Spatial linkage of volatility spillovers and its explanation across G20 stock markets: A network framework

Journal Pre-proof Spatial linkage of volatility spillovers and its explanation across G20 stock markets: A network framework

Weiping Zhang, Xintian Zhuang, Yang Lu, Jian Wang PII:

S1057-5219(19)30538-1

DOI:

https://doi.org/10.1016/j.irfa.2020.101454

Reference:

FINANA 101454

To appear in:

International Review of Financial Analysis

Received date:

29 August 2019

Revised date:

8 January 2020

Accepted date:

12 January 2020

Please cite this article as: W. Zhang, X. Zhuang, Y. Lu, et al., Spatial linkage of volatility spillovers and its explanation across G20 stock markets: A network framework, International Review of Financial Analysis(2020), https://doi.org/10.1016/ j.irfa.2020.101454

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© 2020 Published by Elsevier.

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Spatial linkage of volatility spillovers and its explanation across G20 stock markets: A network framework Weiping Zhanga, Xintian Zhuangb*, Yang Luc and Jian Wangd a

School of Business Administration, Northeastern University, Shenyang, China E-mail: [email protected]

School of Business Administration, Northeastern University, Shenyang, China

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School of Business Administration, Northeastern University, Shenyang, China

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E-mail: [email protected]

School of Business Administration, Northeastern University, Shenyang, China

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E-mail: [email protected]

E-mail: [email protected]

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*Corresponding Author: Xintian Zhuang, Professor. E-mail: [email protected]

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Address: School of Business Administration, Northeastern University, No.195, Innovation road, Hunnan New District, Shenyang, Liaoning, 110167, China.

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Journal Pre-proof Spatial linkage of volatility spillovers and its explanation across G20 stock markets: A network framework Abstract

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This paper empirically estimates the spatial correlation relationship of volatility spillovers and its influencing factors across G20 stock market. We apply GARCH-BEKK model to estimate volatility spillover and construct dynamic volatility networks. The connectedness analysis shows that the spatial linkage of volatility spillover is time varying and has obvious multiple superposition phenomena. As somewhat innovation results, we use the factor analysis method to obtain centrality comprehensive indicators that can clearly depict the risk contagion intensity and risk acceptance intensity. In general, the developed markets are more influential than the emerging markets during periods of turbulence, and the emerging markets are more sensitive to volatility shocks than developed markets during any period. Finally, this paper introduces quadratic assignment procedure (QAP) method to identify the major factors that influence the spatial linkage of volatility spillovers. Results show that geography influences the volatility spatial correlation differently across economic cycles, and the centrality structure factors have greater impact on the spatial correlation than the external economic factors. The QAP regression analysis shows that these influencing factors can explain about 50% of the spatial correlation variation of international financial markets' volatility spillovers.

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1. Introduction

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Key words: Spatial linkage; Volatility spillover networks; G20 stock markets; QAP analysis; GARCH-BEKK model JEL codes: C58, G15, F3

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Measuring co-movements among international financial markets is a widely debated issue since the economy in isolated countries have gradually integrated into an entire global economy system with strong coupling. The rapid development of the global economy not bring the benefits from international portfolio diversification, but also creates opportunities for the spread of volatility and risk in the world. Especially during the frequent occurrence of financial crisis (e.g., 1997 Asian crisis, 2008 Subprime Mortgage, and 2011 Sovereign Debt crisis), a clear need arises among scholars: how to identify the presence of spatial linkage among markets, and contagion mechanism that allows the volatility risk from one economy to the others (see, Kolb, 2011). Moreover, it initiated discussions about whether these spatial linkages are different in turbulence times compared to stable times. There seems to be general agreement on the fact that the global financial markets have become increasing integrated and highly complex, with cross-border spatial connections and spatial dependences (Zhang et al., 2019). Thus, a new strand of research was dedicated to the analysis of broader connections (e.g. volatility and risk) among developed financial markets as well as emerging countries. A thorough analysis about volatility spillover and its spatial linkage could successfully manage systematic risks and keep financial stability, which, in turn, contributes to the smooth functioning of the real economy. Some scholars found that volatility and risk spillover effects generate frequently in the integrated economic system, and the worldwide financial instability mainly comes from of advanced markets and emerging markets (Cardona et al., 2017). 2

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2. Related literature

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The G20 economies, a big group of major developed countries and emerging markets, accounts for approximately 85% of Gross World Product, as well as approximately 80% of the world trading. Therefore, a financial turbulence in G20 represents large changes in global economics, and choosing G20 being the research object is very suitable. It is difficult to identify the complex financial market spillovers by using the economics methods alone. First, a suitable empirical framework is expected to map the various spillover channels simultaneously. This paper estimates the volatility spillover among G20 stock markets, using the GARCH-BEKK model proposed by Engle and Kroner (1995). The BEKK specification can explicitly estimate cross effects of volatilities and perturbation across different markets. Second, the model should take into account markedly contemporaneous spatial correlations, so that the whole structural characteristics of the financial system are observed in a proper way. In order to deal with the problem, we introduce complex network theory which considers markets as nodes and spillover relationships as edges. It can help us grasp the structural features across G20 stock markets and help us to predict the impact of financial crisis. Of course, it is also critical to understand the factors that influence the spatial linkage of volatility spillovers. It can provide the government with suggestions to take the appropriate rescue plans for the financial turbulence, and help investors to formulate investment strategies and diversify investment risks in advance according to the changes of macroeconomic environment. The remainder of this article is organized as follows. Section 2 reviews the related literature. Section 3 presents the econometric modeling framework. Section 4 describes the data. Section 5 discusses the empirical results. Section 6 concludes the paper.

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First, we want to review the existing concerning the volatility spillover that is a phenomenon of information transmission across different financial markets. Seth et al. (2018) argued that volatility related with financial markets is defined as the intensity of price fluctuations in financial instruments. To the best of our known, price volatility could reflect investors’ fear. Therefore, Diebold and Yilmaz (2014) regard volatility connectedness as “fear connectedness”. What’s more, volatility is widely used as a measure of risk. Generally, the higher the volatility, the greater risky the stock market is. When financial market is in depression or uncertainty, its volatility can be high. Thereby, volatility spillover is distress-sensitive. This is the reason that we focus our attention on spatial linkages of volatility spillover among international stock markets. Some researches argued that spillover effects are one kind of common phenomenon of volatility transmission between financial markets (Yu et al., 2015; Rejeb et al., 2016). There is increasing number of studies that analyze volatility spillover in context of developed markets (Golosnoy et al., 2015), emerging markets (Maghyereh and Awartani, 2012; Nishimura et al., 2015), commodity markets (Yoon et al., 2019) and sovereign bond markets (Piljak and Swinkels, 2017). From the econometric methodologies, there are three models to measure the volatility spillover effects. The first is Stochastic Volatility (SV) model (Clark et al., 2010; Keating et al., 2015). Following this method, Tian et al. (2016) used an integrated stochastic volatility (TV-SVAR-SV) to examine the cross-market volatility transmission mechanism in the US financial system. The second is the multivariate generalized autoregressive conditional heteroskedasticty (GARCH) family model (Chege et al., 2014; Asaturov et al., 2015). Baldi et al. (2016) applied GARCH-BEKK model to investigate volatility transmission between SP500 and commodity markets. Apart from the 3

