The stability of Chinese stock network and its mechanism

Accepted Manuscript The stability of Chinese stock network and its mechanism Weiping Zhang, Xintian Zhuang

PII: DOI: Reference:

S0378-4371(18)31277-9 https://doi.org/10.1016/j.physa.2018.09.140 PHYSA 20201

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Physica A

Received date : 18 April 2018 Please cite this article as: W.P. Zhang, X. Zhuang, The stability of Chinese stock network and its mechanism, Physica A (2018), https://doi.org/10.1016/j.physa.2018.09.140 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

The Stability of Chinese Stock Network and Its Mechanism Weiping Zhang*, Xintian Zhuang School of Business Administration, Northeastern University, Shenyang, Liaoning 110167, China

HIGHLIGHTS

The price network, multi-scale network and risk network are built. Risk network is shown to be more robust. Granger causality exists between stock market volatility and network stability. Relationship between international stock market volatility and network stability is investigated. Probit model is applied to forecast abnormal fluctuations of China's stock market network. ABSTRACT Based on the multi-fractal properties of financial time series, Value-at-Risk (VaR) and price fluctuation correlation, we construct China's stock market networks and analyze empirically three networks’ topological features, their stability and their relationship with international stock market indices. The results show that: (1) In the three types of networks, the stock price network does not have scale-free features, multi-scale network and risk network both have small-world and scale-free characteristics; (2) The stock market volatility and the network stability coefficient are Granger causality, the early changes of the stock market volatility can effectively explain the changes of the network stability coefficient, the network connectivity and clustering coefficient are negatively correlated with the stock market volatility, the volatility of the international stock index in Hong Kong, Japan, and the United States has a positive effect on the network stability coefficient, and the international gold or crude oil markets has a negative effect on it; (3) By applying the probit binary selection model test, we found that the network structure is more capable of explaining abnormal fluctuations than international market factors, and that the stock network with higher network stability coefficient and higher eigenvector centrality of financial institutions indicates that downward fluctuation of stock price will intensify in the future. From the empirical results, we see that the risk network is more robust and provides a reference for the analysis of the short-term risks and stability of the Chinese stock market. In order to maintain the stability of the stock market, the regulatory authorities should pay a close attention to internal and external factors, increase network connectivity and integration, and actively guard against the risk spillover effects of financial institutions. Keywords: Price network; Multi-scale network; Risk network; Robustness; Stability coefficient; Probit model

1. Introduction 

The application of complex network methods in financial markets has become increasingly widespread. Due to

the large number of stocks in the stock market, closely related price volatility, and complex network structures [1], the economic and financial fields have attracted network researchers [2-3]. In recent years, the relationship between international financial markets has become increasingly complex. Institutional investors have a close relationship with each other. Any external unfavorable factors will lead to the continuous accumulation of systemic risks in the stock market. When the risk accumulates to a certain limit, it will damage the financial system stability. With the development of globalization, the spillover effects of price fluctuations among international financial



* Corresponding author. E-mail address: [email protected] (W.P. Zhang) 1

markets will also affect the stability of the stock market [4,5]. Therefore, the analysis of the stock market stability and its mechanism is crucial to economic development. The correlation of stock price volatility is the basic law of stock market operation in a certain period. The price fluctuation of a few stocks in the stock market could cause most of the other stocks’ prices to fluctuate [6-7]. Therefore, the correlation based on stock price fluctuations has become the most common method for building stock market network. Chi et al. [8] built a US stock price correlation network. Their results showed that the stock price fluctuations of a few companies, such as Yahoo and Google, would trigger large fluctuations in the overall stock market index. Alkan et al. [9] used the sliding time window to analyze the stock network evolution. Birch et al. [10] built a stock network based on the constituent stocks of the DAX30 index of Germany. The results show that the changes of the stock price volatility during the economic crisis period are significantly weaker than the economic recovery period, and the macroeconomic cycle period is predicted using the stock network structure properties. Huang [11] constructed a dynamic stock network, and found that stocks in the same industry were prone to price fluctuations. Yang et al. [12] established the minimum spanning tree network and hierarchical tree network, and then used the rolling window technique to calculate the average correlation and the variance of the correlation and the distance between these indicators to study the dynamic evolution of network stability. The above studies can well grasp the relationship between stock price volatility and the topological changes in the stock market. Further, many studies have found that the prices of financial assets not only have a price fluctuation correlation, but also generally have a fractal feature [13,14], that is, price volatility has similarity [15] and long memory [16] under different time scales. Onali [17] used fractal theory to explore the effectiveness of eight stock markets in Europe. Shan et al [18] used many local scale indicators to analyze the more detailed structure contained in the actual sequence, which requires considerations of the multifractal nature of price fluctuations when establishing the stock market networks. Ding Zhang [19] combined the fractal theory with the network theory for the first time, and analyzed the fractal characteristics of China's stock network from both time and space perspectives. Ming Yuan [20] constructed the Shanghai-Shenzhen stock network based on the scale curves under different orders and analyzed the multifractal similarities between stocks. The stock market is a high-risk market and the sharp fluctuations in stock prices (such as China's 2008 and 2015 stock crashes are the sudden collapse of stock prices at a certain time [21]) can cause huge losses to investors in a short period, reflecting that the price fluctuation coefficient cannot accurately characterize the long-term equilibrium relationship between stock prices. So, some scholars build a network model based on the VaR model, because the VaR model provides an effective method to measure market risk, and the correlation coefficient of the VaR array reflects the correlation of stock volatility [22], which provide a new perspective to study the stock network systemic risk, contagion, and network stability. Huang [23] calculated the dynamic condition correlation coefficient through the DCC-GARCH model and established the China’s stock market MST network to analyze the financial institutions' contribution to the systemic risk of the dynamic financial network. The decline or collapse of the stock market is related to the critical period of high volatility [24], and this volatility spillover effect may be related to excessive coordination and contagion between the stock market indices [25]. Sudden changes in international stock indexes will give huge instability to emerging markets. Lamounier and Nogueira [26] studied the relationship between the returns of major emerging markets and developed markets, and found that the returns of developed countries could predict emerging market. Park [27] analyzed the close relationship among United States, Japan and ten Asian stock markets and found that China is the only country with a low level of contact with other countries. Hwang et al. [28] established a DCC-GARCH model to analyze the dynamic correlation between the United States and emerging economies stock market during the financial crisis. The above-mentioned literature analyzes the mutual influence of stock markets from the perspective of the stock 2

