JPN KOR KSA LIB MAS MEX NED

Nigeria Norway New Zealand Philippines Poland Portugal Russia Singapore Switzerland

NGA NOR NZL PHI POL POR RSA SIN SUI

Sweden Thailand Turkey Taiwan United States Vietnam

SWE THA TUR TWN USA VEN

L. Changqing et al. / Economic Modelling 51 (2015) 657–671

10 ARG AUS BRA CAN CHI

FRA GBR GER GRE

Stock market indices

CHN ESP FIN

4

KOR KSA

9

BEL

DNK

x 10

661

LIB

8

MAS MEX

7

NED NGA NOR

6

NZL PHI

5

POL

4

POR RSA SIN

HKG

3

HUN IND

SUI SWE

2

THA TUR

INA IRE

TWN

1

ISR JPN

0

USA VEN 0

400

800

1200

1600

2000

2400 Time windows

2800

3200

3600

4000

4400

4700

Fig. 1. Daily stock market indices of 42 countries.

same. This characteristic provides evidence that the risk contagion may exist. To efficiently and precisely examine the risk contagion between Chinese stock market and other international stock markets, it is essential to identify the center nodes as the representative international stock markets. Here, we adopt the topology method provided by Wang and Xie (2015) before the modeling of MRS-Copula models. When matching daily data of all 42 stock markets, nearly one quarter of whole data would be excluded because of holidays, thus, tremendous information would be lost if the daily data is used to model the static correlation structure. While compared to monthly, quarterly and annual data, the weekly data contain more market information. Therefore, we use the weekly data to construct minimum spanning tree and attempt to find the clusters of the international stock markets. The logarithmic rate of return (log return) is used to characterize the rate of return: r t ¼ 100½ logðP t Þ− logðP t−1 Þ

ð24Þ

where rt refers to the return at time t, and Pt and Pt − 1 refer to the price at time t and t − 1 respectively. Before constructing the minimum spanning tree, we have to calculate the static correlation coefficients ρij between stock market indices i and j: n X

  ðr i −r i Þ r j −r j

i; j¼1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#" ρi j ¼ v #: u" n n  X u X  t ðr i −r Þ2 r j −r j 2 i¼1

ð25Þ

j¼1

The correlation coefficient matrix does not satisfy the condition of triangle inequality, we follow Mantegna's (1999) method to calculate the distance matrix: di j ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 1−ρi j :

ð26Þ

The above distance dij fulfills the condition: (1) dij = 0 if and only if i = j; (2) dij = dji; and (3) dij ≤ dik + dki. Distance dij reveals the similarity of two stock markets i and j, the smaller the dij, the markets are more similar. By using the dij, we can construct the minimum spanning tree. Compared to Kruskal's algorithm, the Prim's algorithm is more suitable for large number of nodes (stock markets). In our study, the number of nodes is 42, the graph of minimum spanning tree is relatively dense,

hence, we use Prim's algorithm to determine the network of international stock markets, and minimum spanning tree is displayed in Fig. 2. In Fig. 2, the 42 international markets are divided into several clusters basically by the geographical location and economic linkages. Judging by the minimum spanning tree, we select three categories of samples to empirically study the financial risk contagion between Chinese stock market and international stock markets. The first categories are the red circles. In Fig. 2, the red circles represent the stock markets of Singapore, Great Britain, Australia, France, Netherlands, Norway, Canada and Hungary which connect more than 3 weighted edges. These kinds of stock markets have significant impact on regional or connected stock markets. The second category is the stock market of the United States marked in blue in Fig. 2. The United States connects Singapore and Great Britain. The stock market of Singapore is a developed market in Asia, and Great Britain connects France and Australia which partly represent the location center of Europe and Oceania. Thus, the stock market of the United States probably has a global impact on other markets according to the minimum spanning tree. The third categories are the stock markets of Taiwan and Greece marked in yellow circles which China directly connects. By matching these 12 stock markets and Chinese stock markets respectively, we construct the dynamic MRS-Copula models and detect the financial risk contagion in the following study. 3.2. Data description and unit root test For measuring the risk contagion between Chinese stock market and international stock markets, we apply daily return data which contains more information compared to weekly and monthly data to model the dynamic MRS-Copulas. Meanwhile, when matching the daily data between two countries' stock markets, only few data are excluded because of the holidays. All the index data run from January 1, 1997 to June 30, 2015. The original index data of 12 groups of stock markets are displayed in Fig. 3. To alleviate the impact of the heteroskedasticity, we follow the method provided by P.K. Narayan et al. (2014) and S. Narayan et al. (2014) to standardize the return data. Specifically, we divide the demeaned return data by the SD. Summary statistics of daily returns for international markets are shown in Table 2. The Jarque–Bera test statistics confirm that all returns are significantly non-normal at 1% significance level. The Kurtosis statistics for all markets are larger than 3, implying that the empirical distributions of the frequency are leptokurtic and fat-tailed compared with the normal distribution. Therefore, it is

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NGA

1.19 ARG 0.89 RSA POL

MEX

CHI

0.81 0.78 0.77 0.78 HUN

0.82 BRA

CAN VEN

KSA

0.82

0.64

0.89

1.12

1.22 IND

0.96

1.18 0.71 0.55

0.68 MAS

DNK 0.85

1.73 CHN

0.65

0.91 TWN

0.81

0.62 GBR

USA

IRE 0.42 0.45

0.57 0.63

0.82

0.72 0.87 PHI

SUI AUS

INA 0.90

BEL

0.74

NED GER

SIN

HKG

LIB

NOR

GRE

1.29

ISR

FRA

0.60 0.55

0.71 FIIN

SWE

ESP

0.71

POR

0.93 TUR

0.83

0.84 KOR

0.84

THA

JPN

NZL

Fig. 2. Minimum spanning tree of 42 country indices in international stock markets. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)