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methods above, Diebold and Yilmaz (2009) proposed a new method (DY 2009) to measure return and volatility spillover among different stock markets, based on forecast-error variance decompositions in a VAR framework. Zhou et al. (2012) provided measures of the magnitude and direction of the volatility spillover between Chinese and world stock markets through the Diebold and Yilmaz’s (2011b) framework. Recently, the multivariate GARCH model and DY 2009 are most commonly used by scholars (de Oliveira et al., 2018; Chowdhury et al., 2019), while in this paper, we choose GARCH-BEKK model to test the volatility relationship among financial markets. The reasons are that: as we all know, the financial series have the characteristics of volatility cluster (Bollerslev, 1990; Engle, 1993). In addition, the correlation of assets is time-varying and is higher during high fluctuation periods (Yang, 2005). Compared with DY 2009, GARCH-BEKK model can directly compute the volatility from the variance matrix, which makes up for the shortcoming that the covariance matrix under the generalized decomposition is assumed to be time-invariant. And it has the advantage of avoiding the manual definition of volatility. In addition, the BEKK model does not force any constraints on the correlation structures between variables and automatically ensures that the variance matrix is positive (Gounopoulos et al., 2013; Boldanov et al., 2016; MacDonald et al., 2018). Considering that the high-dimensional interconnected financial systems, the methods mentioned above are not enough to depict the spatial linkages in volatility spillover network of financial markets. Some researchers have more recently focused on combing the econometric model with the network method to build financial network and to assess the spatial linkages among stock markets and the volatility transmission mechanisms (see Billio et al., 2012; Tonzer, 2015; Wang et al., 2018). And, complex network theory has been widely applied in the fields of international trading (Zhong et al. 2014), energy economies (Hao et al., 2016) and financial markets (Berger and Uddin 2016; Zhang and Zhuang, 2018). From information spillover perspective, the existing literatures on financial networks are mainly categorized into three classes: (i) Granger-causality network (that is mean-spillover network), which is proposed by Billio et al. (2012); (ii) volatility spillover network, including the variance decomposition frame-based network (Diebold and Yilmaz, 2014), and the GARCH model-based network (Liu et al., 2017); (iii) tail-risk spillover network (Hautsch et al., 2015; Hardle et al., 2016), and extreme risk network (Wang et al., 2017). Furthermore, many studies have discussed the application of spillover networks. Based on the framework of multivariate GARCH models, Liu et al. (2017) analyzed the features of volatility network in international financial markets, and they were the first to combined GARCH-BEKK model with network method. Huang et al. (2018) built the volatility spillover networks of Chinese financial institutions by VAR-MGARCH in a BEKK form, and then analyzed systemically important institutions and its influential factors. Wang et al. (2019) applied GARCH-BEKK model to capture volatility relationships, and constructed spillover networks of energy stocks to identify influential energy stocks. Besides, considering the multi-scale information, Feng et al. (2018) not only constructed the volatility spillover network under the GARCH-BEKK model, but also designed a research framework through the wavelet analysis method to explore the linkages among sectors in China. The last strand is associated with the influential factors that amplify or mitigate spatial volatility spillover. Buitron and Vesperoni (2015) argued that the financial spillover is related to the current tightening of the Fed’s monetary stance, which is likely to create non-negligible 4

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spillovers for other countries like Japan, Euro area and others. The research of Bernoth and Konig (2016) shows that the pass-through of an appreciated US dollar could result in higher inflation and output growth in other countries, which will, in turn, cause an increase in financial risk spillover. It is important to note that related arguments spillovers may take place independently of the exchange rate regime (Belke and Rees, 2014). Ahmad et al. (2018) used panel regression method to analyze the determinants of net spillover. They found that government debt, current account deficit, and interest rate are major determinants of financial connectedness. Lyocsa et al. (2017) fitted spatial autoregressive probit model to identify the determinants of the connectedness within the equity market networks. Some studies incorporated the impact of financial crisis on the spillover effects among financial markets (Sakthivel et al., 2014). Fern ́ ndez-Rodr ́guez et al. (2016) examined the time-varying behavior of net pair-wise directional connectedness at different stages of the recent financial crisis. Therefore, when analyzing the volatility spatial linkage, it is necessary to consider the time variable of the economic cycle. Our study contributes to the literature in two aspects. First, in prior literature (Baele, 2005) mainly documents the existence of return or volatility spillover across markets. However, in this paper, we are interested in exploring the dynamic spatial linkage of the direction volatility networks, and using factor analysis approach to find the key nodes of the risk contagion intensity and risk acceptance intensity. Second, the analysis about the factors of return or volatility spillovers in the existing literatures mostly use linear regression, ordinary panel regression and spatial autoregressive models. However, these traditional methods cannot solve the correlation problem between relational data, such as the influence of geography adjacency matrix on the spatial correlation matrix of volatility. This is the first study in the literature to use QAP method, a non-parametric method, to test the correlation between relational matrices, and to obtain the factors affecting the spatial linkage of volatility spillovers.

3. Econometric modeling framework

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Logarithmic return series of each stock index are calculated by the following Eq. (1): 𝑅𝑖,𝑡 = [ln(𝑃𝑖,𝑡 ) − ln⁡(𝑃𝑖,𝑡−1 )] ∗ 100 , (1) where 𝑃𝑖,𝑡 is the stock index price for country i at time t; 𝑃𝑖,𝑡−1 is the stock index price for country i at time t-1.

3.1 Volatility spillover modeling

The volatility spillover relations between each stock index are estimated by GARCH-BEKK model (Lin, 2015; Bae et al., 2003). In this model, it does not impose any restriction on the correlation structure between the variables. We choose one lag phase in this paper, because it is sufficient to model the conditional covariance. If we increase the lag order, the number of estimated parameters increase significantly, which brings practical calculation issues. The bivariate GARCH-BEKK model can be expressed as following: 𝑅𝑡 = +φ𝑅(𝑡−1) + 𝜀𝑡 , (2) 𝑅

𝑘

𝜑

𝜀 (𝑡)

𝜑

1/2

where 𝑅𝑡 = [𝑅1𝑡], 𝐾 = [𝑘1 ], φ = [𝜑11 𝜑12 ], 𝜀𝑡 = [𝜀1 (𝑡)] and where 𝜀𝑡 = 𝛴𝑡 2𝑡 2 21 22 2

𝑧𝑡 , with

𝑧𝑡 ~i.i.d. Student-𝑡(𝜈), thus 𝜀𝑡 |Φ𝑡−1 ~ Student-𝑡(0, 𝛴𝑡 , 𝜈) with: ′ 𝛴𝑡 = 𝐶 ′ 𝐶 + 𝐴′ 𝜀𝑡−1 𝜀𝑡−1 𝐴 + 𝐵 ′ 𝛴𝑡−1 𝐵 ,

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(3)

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0

𝑎

𝑎

𝑏

𝑏

where 𝐶 = [𝑐11 𝑐 ] , 𝐴 = [𝑎11 𝑎12 ] and 𝐵 = [𝑏11 𝑏12 ] , where A and B are parameter 21 22 21 22 21 22 matrices and C is a lower triangular matrix constant terms, 𝜀𝑡 represents the error term. Equation (2) and (3) represent the mean equation and the time-varying covariance matrix, respectively. The diagonal elements 𝑎11 , 𝑎22 , 𝑏11 and 𝑏22 measure effects of own stock market’s previous shocks (ARCH effect) and volatility (GARCH effect); 𝑎12 , 𝑎21 , 𝑏12 and 𝑏21 represent cross-stock market (from stock market i to market j) effects of shocks and volatility. The BEKK model can be estimated by the maximum likelihood method (Engle and Kroner, 1995; Li and Giles, 2015). And the log likelihood function is as follows: 1

𝐿(𝜃) = −𝑇𝑙𝑛(2𝜋) − 2 ∑𝑇𝑡=1,𝑙𝑛|𝛴𝑡 (𝜃)| + 𝜀𝑡 (𝜃)′ 𝛴𝑡−1 𝜀𝑡 (𝜃)- ,

(4)

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where T is the number of observations, 𝜃 is the collection of parameters which need to be estimated. Several important steps are needed, like: lag selection, stability test, serial correlation test and ARCH effect test. The extent of aggregation and persistence of volatility can be captured by the absolute value of 𝑎12 and 𝑏12 . So, according to the above studies, the total volatility spillover effect from stock 1 to stock 2 is the sum of |𝑎12 | and |𝑏12 |: 𝑠12 = |𝑎12 | + |𝑏12 |.