market. Gradually, researchers have focused their attention on the relationship between commodity markets or spot and futures markets and stock market volatility. Gold and crude oil physical assets have attracted much attention [29,30]: Baur and McDermott [31] revealed that, during the period when the financial crisis was most severe, gold exhibited a significant negative correlation with stock market in most developed countries; Zhuang et al. [32] analyzed the relationship between ten industry indices and WTI crude oil prices based on multiscale bias correlation; Narayan et al. [33] established a predictive regression model to reveal the role of oil prices in forecasting stock prices. Obviously, these market types or asset types provide additional evidence to explore the stability of the stock market from different perspectives. Later, some scholars tried to explore the deep characteristics such as the network stability [34], robustness [35], and vulnerability [36]. Acemoglu [37] argued that while the complete network is more robust than the ring network, in the extreme cases the stronger the networks are related, the more vulnerable is the financial system; Heiberger [38] confirmed the practicality of the May-Wingner theorem as a stability indicator of the US stock market and found that during the financial turbulence period, the network changed the irrational structural during the boom to a more concentrated topology; Baumöhl et al.[39] analyzed the volatility spillover effects and their determinants among 40 stock markets, and demonstrated how the topological structures of the stock connectivity change with the market fluctuations. Volatility spreads across various markets through spillover effects, and has a greater impact when markets connected more closely [40]. An et al. [41] first established a binary time series network model to analyze the linkage trend between crude oil prices and stock prices. In summary, existing literature mostly use the mothed of stock price volatility to build a network. However, this method cannot reflect the volatility of stock prices in the long run and ignores the similarity characteristics of time series. In addition, most studies use a single statistical method to analyze the volatility spillover effects on the stock market between several stock markets or physical asset markets. Few literatures combine the fluctuations of international stock indexes or market prices of physical assets directly with the stability of financial market networks. The major contribution of this article is threefold. First, we simultaneously consider the similarity features of the stock price series and the high-risk features of the stock market to innovatively construct the scale-free network and risk network of stock market. Based on this, comparing existing stock price network, we analyze the stability of the three networks. Second, we explore the impact of network topology indicators on stock market volatility, and use cross-correlation function to analyze the correlation between international market volatility and network stability coefficient. Last, we apply the probit model to examine the ability of the network structure and external factors such as international market volatility to predict abnormal fluctuations in China’s stock market. The rest of the paper goes as follows. Section 2 outlines the method of constructing three types of networks and gives network stability indictors. Section 3 briefly introduces the statistical regression model. Section 4 is the empirical study and results. Section 5 concludes the paper.