necessary to adopt a non-normal distribution function to describe the distribution of market returns. Since the LM test for the residual series indicates the existence of a high order ARCH effect, we use GARCH models to estimate conditional volatility of different markets. Additionally, before the modeling of marginal distribution, the stationary of the return series should be tested. Referring to Carrion-i-Silvestre et al. (2009) and Narayan and Poop (2013), we take the unit root test with multiple break points and calculate the MZA statistics provided in Carrion-i-Silvestre et al. (2009). In Table 3, the column index and return respectively represent the unit root test results of original market index and standardized return data. The MZA statistics show that the market indices are not stationary, while the standardized data are all stationary at the significance level of 5%. Therefore, the return data can be used to model the dynamic MRS-Copulas. 3.3. Marginal model estimations Because of the asymmetry of stock market return, we adopt an AR(p)–TGARCH(p, q) model to fit the return series. For TGARCH models, p = 1,2, and q = 1,2. In line with Diebold et al. (1998), we conduct the K–S test to examine for the marginal distribution assumption. If the transformation sequences obey the (0,1) uniform distribution, then the distribution of the marginal assumption is appropriate. For most standardized return series of sample markets, the TGARCH(1, 1) models perform better than TGARCH(1, 2), TGARCH(2, 1) and TGARCH(2, 2), and TGARCH(2, 1) models are more appropriate to fit the data of FRA, GBR and USA. Then, we use established TGARCH(p, q) as the marginal

distribution of the dynamic MRS-Copula models. Table 4 reports the results of parameter estimations and the marginal distribution tests. The results in Table 4 show that for all markets δ + 0.5(γ + φ) is close to 1, which represent the strong and continuous stock market volatility. In most cases, the statistics of p-value demonstrate that the parameters estimated are significant. Meanwhile, the bad news exerts a greater impact on the stock market than the good news, where the leverage effect is significant in all sample markets. According to the results from the K–S test, the null hypothesis is accepted at the 5% significance level. Therefore, the marginal distribution assumption of the stock market return is appropriate.

3.4. Parameter estimation and goodness-of-fit test Along with the estimated parameters of marginal distribution, we transform the probability integral by using the standardized residual sequence of each return and the cumulative distribution function of skewed-t or GED distribution, and then the cumulative distribution function value sequence of each stock return is acquired. The transformed data is fitted by the dynamic state transitional Rotated-Gumbel Clayton Copula function and the static state transitional Rotated-Gumbel Clayton Copula function. Herein lies the tests for goodness-of-fit, and the results are reported in Tables 5 and 6. From Tables 5 to 6, the values of p11 and p22 are significantly close to 1 at 1% significance level, which illustrates that there is a strong state persistence in the dependency structure between Chinese stock market and international stock markets in different states.

L. Changqing et al. / Economic Modelling 51 (2015) 657–671 20,000 AUS CHN

Index

Index

10,000

5,000

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2009

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GBR CHN

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5,000

1999

2001

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1999

2001

10000

Index

PHI CHN 5,000

1999

NOR CHN

0 1997

2015

10,000

Index

2001

40,000 GRE CHN

1997

1999

5000

0 1997

2015

10,000

1997

10,000

10000 FRA CHN

1997

CAN CHN

1997

10,000

1997

663

2001

2003

2005 2007 2009 Time(Year)

2011

2013

SIN CHN 5000

0 1997

2015

1999

2001

20,000

Index

Index

20,000 TWN CHN

10,000 1997

1999

2001

2003

2005 2007 Time(Year)

2009

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USA CHN 10,000

1997

1999

2001

Fig. 3. Stock market index series of 12 group countries.

According to Table 5, the values of ωi, αi and βi vary greatly, which signifies that the dependency parameters do not follow the same dynamic process. For dynamic regime switching Rotated-Gumbel Clayton Copula, in the case of states 1 and 2, only a few parameters are not significant, such as ω1, α1 and β1 for state 1 of matchup GRE–CHN. However the results of goodness-of-fit test support that the mixed dynamic state transitional Copula model can still well describe the changes of the dependency structure between Chinese stock market and Greece stock market. From the results of K–S test, the p-values of the two

Copula models are larger than 5%. We can thus conclude that it is appropriate to employ either one of those two models to describe the dependency structure for each stock market matchups. We further test the models by computing the likelihood ratio statistics to test the rationality of this specification. The results are shown in Table 7. This result shows that it is reasonable to set a time varying process for dependency structure. Since the time varying tail dependence could better capture the interaction characteristics between

Table 2 Descriptive statistics of return data series. Series name

Mean

SD

Skewness

Kurtosis

J–B statistics

ARCH(5)

CHN AUS CAN FRA GBR GRE HUN NED NOR PHI SIN TWN USA

0.011 −0.000 0.001 −0.001 −0.005 −0.015 0.013 −0.005 0.013 0.001 −0.007 −0.010 0.003

1.109 0.665 0.796 1.025 0.838 1.371 1.231 1.021 0.967 1.031 0.937 1.007 0.815

−0.219 −0.219 −1.003 −0.030 −0.220 −0.001 −0.565 −0.221 −0.806 0.442 0.551 −0.213 −0.198

7.328 10.245 20.928 8.792 9.780 7.613 13.050 9.624 11.713 17.013 18.735 6.548 11.476

3439.034 (0.000) 9829.377 (0.000) 58,562.000 (0.000) 4721.749 (0.000) 8371.461 (0.000) 3809.807 (0.0000) 18,340.950 (0.000) 8069.680 (0.000) 14,197.79 (0.000) 34,870 (0.000) 45,314.47 (0.000) 2331.082 (0.000) 12,997.32 (0.000)

54.173 (0.000) 205.206 (0.000) 115.036 (0.000) 136.935 (0.000) 185.854 (0.000) 51.204 (0.000) 107.616 (0.000) 170.610 (0.000) 166.727 (0.000) 24.685 (0.000) 57.344 (0.000) 72.190 (0.000) 218.313 (0.000)

Note: In the bracket is the p-value.

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L. Changqing et al. / Economic Modelling 51 (2015) 657–671

Table 3 Unit root test with multiple break points. Series

Index

Return

Series

Index

Return

Series

Index

Return

AUS CAN FRA GBR

−11.111 (−23.163) −15.487 (−22.986) −10.229 (−23.117) −12.529(−23.056)

−302.823 (−22.897) −311.467 (−22.888) −844.433 (−22.913) −742.260 (−22.890)

GRE HUN NED NOR

−5.595 (−22.566) −11.037 (−23.101) −6.434 (−23.126) −9.4545 (−23.064)

−673.094 (−21.276) −849.049 (−22.908) −875.169 (−22.912) −667.682 (−22.886)

PHI SIN TWN USA

−5.622 (−22.666) −11.468 (−21.587) −13.412 (−22.360) −10.090 (−23.172)

−382.911 (−23.589) −753.385 (−22.797) −1829.056 (−22.561) −1411.654 (−22.895)

Note: In the bracket is the critical value at significance level of 5%.

