Discrete Dynamics in Nature and Society

Volume 2018, Article ID 9461870, 13 pages

https://doi.org/10.1155/2018/9461870

## Dynamic Contagion of Systemic Risks on Global Main Equity Markets Based on Granger Causality Networks

^{1}School of Business, University of Shanghai for Science &Technology, Shanghai 200093, China^{2}School of Business, Zhejiang Wanli University, Ningbo 315100, China

Correspondence should be addressed to Qiuhong Zheng; moc.361@0101gnohuiq

Received 1 May 2018; Revised 15 July 2018; Accepted 19 July 2018; Published 7 August 2018

Academic Editor: Ricardo López-Ruiz

Copyright © 2018 Qiuhong Zheng and Liangrong Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A total of 156 Granger causal networks of stock markets are constructed by using the Granger causality test and time series sliding window based on stock index data of 34 major stock markets in the world from 2004 to 2017. The topological structures and evolution characteristics of the Granger causal networks are analyzed from the perspective of complex network theory. Empirical results demonstrate that the network topology has a significant difference during the global financial crisis and other periods. The causal relationships among different global stock markets exhibit a jump growth when each major crisis occurs. The contagion path is also short. A causal relationship between any two stock markets can usually be established with one stock market on average, not by using more than five stock markets. For risk contagion, the American stock markets exerted the largest influence in 12 years, followed by the European stock markets. Stock markets with high intermediate contagion ability play an important role in systemic risk contagion. Despite the crucial markets in Europe and America (e.g., USA, Brazil, and Mexico), stock markets with weak network correlation and strong media ability (e.g., the markets of Japan, Korea, Australia, and New Zealand) play a critical role in risk contagion.

#### 1. Introduction

The American subprime mortgage crisis reignited the intense concern of economists regarding financial systemic risk. The stock market is the most important component in the entire financial system. Risks of a single stock market can spread to other correlated stock markets and even to the whole financial system, thus generating systemic risks. The strong risk contagion of stock markets can cause immense damage to the whole financial system, which may induce a financial crisis. Given the bankruptcy of subprime mortgage institutions and the forced closure of investment funds, the subprime crisis occurred in 2007, which then triggered violent fluctuations of the American stock markets. This subprime crisis swept global major stock markets in the European Union and Japan, which eventually led to the global financial crisis. Therefore, governments worldwide must strengthen supervision over stock markets, maintain market stability, and prevent systemic risk during economic development.

Scholars in the 2008 conference on the “New Progresses on Research of Financial Systemic Risks” in London generally believed that research on financial systemic risks emphasizes the effects of the financial asset price fluctuation possessed by participators in financial market on the whole system. During a crisis, the interaction of such price fluctuation is further intensified by the significant convergence, correlation, and systemic risk contagion in the financial market. Given this background, understanding the complicated correlations among global major financial markets, propagation paths, and key nodes for propagation from the perspective of stock price is imperative to prevent systematic risk contagion. A scientific alarm system and monitoring mechanism are then established.

Systemic risk studies primarily employ three important measurement models: conditional value-at-risk (CoVaR) [1], a method proposed by Adrian and Brunnermeier; the systemic expected shortfall (SES) [2], an approach suggested by Acharya et al.; and the distressed insurance premium (DIP) [3], a technique proposed by Huang, Zhou, and Zhu. These models measure systemic risks based on data at the crisis outbreak. During and after the occurrence of a financial crisis, financial institutions or financial markets show considerable high correlation. However, before the occurrence of a financial crisis, such a correlation seems hardly significant in the measurement of systemic risk. Research on systemic risk in the post-crisis era focuses on the correlation and comovement among financial institutions before and during the crisis.

Complex network [4–6] has been a popular research topic in recent years and has been widely applied in finance. Constructing an economic and financial complex network based on the financial time series (generally stock return series) can systematically and intuitively express mutual dependence among different financial institutions [7–10]. Unconditional direct relationships between all financial institutions and markets can be disclosed by the complex network, especially the relationships among financial institutions that have not suffered similar losses in the crisis. Eryiğit et al. [11] established a complex network of 143 stock indexes of 59 countries by calculating the correlation coefficient among different stock yields by using the minimum spanning tree and the plane maximum filtering graph methods. They found that North American and European markets have the closest relationships, whereas the correlations among countries in Eastern Asian countries as well as between Eastern Asian and Western markets are weaker than other markets. Kenett et al. [12] established a network of well-capitalized 300 stocks in the New York Stock Exchange based on the partial autocorrelation among stock yields. Compared with the correlation coefficient, the partial correlation coefficient can measure the effects of one stock on the correlation coefficient between two other stocks. Such a correlation also has high application values for studying systemic risk contagion. Giuseppe et al. [13] analyzed 49 industries in the American equity markets from 1969 to 2011 and rendered mutation points of the average correlation coefficients of different industries corresponding to the financial crisis points, thus confirming the close correlation between risk contagion and stock correlation. Chunxia Y [14] conducted an in-depth analysis of 789 stocks in the global major stock markets based on the correlation price of stock prices. The systemic risk problem of stock markets in different stages is also studied. The empirical study reveals that a financial crisis may not change the correlation of stocks, but it will increase and then decrease the correlation coefficient.

