Analyzing time-varying volatility spillovers between the crude oil markets using a new method

Journal Pre-proof Analyzing time-varying volatility spillovers between the crude oil markets using a new method

Tangyong Liu, Xu Gong PII:

S0140-9883(20)30050-5

DOI:

https://doi.org/10.1016/j.eneco.2020.104711

Reference:

ENEECO 104711

To appear in:

Energy Economics

Received date:

28 October 2019

Revised date:

4 December 2019

Accepted date:

8 February 2020

Please cite this article as: T. Liu and X. Gong, Analyzing time-varying volatility spillovers between the crude oil markets using a new method, Energy Economics(2020), https://doi.org/10.1016/j.eneco.2020.104711

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© 2020 Published by Elsevier.

Journal Pre-proof

Analyzing time-varying volatility spillovers between the crude oil markets using a new method Tangyong Liua, Xu Gongb* a

College of mathematics and informatics, Fujian Normal University, Fuzhou, 350117,

China b

School of Management, China Institute for Studies in Energy Policy, Collaborative

Innovation Center for Energy Economics and Energy Policy, Xiamen University, Xiamen, 361005, China

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* Corresponding author: [email protected]

Abstract: The spillover effect is an important factor affecting the volatility of crude

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oil price. Basing on the study of Diebold and Yilmaz (2009, 2012, 2014), we propose

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a new method that calculates the time-varying volatility spillover indexes by the generalized forecast error variance decomposition of TVP-VAR-SV model. Then,

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using the new method, we study the time-varying volatility spillovers between four major crude oil markets (WTI, Brent, Oman, Tapis) from November 29, 2002 to July

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13, 2018. By comparing the results of our new method and traditional rolling window method, we verify the superiority of our new method. The results show that the

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volatility spillovers calculated by the new method are clearer, more stable and not outlier sensitive. From the estimated results of time-varying volatility spillovers, we find that the volatility spillover between crude oil markets is slowly increasing, but there are obvious cyclical changes. And from the correlation analysis and the Granger causality test, we find that the volatility and volatility spillovers are positively correlated and are two-way Granger causality, which supported for the market infection hypothesis of King and Wadhwani (1990).

Keyword: Volatility spillovers; Time-varying; TVP-VAR-SV model; Crude oil

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1. Introduction Due to the deregulation, securitization, globalization and rapid development of information technology, there is an increasing integration of major financial markets throughout the world. Especially for the same type of market, the price fluctuation (i.e., volatility) usually has mutual spillover effect which refers to the fact that the volatility of one market can be transmitted to another market. The volatility spillovers may stem from the investors’ behavior of arbitrage. When a large fluctuation occurs in one market, with an arbitrage opportunity arising, the investors can obtain excess

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profits through cross-market transactions until the deviation disappears. Therefore, the

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price change of one market will cause price changes in another market, resulting in volatility spillovers. The volatility spillovers can also be caused by emotional

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contagion. When there is insufficient information in financial markets, the investors are prone to herd mentality (i.e., herd effect). The “herd effect” can cause the prices in

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different markets to change simultaneously, even if there are no direct economic

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connections between them.

Crude oil is an important energy and raw material in modern industry. Its price

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fluctuations can affect many levels of the national economy along the industrial chain, thus having a great impact on the global economy (e.g., Aromi and Clements, 2019;

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Gogolin et al., 2018; Gong and Lin, 2017; Grigoli et al., 2019; Mohaddes and Pesaran, 2017; Nasir et al., 2018; Wei and Guo, 2016; Wen et al., 2016, 2018, 2019; Zhao et al., 2016). Particularly, with the introduction of crude oil as an alternative asset in investment portfolios, and the rapid development of crude oil futures markets, crude oil trading has attracted numerous types of market participants, not just oil-related companies, but also pure speculators. All participants are trying to find the factors that affect crude oil price, among which volatility spillover effect is an important aspect that has been widely studied in economic literatures (e.g., Haigh and Holt, 2002; Jin et al., 2012; Chang et al., 2010; Karali and Ramirez, 2014; Singh et al., 2019; Wen et al., 2019). The econometric methods used are mainly the multivariate GARCH model. However, the multivariate GARCH model has several drawbacks. For instance, it is inconvenient to reveal the directional and time-varying characteristics of the volatility spillovers. Recently, the method of Diebold and Yilmaz (2009, 2012, 2014) which calculate 2

Journal Pre-proof the volatility spillover indexes based on the forecast error variance decomposition of VAR model has been widely used in studying the volatility spillovers between financial markets. To describe the dynamics of volatility spillovers, Diebold and Yilmaz (2009, 2012, 2014) use a rolling window VAR method. Although the rolling window method is simple, there are several drawbacks. Firstly, the result of rolling window method is directly affected by the window size which is artificially given in advance. For different window sizes, the results are always different. And it is difficult to choose the optimal window sizes. Secondly, the rolling window method cannot get

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the results in the first window, which will cause the loss of observations, especially when the window size is large. Lastly, the rolling window method is outlier sensitive.

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The results can jump when the window first contains or rejects an outlier. In this study, we propose a new method calculating the time-varying volatility

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spillover indexes based on the time-varying parameter VAR model with stochastic

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volatility (hereafter, TVP-VAR-SV model). And based on the new method, we investigate the time-varying volatility spillover effect between four major benchmarks

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in the crude oil markets, including WTI (North America), Brent (North Sea), Oman (Middle East) and Tapis (Asia-Pacific). The TVP-VAR-SV model is proposed by

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Primiceri (2005) and has recently been widely used in macroeconomic empirical analysis. The feature of this model is that all parameters in the TVP-VAR-SV model