2. Network construction and network stability indicators 2.1. Methods for constructing stock linkage network 2.1.1. Price relevant network based on stock price At present, the most common method of constructing stock network is based on the correlations of price. This method calculates the correlation coefficient of stock price fluctuations, converts the coefficient matrix into a distance matrix, and obtains the adjacency matrix through the threshold method [42, 43]. There are N stocks in the stock market, a network node represents each stock, and the edges between nodes represent the correlation of stock price fluctuations over time. The main steps are as follows: Let

be the closing price of stock i on day t,

is the return of stock i given by 3

100

∈

,

. We compute the price fluctuation correlation

correlation coefficient. However, ,

,

,

between stock i and j, i.e., the Pearson

does not satisfy the axioms defining an Euclidean metric. We can convert

by an appropriate transformation, as in [44], so that the axioms are satisfied. ,

21

(1)

,

The correlation matrix C can be obtained from the correlation coefficient

,

, and the distance matrix D can

be obtained from formula (1). High positive correlations correspond to small values of

,

. The distance matrix D

is a full-connected matrix, and some specific filtering processes, such as the threshold method, Planar Maximally Graph and Minimum spanning tree, were applied to eliminate redundant edges. 2.1.2. Multiscale network based on multi-scale curves The scaling curves in different orders are fitted according to the fractal features of the stock price series, and the correlation coefficient between the curves are used to construct a multi-scale network through the threshold method. (1) Fit scale curve Kantelhardt [45] proposed an effective method “MFDFA” (Multifractal Detrended Fluctuation Analysis) to verify whether a non-stationary time series has multi-scale, then Ref. [13] extended MFDFA to get MSDFA (Multi-Scale Detrended Fluctuation Analysis). The main steps are as follows: Step 1. Based on a given time series of length N, { ,

∑

y

̅ ,

1,2, ⋯ , }, a new sequence { } is constructed:

1,2, ⋯ ,

(2) ⁄

Step 2. Divide { } into intervals of length s, resulting in a total of

intervals. Since the length

of the time series N divided by s is not necessarily an integer, in order not to discard the last part of the data, this segmentation process is repeated from the tail of the sequence reversely, to give a total of 2 Step 3. Fit the local trend term

on each subinterval v (

1, 2, ⋯ , 2

intervals.

) through OLS method, and

eliminate the local trend term in the subinterval v to obtain the residual sequence. Consider two cases (head-to-tail sequence and tail-to-head sequence): When

1,2, ⋯ ,

When

1,

∑

, 2, ⋯ , 2

(3)

，

∑

(4)

Step 4. Calculate the mean residual squared of the 2



s, v

subintervals after de-trending, that is:

∑

(5)

Given the order q to construct the wave function

∑



s, v

:

⁄

(6)

Step 5. Calculate the local generalized Hurst exponent

[46] from equation (7): （7）

Step 6. The relation between

and

obtained from Eq. (7) is discrete and can become a smooth scale

curve using cubic spline interpolation, so that the continuous variation of closely. (2) Build multi-scale network 4

with s can be analyzed more

By fitting different order scale curves, a multi-scale curve network can be constructed, based on the similarities of sequences’ multifractal feature. The main steps are as follows: 1, 2, ⋯ ,

Step 1. For the financial time series i

,

, given the order

1,2, ⋯ ,

, the MSDFA method

is used to fit the different orders scale curves

. A total of L scale curves are thus fitted, each with K

observations. The set of scale curves can form a

matrix. Stacking the columns of this matrix gives a vector

of length KL. Step 2. For any pair of financial time series i and j, calculate the correlation coefficient corresponding vectors

and

of the

. Note that the contribution of the multifractal similarities of i and j calculated by should be given to the scale exponent

the scale curve under different orders is different, so the weight index sequence in different orders, as shown in Eq. (8): ∑

∙

When

(8)

∙ ∑

∑

is greater than the threshold θ, it is considered that the financial time series i and j are bounded. The 1 if there is an edge between i and j, and 0, otherwise.

element

2.1.3. Risk network based on VaR Jorion [47] gives the authoritative definition of VaR as “the worst expected loss of an asset or a portfolio in the future holding period, given a confidence level”. VaR model can be used to predict the potential risk in the portfolio, calculate the correlation coefficients of VaR arrays, and construct a risk network using a threshold method. Step 1. Select trading time span T as the time span for measuring the short-term risk in a single stock. The daily closing price of stock i in the time span T is

，

1,2 … . At a 95% confidence level, the expected return is 0.

Use the "Historical Simulation Method" to select the Sth smallest number (S=T*(1-95%)) in 1, …, T, and denote it by

. Step 2. Calculate the first VaR value:

⁄ .

Step 3. The time window of T trading days is shifted back one day. Then, repeat the above process to obtain , and so on. Step 4. For all stocks M, repeat step1, 2, 3, to give a VaR array { Step 5. Calculate the VaR array correlation coefficient the given threshold θ and i and j, otherwise no edge.

, if θ is less than or equal to

, ,

} 1,1

.