There were also other good approaches for comovements modeling in finance, such as the technique of hierarchical clustering. Lahmiri S [15] dealt with the problem of Casablanca Stock Exchange (CSE) topology modeling as a complex network using hierarchical clustering linkage technique. The general structure of the CSE topology has considerably changed in 2009(variable regime), 2010 (increasing regime), and 2011 (decreasing regime). Lahmiri S [16] examined short and long-term dynamics in linkages between global major markets during and after financial crisis based on the wavelet presentation of clustering analysis. The empirical results show strong evidence of the instability of the financial system aftermath of the global financial crisis

The approaches in the aforementioned literatures are mainly undirected correlation modeling. However, measuring not only the degree of connectedness between financial institutions but also the directionality of such relationships is important to investigate the dynamic propagation of shocks to the system. Therefore, constructing a directed complex network is necessary to recognize the source of systemic risk and the important nodes in the contagion. However, only a few studies related to the subject are reported. Billio M et al. [17] developed a directed causal network by using the yield data of hedge funds, banks, security traders, and insurance companies in the American financial market. The important role of banks in systemic risk contagion is also uncovered. Mensah J O et al. [18] established the Granger causal network of Asian banks through the CoVaR method and found that the correlations of Asian banks generally increase. The universal systemic risk is higher than those in other emerging markets. Lahmiri S [19] investigated cointegration and causal linkages among five different fertilizer markets during low and high market regimes. Fertilizer markets are closely linked to each other during low and high regimes. Jiang M [20] proposed an algorithm to transfer this evolution process to a complex network. Causality patterns are considered as nodes and the succeeding sequence relations between patterns as edges. The results show that a few types of causality patterns play a major role in the process of the transition and that international crude oil market is statistically and significantly not random. Li L [21] built a co-loan network to research the topological structures and corresponding evolvement characteristics of the Chinese banking system from 2008 to 2016. The co-loan network always displays a core-periphery structure.

Based on the preceding studies, an inference stating that a directed network better reflects systemic risk contagion than its undirected counterpart can be rendered. However, existing literature minimally addresses the directed and dynamic systemic risk contagion among global major stock markets. To this end, we propose using Granger causality measure of connectedness to construct the directed network. We can find Granger causality among price changes of financial assets in the presence of value-at-risk constraints or other market frictions such as transaction costs, borrowing constraints, costs of gathering and processing information, and institutional restrictions on short sales. The degree of Granger causality in asset returns can be viewed as a proxy for return-spillover effects among market participants. As this effect is amplified, connection and integration among financial institutions are tight, heightening the severity of systemic events [17]. Moreover, a Granger causality measure of connectedness can capture the lagged propagation of return spillovers in the financial system. Therefore, Granger causality network can best describe the systemic risk contagion among global major stock markets.

This study collected the stock indexes of 34 global major stock markets from 2004 to 2017. The source of systemic risk contagion and important nodes according to the complicated relationships of yield spillover and systemic risk contagion among research markets in a directed causal network were also identified. Moreover, the evolution characteristics of the stock market network are explored.

The remainder of this paper is organized as follows. Section 2 introduces the research methodology and data source in the empirical analysis and presents the preliminary data processing. Section 3 analyzes and discusses dynamic risk contagion among global major stock markets according to the topological features of the 156 Granger causal networks constructed. Section 4 concludes.

#### 2. Data and Research Methods

##### 2.1. Data Selection and Processing

In this study, index data of 34 major stock markets in Asia, America, Europe, and Oceania from May 3, 2004, to June 30, 2017, were selected. Complete stock market index data are shown in the Appendix. To discuss dynamic contagion of systemic risk among different stock markets, the actual model used sliding window (length: three months and one month for each sliding) to select 156 subsamples and 156 constructed Granger causality network groups. For example, data in May, June, and July 2004 were selected as one subsample to discuss causal relations and systemic risk contagion among different stock markets in July 2004. The rest could be conducted in the same manner.