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are assumed to follow the first-order random walk process, that is, be time-varying. Thus, it enables us to capture the potential time-varying nature of the underlying structure in the economy. And the parameters of TVP-VAR-SV model are estimated under the framework of Bayesian method, which is robust to the outliers. So, compared with the traditional rolling window method, the methodology based on TVP-VAR-SV model (hereafter, TVP method) has several obvious advantages. Firstly, we do not need to set the rolling window size, thus avoiding the impact of window size on the estimated results. Second, the TVP method does not cause observation loss. By the Markov chain Monte Carlo (MCMC) method, we can obtain parameter estimates for all time. Lastly, the TVP method is not outlier sensitive. Therefore, the time-varying pattern of volatility spillovers is clearer and more stable. Our research makes two major contributions to the empirical analysis of crude oil volatility spillovers. On the one hand, this paper proposes a new method calculating the time-varying volatility spillover indexes which can overcome the shortcomings of 3

Journal Pre-proof the traditional rolling window method. Our method can obtain the volatility spillover indexes in the whole sample, and avoid the problem of choosing window size. On the other hand, we study the volatility spillovers between the four major crude oil markets in detail, especially the time-varying volatility spillovers. The results of full-sample analysis show that the volatility spillovers that from WTI crude oil to other markets is strongest. The results of dynamic spillover analysis show that the volatility spillovers between crude oil markets have a slight upward trend, and there are obvious cyclical changes. In addition, we find that volatility and volatility spillovers change in the

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same direction. When the crude oil prices fluctuate greatly, the volatility spillovers will increase accordingly. When the crude oil prices keep stable, the volatility

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spillovers will decrease. To investigate the relationship between volatility and volatility spillovers, we calculate the Pearson correlation and test the Granger

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causality. Results show that volatility and volatility spillovers are positively correlated

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and are two-way Granger causality. This is consistent with the market infection model of King and Wadhwani (1990).

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The rest of this article is organized as follows: Section 2 provides a brief literature review. Section 3 introduces the TVP-VAR-SV model and the calculation of

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time-varying volatility spillovers in detail. Section 4 describes our data. Section 5 reports the empirical results. Section 6 performs the discussions of relationship

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between volatility and volatility spillovers. The last section concludes this paper.

2. Literature review

The common movement (i.e., co-movement) and volatility spillover effect between crude oil markets is an important issue which has been widely studied in the economic literatures. For instance, Weiner (1991) analyzes the relationship between oil prices from different markets using the methods of correlation and regression. The results show that the crude oil prices show a high degree of regionalization, that is, not co-movement. Using the co-integration tests, Gülen (1999) conduct a similar study with Weiner (1991), but confirm the co-movement of different crude oil prices. Ewing and Harter (2000) study the relationships between Alaska North Slope (ANS) and Brent oil prices. They find that the two markets share a long-run common trend, suggesting they are co-movement. Bentzen (2007) analyze the co-movements of WTI, Brent and Dubai using the error correction model (ECM) from 1988 to 2004, and 4

Journal Pre-proof conclude that the global oil markets are integrated. At present, the view that there is a co-movement in different crude oil markets is widely recognized. Most of the recent studies focus on the volatility spillover effect. The econometric methods used are mainly the multivariate GARCH model. For instance, Chang et al. (2010) analyze the volatility spillovers between WTI, Brent, Dubai, Oman and Tapis crude oil spot prices using a constant conditional correlation GARCH (CCC-GARCH) model. They find that WTI and Brent crude oil have the greater influence on Dubai, Oman and Tapis crude oil. Jin et al. (2012) study the

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volatility spillovers between WTI, Dubai and Brent crude oil futures using a vector autoregression BEKK (VAR-BEKK) model. The results show that Brent crude oil

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futures have the greatest influence on other crude oil futures. However, the multivariate GARCH model is mainly applicable to describe or test the volatility

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correlation between pair-wise markets. It is inconvenient to reveal the directional and

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time-varying characteristics of the volatility spillovers, as well as the overall volatility spillovers of all markets. And the estimation of multivariate GARCH model is very

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complicated, especially when the number of variables included is relatively large. In recent years, some researchers have developed methodologies to overcome the

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drawbacks of multivariate GARCH models in capturing time-varying features of volatility spillover effect. Among the many, a notable study is Diebold and Yilmaz

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(2009, 2012, 2014) which conduct the volatility spillover index by the forecast error variance decomposition of VAR model (hereafter, DY method). The DY method can describe the direction of volatility spillovers and the dynamics of volatility spillovers. Due to its ease of use, the DY method has been widely used in studying volatility spillovers between financial markets during the past ten years. For instance, Zhou et al. (2012) analyze the volatility spillovers between China's stock market and major stock markets. Lucey et al. (2014) study the volatility spillovers between the four gold markets in London, New York, Tokyo and Shanghai. Batten et al. (2014) investigate the volatility spillovers between four precious metal markets, including gold, silver, platinum and palladium. Yarovaya et al. (2016) examine the volatility spillovers between 21 stock markets in four regions, including Europe and Asia. Yang and Zhou (2016) study the relationship between quantitative easing in the United States and the volatility spillovers of major international financial markets based on the VIX data of 11 financial markets. Ji et al. (2019) study the volatility spillovers of six large 5

Journal Pre-proof cryptocurrency markets (i.e. Bitcoin, Ethereum, Ripple, Litecoin, Stellar and Dash). There are also a few literatures studying the volatility spillovers between oil markets based on the DY method. For instance, Baruník et al. (2015) study the volatility spillovers between US crude oil futures, gasoline futures and fuel oil futures. Magkonis and Tsouknidis (2017) analyze the dynamic spillover effects across the volatilities of petroleum-based commodities. Wang and Guo (2018) study the volatility spillovers between carbon, WTI oil, Brent oil, and natural gas market. The DY method has many advantages, but it still has several drawbacks that we

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have discussed in the first section, including: (1) affected by the window size; (2) lose observations; (3) outlier sensitive. To overcome the shortcomings above, we

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extend the DY method. Instead of using rolling window VAR method, we calculate the time-varying volatility spillover indexes directly using the TVP-VAR-SV model