,



,

, there is an undirected edge with no weight between

,

∈

1,2, ⋯ ,

of any node pair i and j, comparing

are computed as: (9)

2.2. Network stability indicators The stability of the stock market is affected by "external factors" and "intrinsic factors." The former mainly refers to macroeconomic policies and international stock market volatility and the latter mainly refers to the stock market's own factors such as investor sentiment, market operating situation and network structure. This article uses the robustness, stability and volatility of the stock network to measure the stability of the stock market. The first two indicators are based on the stock network topology structure, and the latter indicator focuses on the stock market volatility. (1) Network crash index 5

Robustness is the resistance of the network structure to external destruction. The network with poor robustness is disturbed or destroyed during operation, and the performance will be greatly reduced or even completely lost. The network with better robustness is more robust. We defined network crash index

to reflect the robustness of

the network:

(10) Where N is the number of all nodes in the target network, and

is the number of nodes in the largest

connected subgraph after the network is subjected to random or deterministic attacks. When

1, the network is

0.1, the network is completely destroyed.

complete; when

(2) Stability coefficient Network structure stability coefficient measure the stability of the stock network. On the basis of section 2.1 network modeling and referring to Heiberger's literature, define the stability coefficient NS as follows:

√

(11)

Where N is the number of network nodes, D is the network connection density, and S is the average interaction strength between nodes.

1 indicates that the network is a stable network, and the larger the value is, the

more unstable the network is [48]. (3) Volatility In order to compare with the stability coefficient of the network, we select the square of returns as the proxy variable of volatility. The formula of

is :

100

.

3. Regression models We further explore the features of network stability, the influence of network topology indicators on stock market volatility, and the linkage mechanism between the fluctuation of international stock market and network stability. The network topology indicator is calculated through the risk network. We use multiple linear regression model to analyze the impact of the risk network topological indicators on the stock market volatility. The model is as follows: (12)

where

、

、

、

risk at time t, respectively.

represent network connectivity, clustering coefficient, network density and finance is the stock market volatility,



0,1, … ,4 is correlation parameter, and

is a random error term. Notice that in order to spurious regression, we first perform a unit root test on the stationarity of all explanatory variables. This article chooses ADF inspection method. If a sequence contains a unit root, it is considered to be a nonstationary time series. On the contrary, it is considered as a stationary sequence. When the returns of most stocks have fallen significantly, they can generate strong trading signals of stock market, which will cause a certain impact on institutions and the stock market. From the perspective of maintaining the stability of the stock market, it is very important to explore the abnormal fluctuations mechanism of the stock market. So, we proposed three hypotheses as follows: Hypothesis 1: The stability of the stock market network structure and financial risks have an important impact 6

on the abnormal fluctuations of China's stock market. Hypothesis 2: The fluctuation of the international stock market price has an important impact on the abnormal fluctuations of China's stock market. Hypothesis 3: The volatility of the international gold market and the international crude oil market has an important impact on the abnormal fluctuations of China's stock market. Next, we define negative values with abnormal returns as abnormal fluctuations of stock market. And using binary discrete selection model (probit model) examine the prediction mechanism that the network structure and international market have an effect on the China's stock market volatility. The probit models for three hypotheses are as follows: Model 1. P

1

Model 2. P

1

Model 3. P

1

where

⁄ √

denotes parameter.



(13)



(14) (15)

, and it is the probability density function of the standard normal distribution.

represents the abnormal fluctuation, and it is dichotomous assuming value one if the

market suffers from a negative return larger than three standard deviations (computed from the entire dataset) and zero otherwise. Since this article focuses on the stock market crisis ("accidental fall"), abnormal fluctuations only take into account different levels of negative returns. The maximum likelihood method is usually used to estimate the probit model. It is worth noting that since the probit model is actually a nonlinear regression, the regression coefficient cannot be interpreted as a marginal effect on the dependent variable. However, it can be judged from the sign that an increase of the explanatory variable causes an increase or decrease in the probability that the dependent variable has a certain result. For example, coefficient

is positive, it indicates that the increase in explanatory variables will lead to an increase

in the probability of making certain choices. The sign of the coefficient

is negative, indicates the opposite.

4. Empirical analysis and results 4.1. Data The daily closing price of constituent stocks of China’s Shanghai-Shenzhen 300 Index was selected for the period from July 2006 to December 2017 (2801 trading days). The stocks ultimately selected should satisfy: (i) always in the listing status during the periods of study, and (ii) the duration of consecutive suspension does not exceed 60 days. In the end, we got 83 eligible sample stocks. The data comes from Wind Info. 4.2. Network topology To construct a network based on the multi-scale curve method, we first calculate the daily logarithmic returns, and then use MSDFA to analyze the stock multi-fractal features. In the calculation process, the time scale S is set to [9: 9: N/5] (N is the total of trading days), the order q is set to [-10,10], the de-trended polynomial order is 1. To build a network based on the value-at-risk method, a time span is 100 trading days. Using the three network methods given in Section 2.1., finally choose threshold method to construct adjacency matrix. The average value of the stock price correlation is 0.4, the average value of the multi-scale curve correlation is 0.4 and the average value of the risk value correlation is 0.6, so the thresholds for the three networks are 0.4, 0.4, and 0.6, respectively. Using Ucinet and Pajek software, three associated network topologies are shown in Fig.1. 7

The bbetweenness off the node is a global featuree and reflects th he force and in nfluence of the node or edge in the entiree network. A node n may play a role in conneecting two comm munities. If thee node is attackked, it may cau use the interrruption of the connection c betw ween the two coommunities, wh hich seriously th hreatens the stabbility of the nettwork. As cllearly seen in Fig. F 1 (b), there is one node wiith the largest betweenness b cen ntrality, so com mpared with thee other two nnetworks, the multiscale curv ve network undder the outsidee influence is first f destroyed and shows a strong s instabbility. Fig.1 (a) and (c) have a higher numberr of nodes with higher centrality and its netw work structure iss more compplex.