Stationary test of data is necessary before Granger causality test. Unit-root test is generally applied. Certain financial time series, such as volatility and return series, are stationary in most cases. Therefore, logarithmic returns of closing price were selected as research data. The formula is as follows:where is the closing price of index on .

Before the analysis, return series of all 34 stock market indexes were examined by unit-root test [22]. The results show that each return series possessed no unit-root. All return series were stationary.

##### 2.2. Granger Causality Test

In this study, mutual overflow relations among global major stock markets were recognized by Granger causality test [23, 24]. Time series is said to “Granger cause” time series if past values of contain information that helps predict above and beyond the information contained in the past value of alone[17]. In one binary -order VAR model,

Variable is not the Granger cause of if and only if all coefficients in the coefficient matrix are zero. This condition indicates that cannot cause changes in . Granger causality test examines whether lagged variable of one variable can be introduced into other variable equations. If variable can help interpret variable , then variable is the Granger cause of variable . F-test is a direct method to judge Granger cause:

The statistics are as follows:

They conform to the F distribution. If is higher than the critical value of F, then the null hypothesis is rejected. Variable is the Granger cause of variable . Otherwise, the null hypothesis is accepted. is the residual sum of squares in the equation (2): , and is the residual sum of squares in the equation when .

Granger causality test results are closely related to the number of lag orders. Therefore, the appropriate in VAR model should be selected appropriately. On the one hand, is expected to be sufficiently large to reflect dynamic features of the constructed model completely. On the other hand, large bring several parameters for estimation, thereby reducing degrees of freedom (DOFs) of the model. Thus, comprehensive considerations to the quantity of lag items and DOFs are necessary when selecting the number of lag orders. In actual studies, Akaike Information Criterion (AIC) and Schwarz Criterion (SC) are common methods. The calculation formula is as follows:where is the total number of estimated parameters, is the number of endogenous variables, is the sample length, is the number of exogenous variables, and is the number of lag orders. The logarithmic likelihood value can be calculated by hypothesizing that the multivariate normal distribution is obeyed, as follows:Low values of AIC and SC are ideal [24].

##### 2.3. Granger Causality Network

The causality network at t was established through Granger causality test. Among them, the point set is listed companied (number of points is 34 at this moment), and the side set contains all sides between any two points . If is the Granger cause of , then one directed side from to () exists. Otherwise, no side exists. The following causality index function is defined:

#### 3. Empirical Results

The reasonable number of lag order was determined by AIC and SC in (5). The VAR model was established to test F statistics in (4), thereby obtaining causalities of daily return rates of 34 stock markets. The stock market indexes were used as network nodes. The Granger causality network groups (156 in total) that represent monthly causality from July 2004 to June 2017 were established in accordance with (7) by using 156 subsample data. Therefore, studying complicated causal relations of different stock markets is equal to studying causality network characteristics of stock market, including time-varying and important node characteristics.

Based on the established 156 Granger causality networks that change with time, important network nodes, and the differences of network topology before, during, and after financial crisis were analyzed by calculating out-degree and in-degree of network nodes. Other topological properties of each network were also examined.

##### 3.1. Out-Degree and In-Degree of Granger Causality Networks

Given that Granger causality networks are directed networks, the following out-degree and in-degree of causality networks are defined:where is the point set in the entire network.

The definition shows that the out-degree and in-degree of nodes in causality networks can reflect the influence of different stock markets in the network. If one node possesses high out-degree, then it possesses more one-way causations than other nodes and the stock market is influential. If one node possesses high in-degree, then other nodes possess more one-way causations than this node and the stock market can be easily influenced by other markets.

Figure 1 shows the color code representations of the time evolution of out-degrees. Out-degrees of the American stock market were kept at a high level in most time, whereas out-degrees of the Asian and Oceania stock markets were low in most time. From the data statistics in Table 1, the American stock market was found with the highest average out-degrees and standard deviation (SD) in the past 14 years. Specifically, the New York Stock Exchange showed the maximum out-degree (17.11 in average). In terms of skewness and kurtosis of out-degrees, some values of most American and European markets were close to 0 and approached to normal distribution with time. Monthly out-degree values are distributed at the two sides of the average symmetrically. Time-related distribution of out-degree values of the Asian and Oceania stock markets presented “peaks” and “right avertence.” This condition indicated that out-degrees concentrated at peak values and that such values were smaller than average values. The minimum out-degree was contributed by Shanghai Securities Composite Index in China. From the given statistical features, the American stock market was found to be the most influential in global stock markets, whereas the Asian and Oceania stock markets were the least influential. Out-degrees of the Asian and Oceania stock markets were lower than the average level in most time.