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of Primiceri (2005). The advantage of this model is that all parameters are assumed

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to follow the first-order random walk process, that is, the dynamic relationship between variables is allowed to change at any time. In recent years, the

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TVP-VAR-SV model has been widely used in economic empirical analysis, especially in macroeconomic. For example, Nakajima et al. (2011) study the

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time-varying relationship between Japan's monetary policy and economic output from 1981 to 2008. Baumeister and Peersman (2013) study the impact of oil supply

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shocks on the US economy. Jebabli et al. (2014) study the impact of stock market and oil price shocks on food prices. Simokengne et al. (2015) study the impact of house prices and stock prices on consumption. Bianchi and Civelli (2015) explore the relationship between globalization and inflation. Prieto et al. (2016) studied the linkage between financial markets and macroeconomics. Particularly, Antonakakis and Gabauer (2017) proposed the refined connectedness measures based on the TVP-VAR-SV model of Koop and Korobilis (2013), and empirically analyze the exchange rate volatility connectedness. Based on the approach of Antonakakis and Gabauer (2017), Gabauer and Gupta (2018) investigate the internal and external categorical economic policy uncertainty (EPU) spillovers between the US and Japan. Korobilis and Yilmaz (2018) introduce a connectedness measures based on the large TVP-VAR model, and study the dynamic connectedness between 35 U.S. and European financial institutions. However, to the best of our knowledge, there are no literatures that are published to combine the DY method and TVP-VAR-SV model to 6

Journal Pre-proof study the time-varying volatility spillovers between the crude oil markets.

3. Methodology 3.1. TVP-VAR-SV model To describe the dynamics of volatility spillovers, Diebold and Yilmaz (2009, 2012, 2014) use a rolling window VAR method. Although it is simple, it still retains the drawbacks of fixed-parameter VAR model, that is, assume the parameter is unchanged in the window. Parameters at a certain time are only estimated by several samples in

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the window, which may not be efficient enough. We assume that the data is generated

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by the TVP-VAR-SV process, and the parameters are directly estimated by the MCMC method.

where t  p  1, p  2,

  p ,t yt  p  At11/t 2 t ,  t  N (0, I k )

(1)

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yt   0,t  1,t yt 1   2,t yt  2 

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According to Primiceri (2005), we specify the TVP-VAR-SV model as

, T . yt is a k  1 vector of observed variables.  0,t is a ,  p ,t are k  k matrices of time-varying VAR

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k  1 vector of constant. 1,t ,  2,t ,

coefficients. At is the lower triangular matrix which diagonal elements are all 1.  t

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is the diagonal matrix of time-varying variances, which diagonal elements are

 12t ,  22t , ,  kt2 . Stacking the elements of  0,t , 1,t ,  2,t , ,  p ,t by rows, and

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defining X t  I k  (1, yt-1 ,, yt- p ) , where  denotes the Kronecker product, the model can be rewritten as

yt  X t t  At11/2 t t .

(2)

Following Primiceri (2005), let  t be the stacked vector of lower triangular elements in At , and ht  (h1,t , h2,t ,

, hp ,t ) with h jt  log  2jt , for j  1, 2,

, k . We

assume that the parameters in Eq. (2) all follow a random walk process as follows:

t +1  t   ,t , t +1  t   ,t , ht +1  ht  h ,t

(3)

with   Ik  t        ,t   N  0,  0  0    ,t        h ,t   0

for t  p  1, p  2,

0  0 0

0 0  0

0   0  0    h  

, T , where  ,   ,  h are time-invariant diagonal matrices; 7

Journal Pre-proof  , p 1  N (  ,  ) ,  , p 1  N (  ,   ) , h, p 1  N ( h , h ) , and we assume 0

0

0

0

0

0

that the disturbances  t ,  t ,  t ,  t are all uncorrelated. The TVP-VAR-SV model above can be estimated by the MCMC method which is discussed in detail in Primiceri (2005). With the MCMC sample, we use the posterior mean as the estimated value. Thus, we can get the estimated parameter matrices ˆ , ˆ , , ˆ , Aˆ t and ˆ t . Basing on the estimated parameters of TVP-VAR-SV  1,t 2,t p ,t model, we can calculate the time-varying impulse response function (IRF) and the generalized forecast error variance decompositions. Then, we construct time-varying

decompositions. Below we will introduce in detail.

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3.2. Time-varying volatility spillovers

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volatility spillover indexes based on the generalized forecast error variance

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Diebold and Yilmaz (2009, 2012, 2014) define the spillovers as the fractions of the H-step-ahead error variances in forecasting yi due to the shocks to y j for i  j .

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In order to calculate the impulse response function, we transform Eq. (1) to its vector

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moving average (VMA) representation:



yt*   Bi ,t ut i

(4)

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i 0

where yt*  yt  yt , and ut  At11/2 t  t . The k  k coefficients matrices Bit can be

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calculated by the recursion:

Bh,t  1,t Bh 1,t   2,t Bh  2,t 

for t  p  1, p  2,

  p ,t Bh  p ,t

(5)

, T , where B0,t  I k and B j ,t  0 if j  0 . Then, the impulse

response function of orthogonal innovations  t at time t is IRFh  Bh ,t At11/t 2 . Because the forecast error variance decompositions based on IRFh depend on the ordering of the variables, following Diebold and Yilmaz (2012), we use the generalized forecast error variance decompositions which are invariant to the ordering. Denoting ij ,t ( H ) as the H-step-ahead forecast error variances decompositions at time t, we have H 1

ij ,t ( H ) 

 jj1,t  (eiBh ,t u ,t e j ) 2 H 1

h 0

 (eB h0

i

8

h ,t

(6) u ,t Bh,t ei )

Journal Pre-proof where ut  At1t ( At1 ) is the variance matrix of the error vector ut ,  jj ,t is the jth diagonal element in u ,t (the same as  t ). ei is a selection vector, where the ith element is 1 and the others are 0. Because  kj 1ij ,t ( H )  1 , in order to match the traditional variance decomposition, we normalize each element of the generalized variance decomposition matrix by the row sum as follows:

ij ,t ( H ) 

ij ,t ( H )

(7)

N

 j 1

ij ,t

(H )

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where kj 1ij ,t ( H )  1 , and ik, j 1ij ,t ( H )  N . Basing on ij ,t ( H ) , we can construct

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time-varying volatility spillover indexes described in Table 1.