Table 4 Results of parameter estimation and K–S test results for marginal distribution.

CHN AUS CAN FRA GBR GRE HUN NED NOR PHI SIN TWN USA

μ

c

φ1

φ2

δ1

γ

υ(ς)

η

KS test (p-value)

0.007 (0.358) 0.032 (0.002) 0.030 (0.009) 0.035 (0.033) 0.021 (0.095) 0.041 (0.009) 0.042 (0.051) 0.019 (0.105) 0.070 (0.000) 0.024 (0.092) 0.014 (0.148) 0.037 (0.042) 0.039 (0.002)

0.004 (0.000) 0.001 (0.000) 0.001 (0.000) 0.003 (0.000) 0.002 (0.000) 0.039 (0.000) 0.009 (0.000) 0.002 (0.000) 0.005 (0.000) 0.011 (0.000) 0.001 (0.000) 0.011 (0.010) 0.018 (0.000)

0.067 (0.000) 0.011 (0.045) 0.011 (0.076) −0.015 (0.004) −0.018 (0.000) 0.068 (0.000) 0.047 (0.000) 0.009 (0.063) 0.047 (0.000) 0.079 (0.000) 0.045 (0.000) 0.018 (0.008) −0.031 (0.000)

–

0.905 (0.000) 0.910 (0.000) 0.923 (0.000) 0.913 (0.000) 0.911 (0.000) 0.898 (0.000) 0.880 (0.000) 0.910 (0.000) 0.866 (0.000) 0.814 (0.000) 0.908 (0.000) 0.946 (0.000) 0.901 (0.000)

0.038 (0.006) 0.128 (0.000) 0.103 (0.000) 0.146 (0.000) 0.155 (0.000) 0.060 (0.000) 0.087 (0.000) 0.143 (0.000) 0.119 (0.000) 0.127 (0.000) 0.090 (0.000) 0.059 (0.000) 0.159 (0.000)

4.743 (0.000) 9.476 (0.000) 6.594 (0.000) 8.446 (0.000) 8.964 (0.000) 1.237 (0.000) 6.203 (0.000) 8.840 (0.000) 7.268 (0.000) 4.703 (0.000) 6.342 (0.000) 1.284 (0.000) 7.180 (0.000)

−0.065 (0.000) −0.124 (0.000) −0.167 (0.000) −0.099 (0.000) −0.118 (0.000) –

0.320

−0.001 (0.413) −0.101 (0.000) −0.091 (0.000) −0.032 (0.083) −0.052 (0.003) –

0.233

−0.091 (0.000)

0.624

– – 0.027 (0.002) 0.013 (0.079) – – – – – – – 0.036 (0.001)

0.415 0.435 0.587 0.422 0.494

0.452 0.715 0.556 0.620 0.599

Note: The p-value from Kolmogorov–Smirnov (K–S) test less than 0.05 shows a rejection of the null hypothesis that this marginal distribution is well specified. For GRE and TWN, the residuals are assumed to obey the GED distribution, and the parameter is ς.

international stock markets and Chinese stock market, we employ the mixed dynamic state transitional Copula model to describe the dependency structure for the investigation of financial risk contagion for Chinese stock markets.

3.5. Analysis of financial risk contagion In order to measure the financial risk contagion, we calculate the time varying lower dependence coefficient and smooth probability of all Chinese stock markets and other international stock market portfolios.2 The time-varying path chart displays the size and direction of dependency structure, and the smooth probability figure is used to judge the points of abrupt change in crises.3 Concerning the state transitional Copula models, the dependency coefficient of the left tail can be defined as the conditional expectation of the dependency coefficient under different conditions. Thus, the left tail dependency is expressed as follows: τL = π1tτLRG(t) + (1 − π1t)τLC(t). The smooth probability and dynamic lower tail dependence of all market groups in regime 1 are presented in Figs. 4 and 5.

2 A detailed description of the calculation method for the smooth probability of the Markov state transition model is reported in Hamilton, Time Series Analysis. 3 At time t, if the smoothed probability of the state 1 is greater than that of the state 2, it suggests that the time series of dependency is in the state 1. In this paper, we perceive it as a basis to determine the structural change point of the lower tail dependencies during the financial crisis.

As revealed in Fig. 5, the tail dependence coefficients between Chinese stock market and other international stock markets are relatively low. Even for the highest matchup (SIN–CHN), the average daily tail dependence is lower than 0.2. The Chinese stock market is slightly affected by the external financial risk for most of the time from January, 1997 to June, 2015. Our results are partly consistent with Hwang et al. (2013). Hwang et al. (2013) calculate the dynamic conditional correlations (DCCs) based on GARCH models, and find that the DCC between China and US is lower than most of the other matchups. Although the supervisors have increased the openness of stock markets, the liquidity linkages between China and international markets are weakened by the capital control. In 2015, according to the annual report released by the National Bureau of Statistics of China, the debit side of capital account of China is 19.39 billion US dollars, which is only 0.187% of GDP. Thus, the probability of financial risk contagion may be decreased by the capital control. According to the break dates and the lower tail dependence variation, we can find several time varying characteristics about the financial risk contagion. In 2003, for some matchups, such as CAN–CHN, GRE–CHN, HUN– CHN, NED–CHN and USA–CHN, the tail dependence gradually increases. We can partly attribute this increase to the implementation of QFII (Qualified Foreign Institutional Investors) system in China. In December 2002, China began implementing the QFII system, marking Chinese stock market and opening up into a new state. In 2003, UBS AG, Nomura Securities Co., Ltd., Morgan Stanley & Co. International Limited, Citigroup Global Markets Limited, Goldman, Sachs & Co., Deutsche Bank Aktiengesellschaft, The Hong Kong and Shanghai Banking