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3.2 Network construction

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Complex network theory considers the relationship among various parts of real complex financial system as a network. A network graph 𝐺(𝑉, 𝐸) is defined as a collection of nodes connected by edges. 𝑉 = *1,2, … , 𝑁+ is the set of nodes, and 𝐸 is the edges between them. The set 𝑉 represents the stock index (like, 𝑖, 𝑗 ∈ N) and E = 𝑒𝑖,𝑗 ∈ *0; 1+𝑁×𝑁 represents volatility

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spillover relationships between i and j, where N is the number of stock index. If the estimated parameters of 𝑎ij and 𝑏𝑖𝑗 in formula (3) is highly significant and positive, it implies that there is

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a volatility spillover from market j to market i. In this paper, we choose the significance level of Wald Test as 1%. The spillover direction is the edge direction. If there is a volatility spillover from market i to market j, the 𝑒𝑖,𝑗 = 1; otherwise, 𝑒𝑖,𝑗 = 0. 𝐴2𝑁 GARCH-BEKK models should be estimated so that we can get the spatial matrix E and construct the volatility spillover network. 3.2.1 Connectedness analysis Connectedness can reflect the robustness and vulnerability of the network (Krackhardt, 1944). If the relationship between nodes connects the financial markets into a whole, and there are indirect or indirect paths between any two nodes. It indicates that the network has a good correlation. If many paths are connected through one node, then the network has a great dependence on the node, and once the node is shocked, the network may crash. So the network is not robust and its connectedness is low. Connectedness (GC) can be measured by reachability. Let N is the number of nodes, and V is the number of unreachable pairs that are in the network. GC is computed as: 𝑉

𝐺𝐶 = 1 − [𝑁(𝑁−1)⁄2] ,⁡⁡⁡𝐺𝐶 ∈ ,0,1- .

(5)

Network efficiency (GE) is another indicator of network connectedness. In the volatility spillover network, the lower the network efficiency, the more spatial spillover channels of fluctuation, which indicates that the network has multiple superposition phenomena of spillover effects and network structure is more stable. Let L is the number of excess links, and max(L) is the 6

Journal Pre-proof maximum possible number of excess links. GE is calculated as: 𝐿

𝐺𝐸 = 1 − max(𝐿) ⁡,⁡⁡⁡⁡⁡𝐺𝐸 ∈ ,0,1- .

(6)

For directed networks, the network hierarchy (GH) is the degree to which nodes are asymmetrically reachable. It reflects the dominance of various stock markets in international financial networks. Let K is the number of symmetrically reachable pairs, and max(K) is the maximum pairs. GH is expressed by: 𝐾

𝐺𝐻 = 1 − max(𝐾) ⁡,⁡⁡⁡⁡⁡𝐺𝐸 ∈ ,0,1- .

(7)

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3.2.2 Centrality analysis Centrality is often used to compare the strength of influence between nodes and is widely applied in the spread of infectious diseases or messages. Centrality of directed networks usually includes degree centrality (𝐷𝐶), closeness centrality (𝐶𝐶), beta reach centrality (𝛽𝐶) and Bonacich centrality (𝐵𝐶). Measuring the node centrality is essential for deep analysis of volatility spillovers and systemic importance in a single financial market. (1) Degree centrality The degree of a node is the number of edges connected to the node. In the directed volatility spillover network, a node that has a larger degree has a greater ability to overflow or to be affected. The two types of degree centralities are defined as: 1

𝐷𝐶𝑖,𝑖𝑛 =

1 𝑁−1

∑𝑁 𝑗=1 𝑒𝑗𝑖 ,

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𝐷𝐶𝑖,𝑜𝑢𝑡 = 𝑁−1 ∑𝑁 𝑗=1 𝑒𝑖𝑗 ,

(8) (9)

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where 𝐷𝐶𝑖,𝑜𝑢𝑡 and 𝐷𝐶𝑖,𝑖𝑛 represent the out-degree centrality and in-degree centrality, respectively. 𝑒𝑖𝑗 denotes the amount of links from node i to node j. The larger the out (in) degree

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centrality is, the stronger is the ability to transmit (withstand) risks. (2) Closeness centrality Degree centrality only takes into account the direct volatility spillover at the local structure around nodes, rather than indirect spillover to all others. The closeness centrality (CC) indictor is based on the whole network, and the nodes have a short distance to other nodes and consequently are able to contagion volatility effects on the network very quickly. The out (in) closeness centrality of node i is defined as: 1

CC𝑖,𝑜𝑢𝑡 = ∑𝑁

,

𝑗=1,𝑗≠𝑖 𝑑𝑖𝑗

CC𝑖,𝑖𝑛 = ∑𝑁

1

𝑗=1,𝑗≠𝑖 𝑑𝑗𝑖

(10)

,

(11)

where 𝑑𝑖𝑗 represents the length of a shortest directed path from i to j. The larger the out-closeness centrality is, the smaller is the distance from the node to other nodes, and the volatility spillover of the node is faster. (3) Beta reach centrality The beta reach centrality focuses on measuring the breadth of one node associated with other nodes. The greater beta centrality of a node, the wider its impact on other nodes, that is, the more volatility spillover paths of this node. The out-beta reach centrality (𝛽𝐶𝑖,𝑜𝑢𝑡 ) and in-beta reach centrality (𝛽𝐶𝑖,𝑖𝑛 ) are defined as: 7

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(𝑗−1)𝑎𝑗𝑖 𝛽𝐶𝑖,𝑜𝑢𝑡 = 𝑁−1 ∑𝑁 , 𝑗=1 𝛽

𝛽𝐶𝑖,𝑖𝑛 =

1 𝑁−1

(12)

(𝑗−1)𝑎𝑖𝑗 ∑𝑁 , 𝑗=1 𝛽

(13)

where 𝛽 is a constant, 𝑎𝑖𝑗 represents that node i reaches other nodes through j links. (4) Bonacich centrality The Bonacich centrality (BC) of a node not only considers its location in the network, but also considers the importance of its neighbors. Bonacich key nodes calculated under the spatial volatility correlation matrix can simultaneously capture the direct and spatial effects between nodes. And it is measured as follows: 𝜆 𝐵𝐶𝑖,𝑜𝑢𝑡 𝜆 = ∑𝑁 𝑗=1 𝐴𝑖𝑗 𝐵𝐶𝑗,𝑜𝑢𝑡 ,

(14)

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𝜆 𝐵𝐶𝑖,𝑖𝑛 𝜆 = ∑𝑁 𝑗=1 𝐴𝑖𝑗 𝐵𝐶𝑗,𝑖𝑛 ,

(15)

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where A is the network matrix, λ is the largest eigenvalue of the network. The larger the out-Bonacich centrality, the stronger is the ability of node’s spatial spillovers. 4. Data and summary statistics

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We use daily closing spot price for G20 stock index (expect European Union). Closing price of any stock index should be pairwise synchronized; i.e., if there is missing observation on one index on a given day, observations on the other index of the same day are deleted. The return series are calculated through Eq. (1) and apply two-day rolling-average returns. The countries and the corresponding stock indexes are shown in Appendix A (Table A1). The time spans of this study is from January 2, 2006 to December 31, 2018. The whole sample period covers multiple market conditions as much as possible (such as bull markets, bear markets, and recovery periods), and has good research representation. What’s more, using the breakpoint tests method first proposed by Contessi et al. (2014), we construct the five stages as follows: stage 1 represents the period of lead-up to the global financial crisis (GFC), stage 2 covers the recognized the US subprime crisis, stage 3 is the usually accepted the European debt crisis, stage 4 is to some extent recovery period, and stage 5 contains recent world trade friction (WTF); see also Dungey et al. (2015), and Chowdhury et al. (2019). Actually, the stage 2 and stage 3 are the periods of crisis, and it can be divided into two stages based on the timeline of the GFC. The purpose of this study is to analyze not only the network topology in these stages, but also the transitions and changes that occur in volatility networks between the different stages. In addition, our area of interest is to evaluate the impact of influencing factors such as network structure, geographical distance, and macroeconomic variables on volatility spillover under different economic policy uncertainties. The detailed division process of the five stages is shown in Appendix Table A2. Table 1 depicts the detailed statistic results of stock return series. The average daily return series are positive for all stock indexes except for those of Italy, Russia and Saudi Arabia. The standard deviation of G19 stock return series are very high and this correspond to their economic, credit and political related risk spatial transfers. Also, we find that except for India, Korea and Mexico all sample stock returns exhibit negative skewness, and the kurtosis values are above three for all return series. This indicates that severe impact of crisis on these stock markets, and suggests that the distributions of these stock returns are non-normal but with leptokurtic. The 19 return 8