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Table 1

Index

Meaning

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Time-varying volatility spillover indexes.

Measuring the volatility spillovers

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Total spillovers

across all markets

Measuring the spillovers received by

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market i from all other markets

Calculation SPt ( H ) 

1 N

N



i , j 1, i  j

Fromi.t ( H ) 

N



j 1, j  i

ij ,t ( H ) 100 ij ,t ( H ) 100

Measuring the volatility spillovers Directional spillovers

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transmitted from market i to all other

Toi ,t ( H ) 

N



j 1, j  i

 ji ,t ( H ) 100

markets

Measuring the net volatility spillovers

Neti.t ( H )  Toi.t ( H )  Fromi ,t ( H )

from market i to all other markets

Net pairwise spillovers

Measuring the volatility spillovers between markets i and j

NPij ,t ( H )  [ ji ,t ( H )  ij ,t ( H )] 100

4. Data and descriptions We will examine volatility spillovers between the crude oil markets, including the four major pricing benchmarks for the international crude oil markets, namely WTI (North America), Brent (North Sea), Oman (Middle East) and Tapis (Asia-Pacific). These four markets account for more than half of the world's crude oil trading volume 9

Journal Pre-proof and their prices are likely to be highly correlated. For the crude oil prices, there are two choices, that is, futures prices and spot prices. Because the Oman crude oil futures contract just began trading on June 1, 2007, in order to get more samples, we choose the daily crude oil spot prices. Our data cover the period from November 29, 2002 to July 13, 2018. All prices are quoted in US dollars per barrel and are downloaded from the Bloomberg database. Because the market volatility cannot be directly observed, we use the realized volatility (RV) as an agent variable of volatility, which is broadly used in financial

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research during the past 20 years (Andersen and Bollerslev, 1998; Boldanov et al., 2016; Gong and Lin, 2018; Hsu and Murray, 2007; Liu et al., 2018; Ma et al., 2017,

calculated by M

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RVt =  rt 2i

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2018; Prasad et al., 2018; Wang et al., 2017). The weekly realized volatility ( RVt ) is

(8)

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i 1

where rti is the ith daily return in week t; M is the number of trading days

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(normally, M  5 ). The daily return rt i is calculated by rti  100 * (ln pti  ln pt , i 1 ) , where pt i is the ith closing price in week t and pt , i 1 is the previous day’s price.

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According to the calculation above, we get a total of 810 observations for each series. Fig. 1 shows the weekly realized volatility of four crude oil markets, and Table

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2 provides the summary statistics. Several interesting facts emerge, including: (1) the WTI oil has been the most volatile, followed by Brent oil, with the Tapis oil being the least volatile; (2) the volatility appear to be highly persistent, that is, there is volatility clustering which is common in financial data; (3) all four volatilities are high during the period of 2007-2009, which is results from the global financial crisis in 2008; and (4) volatilities increased simultaneously during the period of 2013-2016 during when crude oil price fell sharply.

10

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400 WTI

Brent

300

300

200

200

100

100

2005

2010

2015

2005

400

2010

2015

400 Tapis

200

200

100

100

2010

2015

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2005

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300

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300

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Oman

2005

2010

2015

Table 2

Median Max Min Std.dev Skewness Kurtosis

WTI

Brent

Oman

Tapis

26.83

21.82

19.63

18.98

15.27

13.30

11.33

10.57

289.81

399.02

378.25

234.86

0.23

0.31

0.14

0.28

37.35

30.94

30.11

25.34

3.39

5.31

5.19

3.69

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Mean

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Descriptive statistics of volatilities.

16.21

46.15 ***

J-B

7.44e+3

Q(1)

284.98***

ADF

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Fig. 1 Weekly realized volatility of four crude oil markets.

-5.20

***

6.67e+4

43.02 ***

163.07*** -5.65

***

5.77e+4

20.92 ***

109.64*** -7.24

***

1.27e+4*** 152.21*** -6.91***

Notes: ***denote significance at 1% level. J-B is the normality test of Jarque and Bera (1980). Q (1) is the autocorrelation test of Ljung and Box (1978) for 1th order. ADF is the unit root test.

In Table 2, the J-B statistics of four series reject the null hypotheses of normal distribution, which indicate that all volatilities are not normal distributed. The Q statistics reject the null hypotheses of no autocorrelation, which show that volatility series have significant serial autocorrelations. At last, the ADF tests show that all 11

Journal Pre-proof volatilities are stationary. In addition, we compute the Pearson correlation of the four volatilities. Table 3 displays the results. It can be found that the correlations between the four volatilities are all significantly positive at 1% level, indicating that there may be mutual volatility spillovers between the four crude oil markets. In particular, the results show that the correlation between Oman and Tapis is much higher than other correlations, which means that there is a closer relationship between these two markets.

Table 3

WTI

Brent

1

0.71 1

Oman

Oman Tapis

0.64***

0.56***

0.62***

1

0.78***

0.59

1

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5. Empirical results

Tapis

***

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Note: ***denote significance at 1% level.

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Brent

***

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WTI

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Correlation matrix between four volatilities.