L. Changqing et al. / Economic Modelling 51 (2015) 657–671

665

Table 5 Parameters of dynamic regime switching Rotated-Gumbel Clayton Copula. Matchups

p11

p22

ω1

α1

β1

ω2

α2

β2

K–S test (p-value)

AUS–CHN

0.999 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 0.999 (0.000) 0.999 (0.000) 1.000 (0.000) 0.999 (0.000) 0.999 (0.000) 1.000 (0.000) 0.999 (0.000) 0.984 (0.000)

1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 0.999 (0.000) 1.000 (0.000) 0.999 (0.000)

−0.619 (0.000) −1.125 (0.000) 1.737 (0.000) 1.263 (0.000) −0.839 (0.184) −0.187 (0.171) 1.999 (0.000) 0.885 (0.176) −0.863 (0.000) −0.971 (0.045) −0.386 (0.000) 0.596 (0.000)

0.904 (0.000) 1.333 (0.000) −1.374 (0.000) −0.869 (0.009) 0.450 (0.294) 0.626 (0.000) −1.863 (0.000) −0.471 (0.286) 1.103 (0.000) 1.208 (0.007) 0.746 (0.000) −0.687 (0.000)

−0.046 (0.463) −0.104 (0.016) 0.172 (0.013) 0.113 (0.015) 0.108 (0.191) −0.919 (0.000) −0.266 (0.255) 0.089 (0.323) −0.062 (0.000) −0.279 (0.161) −0.162 (0.000) −0.522 (0.000)

−0.459 (0.000) −0.133 (0.005) 0.032 (0.336) 0.119 (0.029) −0.071 (0.000) 0.267 (0.000) 0.563 (0.000) −0.055 (0.000) −0.262 (0.000) 1.067 (0.000) −0.348 (0.000) −0.183 (0.009)

1.385 (0.000) −2.109 (0.000) 2.989 (0.000) 1.188 (0.121) −0.575 (0.000) −2.830 (0.000) −1.148 (0.000) 2.995 (0.000) 1.636 (0.000) −0.625 (0.000) −0.041 (0.471) −1.143 (0.000)

0.151 (0.071) 0.244 (0.000) −0.376 (0.000) −0.359 (0.039) 0.617 (0.000) 0.149 (0.000) 0.154 (0.031) −0.374 (0.000) 0.117 (0.050) 0.116 (0.035) 0.224 (0.054) 0.264 (0.055)

0.410

CAN–CHN FRA–CHN GBR–CHN GRE–CHN HUN–CHN NED–CHN NOR–CHN PHI–CHN SIN–CHN TWA–CHN USA–CHN

0.514 0.410 0.379 0.242 0.362 0.379 0.241 0.566 0.311 0.401 0.551

Note: The p-value from K–S test less than 0.05 shows a rejection of the null hypothesis that this Copula model is well specified.

Table 6 Parameters of static regime switching Rotated-Gumbel Clayton Copula. Matchups

p11

p22

κ1

κ2

K–S test (p-value)

AUS–CHN CAN–CHN FRA–CHN GBR–CHN GRE–CHN HUN–CHN NED–CHN NOR–CHN PHI–CHN SIN–CHN TWA–CHN USA–CHN

0.999 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 1.000 (0.000) 0.999 (0.000) 0.999 (0.000) 1.000 (0.000)

1.255 (0.000) 1.001 (0.000) 1.001 (0.000) 1.001 (0.000) 1.009 (0.000) 1.004 (0.000) 1.001 (0.000) 1.017 (0.000) 1.023 (0.000) 1.287 (0.000) 1.289 (0.000) 1.009 (0.000)

1.000 (0.000) 1.000 (0.000) 0.999 (0.000) 1.000 (0.000) 0.999 (0.000) 1.000 (0.000) 1.000 (0.000) 0.999 (0.000) 0.999 (0.000) 1.000 (0.000) 1.000 (0.000) 0.997 (0.000)

0.091 (0.000) 0.163 (0.000) 0.189 (0.000) 0.210 (0.000) 0.179 (0.000) 0.150 (0.000) 0.178 (0.000) 0.253 (0.000) 0.291 (0.000) 0.055 (0.013) 0.082 (0.000) 0.113 (0.000)

0.485 0.657 0.505 0.451 0.358 0.484 0.439 0.367 0.566 0.350 0.465 0.574

Note: The p-value from K–S test less than 0.05 shows a rejection of the null hypothesis that this Copula model is well specified.

Corporation Limited, ING Bank N.V., JPMorgan Chase Bank, National Association, Credit Suisse (Hong Kong) Limited, Standard Chartered Bank (Hong Kong) Limited, and Nikko Asset Management Co., Ltd. became the first group of companies approved by the State Administration of Foreign Exchange (SAFE) to trade in the A-share market with a total quota of $1.7 billion. Although the total quota is relatively small, with the entrance availability of foreign capital, the Chinese stock markets attracted more and more attention from international investors, and gradually became a part of regional and international stock markets. For most of the market matchups (CAN–CHN, FRA–CHN, GBR–CHN, GRE–CHN, HUN–CHN, NED–CHN, NOR–CHN, SIN–CHN and USA–CHN), the smooth probability and tail dependence start to change after

the July, 2005. We can partly attribute this phenomenon to the RMB exchange rate system reform in 2005. On July 21, 2005, China's central bank, the People's Bank of China (PBC) launched the reform of the RMB exchange rate system by allowing the RMB exchange rates to float within a daily band of 0.3% and linking to a basket of currencies. The reform of RMB exchange rate system has actually promoted the internationalization of China's capital markets. And the co-movement between Chinese stock market and other international stock markets has significantly been strengthened since the RMB exchange rate system reform. Meanwhile, in August 2005, the government opened the non-tradable share reform to all listed companies. To avoiding the failure of this reform, the administrators approved foreign strategic

Table 7 The likelihood ratio test of the regime switching Copula models. Matchups

AUS–CHN

CAN–CHN

FRA–CHN

GBR–CHN

GRE–CHN

HUN–CHN

LR test (p-value)

0.019

0.008

0.000

0.032

0.002

0.019

Matchups

NED–CHN

NOR–CHN

PHI–CHN

SIN–CHN

TWA–CHN

USA–CHN

LR test (p-value)

0.038

0.005

0.007

0.007

0.007

0.033

Note: The p-value from likelihood ratio (LR) less than 0.05 shows a rejection of the null hypothesis that the dependence parameters in the different states do not follow the time-varying process.