Journal Pre-proof series are net Gaussian distribution as shown by Jarque-Bera, kurt and skewness. All stock returns exhibit a significant serial correlation except for Italy and Korea as shown in Ljung-Box test. It is further confirmed by the ARCH test which shows significant clustering effect in each stock returns. The ADF test results with negative values at 1% significance level, indicating that all returns series are stationary. Hence, all statistics of the return series support to build GARCH-BEKK model for estimating their spillover relationship. Table 1 Descriptive statistics of G19 return series. Std. dev

Kurt

Skew

JB

ARCH

ADF

LBQ

0.033

2.066

8.027

-0.369

2332.55***

237.06***

-47.67***

42.58***

***

***

***

30.29***

-50.17***

52.09***

Argentina

0.134

2.521

10.977

-0.589

5876.76

Australia

0.007

1.297

8.690

-0.509

3014.99***

Brazil

0.042

2.066

12.173

-0.230

7623.93***

526.63***

-49.63***

101.65***

Canada

0.009

1.288

24.713

-1.385

43300.70***

414.87***

-47.94***

81.34***

France

0.000

1.640

8.119

-0.284

2397.10***

428.00***

-49.49***

41.11***

Germany

0.031

1.640

10.215

-0.048

4705.48***

416.53***

-47.31***

39.31***

India

0.062

1.747

11.844

0.046

7068.89***

298.39***

-47.66***

52.17***

Indonesia

0.076

1.637

15.280

-0.460

13705.00***

155.48***

-46.76***

53.53***

Italy

-0.025

1.822

8.332

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China

Mean

-0.649

2721.45***

311.21***

-48.15***

15.92

Japan

0.008

1.774

10.508

-0.533

5197.06***

380.86***

-50.22***

42.30***

Korea

0.017

1.488

18.155

0.129

20763.12***

543.79***

-47.15***

21.32

***

***

***

138.01***

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390.16***

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291.31

0.037

1.529

23.398

0.208

37620.20

Russia

-0.007

2.835

37.015

-0.523

10466.24***

547.47***

-47.28***

129.77***

Saudi Arabia

-0.037

2.004

33.619

-2.374

86767.94***

325.77***

-43.24***

88.86***

South Africa

0.048

1.982

15.087

-0.796

33348.22***

823.97***

-59.31***

149.77***

Turkey

0.032

2.005

9.389

-0.379

3741.42***

243.97***

-46.17***

50.79***

UK

0.008

1.383

11.989

-0.117

7307.19***

580.51***

-50.00***

74.12***

USA

0.031

1.389

15.221

-0.778

13716.15***

539.09***

-51.57***

161.63***

na

Mexico

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508.08

-44.49

-45.38

Notes: LBQ denotes the Ljung-Box test, which is used to check for the serial correlation in returns up to the 20th order. Augmented Dickey-Fuller (ADF) test is used to check time series’ stationarity. JB represents the Jarque-Beta test for normality. The asterisk *** denotes the level of significant at 1%.

5. Empirical results To gain the volatility spillover linkage and construct the spatial network among G20 stock markets, we use BEKK (1,1)-GARCH model to estimate the volatility relationship in five periods. In the estimation process, we select the significance level of Wald Test at 1%, which indicates that spillover effects exist any two stock return series. During five sample stages, the relationships determined by the Wald test are 229, 273, 209, 233 and 215, respectively. The five volatility spillover networks are constructed (Appendix B-Fig. B). Fig. B is a visual expression of time-varying volatility spillover linkages in the G20 stock markets. 5.1 Connectedness analysis of volatility spillover network Connectedness, efficiency and hierarchy are effective indicators for measuring network connectedness. First, we calculate the connectedness of volatility spillover network through 9

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formula (5). The connectedness values of networks in five periods are always 1, showing that the spatial correlation of volatility spillover is very high. And this result indicates that the volatility network has a good reachability, and there is a general volatility spillover effect among G20 financial markets. Furthermore, Fig.1 shows dynamic evolution of network hierarchy, efficiency, clustering coefficients and small path length in five periods. As time varying, the network hierarchy is relatively small from 0.11 to 0.28, indicating that the spillover effects between stock markets are not hierarchical, and there is a possibility of spillover effects at different levels of financial development. The network efficiency varies from 0.611 to 0.798, thereby there are more redundant edges in the volatility spillover network, indicating that spatial spillover of volatility has obvious multiple superposition phenomena and further increases the stability of the network. The network clustering coefficient and small path length are also the important indicators. A high clustering coefficient indicates a small-world effect of the volatility network. The clustering coefficient vary from 0.679 to 0.850, which is much larger than 0. This proves that the volatility spillover network has small-world characteristics. The small path length represents the minimum distance that the volatility transmission from one node to other. Noticeably, the network during the period 2 has the smallest network hierarchy and small path length, while has the largest network efficiency and clustering coefficient. The finding shows that a volatility can spread more rapidly and directly in G20 stock markets during the financial crisis.

Fig.1. Connectedness and tightness of volatility spillover networks in different periods

5.2 Centrality analysis of volatility spillover network 5.2.1 Out-centrality analysis The status and influence of a node is largely affected by the topological structure of the network. Thereby, we now analyze the four out-centrality measures, out-degree centrality, out-closeness centrality, out-beta reach centrality and out-Bonacich centrality, which reflect the status and influence of a node in the volatility network. Fig.2 displays four out-centrality values for all nodes of networks during the five stages. A close inspection of this figure shows that the centrality values of different nodes are quite different in the same stage. Interestingly, in the same network, the relative sizes and variation trends of the different centralities of the same node are consistent. Each centrality indicators are positively related to each other. For instance, in the second network during US subprime crisis, the node 19 (USA) and node 6 (France) have the largest out-Bonacich centrality value, and node 1 10

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(China) and node 2 (Argentina) have the smallest out-Bonacich centrality value. In fact, a similar finding is behind other out-centralities. This indicates that the USA has the largest ability of spatial spillovers, which is an indicative evidence of risk contagion, because the Lehman collapse in the USA stock market is a source of volatility transmission. Regarding the node 1 and node 2, we can see from Appendix Fig. B that the nodes of China and Argentina are at the edge of the network, therefore they have the smallest spatial spillover effects. Another important explanation is that the openness of China's financial market is low and is heavily dominated by macroeconomic policies. In addition, we perform a correlation test between the centrality indicators and find that, the correlations of any two centrality indicators are greater than 0.9 at the 1% significance level. Analysis process of other stages are similar to those of the above stage 2, and they are not listed due to the space constrains.

Fig.2. Four out-centrality values for nodes of volatility networks during sub stages: (a) stage 1, (b) stage 2, (c) stage 3, (4) stage 4, (e) stage 5. Notes: The X is the 19 stock market nodes (the corresponding name of each number is presented in Appendix A, Table A1), and Y axis is the value of different centrality measures.

5.2.2 In-centrality analysis Corresponding to the out-centrality, the in-centrality is another important indicator of networks, which can reflect the sensitive to the volatility in other markets. Fig.3 depicts four in-centrality values for all nodes of networks during the five stages. From fig.3, we can find that the in-centrality value of the same stock market is also time varying. We take node 1 (China stock market) as an illustration example. The in-beta reach centrality value of China is ranked last with value 0.844 in period 2, while it is ranked seventh with value 0.944 in stage 5. It indicates that China is the least sensitive to the volatility spillover effects from other international markets in the financial crisis period. Due to the low matureness of the stock market, the China’s stock market is affected through the narrowest path range of spatial 11

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volatility spillover. From the perspective of network topology (Appendix B-Fig. B. (e)), we find that China is not at the edge of the network, whereas moves toward the center, which has increased its path of volatility. In fact, trade frictions broke out between China and the United States in stage 5, and China’s counterattack are the main factors that enhanced China’s sensitive to the risks. Further, it is observed that in five stages, node 8 (India), node 12 (Korea), node 16 (South Africa), node 19 (USA), and node 4 (Brazil) have the highest centrality value, respectively, and almost all centrality measures attribute the highest centrality to the same stock market. The results demonstrate that in general, the four in-centrality measures are consistent with each other in identifying the stock market’ sensitive. Another important evidence comes from the Pearson correlation coefficient of any two central indicators. E.g., in stage 3, the correlation coefficient between in-degree centrality and in-closeness centrality is 0.98 at 1% significance level.