5.1. Full-sample spillover analysis

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Following Diebold and Yilmaz (2009, 2012, 2014), we analyze the volatility spillovers of full sample using a four-variable fixed-parameter VAR model. Before the process of estimate, we must choose the lag length (p) of VAR model and the forecasting horizon (H) for the forecast error variance decomposition. According to the Schwarz information criterion (SIC), we choose the optimal lag length of VAR model as p  2 . Next, to choose the optimal ahead forecasting horizon, we calculate the total volatility spillover indexes, with H changes from 1 to 32. The results show that the total volatility spillover index increase with the increase of H, but when H increases to 16, the total volatility spillover index almost reaches the maximum value. So we set the ahead forecasting horizon as H  16 . Table 4 shows the variance decomposition matrix and the various volatility spillover indexes. The ijth entry is the forecast variance contribution to market i from market j. The off-diagonal row sums are the directional spillovers “From others”, and 12

Journal Pre-proof the off-diagonal column sums are the directional spillovers “To others”. The “To others” minus “From others” are the “Net” spillovers. The “Total spillover” is the average of “From others” (or “To others”). In addition, Table 5 shows the estimated net pairwise volatility spillovers between two specific crude oil markets. Its ijth entry is the forecast variance contribution of market i to market j minus the forecast variance contribution of market j to market i, that is, subtracted from the elements of the symmetric position in Table 4. For instance, the number 4.94 of Row 2, Column 2 in Table 5 is calculated by 27.34-22.44=4.94, which indicates that the net volatility

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spillover of WTI oil to Brent oil is 4.94%.

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Table 4 Volatility spillover indexes of crude oil markets.

Tapis

From others

WTI

51.90

22.44

11.82

13.84

48.10

Brent

27.37

47.50

11.08

14.05

52.50

Oman

18.47

12.76

44.87

23.90

55.13

Tapis

20.85

15.91

22.34

40.90

59.10

To others

66.69

51.11

45.24

51.79

Total Spillover:

Net

18.59

-1.39

-9.89

-7.31

53.71

-p

Oman

re

Brent

lP

WTI

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Notes: The ijth entry is the forecast variance contribution to market i from market j. The column “From others” are off-diagonal row sums. The row “To others” are off-diagonal column sums. The “Net” column is “To others” minus “From others”. The “Total spillover” is the average of “From others” (or

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“To others”). For example, the number in Row 1, Column 2 indicates the forecast variance contribution from Brent oil to WTI oil is 22.44%. The number in Row 1, Column 5 denotes the spillover from other three markets to WTI is 48.10%. The number in Row 5, Column 1 denotes the spillover from WTI to other three markets is 66.69%. The number in Row 6, Column 1 denotes the net spillover from WTI to other three markets is 18.58%.

Table 5 Net pairwise volatility spillover indexes of crude oil markets. WTI

Brent

Oman

Tapis

WTI

0

-4.94

-6.65

-7.00

Brent

4.94

0

-1.69

-1.86

Oman

6.65

1.69

0

1.56

Tapis

7.00

1.86

-1.56

0

Notes: The values in the table are net volatility spillovers. Taking 4.94 in Row 2, Column 2 as example, it indicates that the net volatility spillover of WTI oil to Brent oil is 4.94%.

13

Journal Pre-proof According the results in Table 4, we can generally find that the WTI crude oil is the market that: (1) produces the most volatility spillovers (66.69%) to other markets; (2) receives the least volatility spillovers (48.10%) from other markets; (3) produces the most net volatility spillovers (18.59%) to other markets. It indicates that WTI oil is dominant in the international crude oil markets, which is in line with the actual situation. As the world's largest economy and the largest consumer of crude oil, the United States consumes about 900 million tons of oil in 2017, accounting for about a quarter of the world's total. The WTI crude oil which is traded in New York

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Mercantile Exchange (NYME) is the world's most traded commodity futures and one of the most important pricing benchmarks in the global crude oil markets. Therefore,

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the change of WTI oil price attract the most attention from the investors all over the world, thus cause an important impact on the international crude oil markets.

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In addition, Table 4 shows that Brent crude oil produces 51.11% volatility

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spillovers to other markets, a little lower than Tapis crude oil (51.79%). This can be explained by the high correlation between the Oman and Tapis markets. The results in

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Table 3 show that the correlation between Oman and Tapis is 0.775, much higher than other correlations. Table 4 shows that the Tapis produces 23.90% volatility

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spillovers to Oman, with Brent producing only 12.76%. Oman produces 22.34% volatility spillovers to Tapis, with Brent producing 15.91%. But according to Table 5,

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we can find that Brent has net volatility spillovers to Oman, as well as to Tapis. It indicates that Brent crude oil market also dominates the international crude oil markets, which is second only to WTI crude oil market. Finally, we consider the total spillover index which is presented in the lower right-hand corner of Table 4. The total spillover index effectively translates the various directional spillovers into a single index. It measures the contribution of spillovers of volatility shocks across all markets to the total forecast error variance. According to the estimated results, we find that the total spillover between all four crude oil markets is 53.71%. This implies that approximately half of the forecast error variances are due to spillover effect among different markets. So we can conclude that the volatility spillover effect is an important factor which must be considered in analyzing crude oil price fluctuations. 5.2. Dynamic spillover analysis 14

Journal Pre-proof However, during the years in our sample of 2002 to 2018, many changes took place. Some are described as continuous evolution, such as globalization, the development of electronic technology and the rise of hedge funds. Others are described as bursts, such as the global financial crisis around 2008. Hence, at any given point in time, the intensity of the volatility spillovers may be very different. The previously obtained full-sample volatility spillovers, although providing a useful summary of “average” behavior, are likely to miss the potential important patterns in spillover effects, such as the dynamic evolution of volatility spillovers. To solve this problem, we are now

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moving from static analysis to dynamic analysis. In the literatures of Diebold and Yilmaz (2009, 2012, 2014) and other related

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researches (e.g., Batten et al., 2014; Lucey et al., 2014; Wang and Guo, 2018; and Zhou et al., 2012;), the method used to describe the dynamics of volatility spillovers

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is a rolling window VAR method, that is, estimates a fixed-parameter VAR model

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using the samples in a given window. When the window is rolling as the time, the estimated parameters change. Although the rolling-window method is simple, there

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are some drawbacks which we have discussed in detail in Section 1, that is, affected by the window size; cannot get the results in the initial window; and outlier sensitive.

window VAR method.