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L. Changqing et al. / Economic Modelling 51 (2015) 657–671 1 AUS-CHN

Probability

Probability

1

0.5

0 1997

1999

2001

2003

1999

2001

2003

2009

2011

2013

2001

2003

1

2005 2007 Time(Year)

2009

2011

2013

2001

1

2005 2007 Time(Year)

2009

2011

2013

2001

Probability 2003

1

2005 2007 2009 Time(Year)

2011

2013

2001

Probability 2003

2005 2007 Time(Year)

1999

2001

2003

2005 2007 Time(Year)

2009

2011

2013

2015

1999

2001

2003

2005 2007 2009 Time(Year)

2011

2013

2015

2003

2005 2007 Time(Year)

2009

2011

2013

2015

2003

2005 2007 Time(Year)

2009

2011

2013

2015

2003

2005 2007 Time(Year)

2009

2011

2013

2015

NOR-CHN 0.5

1999

2001

2009

SIN-CHN 0.5

1999

2001

1

0.5

1999

2015

0.5

0 1997

2015

TWN-CHN

0 1997

2013

1

0.5

1999

2011

HUN-CHN

0 1997

2015

PHI-CHN

0 1997

2009

1

Probability 2003

2005 2007 Time(Year)

0.5

0 1997

2015

0.5

1999

2003

1

NED-CHN

0 1997

2001

GBR-CHN

0 1997

2015

0.5

1999

1999

1

Probability

Probability

2005 2007 Time(Year)

0.5

0 1997

2015

GRE-CHN

0 1997

Probability

2013

0.5

1

Probability

2011

FRA-CHN

0 1997

Probability

2009

Probability

Probability

1

2005 2007 Time(Year)

CAN-CHN

2011

2013

2015

USA-CHN 0.5

0 1997

1999

2001

Fig. 4. The smooth probability of different market groups in regime 1. Note: The structural break date of smooth probability for each market group is reported as follows: AUS–CHN: 200510-20, 2007-8-15, and 2013-9-18; CAN–CHN: 2001-7-5, 2003-4-21, 2007-1-23, and 2013-9-18; FRA–CHN: 2003-9-15, 2006-6-22, 2007-3-19, and 2012-9-21; GBR–CHN: 2003-9-17, 2006-6-28, 2009-3-25, and 2012-9-19; GRE–CHN: 2001-4-25, 2005-6-17, 2007-11-14, 2009-12-10, and 2013-9-11; HUN–CHN: 2001-2-27, 2003-12-12, 2005-4-11, 2007-12-12, and 2013-9-10; NED–CHN: 2003-9-24, 2006-7-4, 2009-4-1, and 2013-9-23; NOR–CHN: 2005-9-29, 2007-8-27, and 2013-9-13; PHI–CHN: 2005-10-27, 2007-8-14, and 2013-9-9; SIN– CHN: 2003-11-7, 2005-8-16, 2007-5-18, and 2013-9-17; TWA–CHN: 2007-8-13, and 2013-10-31; and USA–CHN: 2000-4-24, 2003-1-29, 2005-11-10, 2007-7-11, and 2013-9-12.

investors to participate in the non-tradable share reform, for example, Bank of America Corp., Temasek Holdings and other qualified foreign institutional investors became the strategic investors of China Construction Bank and other state owned companies. With the reform of foreign exchange market and stock market, China gradually opens the financial markets. As a result, the financial risk contagion which is represented by the tail dependence occurs more obviously in Chinese stock market since 2005. In 2007, the tail dependences increase to historical highest position almost for all matchups. The evidence presented in Figs. 4 and 5 suggests that the change point of the dependency structure between Chinese stock markets and other international stock markets is rather consistent. At the change points, the state of market condition switches and these points are in the year of 2007. On March, 2007, New Century Financial Corporation, the second largest sub-prime mortgage enterprise, declared bankruptcy. This is the sign of the eruption of the U.S. Sub-prime mortgage crisis. According to Fig. 5, in the aftermath of the U.S. subprime mortgage crisis, the lower tail dependence between Chinese stock market and other international stock markets increases sharply. This result indicates that after the crisis, there is an increasing probability of financial risk contagion between Chinese stock market and other global stock markets. In the period of crisis, the lower tail

dependences between Chinese stock market and other markets increase firstly, then followed by a decrease, then finally rise again and gradually hold stable in a high correlation level until 2013. The high level of lower tail dependence from 2007 to 2013 shows that the Chinese stock market is infected by the international markets during the US subprime mortgage crisis and European debt crisis. From 2013 to 2015, the lower tail dependence between Chinese stock market and international stock markets gradually declines. Judging by Fig. 5, we can find that the Chinese stock market performs worse than most of the international stock markets during this period. In 2013, the money crunch occurs in China's financial market, overnight interbank interest rate rises to 30%, and the liquidity is not adequate enough to support the stock market. Meanwhile, the China Securities Regulatory Commission releases a plan in November to overhaul the nation's IPO system which tenses the liquidity condition of the stock market, and the market becomes more vulnerable accompanied with the panic sensation of the investors while the Chinese stock market experiences a sharp increase between the middle of 2014 to 2015. Comparing the index on July 22, 2014, the Shanghai Stock Exchange index rises from point 2077 to 5178 on June 12, 2015, increasing by 149% because of the margin trading and state owned enterprise reconstruction, and the rising degree is higher than most of the international stock

L. Changqing et al. / Economic Modelling 51 (2015) 657–671 0.015 AUS-CHN

Coefficient

Coefficient

0.4

0.2

0 1997

1999

2001

2003

1999

2001

2003

2005 2007 Time(Year)

2001

2003

2005 2007 Time(Year)

2009

2011

2013

1999

2009

2011

2013

2003

0.1

2005 2007 Time(Year)

2009

2011

2013

2003

0.4

2005 2007 2009 Time(Year)

2011

2013

2003

2005 2007 Time(Year)

2003

2005 2007 Time(Year)

2009

2011

2013

2015

1999

2001

2003

2005 2007 2009 Time(Year)

2011

2013

2015

2003

2005 2007 Time(Year)

2009

2011

2013

2015

2003

2005 2007 Time(Year)

2009

2011

2013

2015

NOR-CHN 0.04 0.02 1997

1999

2001

SIN-CHN

2009

0.2

0 1997

1999

2001

0.1

Coefficient

2001

2015

0.06

2015

0.2

1999

2013

0.4

TWN-CHN

2011

2013

USA-CHN 0.05

0 1997

2015

1999

2001

2003

2005 2007 Time(Year)

2009

2011

Fig. 5. The daily lower tail dependence of different market groups.