Fig.3. Four in-centrality values for nodes of volatility networks during sub stages: (a) stage 1, (b) stage 2, (c) stage 3, (4) stage 4, (e) stage 5. Notes: The X is the 19 stock market nodes (the corresponding name of each number is presented in Appendix A, Table A1), and Y axis is the value of different centrality measures.

5.2.3 Factor analysis Factor analysis is used to find a few major factors from the intricate relationship problem, thus achieving the goal of reducing the dimension of datasets. We attempt to measure centrality comprehensive values of each node in every volatility networks through factor analysis. In the factor model, the idea of regression is used to find the factor score, that is the centrality comprehensive indicator. Suppose the common factor F is a linear combination of the variable X: 𝐹𝑗 = 𝛽𝑗1 𝑋1 + 𝛽𝑗2 𝑋2 + 𝛽𝑗3 𝑋3 + 𝛽𝑗4 𝑋4 ,⁡⁡⁡𝑗 = 1,2, , … , 𝑛

(16)

where 𝑋1, 𝑋2, 𝑋3, 𝑋4 represents degree centrality (DC), closeness centrality (CC), beta reach centrality (𝛽𝐶) and Bonacich centrality (BC), respectively, j represents the node label of 19 stock 12

Journal Pre-proof markets, 𝛽𝑗𝑖 denotes the coefficient of factor score. Because F and X are all normalized vectors, there are no constant terms in the model. In this way, using a set of sample values, the coefficient 𝛽𝑗𝑖 can be estimated by the least square method or the maximum likelihood function method. Next, we make the estimated factor score coefficient 𝛽𝑗𝑖 and the value of the original variable into the Eq. (16) to gain the factor score 𝐹𝑗 , and 𝐹𝑗 represent 𝐶𝑜𝑛𝑡𝑎𝑔𝑖𝑜𝑛𝑗 and 𝐴𝑐𝑐𝑒𝑝𝑡𝑎𝑛𝑐𝑒𝑗 ,

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respectively. It is worth mentioning that, based on the attributes of the sample variables, we will perform two types of factor score calculations: four out-centrality indicators (DC𝑜𝑢𝑡 ,⁡CC𝑜𝑢𝑡 , 𝛽𝐶𝑜𝑢𝑡 and 𝐵𝐶𝑜𝑢𝑡 ), and four in-centrality indicators (DC𝑖𝑛 , CC𝑖𝑛 , 𝛽𝐶𝑖𝑛 , BC𝑖𝑛 ). The out-degree centrality, out-closeness centrality, out-beta reach centrality and out-Bonacich centrality measure the volatility contagion feature of financial markets from different perspectives. Therefore, we first define the out-centrality factor score based on four out-centrality indicators (DC𝑜𝑢𝑡 ,⁡CC𝑜𝑢𝑡 , 𝛽𝐶𝑜𝑢𝑡 and 𝐵𝐶𝑜𝑢𝑡 ), that is, comprehensive volatility contagion intensity indicator Contagion. The stock market with the greater volatility contagion intensity, the more the number of other stock markets affected by its direct and indirect volatility spillovers, and the faster the contagion. Now, we take computations of out-centrality factor evaluation values of each node in the network during stage 1 as an example to illustrate the factor analysis process. Correlation analysis between the various out-centrality (DC𝑜𝑢𝑡 ,⁡CC𝑜𝑢𝑡 , 𝛽𝐶𝑜𝑢𝑡 and 𝐵𝐶𝑜𝑢𝑡 ) of nodes shows that there is a significant positive correlation between them (correlation coefficients are 0.92, 0.96, 0.93, 0.98, 0.97, 0.96, respectively). The value of KMO is 0.7411, and the Bartlett spherical test is 289.61 with p value 0.000. Therefore, these four factor indicators are suitable for factor analysis. It is known from Table 2 that a factor is finally extracted, which can explain about 99% of the total variability. Similarly, the in-degree centrality, in-closeness centrality, in-beta reach centrality and in-Bonacich centrality depict the volatility acceptance feature of financial markets from different perspectives. We define the in-centrality factor score based on four in-centrality indicators (DC𝑖𝑛 , CC𝑖𝑛 , 𝛽𝐶𝑖𝑛 , BC𝑖𝑛 ), that is, comprehensive volatility acceptance intensity indicator Acceptance. The stock market with the greater volatility acceptance intensity, the more affected by the direct and indirect volatility spillovers from other markets, and the faster the impact. In stage 1, correlation analysis between the various in-centrality (DC𝑖𝑛 , CC𝑖𝑛 , 𝛽𝐶𝑖𝑛 , BC𝑖𝑛 ) of nodes shows that there is a significant positive correlation between any two centrality indicators at the 1% significance level. The KMO value is 0.7424, and the Bartlett spherical test is 300.96 with p value 0.000. Therefore, these four centrality indicators are suitable for factor analysis. From Table 3, we can extract two factors, and the first factor retains 50.87% information of the original variables, the second factor retains 49.13% information of the original variables. Table 2 Total variance explained. Factor

Initial eigenvalues

Extraction sums of squared loading

Total

% of variance

Cumulative %

Total

% of variance

Cumulative %

1

3.964

99.001

99.001

3.964

99.001

99.001

2

0.027

0.669

99.670

3

0.009

0.230

100.000

4

0.000

0

100.000

Table 3 Total variance explained. 13

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Initial eigenvalues

Extraction sums of squared loading

Total

% of variance

Cumulative %

Total

% of variance

Cumulative %

1

2.020

99.001

50.870

2.02

50.870

50.870

2

1.951

0.669

49.130

1.051

49.130

100.000

3

0.029

0.230

100.000

4

0.000

0

100.000

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Accordingly, we can gain the volatility contagion intensity and acceptance intensity of financial markets, which are represented by the Contagion and Acceptance, respectively. The process of node centrality factor analysis in other periods is the same as above, whereas we have not shown detailed process because of space limitations. In addition, Table 4 and Table 5 give the ranking of the intensity of volatility contagion and acceptance of each node in each period, respectively. It can comprehensively measure the influential ability and the sensitive to volatility in all financial markets from the perspective of the node centrality. Table 4 clearly shows that USA has the highest value of contagion intensity only in stage 2. UK has the highest value of contagion intensity in stage 5. India, Brazil and South Africa is the largest in stage 1, 3 and 4, respectively. This means that the risk contagion capacity of the stock market is different across different market or economic cycles. From table 5, we also find that Korea is the most sensitive market to external volatility shocks in stage 1 and 2. South Africa, USA and Brazil is the most sensitive market to volatility in stage 3, 4 and 5, respectively. The ranking changes can be recognized as an evidence of the time-varying spillover effects. This indicates that the sensitivity of the stock market to volatility shocks is vary across different economic cycles. Additionally, the results prove that the variations of centrality comprehensive value are consistent with the general variations of different centrality values. Comparing Table 4 and Table 5, we find that there is no significant correlation between the contagion intensity and acceptance intensity of the same stock market in the global financial system. For instance, in stage 5, the top 5 countries of contagion intensity are the UK, France, Canada, India and Italy, where developed countries account for 4/5; the top 5 countries acceptance intensity are Brazil, Saudi Arabia, India, South Africa and the United States, where emerging markets account for 4/5. This indicates that at this stage, the possibility of volatility transmission from developed markets to emerging markets is more acute, which is similar with results of Yarovaya et al. (2016). Additionally, we also find that developed markets are more influential than emerging markets. Likely, Indonesia, Turkey and Argentina, the emerging markets, are less influential on other markets in the stage 5. Furthermore, the emerging markets are more sensitive than developed markets to volatility shocks. Likely, Brazil and Saudi Arabia, the emerging markets, are the larger volatility acceptances, while the Germany and France are less acceptances to the volatility shocks in stage 5. Taking these results together (the ranking of the comprehensive indicators for all stages), we are clearly aware that the developed markets are more influential than the emerging markets during stages of turbulence, and the emerging markets are more sensitive to volatility shocks than developed markets during any stage. Table 4 Rank of volatility contagion intensity of nodes in different periods. China