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To verify it, we firstly estimated the dynamic volatility spillovers using the rolling

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Fig. 2 shows the total volatility spillover indexes, where RW50, RW100, RW150 and RW200 denote that the window size used is 50,100,150 and 200 weeks. Fig. 3 shows the total spillovers using log volatility. It can be found that the estimated results of total volatility spillovers are very different for different window sizes. The total spillovers change very sharply when the window size is set to 50 weeks. As the window size increases, the total spillover index changes more gently. In addition, the rolling window method cannot get the results in the initial window, thus cause the loss of observations, especially when the window size is large. Furthermore, we find the results are significantly affected by outliers. The total spillover index in Fig. 2 jumps when the window first contains or rejects an outlier. Take the March, 21, 2008 for example, the total spillover index jumps from 45.80% to 75.31% when the window size is set to 50 weeks. Although, this feature may be used to examine the impact of extreme events, it makes the trend of volatility spillovers unclear. As shown in Fig. 3, logarithmic transformation of data can weaken the impact 15

Journal Pre-proof of outliers to some extent. But it still has the problems that the result is affected by the window size, and cannot get the results in the initial window. SP-RW50

SP-RW100

60

60

40

40

20

20 2010

2015

2005

2010

2015

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2005

SP-RW200

60

-p

60

ro

SP-RW150

40

20

re

40

20

2010

2015

2005

2010

2015

lP

2005

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Fig. 2 Total spillover indexes using rolling window method.

70 Log-SP-RW50

Log-SP-RW100

60

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60

70

50 40 30

50 40 30

20 2005

2010

2015

2005

Log-SP-RW150

2010

2015

2010

2015

Log-SP-RW200

60

60

50

50

40

40

30

30 2005

2010

2015

2005

Fig. 3 Total spillover indexes of log volatility using rolling window method. 16

Journal Pre-proof Next, we estimate the time-varying volatility spillover indexes based on the TVP-VAR-SV model (hereafter, we call it “TVP method”). The calculation formulas are described in detail in Section 3. The parameters in TVP-VAR-SV model are all assumed to be different at each point in time, so we do not need to set the rolling window size. But we still need to choose the lag order. Following Nakajima et al. (2011), we calculate the marginal likelihood for different lag orders (1 to 6). Based on the highest marginal likelihood, we set the lag order as p  2 , which is the same as fixed-parameter VAR model. As for the ahead forecasting horizon, we set it as

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H  16 , consistent with the full-sample analysis. In addition, we need to set the priori distribution of the hyper-parameters in the MCMC algorithm. Following Nakajima et

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al. (2011), the prior distributions of the ith diagonal elements in covariance matrices  ,   ,  h are set as

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(  )i IG (40, 0.02) , ( )i IG(4,0.02) , (h )i IG(4,0.02)

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where IG () is inverse-Gamma distribution. The initial states of the time-varying parameters are set as flat priors:

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u0  ua0  uh0  0 ,  0  0   h0  10  I .

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To compute the posterior mean, we draw 11,000 samples and “burn-in” the initial 1,000 samples. Table 6 gives the estimated results of selected parameters, including

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posterior means, standard deviations, the 95 percent credible intervals, the convergence diagnostics (CD) of Geweke (1992) and inefficiency factors. They are calculated using the remaining 10,000 samples (see, Nakajima et al., 2011). Table 6

Estimated results of TVP-VAR-SV model. Parameter

Mean

Std.

95% interval

CD

Inefficient

 )1

0.0229

0.0027

[0.0183, 0.0290]

0.92

31.33

 )2

0.0214

0.0023

[0.0176, 0.0265]

0.19

36.55

 )1

0.0537

0.0110

[0.0110, 0.0363]

0.67

112.14

 ) 2

0.0537

0.0110

[0.0110, 0.0363]

0.67

112.14

 h )1

0.2820

0.0443

[0.2084, 0.3779]

0.11

93.81

 h ) 2

0.3859

0.0494

[0.2955, 0.4850]

0.85

62.89

17

Journal Pre-proof According to the CD statistics in Table 6, we can conclude that the MCMC samples convergent to the posterior distribution. In addition, we find the maximum of inefficiency factors is about 112, which implies that we obtain about 10000 / 112  89 uncorrelated samples. It is sufficient for our posterior inference. Then, based on the MCMC samples, we can get the estimated matrices ˆ , ˆ , , ˆ , Aˆ and ˆ . Using the estimated parameters, we can calculate the  1,t 2,t p ,t t t time-varying volatility spillover indexes (see, Table 1). Fig. 4 shows the time-varying total spillover index. We can find that the trend in Fig. 4 is similar to that in Fig. 2

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(and Fig. 3), but it is more stable and clearer, and we can get the estimates for all weeks (except for the first two weeks). So, we can conclude that our TVP method can

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overcome the shortcomings of traditional rolling window method.

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SP-TVP

re

60

lP

55

na

50

40

35 2002

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45

2004

2006

2008

2010

2012

2014

2016

2018

Fig. 4 Total volatility spillover using the TVP method.

According to Fig.4, we can find there is a slight upward trend in the volatility spillover effect between the crude oil markets. Starting at a value approximately 45% in the first week of 2003, the total spillovers reach approximately 55% in the last week of our sample (July 13, 2018), and exceed 60% during 2015-2016. It can be explained by the globalization and rapid development of information technology. Economic globalization increases the correlation between different economies. The development of information technology enables the rapid transmission of transaction 18

Journal Pre-proof information in different markets. As a result, the integration of major global financial markets has been promoted, including the crude oil markets. In addition, we can identify several cycles in the total volatility spillover. The first cycle started from mid-2003 to mid-2004 during when the crude oil prices rise sharply, with total volatility spillover index climbed from 38% to 52%, an increase of 14%. The second cycle began in April 2006 and went to February 2010, during when the famous US subprime mortgage crisis broke out. The crude oil prices fell sharply and the total volatility spillover index climbed from 34% to 60%, an increase of 26%,

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almost twice the previous cycle. The third cycle started from August, 2013 to mid-2016, during when the crude oil prices (Brent) fell from more than $100 to less

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than $30. Due to severe price fluctuations, the total volatility spillover index climbed from 38% to 63%, an increase of 25%, almost equal to the impact of the US subprime

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na

lP

re

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mortgage crisis.