1000 600

Credit side of Current Accounts Debit side of Current Accounts

200 -200

Amounts(Unit: Billion US Dollars

0 1997

2011

6

Coefficient

2001

2009

0 1997 -3 1999 2001 x 10 10 HUN-CHN 8

4 1997

2015

0.05

1999

2005 2007 Time(Year)

0.08

PHI-CHN

0 1997

2003

GBR-CHN

2015

Coefficient

2001

2001

0.02

2015

0.01 0 1997

1999

0.04

NED-CHN

0.02

0 1997

2015

0.015 1999

0.005

Coefficient

Coefficient

2013

GRE-CHN

0.02

0.03

Coefficient

2011

0.01

0.01 1997

Coefficient

2009

FRA-CHN

0.02

0.025

Coefficient

2005 2007 Time(Year)

CAN-CHN

0.01

Coefficient

Coefficient

0.03

0 1997

667

-600 1998Q1

2000Q2

2002Q4

2005Q2

2007Q4

2010Q2

2012Q4

2015Q1

2005Q2

2007Q4

2010Q2

2012Q4

2015Q1

2007Q4

2010Q2

2012Q4

2015Q1

2 Credit side of Capital Accounts Debit side of Capital Accounts

0

-2 1998Q1

250

2000Q2

2002Q4

Assets of Financial Accounts Debts of Financial Accounts

0 -250 1998Q1

2000Q2

2002Q4

2005Q2 Time

Fig. 6. Balance of China's payment accounts (1998Q1–2015Q1).

2013

2015

668

L. Changqing et al. / Economic Modelling 51 (2015) 657–671

markets. Thus, the relative independent trend of the Chinese stock market causes a decline of the tail dependence between China and other nations during this period. 3.6. Further discussion of risk contagion channels Since the capital flow is still controlled, the fundamental economy works as a basic channel for financial risk contagion in Chinese stock market. Fig. 6 illustrates the balance of China's payment accounts from first quarter of 1998 to first quarter of 2015. Compared to the amount of credit/debit side of current accounts, the credit/debit side of capital accounts and the assets/debts of financial accounts are relatively small, thus the economic integration to world economy is a result of the internationalization of fundamental economy rather than the financial market. This characteristic of economy protects the financial market from the Asia financial crisis in 1997. While in the subprime crisis, the tail dependence significantly increases and this result shows that the financial contagion occurs. The source of the crisis, US is one of the important economic and financial centers in the world, and it is also the largest trading partner of China. The depression of US caused by the financial crisis directly slows down the export and growth of China's economy. Some industries, such as the manufacture industry face serious external environment, and the downturn of economy leads to a decline of the stock markets. Additionally, the tail dependence between China and the top largest trading partner in our sample is relatively high. Our empirical result shows that the top 4 groups of average tail dependence are SIN–CHN, TWN–CHN, AUS–CHN and USA–CHN, and the average trading volumes between Singapore, Taiwan, Australia and USA from January 2008 to July 2015 are 5410.5, 13,326.3, 8638.7 and 36,690.1 million US dollars, and their trading volume against China rank fourth, second, third and first among our sample countries and regions. Thus, the exporting data and lower tail dependence result also provide indirect evidence that the fundamental economy plays an important role in financial risk contagion. Judging by Figs. 4–6, we can find that liquidity or capital linkage begins to play a more important role in the process of financial risk contagion for Chinese stock market since 2005. As shown in Fig. 6, although the total amount of capital flow is low in China, it increases significantly around 2005. Accompanying with this increase, the smooth probability and tail dependence both gradually rise to the highest level until the

burst of the global financial crisis in 2007. The result provides evidence that the liquidity or capital flow channel does exist when Chinese stock market is infected from external financial markets. Financial contagion is often amplified by investors' sentiment and behavior. During the financial crisis period, the panic sentiment will spread from the crisis sources to other countries. The burst of one country's crisis attracts global attentions in a short time, and the expectation to the stock market will change, thus the investors will reallocate their asset portfolios and panic selling will strengthen the influence of financial risk contagion. Sharma et al. (2015) point out that herding behavior is present on Chinese market in crisis and calm period. This empirical evidence could help us to explain why the financial contagion occurs in subprime crisis period even though there is relatively weak capital linkage between Chinese and other international stock markets. Moreover, according to the 35th Statistical Report on Internet Development in China released by the China Internet Network Information Center (CNNIC), as of the end of 2014 China had 649 million Internet users and the Internet penetration rate reached 47.9%. With the quick development of the information technology, the investors tend to accept worldwide information from the Internet and make the homogeneous investment decisions, thus the investor sentiment or behavior functions as a more important channel in financial risk contagion under the condition of quick information flows.

4. Robustness test In this section, we conduct robustness tests to verify the financial risk contagion. Narayan et al. (2013), P.K. Narayan et al. (2014), S. Narayan et al. (2014), Narayan and Sharma (2015), and recent literature point out that using multiple data frequencies could increase the robustness of result, and data frequency could affect the empirical results of time series models, thus we deal with the robustness test of financial risk contagion with respect to weekly and monthly return data. The parameters of dynamic MRS-Copula models based on weekly data are reported in Table 8, and the corresponding tail dependences are illustrated in Fig. 7. Based on monthly data, we can also get the parameters of financial contagion models and the tail dependence (as shown in Table 9 and Fig. 8).

Table 8 Parameters of dynamic MRS-Copula with weekly return data.