Period 1

Period2

Period 3

Period 4

Period 5

4

19

13

8

8

14

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11

17

18

Australia

19

17

3

3

17

Brazil

7

12

1

16

9

Canada

17

1

12

5

3

France

8

2

4

9

1

Germany

9

16

16

11

7

India

1

13

5

10

4

Indonesia

2

15

17

15

15

Italy

6

8

14

4

5

Japan

18

11

8

6

11

Korea

13

7

15

14

12

Mexico

14

6

6

19

14

Russia

3

9

7

18

10

Saudi Arabia

11

5

19

7

13

South Africa

15

14

10

2

6

Turkey

12

10

18

1

16

UK

16

4

9

12

1

USA

5

3

13

19

Period 3

Period 4

Period 5

19

16

18

8

17

18

14

16

3

3

6

7

6

16

4

7

1

8

13

7

17

12

13

8

8

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2

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Table 5 Rank of volatility acceptance intensity of nodes in different periods. 18

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11

Australia

3

Brazil Canada

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Germany

10

15

19

13

17

India

2

14

9

4

3

Indonesia

14

2

17

5

15

Italy

7

6

15

16

13

15

4

12

9

9

Japan Korea

1

1

5

11

10

Mexico

12

10

11

19

11

Russia

16

9

6

15

6

Saudi Arabia

19

18

10

10

2

South Africa

9

12

1

2

4

Turkey

5

11

14

8

19

UK

17

5

13

3

14

USA

4

7

2

1

5

5.3 Influencing factors of volatility spillover spatial linkage——based on QAP method 5.3.1 Theoretical assumption 15

Journal Pre-proof As mentioned above, we have a clear understanding of the spatial spillovers of volatility in international stock markets. The next part is to explore the factors that influence the spatial linkage of volatility spillover between stock markets. These factors include network structure factors and external economic factors. Now, we select the centrality comprehensive indicator “Contagion and Acceptance” obtained from the factor analysis in section 5.2.3 being proxy indicators for the network structure. With the reference to the previous researches (Debarsy et al., 2018),we choose the growth of GDP (𝐺𝐷𝑃𝑔 ), the exchange rate (𝐸𝑅), and the ratio of total capital formation to GDP

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(𝑇𝐶𝐹) being the external macro variables. The ratio of total capital formation to GDP can reflect the similarity of development patterns among countries. Besides, it is well known that the initial measurement of spatial effects is based on binary adjacencies of spatial units (Moran, 1948). Therefore, there may be more significant correlations and spatial spillover effects between neighboring countries. Based on this, we can set up the following model: 𝑊 = 𝑓(𝐶𝑐𝑒𝑛 , 𝐴𝑐𝑒𝑛 , 𝐺𝐷𝑃𝑔 , 𝐸𝑅, 𝑇𝐶𝐹, 𝐷) ,

(17)

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indicators of each country in the sample interval, and then use the absolute difference of average values of each country to form the difference matrix, and 𝐷 represents a spatial neighbor relationship determined by geographical relationship (in the same continent its value is 1, otherwise 0). Time range of data covers 2006-2018 (five sub stages) and 19 countries. In general, we can’t use conventional statistic tests to verify the existence of relationships between relational data. A specific method is required when testing the hypothesis test at the relationship-relationship level. This paper chooses the QAP (quadratic assignment procedure) method, which is a non-parametric method. In particular, it is not necessary to assume that the independent variables are independent of each other, so that the “multi-collinearity problem” between the relational data can be avoided, which is more robust than the parameter method (Barnett, 2011). Hence, in the next part, we use QAP correlation analysis and QAP regression analysis to explore the influencing factors of volatility spillover between international stock markets. 5.3.2 QAP correlation analysis The QAP correlation analysis is based on the matrix permutation. The correlation coefficient is given by comparing the similarity of each lattice value in the two square matrices, and it is subjected to non-parametric test (Everett, 2002). The specific steps of QAP correlation analysis are as follows: Firstly, calculating the correlation coefficient between long vectors formed by the known matrices. Each vector contains 𝑛(𝑛 − 1) elements (elements on the diagonal are ignored). To briefly understanding, we assumed that two matrices represent the relationship of friends and suggestions between five people (A, B, C, D, E), as shown in Fig. 4 (“1” means related, while “0” means not related). 16

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B

C

D

E

A

B

C

D

E

A

--

1

0

0

0

A

--

0

1

1

1

B

1

--

1

0

0

B

0

--

0

1

1

C

0

1

--

1

1

C

1

0

--

0

1

D

0

0

1

--

1

D

1

1

0

--

0

E

0

0

1

1

--

E

1

1

1

0

--

(a) Friends matrix

(b) Suggestions matrix

Fig. 4 Two matrices of friend relationship and suggestion relationship

C

D

E

A

--

1

0

0

1

B

1

--

1

0

0

C

0

1

--

1

1

D

0

0

1

--

1

E

1

0

1

1

--

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Secondly, the rows of one matrix and the corresponding columns are simultaneously replaced, then compute the correlation coefficient between the replaced matrix and the other matrix, and save the computing results. Repeating this calculating process enough times to get a distribution of correlation coefficients, from which it can be seen that the multiple correlation coefficients calculated after the random permutation are greater than or equal to the ratio of the observed correlation coefficients calculated in the first step. In the example, we permutated the “label” in one of the matrices (e.g. Suggestions matrix). For a matrix with five actors, there may be up to 5! = 120 permutations. We randomly give two kinds of permutation matrices for the suggestion relationship, as shown in Figure 5. And calculating the correlation coefficient between “Friends matrix” and “Permutation matrix 1”, “Permutation matrix 2”, respectively.

(a) Permutation matrix 1 (WP1) B

C

D

E

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A A

--

0

0

1

1

B

0

--

1

1

0

C

0

1

--

0

1

D

1

1

0

--

1

E

1

0

1

1

--

(b) Permutation matrix 2 (WP2)

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Finally, comparing the actual observed correlation coefficient in the first step with the distribution of the correlation coefficient calculated by the random rearrangement, and examining whether the correlation coefficient falls into the rejection domain or the acceptance domain, and then judging the correlation. Now, we apply QAP correlation analysis to test the correlation between spatial correlation matrix and influencing factors. Selecting 5000 random permutations, and the results of sub stages are shown in Table 6. The results of the correlation analysis indicate that there are five independent variables that significantly influence the spatial correlation matrix W, i.e. 𝐶𝑐𝑒𝑛 , 𝐴𝑐𝑒𝑛 , 𝐺𝐷𝑃𝑔 , 𝐸𝑅 and 𝐷. The correlation coefficient of 𝐶𝑐𝑒𝑛 is positive at 1% significance level,

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meaning that the contagion strength of nodes has a significant role in promoting the spatial correlation of volatility spillovers. While 𝐴𝑐𝑒𝑛 is significantly negative affecting spatial correlation matrix W. Importantly, variables of 𝐺𝐷𝑃𝑔 and 𝐸𝑅 have negative correlation

Table 6

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coefficients at 5% significance level only in stage 2 and 3. This means that, in times of crisis, the improvement of economic development can significantly inhibit spatial linkages of volatility spillovers, and the appreciation of an economy’s currency relative to US dollar can significantly promote volatility spillovers. Besides, the geography variable D has a positive correlation coefficient at 10% significance level only in stage 1,4 and 5. It indicates that geography factor has different effects on the spatial correlation of volatility spillovers in the economic cycles. In normal times, geography factor does have a positive impact on the volatility spatial linkage; in times of crisis, geography factor no longer have an impact on the volatility spatial linkage. Unfortunately, variables of 𝑇𝐶𝐹 gives no significant impact on spatial correlation matrix W. This suggests that the similarity of development patterns does not highlight the impact on the volatility spatial correlation. It is worth noting that variables of 𝐶𝑐𝑒𝑛 and 𝐴𝑐𝑒𝑛 are the most influential factors, indicating that the centrality structure of the volatility network has a greater impact on the spatial correlation than the external economic factors.