Fig. 5 Directional spillovers using the TVP method.

Next, we will discuss the directional volatility spillovers. Fig. 5 shows time-varying 19

Journal Pre-proof directional volatility spillovers using the TVP method. It can be found the directional spillovers vary greatly over time. The change pattern of the directional spillovers of WTI and Brent are basically the same as the total volatility spillover in Fig. 4. During the period of US subprime mortgage crisis, the “To others” and “From others” spillover indexes increase significantly. The change in the directional spillovers of Oman and Tapis is relatively small. 5.3. Robustness test In the previous analysis, we use the realized volatility as the proxy of unobservable

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volatility. Now, we use the range volatility of Parkinson (1980) as the proxy of

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volatility to do the similar analysis, which can be used as the robustness test of our empirical results. Following Alizadeh et al. (2003) and Diebold and Yilmaz (2012),

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the weekly range volatility can be calculated by

(9)

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Ranget  0.361  [ln( Pt max )  ln( Pt min )]2

where Pt max is the maximum price on week t, and Pt min is the minimum price.

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In addition, we use the conditional variance extracted from GARCH model of Bollerslev (1986) as another proxy of volatility, named as GARCH volatility. Fig. 6

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shows the total volatility spillovers of range volatility, GARCH volatility and realized volatility using the TVP method. It can be found that the results of three volatilities

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are very similar. Thus, we believe our empirical results are credible.

20

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Fig. 6 Total volatility spillovers of three volatilities using the TVP method.

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6. Relationship between volatility and volatility spillover

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According to Fig. 4, we can find that the volatility and volatility spillovers change in the same direction. When the crude oil prices fluctuate greatly, the volatility spillovers will increase accordingly. When the crude oil prices keep stable, the

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volatility spillovers will decrease. So what is the relationship between volatility and volatility spillovers? And what is the mechanism that leads to this relationship? To explain the volatility spillovers between financial markets, King and Wadhwani (1990) propose a market infection model. In their model, the intrinsic cause of market infections is the incomplete information, that is, the investor can observe the price change, but cannot know all the information that causes the price change. Therefore, investors need to make subjective inferences when making actual investment decisions. Once the price of a market fluctuates greatly, investors may think that there is a major change in the economic fundamentals behind it and make a corresponding buy or sell decision in another market. This makes price fluctuations in the market and even "unexpected errors" can infect another market. According to the market infection model of King and Wadhwani (1990), the market contagion coefficient which reflects the volatility spillover effect is an increasing 21

Journal Pre-proof function of volatility. It means that if we observe the volatility and volatility spillovers change in the same direction, then we can conclude the market contagion mechanism has worked. To verify it, we calculate the Pearson correlation between the total volatility spillover index and the four realized volatilities. Table 7 shows the results. It can be found that all four Pearson correlations are positively significant at 1% level, that is, high volatility corresponds to high volatility spillovers, which is consistent with the market infection model of King and Wadhwani (1990).

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Table 7

Brent

0.23***

0.19***

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TVSI

WTI

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Correlation between volatility spillover and volatility.

Oman

Tapis

0.26***

0.23***

Notes: TVSI denotes total volatility spillover index. WTI, Brent, Oman and Tapis denote the volatility ***

denotes significance at 1%

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series of WTI, Brent, Oman and Tapis crude oil markets, respectively.

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level.

Table 8

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Granger causality between volatility and volatility spillover. Lags

Obs

F (LS)

sup WT (QR)

WTI does not Granger Cause TVSI

3

805

19.56***

158.67***

TVSI does not Granger Cause WTI

3

805

3.05**

13.50*

Brent does not Granger Cause TVSI

4

804

10.47***

149.48***

TVSI does not Granger Cause Brent

4

804

2.41**

23.65***

Oman does not Granger Cause TVSI

4

804

8.96***

638.68***

TVSI does not Granger Cause Oman

4

804

3.34**

26.78***

Tapis does not Granger Cause TVSI

4

804

17.98***

324.13***

TVSI does not Granger Cause Tapis

4

804

3.20**

52.59***

Panal A:

Panal B:

Panal C:

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Null hypothesis ( H 0 )

Panal D:

Notes: TVSI denotes total volatility spillover index. WTI, Brent, Oman and Tapis denote the volatility series of WTI, Brent, Oman and Tapis crude oil markets, respectively. F (LS) denotes the F-statistics from the least square (LS) two-variable VAR model. The lags are set based on the Schwarz information criterion (SIC). sup WT (QR) denotes the sup-Wald statistics of Chuang et al. (2009), where the quantile range is [0.05, 0.95]. The critical values of sup-Wald test can be found in Andrews (1993). *, ** and *** denote significance at 10%, 5% and 1% level. 22

Journal Pre-proof

Furthermore, we use the Granger causality test to examine the dynamic relationship between the volatility and the volatility spillover. Table 8 shows the results. F (LS) is the F-statistics from the least square (LS) estimation, which is used to test the Granger causality in conditional mean. Considering the volatility series are non-normal distributed (see Table 2), we also apply the sup-Wald statistics of Chuang et al. (2009) to test the quantile Granger causality. sup WT (QR) denotes the estimated sup-Wald statistics from quantile regression (QR), where the quantile range is [0.05, 0.95]. According to the estimated results, we can find that the two tests reach a consistent

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conclusion, that is, the total volatility spillover and four volatilities are all two-way

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Granger causality. It means that high volatility may lead to an increase in volatility spillover effect, while high volatility spillovers may further lead to increased volatility.