(aa) Stock price network n

(b) M Multiscale curv ve network

(c) Risk netw work Fig. 1. Threee types of netw work topologicaal structure in China stock markket Inn order to fullyy understand an nd compare toppological structu ure features of three networkss, we gave stattistical charaacteristics of thhree networks in i Table 1. Thee scale index λ is calculated from the doubl ble logarithmic graph (ln P

∗

~ln

∗

of the degree diistribution. Froom Table 1, it can be seen that the scale inddex λ of stock k price

network is 1.67, annd it is less than n 2. While the scale index λ of multiscale curve c network aand risk network are 8

range from 2 to 3. This implied that multiscale curve network and risk network both have scale-free features, while price network does not have scale-free features. From the perspective of the clustering coefficient, we found that they are all much greater than 0. So all three networks have a small world nature. Simultaneously, we noted that the average path length of the risk network is the smallest. To the best of our knowledge, information transmission generally chooses the shortest path in the network. The information travels fewer path, the reliability is stronger and the network robustness is stronger. Therefore, further exploring the robustness of risk network is a very meaningful part to analyze the stability of the risk network. Table 1. Three kinds of network statistical description Scale index λ

Cluster coefficient

Average path length

Average degree

links

price network

1.67

0.808

1.542

38.561

1606

Multiscale curve network

2.43

0.745

2.282

23.045

1440

Risk network

2.40

0.776

1.477

38.463

1631

4.3. Robustness of risk network This part mainly studies the network static robustness. The network static robustness means that when some nodes in the network are deleted, no redistribution of traffic occurs in the network, but the network can still maintain its normal capabilities. The network crash indicators are commonly used to measure the network robustness. For the sake of comparison, we also give the robustness analysis of the stock price correlation network and the multi-scale curve network, and then highlights the more robust properties of the risk network. The attacks of the network are roughly classified into two types in the current research: one is the random removal of nodes or edges of the network, which is called a random attack; the other is to remove the nodes with higher degree or higher betweenness, which is called deliberate attack. The analysis results are shown in Fig. 2.

(a) Stock price network

(b) Multiscale curve network

9

(c) Risk network Fig.2. The relationship between p and G in different networks under two attacks As can be seen from Fig.2., the damage caused by random attacks on the network is significantly weaker than the deliberate attacks. Under random attack conditions, destroying 70% of the nodes in the multiscale curve network means that the network is in a state of near collapse. For the price network and the risk network, only when the proportion of destroyed nodes exceeds 90%, i.e., the network crash index is less than 0.1, they are in a paralyzed state. Under deliberate attacked conditions, the proportion of node deletion in the price network reaches 63%, the ability of the network structure to resist external interference is greatly reduced, and it is nearly collapsed. In a multi-scale curve network, when the proportion of destroyed nodes is 42%, the anti-attack ability of the structure is broken down (the crash index is 0.2), and the network performance is greatly reduced. When 63% of the nodes were destroyed, the network structure collapsed. In the risk network, attacking 70% of the nodes would cause the network structure to collapse. Comparing the anti-random attack and anti-intentional attack ability of the three networks, we find that the risk network has strong robustness both in random attack and deliberate attack, and the network structure is more robust. Judging from the factors affecting the stock market, the international financial market environment, monetary policy, and macro-control will affect the structure of the stock network and the fluctuations of the stock price. Therefore, our analyses on the stock network stability of China's stock market under fluctuations of international stock indexes is particularly important. 4.4. Risk Network Stability Mechanism 4.4.1. Stability analysis In order to explore the dynamic evolutionary process of the risk network stability, we divided the time interval (July 2006 to December 2017) into 46 windows by quarter, and established 46 networks. Under the T-th risk network, the daily return rate sequence of stock i is

, the volatility sequence of stock i is

the average volatility of all stocks in this time window is

∑

, So

. Fig.3 shows the changes in the

stability coefficient of the risk network and the volatility of China's stock markets.