AUS–CHN CAN–CHN FRA–CHN GBR–CHN GRE–CHN HUN–CHN NED–CHN NOR–CHN PHI–CHN SIN–CHN TWA–CHN USA–CHN

p11

p22

ω1

α1

β1

ω2

α2

β2

K–S test

0.994 (0.000) 0.991 (0.000) 0.980 (0.000) 0.980 (0.000) 0.998 (0.000) 0.993 (0.000) 0.952 (0.000) 0.973 (0.000) 0.995 (0.000) 0.996 (0.000) 0.996 (0.000) 0.980 (0.000)

0.995 (0.000) 0.998 (0.000) 0.998 (0.000) 0.998 (0.000) 0.996 (0.000) 0.993 (0.000) 0.993 (0.000) 0.996 (0.000) 0.998 (0.000) 0.998 (0.000) 0.995 (0.000) 0.998 (0.000)

−0.549 (0.000) −0.229 (0.000) −1.164 (0.000) 1.286 (0.002) 1.048 (0.144) 0.513 (0.071) −0.101 (0.198) −0.332 (0.000) −0.242 (0.338) −0.429 (0.000) 0.113 (0.247) 0.677 (0.140)

0.870 (0.000) 0.530 (0.000) 0.328 (0.014) −0.325 (0.144) −1.009 (0.153) 0.004 (0.013) 0.273 (0.000) 0.615 (0.000) 0.615 (0.090) 0.693 (0.000) 0.246 (0.268) −0.047 (0.144)

−0.186 (0.026) 0.398 (0.000) 0.548 (0.036) −0.296 (0.248) −0.392 (0.064) −0.591 (0.008) 1.518 (0.000) 0.276 (0.000) −0.192 (0.215) 0.236 (0.028) 0.158 (0.169) 0.584 (0.162)

−0.191 (0.178) 0.258 (0.000) 0.221 (0.003) 0.348 (0.016) −0.926 (0.000) −0.332 (0.000) 0.040 (0.225) −0.386 (0.000) −0.317 (0.278) −0.294 (0.003) −0.428 (0.000) 0.080 (0.285)

0.150 (0.049) −2.466 (0.000) −1.668 (0.135) −3.245 (0.000) 0.827 (0.006) 1.339 (0.000) 1.174 (0.000) −0.339 (0.254) −0.796 (0.158) −1.117 (0.000) −0.847 (0.000) −1.290 (0.026)

0.154 (0.171) 0.389 (0.000) 0.236 (0.007) 0.043 (0.253) 1.468 (0.000) −0.765 (0.000) 0.352 (0.011) 0.896 (0.000) 0.705 (0.236) 1.025 (0.000) 0.762 (0.000) 0.354 (0.000)

0.323

Note: The p-value from Kolmogorov–Smirnov test less than 0.05 shows a rejection of the null hypothesis that this Copula model is well specified.

0.586 0.418 0.342 0.430 0.503 0.298 0.482 0.137 0.512 0.306 0.196

L. Changqing et al. / Economic Modelling 51 (2015) 657–671

0.4

0.5

0 1997 2000 2003 2006 2009 2012 2015 Time(Year) 0.2 NED-CHN 0.1

0 1997 2000 2003

2006 2009 Time(Year)

2012 2015

SIN-CHN

Coefficient

Coefficient

0.4

0.2

Coefficient

0 1997 2000 2003 2006 2009 2012 2015 Time(Year) 0.06 GRE-CHN 0.04 0.02 0 1997 2000 2003 2006 2009 2012 2015 Time(Year) 0.2 NOR-CHN 0.1

0 1997 2000 2003 2006 2009 2012 2015 Time(Year)

0 1997 2000 2003

2006 2009 Time(Year)

2012

0.2

HUN-CHN 0.1

0 1997 2000 2003 2006 2009 2012 2015 Time(Year) 0.2

0 1997 2000 2003 2006 2009 2012 2015 Time(Year) 0.4 TWN-CHN 0.2

FRA-CHN

0 1997 2000 2003 2006 2009 2012 2015 Time(Year) 0.2

Coefficient

Coefficient

0 1997 2000 2003 2006 2009 2012 2015 Time(Year) 1 GBR-CHN

0.2

Coefficient

0.2

0.4 CAN-CHN

PHI-CHN 0.1

0 1997 2000 2003 2006 2009 2012 2015 Time(Year) 0.4 USA-CHN

Coefficient

Coefficient

AUS-CHN

Coefficient

Coefficient

Coefficient

Coefficient

0.4

669

2015

0.2

0 1997 2000 2003 2006 2009 2012 2015 Time(Year)

Fig. 7. The weekly lower tail dependence of different market groups.

The K–S test and significance of parameters show that the dynamic MRS-Copula models with different frequencies have high goodnessof-fit. Therefore, the MRS-Copula models with daily return data and the corresponding results are robust on the whole. For matchups of FRA–CHN, NED–CHN and TWN–CHN, the monthly data do not satisfy the modeling requirement of MRS-Copula models, thus the data frequency has slight impact on the robustness of the empirical results, and dynamic MRS-Copula models have good performance in capturing the tail dependence of different groups of financial markets. The tail dependences based on different data sets are similar when comparing the

volatility trace. The major difference of tail dependence based on daily, weekly and monthly data is the numerical value, it is interesting to find that the average tail dependence coefficient increases with the prolonged time frequency, and the monthly tail dependence is larger than the result of other two kinds of models. Since the monthly tail dependence represents the contagion in a long time horizon, the empirical results reflect that the contagion effect in Chinese stock markets is mostly driven by the fundamental factors, such as international trade and economic linkages, rather than the short term capital flows, and these results are in line with the contagion channel analysis in

Table 9 Parameters of dynamic MRS-Copula with monthly return data.