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The results are also consistent with the market infection model of King and Wadhwani (1990). Therefore, basing on the correlation analysis and the Granger

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causality test, we can conclude that the market contagion mechanism is an important

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factor that causes volatility spillovers in crude oil markets. Table 9

Null hypothesis ( H 0 )

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Granger causality in quantiles between volatility and volatility spillover.

Wald –test (QR) 0.25

0.5

0.75

0.95

WTI does not Granger Cause TVSI

1.87

13.50***

27.43***

2.42*

2.85**

TVSI does not Granger Cause WTI

3.15**

2.97**

1.48

1. 28

0.35

3.59***

9.20***

4.41***

7.01***

0.59

Panal A:

Panal B:

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0.05

Brent does not Granger Cause TVSI

TVSI does not Granger Cause Brent

0.28

2.53

**

1.11

3.33

**

5.52***

Panal C: Oman does not Granger Cause TVSI TVSI does not Granger Cause Oman

11.62***

0.67

4.20***

0.57 7.86

***

0.95 4.39***

1.50

0.66

1.41

Tapis does not Granger Cause TVSI

12.56***

10.08***

12.85***

3.38***

19.06***

TVSI does not Granger Cause Tapis

1.44

4.70***

0.96

7.19***

8.16***

Panal D:

Notes: TVSI denotes total volatility spillover index. WTI, Brent, Oman and Tapis denote the volatility series of WTI, Brent, Oman and Tapis crude oil markets, respectively. Wald-test (QR) denotes the Wald statistics from the quantile regression estimation of two-variable VAR model. The lags are the same as in Table 8. *, ** and *** denote significance at 10%, 5% and 1% level.

23

Journal Pre-proof Lastly, we investigate the asymmetric behavior between the volatility and volatility spillovers. Table 9 shows the Wald-test of Hendricks and Koenker (1992) based on quantile regression in some selected quantiles. The results show that the Granger causal relations are different in different markets and in different quantiles. And Granger causal relations of volatility on volatility spillovers and volatility spillovers on volatility are different. For the WTI crude oil market, we find that volatility has Granger causal effects on volatility spillovers at most selected quantiles (except 0.05), but volatility spillovers have Granger causal effects on volatility only at the lower quantiles (0.05 and 0.25). The Brent and Tapis crude oil markets exhibit a different

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pattern from the WTI crude oil market. For these two markets, the volatility spillovers

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have Granger causal effects on volatility spillovers at the two higher quantiles (0.75 and 0.95), as well as at median (0.5). The Oman crude oil market is very different

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from other three markets. The volatility has Granger causal effects on volatility

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spillovers only at quantiles 0.25 and 0.5, and the volatility spillovers have Granger

7. Conclusions

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causal effects on volatility at the higher quantiles (0.75 and 0.95).

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Understanding the transmission mechanism of volatility between crude oil markets is crucial for the purposes of asset allocation, risk management and dynamic hedging.

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The market infection or spillover effect has been studied in many literatures, but few analyses the time-varying features, especially based on the TVP-VAR-SV model. Based on the TVP-VAR-SV model, this paper extends the methodology of Diebold and Yilmaz (2009, 2012, 2014) which analyze the volatility spillovers by the forecast error variance decomposition of VAR model. We propose a new method that calculates the time-varying volatility spillover indexes directly. Then, using the new method, we study the time-varying volatility spillovers between four major crude oil markets (WTI, Brent, Oman, Tapis) from November 29, 2002 to July 13, 2018. Firstly, following the study of Diebold and Yilmaz (2009, 2012, 2014), we analyze the volatility spillovers of full sample using a four-variable VAR model. The results show that WTI transmits the strongest spillovers to other three crude oil markets. This implies that investors, relevant companies and policy makers should pay the most attention to the price fluctuations of WTI crude oil. Secondly, we compare our new method and the traditional rolling window method. The results show that the volatility 24

Journal Pre-proof spillovers calculated by the new method are clearer, more stable and not outlier sensitive, that is, our new method is better than the traditional rolling window method. This provides a new way for us to study the dynamic volatility spillovers between financial markets. Thirdly, based on the time-varying volatility spillovers indexes, we analyze the dynamic features of volatility spillovers between the crude oil markets. We find that the spillover effect is slowly increasing which can be explained by the globalization and rapid development of information technology. And, the volatility spillovers show obvious cyclical changes. During the period of great changes in oil

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prices, such as the US subprime mortgage crisis, the volatility spillovers increased significantly. Lastly, we investigate the relationship between volatility and volatility

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spillover using the Pearson correlation and Granger causality test. The results show that volatility spillover and volatility are positively correlated and are two-way

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Granger causality, which confirm the market infection hypothesis of King and

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Wadhwani (1990). It implies that when we analyze the fluctuations in the crude oil market, it is necessary to consider the impact of market contagion.

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Data availability statement

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The data that support the findings of this study are available from the

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corresponding author upon reasonable request.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (No. 71701176), Fundamental Research Funds for the Central Universities (No. 2072019029), and Social Science Planning Project of Fujian Province (No. FJ2017C075).

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Author contribution Tangyong Liu: Developed model, Performed the analysis (including performed the statistical analysis and performed the empirical analysis), Wrote the paper, and Revise the paper.

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Xu Gong: Conceived and designed the analysis, Collected the data, Processed the data, and Revise the paper.

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Research Highlights

1. We propose a new method to calculate the time-varying volatility spillover indexes. 2. We study the time-varying volatility spillovers between four major crude oil markets. 3. The volatility spillover between crude oil markets is slowly increasing. 4. The volatility spillover between crude oil markets indicates obvious cyclical

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5. The volatility spillover and volatility are positive correlation.

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