10

100

Fig. 3. The changes of stability index in the China’s stock market As can be seen from Fig.3, the trend of changes in the stability coefficient of the risk network and volatility of stock market is roughly the same, and can be divided into two major stages: the first stage from the T1 to the T22 (2006q3~2011q4), and the second stage from the T23 to the T46 (2012q1~2017q4). Especially on T12 and T40 windows, the network stability coefficient reached the maximum value at each stage, and the network stability was the worst. Simultaneously, the stock market volatility also showed high intensity fluctuations. This is in line with reality  the Chinese stock market happens to be in the 2008 financial crisis and 2015 stock market turmoil. In Fig.3 (the left), the stability coefficient shows an overall upward trend. The stability coefficient around 2015 is significantly higher than the vale in 2008, and the stock market volatility (the right figure of fig.3) at this time also reaches a maximum, indicating that the stock network structure has shown higher instability in recent years, and that there is a certain relationship between the instability and the abnormal fluctuations. In order to distinguish this relationship, it is necessary to test Granger causality between them. The premise of the Granger causality test is that the time series is stable. We first conduct stationary tests on NS and RV series. The unit-root hypothesis can be rejected for two variables. Thus NS and RV are stationary sequence. The Granger causality test results are shown in Table 2. Table 2. Grainger causality test of Stability coefficient and Volatility Null Hypothesis NS is not RV Granger Cause

Obs

F-Statistic

p-value.

45

9.1145

0.0043

16.8805

0.0002

RV is not NS Granger cause

From Table 2, we see that the p-value of the null hypothesis "NS is not the Granger cause of RV" is 0.0043, and it is less than 0.05. So we reject the null hypothesis. It indicates that the stability coefficient of China's stock risk network is the cause of stock market volatility. Furthermore, the p-value of the null hypothesis "RV is not a Granger cause of NS" is 0.002. Therefore, the null hypothesis is rejected, which indicates that the returns volatility of China's stock market is the reason for the stability coefficient of the network structure. The early changes of the returns volatility can effectively explain the changes in the stability coefficient of the stock market network. 4.4.2. Relationship between risk network topology and stock market volatility The stock market risk network portrays the associated structural pattern of risk fluctuations among stocks, we measure the stability by robustness, stability coefficient, and volatility. In order to comprehensively analyze the 11

mechanism of risk network stability of China's stock market, this paper analyzes the effects of network topology structure on stock market volatility and the dynamic relationship between international stock index volatility and China's stock market. The dependent variable is the volatility of China's stock market during the period of （t

1,2,3 … 46. The independent variable include the connectivity of risk network (

clustering coefficient (

), network density (

) and financial risk (

), the ) (We define the

financial risk as the natural logarithm of the average of the eigenvector centrality of financial institutions in the network). To avoid spurious regression, we conduct stationary tests on variables and find that the all null hypothesis is significant at the 1% significance level. The regression model as follows: 1

2

3

(19)

Table3. OLS regression estimation results Dependence variable： Independence variables

Coef

Std.Err

t

P

*

0.0432

5.0495

0.0719

**

0.0526

3.8415

0.0202

***

0.0475

3.2314

0.0024

0.0468

0.3111

0.2057

0.02959

-2.2653

0.0810

-0.0169 -0.0405 0.1535

    _cons R-squared=0.8952

0.7169 -0.0374

*

F=129.1942

Note: *、**、*** represent 0.1、0.05 and 0.01 significance level respectively.

Table 3 reports the impact of network topology indicators on the volatility of China’s stock market, where the connectivity of the risk network is significantly negative (-0.0169) at a significance level of 1%. It indicates that increased network connectivity will reduce stock market volatility. Since a highly-connected network is more conducive to risk dispersion, and further to reduce the volatility of stock returns and maintain network stability. The network clustering coefficient is significantly negative for stock market volatility at a 5% significance level, indicating that the clustering coefficient of the network is greater, the smaller the stock market volatility is. The approximate reciprocal relationship exists the cluster-degree correlation of the nodes in the network [49]. That is, the local clustering coefficient of the network is larger, the node with high degree in the local network is fewer. So it is not easy to be the target of the impact in crisis period, and could reduce the stock market volatility. The network density is significantly positive at a significance level of 1%, which means that the rise of the links in the network will increase the stock market volatility. Financial risk is positively correlated with stock market volatility, but it does not significantly. 4.4.3. Relationship between international stock indexes and risk network stability The stability of China's stock market risk network is not only affected by the network structure and domestic macroeconomic policies, but also affected by the changes of international stock market. The fluctuations of international stock market can generate stronger market trading signals, and cause a certain impact on domestic companies and stock markets. When the impact of stock volatility is large enough, the entire stock market will encounter greater volatility risks, and even collapse. From the perspective of maintaining the stability of stock market, it is extremely important to explore the relationship between international stock market volatility and the stability of China's stock market network. Five international markets are as follows: US S&P 500 Index, Hong Kong Hang Seng Index, Japan's Nikkei 225 Index, International gold market (London gold spot daily closing 12

price), International crude oil market (Brent crude oil spot daily closing price). In order to quantitatively analyze the inner relationship between the stability coefficient of China’s stock market network and the fluctuations of international stock indexes. We established a cross-correlation function between network stability and international stock market volatility. The analysis results of cross-correlation are shown in Fig. 4.