AUS–CHN CAN–CHN GBR–CHN GRE–CHN HUN–CHN NOR–CHN PHI–CHN SIN–CHN USA–CHN

p11

p22

ω1

α1

β1

ω2

α2

β2

K–S test

0.977 (0.000) 0.950 (0.000) 0.954 (0.000) 0.976 ((0.000)) 0.991 ((0.000)) 0.969 (0.000) 0.991 (0.000) 0.994 (0.000) 0.981 (0.000)

0.950 (0.000) 0.994 (0.000) 0.988 (0.000) 0.986 (0.000) 0.992 (0.000) 0.996 (0.000) 0.993 (0.000) 0.993 (0.000) 0.992 (0.000)

0.615 (0.019) −0.918 (0.000) −0.809 (0.000) −0.353 (0.000) −0.299 (0.000) −0.739 (0.000) −0.457 (0.000) −0.296 (0.000) −0.069 (0.250)

−0.273 (0.027) 0.224 (0.000) 0.665 (0.003) 0.705 (0.000) 0.570 (0.000) 0.571 (0.000) 0.649 (0.000) 0.651 (0.000) 0.391 (0.000)

−0.099 (0.441) 3.553 (0.000) 1.042 (0.002) 0.008 (0.000) 0.401 (0.134) 1.245 (0.003) 0.472 (0.026) 0.084 (0.063) 0.675 (0.004)

−1.105 (0.000) −0.773 (0.000) −0.565 (0.000) −0.459 (0.000) −0.810 (0.000) −0.587 (0.000) −0.385 (0.080) −0.133 (0.163) −0.665 (0.000)

−0.247 (0.000) −0.454 (0.000) −0.593 (0.000) −0.517 (0.063) 1.589 (0.000) −0.529 (0.000) −0.194 (0.193) 0.147 (0.016) 1.229 (0.000)

1.693 (0.000) 1.431 (0.027) 0.846 (0.000) 1.364 (0.000) 0.544 (0.000) 0.697 (0.038) 0.506 (0.068) 0.177 (0.192) 1.396 (0.000)

0.299 0.439 0.506 0.403 0.472 0.533 0.294 0.315 0.342

Note: The p-value from Kolmogorov–Smirnov test less than 0.05 shows a rejection of the null hypothesis that this Copula model is well specified. For matchups of FRA–CHN, NED–CHN and TWN–CHN, the parameters are not significant for static MRS-Copula models, thus the dynamic MRS-Copula cannot be well estimated.

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0.2 0 1997 2000 2003 2006 2009 2012 2015 Time(Year)

0.3 0.2 0.1

Coefficient

0.2 0.1 0 1997 2000 2003 2006 2009 2012 2015 Time(Year)

0.3

0.4 0.2

0.4

NOR-CHN

0.3 0.2 0.1 1997 2000 2003 2006 2009 2012 2015 Time(Year)

0.4 PHI-CHN

GBR-CHN

0.5 HUN-CHN

0 1997 2000 2003 2006 2009 2012 2015 Time(Year)

0.4

0.6

0 1997 2000 2003 2006 2009 2012 2015 Time(Year)

Coefficient

Coefficient

Coefficient

0.1 0 1997 2000 2003 2006 2009 2012 2015 Time(Year)

Coefficient

0.2

0.4 GRE-CHN

0.2

0.3

0.3

0.1 1997 2000 2003 2006 2009 2012 2015 Time(Year)

0.4 0.3

Coefficient

0.4

0.4

0.8 CAN-CHN

0.4 SIN-CHN

Coefficient

0.6

0.5 AUS-CHN

Coefficient

Coefficient

0.8

0.2 0.1 0 1997 2000 2003 2006 2009 2012 2015 Time(Year)

0.3

USA-CHN

0.2 0.1 0 1997 2000 2003 2006 2009 2012 2015 Time(Year)

Fig. 8. The monthly lower tail dependence of different market groups.

Section 3.6. Additionally, considering the fact that the higher frequency data contains more volatility information, we prudentially suggest to construct the contagion models by using daily return or higher frequency return data.

5. Conclusions This paper measures the financial risk contagion by developing a dynamic Markov Regime Switching Copula model to depict the contagion characteristics. For this purpose, we select an appropriate model by goodness-of-fit testing to analyze the cross-market lower tail dependency between Chinese stock market and other international stock markets. The results in our study point to several conclusions as follows. Firstly, the dynamic MRS-Copula model can clearly distinguish the different states of market correlation structure, and the financial risk contagion can be judged by the lower tail dependence and smooth probability. Most of the dynamic MRS-Copula models with different time frequencies are statistically significant, and the empirical results of each model are basically consistent. Our results also show that the data frequency do have slight impact on the goodness-of-fit of the models. Secondly, on the whole, the lower tail dependences between Chinese and international stock markets are relatively low. Even in the financial crisis period, the coefficients are less than 0.5. Judging by lower tail dependence and smooth probability, the Chinese stock market is infected by the international stock markets in the period of subprime mortgage crisis and European debt crisis. Thirdly, since the capital is not completely transmitted between international stock markets and Chinese stock market, the economic linkages, such as international trade function as the fundamental channel of financial risk contagion. Meanwhile, with the implementation of QFII system in 2003 and RMB system reform in 2005, the openness of Chinese financial market gradually increases, and the capital flow or liquidity begins to play a more and more important role in the channel of financial contagion. Additionally, investors' sentiment and behavior

also indirectly have impact on financial risk contagion in Chinese stock markets. Our conclusion could be conducive to the international investments of world's financial institutions in financial markets and especially in Chinese stock market. Nevertheless, our paper points to some limitations to be dealt with in the future study. Firstly, the cause and the path of the financial risk contagion channel have not been fully investigated in the paper, the future research could provide more detailed theoretical and empirical explanations about the financial risk contagion. Secondly, dynamic Markov Regime Switching Copula models depicted in our paper are constructed under the 2-dimensional Copula framework. Furthermore, the financial risk contagion among different financial markets could be investigated using dynamic multivariate MRS-Copula models. Acknowledgements We are grateful to the editors and the anonymous referees for their very helpful comments and suggestions which greatly improved the previous version of our paper. This work is supported by the innovative research groups of the National Natural Science Foundation of China (grant no. 71221001), the National Natural Science Foundation of China (grant no. 71373072, no. 71503078), the Ministry of Education of Humanities and Social Science Project (grant no. 13YJCZH123), the program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province (grant no. [2014]207) and program for Key Research Base of Philosophy and Social Sciences in Higher Educational Institutions of Hunan Province (grant no. [2014]228). We are also grateful for the suggestions and warm help from Dr. Gang-jin Wang. References Alotaibi, A., Mishra, A.V., 2015. Global and regional volatility spillovers to GCC stock markets. Econ. Model. 45 (2), 38–49. Aloui, C., Hkiri, B., 2014. Co-movements of GCC emerging stock markets: new evidence from wavelet coherence analysis. Econ. Model. 36 (1), 421–431.

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