QAP correlation analysis results between spatial correlation matrix and influencing factors (five stages). Influencing factors

𝐶𝑐𝑒𝑛

𝐴𝑐𝑒𝑛

𝐺𝐷𝑃𝑔

Correlation actual value

Stage 1

Stage 2

Stage 3

Stage 4

Stage 5

0.2123

0.2292

0.1137

0.2371

0.2950

(0.000)

(0.000)

(0.007)

(0.021)

(0.000)

-0.2252

-0.0386

-0.0738

-0.0366

-0.2503

(0.000)

(0.017)

(0.052)

(0.000)

(0.002)

-0.0420

-0.2747

-0.1953

-0.2150

-0.0659

(0.263)

(0.046)

(0.014)

(0.362)

(0.244)

0.0333

0.0167

0.0594

0.0439

0.0442

(0.332)

(0.382)

(0.321)

(0.254)

(0.305)

-0.0723

-0.0917

-0.0225

-0.0776

-0.0235

(0.204)

(0.017)

(0.023)

(0.134)

(0.437)

0.0528

0.0260

0.0409

0.0330

0.0882

(0.071)

(0.508)

(0.289)

(0.069)

(0.083)

𝑇𝐶𝐹

𝐸𝑅

𝐷

Note: The value in bracket is significant level. 18

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5.3.3 QAP regression analysis The purpose of QAP regression analysis is to study the regression relationship between multiple matrices and one matrix, and to evaluate the significance of the adjustment coefficient. The estimation process is divided into two steps: first, the ordinary multiple regression is performed on the long vector elements corresponding to the explanatory variable matrix and the interpreted variable matrix; second, the rows and columns of the interpreted variable are randomly replaced at the same time, and the regression calculation is performed again. Repeating this calculating process enough times. Coefficient estimation method and test method are consistent with QAP correlation analysis. Selecting 5000 random permutations, and the results are shown in Table 7 and Table 8. The adjusted judgment coefficients in five stages of Table 7 are 0.488, 0.549, 0.534, 0.522 and 0.544, respectively, indicating that the five variables can explain approximately 50% of the spatial correlation variation of international financial markets' volatility spillovers. The probability in the model means that the probability coefficient generated by the random permutation is not less than the probability of the actually observed judgment coefficient. It is the one-tailed probability with a value of 0, indicating that the adjusted judgement coefficient (R2) is significant at the 1% level, and the model fits the relationship data very well. The sample size is 342, because 19 countries form a 19 × 19 square matrix (ignoring the diagonal elements).

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Table 7 Model fitting results.

𝑅2 Adj-𝑅

0.488 2

0.474 0.000

Sample volume

342

Stage 3

Stage 4

Stage 5

0.549

0.534

0.522

0.544

0.525

0.511

0.503

0.513

0.000

0.000

0.000

0.000

342

342

342

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Probability

Stage 2

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Stage 1

342

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In order to consider the impact of economic cycles (e.g. financial crisis), we perform QAP regression analysis in five time intervals (including normal times and turbulent times). The coefficients and test indicators of each variable matrix obtained are presented in Table 8. The P-value 1 represents the probability that the regression coefficient produced by the random permutation is not less than actually observed regression coefficient; the P-value 2 indicates that the regression coefficient generated by the random permutation is not greater than the probability of the actual observed regression coefficient (two-tailed probability is performed). As can be seen from five stages in Table 8, regarding the network structure factors, the regression coefficient of the node contagion intensity (𝐶𝑐𝑒𝑛 ) is statistically positive at 1% significance level. It means that node with larger 𝐶𝑐𝑒𝑛 tend to has a larger volatility spillover ability. In contrast, the coefficient of the node acceptance intensity (𝐴𝑐𝑒𝑛) is significantly negative at 1% level (except for stage 2 is at 5% significance level), indicating that the 𝐴𝑐𝑒𝑛 indicator can suppress the spatial linkage of volatility spillover. As for external macroeconomic factors, the regression coefficients of GDP growth rate and exchange rate are significantly negative at the level of 10% only in stage 2 and 3. This results show that the volatility spatial spillovers have the features of inverse periodicity. And during periods of turbulence, the appreciation of local currency can promote the volatility spillovers of global stock markets. However, during normal times, the factors of GDP growth rate and exchange rate have no impact on the volatility spatial linkages. Last, we find that the regression coefficients 19

Journal Pre-proof of the spatial adjacency matrix are significantly positive at the level of 10% during the stage 1, 4, and 5, indicating that adjacent geographical locations only play an important role in the volatility spatial linkages during the normal periods. Table 8 The coefficients and test indicators of each variable matrix (five periods). Variables

Un-Stdized Coef

Stdized Coef

Significa

P-value 1

P-value 2

𝐶𝑐𝑒𝑛

0.143

0.189

0.000

0.000

1.000

Stage 1

-0.202

-0.203

0.000

1.000

0.000

-0.053

-0.047

0.496

0.504

0.496

𝐸𝑅

-0.022

-0.036

0.218

0.782

0.218

𝐷

0.045

0.042

0.084

0.084

0.916

Intercept

0.658

0.000

0.000

𝐶𝑐𝑒𝑛

0.245

0.333

𝐴𝑐𝑒𝑛

-0.101

-0.173

𝐺𝐷𝑃𝑔

-0.044

-0.039

𝐸𝑅

-0.018

-0.035

𝐷

0.019

0.024

Intercept

0.797

0.000

0.000

0.000

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𝐴𝑐𝑒𝑛 𝐺𝐷𝑃𝑔

0.000

0.000

1.000

0.023

0.977

0.023

0.075

0.925

0.075

0.086

0.914

0.086

0.492

0.492

0.608

0.000

0.000

0.000

0.212

0.000

0.000

1.000

-0.188

0.000

1.000

0.000

-0.019

0.066

0.934

0.066

-0.041

0.078

0.782

0.218

0.046

0.232

0.232

0.768

0.000

0.000

0.000

0.000

0.152

0.132

0.000

0.000

1.000

-0.154

-0.282

0.000

1.000

0.000

-0.086

-0.058

0.132

0.868

0.132

𝐸𝑅

-0.013

-0.015

0.202

0.798

0.202

𝐷

0.031

0.039

0.051

0.051

0.949

Intercept

0.692

0.000

0.000

0.000

0.000

𝐶𝑐𝑒𝑛

0.259

0.285

0.000

0.000

1.000

𝐴𝑐𝑒𝑛

-0.152

-0.241

0.000

1.000

0.000

𝐺𝐷𝑃𝑔

-0.040

-0.033

0.468

0.532

0.468

𝐸𝑅

-0.043

-0.051

0.187

0.813

0.187

𝐷

0.086

0.078

0.085

0.085

0.915

Intercept

0.606

0.000

0.000

0.000

0.000

𝐴𝑐𝑒𝑛

-0.149

𝐺𝐷𝑃𝑔

-0.025

𝐸𝑅

-0.038

𝐷

0.051

Intercept

0.625

Stage 4 𝐶𝑐𝑒𝑛 𝐴𝑐𝑒𝑛 𝐺𝐷𝑃𝑔

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0.149

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𝐶𝑐𝑒𝑛

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Stage 3

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Stage 2

Stage 5

5.3.4 Robustness checks 20

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Table 9

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We conduct a number of robustness tests in order to check whether our results are sensitive to the model specification and the choice of model parameters. In our baseline model, presented above, we have adopted the network structure as internal influence factors and macroeconomic variables as external factors. However, we ignore one important issue that is the impact of the capital liquidity on the global stock markets’ volatility spillovers. Thereby, the additional inclusion of associated variable in the model poses some challenges. First, we choose the ratio of foreign direct investment (FDI) to GDP being the indicator of capital liquidity (CL). Second, we choose substitution variable for geography adjacency matrix, and redefine the adjacency criteria based on the regional economic organization1 (stock market within the same regional economic organization is 1, otherwise is 0, and the variable symbol is abbreviated as REO). Finally, in order to test the impact of time dummy variable (financial crisis) on regression results, we perform regressions in each subsample interval and only present regression results for the global financial crisis period (stage 2) and the post-financial crisis (stage 4)2. From Table 9, we do see that the additional variable of capital liquidity (CL) and the substitutional variable of regional economic organization (REO) do not disturb spatial linkages of international stock markets’ volatility. Comparing the results in panel A and B (Table 9), we also find that regional economic organization influences the volatility spatial spillovers differently across economic cycles. During turmoil periods, regional economic organization variable does not drive the spatial linkages of volatility, however, during normal periods, economic organization factor promotes the linkages. This result is consistent with the effect of the geographic adjacency matrix on volatility spillovers, indicating that the initial regression results are robust. In addition, the correlation coefficient of CL is statistically significant positive (at 5% and 10% level, respectively), indicating that the stronger is the capital liquidity, the stronger is the spatial correlation between international stock markets, and the more significant is the spatial spillover effect. As for the remaining initial variables, we find that the effects of independent variables on the dependent variable in stage 2 and 4 are roughly consistent with the results in the basic model (Table 8). Therefore, we believe that our results obtained in Section 5.3.3 are very robust.