Fig.4 Cross-correlation coefficient between network stability and international market returns It is evident from the fig.4 that the Hong Kong Hang Seng Index, the Nikkei 225 Index and the US S&P 500 index have a positive cross-correlation relationship with China's stock market network stability coefficient during the lagging period of the first seven periods. Subsequently, the positive and negative changes in correlation coefficient appear alternately. However, the effect of the smaller correlation coefficient is not significant. It indicates that the sharp fluctuations of the international stock market can cause larger stability coefficient of 13

China’s stock market network, further leading to unstable in China’s stock market. The correlation coefficient between network stability coefficient and current Hong Kong Hang Seng Index's return fluctuation is the highest (0.4439), and with Nikkei 225 index's return fluctuations reaches the maximum (0.3958) after lagging second-period. Compared with the US, the correlation coefficient between network stability coefficient and the current US S&P500 index's return fluctuations is the smallest (0.2819). So we believe that the stability of China's stock market network is least affected by the US market, and it is most closely related to the Hong Kong market fluctuations. Since the stock market of China’s stock market has been relatively closed to for a long time, and it is more severely constrained by national policies. With the implementation of the QFII system and the Shanghai-Hong Kong Stock Connect system, the degree of capital opening between China and Hong Kong is relatively high. Additionally, both the mainland stock market and the Hong Kong stock market are affected by the country's macroeconomic policies, so they have a more closely relationship. The price fluctuations in the international gold market and the international crude oil market have negative correlations with the stability coefficient of China’s stock market network for most of periods. As the lag period prolongs, the correlations gradually decrease. This conclusion is consistent with the view of DIAZ et al. [50], who thinks that the increase of oil price volatility has a negative impact on the G7 countries' stock markets. The maximum negative correlation between the network stability coefficient and the international gold market or the international crude oil market are -0.3169 and -0.2884, respectively. Compared with the international crude oil market, the relationship between the international gold market and the stability coefficient of China's stock network is even greater. It means that the volatility of the international gold market is more likely to affect the structure of the China’s stock market network and make it turbulent. 4.4.4. Probit test results To further analyze the ability of the network structure and international market factors to predict abnormal fluctuations in China's stock market, we use probit model to estimate the severe negative market changes (outlier measured by choosing less than three standard deviations of negative returns). The dependent variable is "accidentally dropped" which is assigned a value of 1, otherwise it is 0. For independent variables: Model 1 selects risk network indicators (such as network stability coefficient and eigenvector centrality of financial institutions), Model 2 selects index prices of international stock market (such as Hang Seng Index, S&P 500 Index and Nikkei 225 index, taking the logarithm of the price), and Model 3 selects the international gold price and international oil price (logarithm) which have the attributes of financial and physical assets. We use bootstrap○1 methods mentioned by Gustavo [51], it is a method of ‘resampling’ the original sample (without relying on additional samples or data). Efron [52] has proven to be a powerful nonparametric tool for approximating the sampling distribution and variance of complicated statistics based on iid observations. 1

Table 4 shows probit test regression results with three different types of independent variables. The results of

Model 1 indicates that the increase of the risk network stability coefficient will increase the probability of China’s stock market facing a sharp decline, and this possibility is significantly at a significance level of 1%. The eigenvector centrality of financial institutions has a significant positive effect on the abnormal fluctuation, indicating that the significant increase in the eigenvector centrality of financial institutions could also increase the probability of a sharp decline in China's stock market. The results of Model 2 shows that the regression

○1

The original meaning of “Bootstrap” is shoelace. There is a saying that "people should tie their own laces, that is, people should be 'self-help'". Suppose that a random sample with a capacity of N is extracted from the overall datasets. To a certain extent, this sample can be regarded as a whole, and then the sample is ‘with replacement’. The sample size is still N. This sample is called ‘bootstrap sample’. The operating procedures of the bootstrap method are roughly divided into three categories, namely, nonparametric bootstrap, parametric bootstrap and residual bootstrap. Detailed algorithm process reference literature [52]. 14

coefficients of the Hong Kong Hang Seng Index price, the US S&P 500 index price and the Nikkei 225 index price are all negative, but only the Hong Kong Hang Seng Index has a significant negative effect on the stock market abnormal fluctuations at the 10% significance level. This means that the rising of Hong Kong stock prices will reduce the frequency of abnormally falling volatility in China's stock market. Therefore, the sharp drop of stock price in Hong Kong stock market has a certain warning effect on forecasting the decline of China's stock market, while the decline of stock price in the US and Japan does not have the ability to explain the severe abnormal fluctuations in China's stock market. At least for the current period, the decline of the international stock market is not enough to predict the abnormal fluctuations of China’s stock market. The results of Model 3 shows that the international gold price and abnormal fluctuations of stock market are negatively correlated at the 1% significance level. It indicates that in crisis period, the more serious stock market falls, the more it will increase the possibility of decline in international gold prices. That is, when the market is in serious condition, the hedging effect of gold is not significant, and it shows the same changes with the stock market. The regression coefficient of international crude oil prices is negative, while it is not significant. Comparing the three models, we believe that during the period of abnormal decline in China's stock market, the ability of the stock network structure to predict abnormal fluctuations is much higher than the external factors such as the international stock market. Secondly, during the severe crisis of the stock market, the role of gold as a hedging asset is no longer significant. Instead, it has the same changes as the stock market. Table 4. Probit model estimation results. Model 1 NS