Results of the robustness test. Variables

Un-Stdized Coef

Stdized Coef

Significa

P-value 1

P-value 2

Panel A: Satge 2 (US subprime crisis) 𝐶𝑐𝑒𝑛

0.248

0.337

0.001

0.001

1.000

𝐸𝑐𝑒𝑛

-0.103

-0.176

0.022

0.978

0.022

𝐺𝐷𝑃𝑔

-0.072

-0.083

0.063

0.937

0.063

𝐸𝑅

-0.012

-0.023

0.052

0.948

0.052

𝐶𝐿

0.148

0.205

0.037

0.037

0.963

𝑅𝐸𝑂

0.061

0.054

0.296

0.296

0.704

Intercept

0.796

0.000

0.000

0.000

0.000

Model fit

1 2

2

𝑅 =0.551

2

Adj-𝑅 =0.525

Probability=0.000

Representative regional economic organizations: European Union, BRICS, Asia-Pacific Economic Cooperation, North American Free Trade Area The results in other periods are available upon request. 21

Journal Pre-proof Panel B: Stage 4 (post-crisis of global financial crisis) 𝐶𝑐𝑒𝑛

0.151

0.132

0.001

0.001

1.000

𝐸𝑐𝑒𝑛

-0.152

-0.278

0.001

1.000

0.001

𝐺𝐷𝑃𝑔

-0.035

-0.057

0.074

0.926

0.074

𝐸𝑅

-0.007

-0.013

0.081

0.919

0.081

𝐶𝐿

0.190

0.215

0.066

0.066

0.934

𝑅𝐸𝑂

0.098

0.083

0.091

0.091

0.909

Intercept

0.681

0.000

0.000

0.000

0.000

Model fit

2

2

𝑅 =0.582

Adj-𝑅 =0.570

Probability=0.000

Notes: Models of panel A and B are extended by additional variable (CL), and the original geographic adjacency variable is redefined as ROE.

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6. Conclusions

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In this paper we have estimated the spatial linkage characteristics of volatility spillovers and its influencing factors across G20 stock markets. The time interval is from January 2, 2006 to December 31, 2018, and it is divided into five sub-periods to understand the dynamic properties of volatility spillovers. For this purpose, we apply identification through generalized autoregressive conditional heteroscedastic models (GARCH) in a BEKK form to estimate volatility spillover, and use QAP method to reveal the main influencing factors for international stock markets’ volatility linkages. We first analyze the volatility spillover effects among G20 stock markets by GARCH-BEKK model and construct dynamic volatility spillover networks. Empirical results show that the volatility network has a good reachability since the spatial correlation of volatility spillover is very high. Besides, volatility spatial spillovers have obvious multiple superposition phenomena and, further increase the stability of the network. Dynamic network connectedness analysis provides evidence that spatial correlations do not remain constant over time. And during the financial crisis, the network connectivity was the strongest and the speed of volatility propagation was faster. Moreover, we calculate four network centralities, including degree centrality, closeness centrality, beta reach centrality and Bonacich centrality, and analyze the status and influence of each stock market. The results prove that the four centrality measures are consistent with each other in distinguishing the central features of network nodes. Furthermore, we use the factor analysis method to obtain centrality comprehensive estimation indicators, which can measure the influential ability and the sensitive to volatility in financial markets. The ranking changes of centrality comprehensive indicators can be recognized as an evidence of the time-varying spillover effects. We believe that the developed markets are more influential than the emerging markets during periods of turbulence, and the emerging markets are more sensitive than developed markets to volatility shocks during any period. Finally, we apply QAP method to identify various conditions, which influence the propagation of volatility shocks that can either strengthen or diminish spillover effects. It is observed that the centrality comprehensive indicators are significantly impact on spatial linkages. The GDP growth rate and exchange rate have a significant negative effect on spatial linkages only in periods of crisis, while the capital liquidity is significantly positive impact on spatial linkages during any period. Importantly, the geography factor (or regional economic organization) influences the 22

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spatial linkage of volatility spillovers differently across economic cycles. During turmoil periods, geography factor does not drive the spatial linkage of volatility; and during normal periods, geography factor promotes the linkages. Furthermore, the QAP regression analysis shows that these influencing factors can explain approximately 50% of the spatial linkage variation of international financial markets' volatility spillovers. In terms of policy conclusions, our estimates give some hints at potential contagion channels of volatility risks. They therefore have a bearing on the construction of global financial stability safety that take into account spatial spillover, since the volatility spillovers are time varying. In addition, it is important and meaningful for regulators and policy-makers to identify the factors affecting volatility spillover linkage of stock markets. These can help investors to prevent risks or adjust investment strategies in advance based on the changes of internal or external influencing factors.

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Acknowledgements

Reference

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This paper was supported by grants from the National Natural Science Foundation of China (Grant No. 71571038, 71671030 and 71971048)

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Journal Pre-proof Appendix A Table A1 The 19 countries and the corresponding stock indexes. China

SHCI

Argentina

Number

Country

Stock index

Number

1

Japan

Nikkei225

11

MERV

2

Korea

KOSPI

12

Australia

AUS200

3

Mexico

IPC

13

Brazil

BOVESPA

4

Russia

RTS

14

Canada

GSPTSE

5

Saudi Arabia

SASEIDX

15

France

CAC40

6

South Africa

TOP40

16

Germany

DAX30

7

Turkey

ISE100

17

India

SENSEX

8

UK

FTSE100

18

Indonesia

JKSE

9

USA

SP500

19

Italy

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10

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Stock index

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Table A2 Time series observation in each sub stage.

data

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stage

pre-crisis of global financial crisis

2006/01/02 to 2007/08/09

Stage 2

US subprime crisis

2007/08/10 to 2009/12/31

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Stage 1

European debt crisis

2010/12/08 to 2012/12/31

Stage 4

post-crisis of global financial crisis

2013/01/02 to 2017/12/31

Stage 5

world trade friction

2018/01/02 to 2018/12/31

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Stage 3

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Appendix B

SA frica

Japan

USA Turkey

Korea

A ustralia

India

Indonesia

Canada

Russia

China

France

A rgentina Brazil

UK Mexico Italy

Germany

SA rabia

(a) Volatility spillover network in Stage 1

27

Journal Pre-proof Mexio Indonesia

Japan India

UK Brazil

China Russia

Canada USA

A ustralia

SA rabia

Turkey

SA frica

Italy

Korea France Germany

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A rgentia

(b) Volatility spillover network in Stage 2 Italy

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A rgentina

Mexico

Japan

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Germany

Korea

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A ustralia

re

Brazil

SA frica

UK

Turkey

India Indonesia

USA

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Russia

China

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Canada

France

SA rabia

(c) Volatility spillover network in Stage 3 Russia SA rabia

India

Mexico

Brazil A rgentina

USA

UK

Korea Japan

Turkey

Canada

China

Italy

SA frica Germany

Indonesia

France

A ustralia

(d) Volatility spillover network in Stage 4

28

Journal Pre-proof Mexico A ustralia

Canada Turkey

SA frica

France

Germany India

Indonesia

UK

Brazil Japan SA rabia

USA

Russia

China Italy

Korea

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(e) Volatility spillover network in Stage 5

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Fig. B. Dynamic spatial connectedness of volatility spillover network during five stages (a, b, c, d, e) Notes: the size of the node represents the betweenness centrality, which is located on the volatility spillover

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paths linking pairs of others.

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Journal Pre-proof Spatial linkage of volatility spillovers and its explanation across G20 stock markets: A network framework Highlight

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1. Dynamic volatility spillover networks are constructed by GARCH-BEKK model. 2. Spatial linkage of volatility spillover has obvious multiple superposition phenomena. 3. Using the centrality structure characteristics of network nodes to depict the risk contagion and acceptance intensity of stock market. 4. Applying factor analysis method to obtain the centrality comprehensive indicators of stock market. 5. Analyzing the correlation between relational matrices and identifying the major factors affecting spatial linkage through the QAP method.

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