0.6898

Model 2

Model 3

***

(3.59) lnFinance

13.5989*** (3,89) -5.7615*

lnPHIS

(1.81) lnPBP500

-1.9638 (-0.06)

lnPN225

3.4843 (0.17) -7.5859***

lnPgold

(-4.65) lnPoil

-2.5558 (-0.99)

_cons

Log likelihood 2

R

-2.5313

***

-1.5209

***

26.9063***

(-3.65)

(-3.35)

(4.74)

-22.9615

-21.2285

-24.2041

0.6418

0.2825

0.2776

Note: *、**、*** represent 0.1、0.05 and 0.01 significance level, respectively.

5. Conclusions The analysis of the stability characteristics of the stock market network structure and linkage analysis with international stock markets is of great significance for effectively identifying stock market risk, risk warning and stock investment. From the perspective of network modeling, this paper selects constituent stocks of the China 15

Shanghai-Shenzhen 300 Index from July 2006 to December 2017, and uses three modeling methods to establish three networks (stock price network, multi-scale curve network, and risk network). We analyzed the stock market stability including stock market network robustness, stability coefficient and volatility. Then using OLS linear regression empirically analyzed the effect of risk network structure index on stock market volatility and using cross-correlation function observed lagging relationship between the stability coefficient of risk network and the fluctuations in the international market. Finally using the probit model we tested the ability of the network structure and international market factors for predicting abnormal fluctuations in China's stock market. The following results are obtained: (1) Three networks of the China stock market constructed based on stock prices, multi-scale curves and risk values three different methods. By network structure, it is found that all three are typical small-world networks, indicating that stocks in the stock market have smaller average paths, so stocks inter risks can easily be transmitted to other stocks via the stock-linked network. However, the stock price network does not have scale-free features. Multi-scale curve network and risk network all have scale-free features, indicating that the two networks have a “Matthew effect”. There are many Hub nodes of multi-scale curve network and risk network, and these stocks will have a greater impact on other stocks in risk contagion and stock trading. (2) From the robustness perspective of network stability, it is found that the risk network shows strong robustness both against random attacks and deliberate attacks. Comparing the clustering coefficients of the three networks, it is found that the price network with the highest clustering coefficient has the lowest robustness, and the risk network with the second highest clustering coefficient has the best robustness, contrary to the view of Stefania [53] who thinks the higher the clustering coefficient is, the better the robustness of the network is. This is because the shortest path length of risk network is smallest. In the network communication process, nodes often transmit with a certain shortest path. If it across fewer paths, the network reliability is stronger and the network is more robust. (3) Analyzing the relationship between stock market volatility and stock market stability coefficient from the point of view of network stability, and applying Granger causality test to test it. We show that the stability coefficient of risk network in China stock markets is the cause of stock market volatility, while the stock market volatility is the reason for the stability coefficient. And the early changes in the stock market volatility can effectively explain the changes in the stability coefficient of the stock market network. Finally, we believe that reducing the connectivity of the network and enhancing the network clustering coefficient are conducive in reducing the volatility of the stock market, and thus maintaining the stability of the stock market. (4) Considering the impact of the international stock market, we found that the volatility of the Hong Kong stock market, the Japanese stock market and the US stock market are all positively related to the stability of China's stock market, but only the current Hong Kong stock market volatility is most closely related to the stability of China's stock market. The price fluctuations of international gold and oil with physical and financial attributes are negatively related to the stability of China's stock market network, indicating that in the long-term, the gold market and crude oil market have a “seesaw” effect on the stability of China’s stock market. However, it is worth noting that the probit test results also prove that the "seesaw" effect of gold and crude oil on the stability of China's stock market has disappeared in the crisis period of abnormal fluctuations (strikingly down) in China's stock market. Instead, it presents the same trend of stock market changes and the functions of gold as a safe-haven asset is no longer obvious. (5) In the forecast of abnormal fluctuations, the significant increase in the stability coefficient of the network and the significant increase in the eigenvector of financial institutions will indicate that the stock market's future negative volatility will intensify. The ability of the network structure to interpret the abnormal fluctuations in China's stock market is more effective than the external factors such as fluctuations in the international stock 16

market.

Acknowledgment This research was supported by the National Nature Science Foundation of China (Grant No. 71671030, and 71571038).

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