How to effectively estimate the time-varying risk spillover between crude oil and stock markets? Evidence from the expectile perspective

Journal Pre-proof How to effectively estimate the time-varying risk spillover between crude oil and stock markets? Evidence from the expectile perspective Yue-Jun Zhang, Shu-Jiao Ma

PII:

S0140-9883(19)30357-3

DOI:

https://doi.org/10.1016/j.eneco.2019.104562

Reference:

ENEECO 104562

To appear in: Received Date:

28 June 2019

Revised Date:

20 September 2019

Accepted Date:

15 October 2019

Please cite this article as: Zhang Y-Jun, Ma S-Jiao, How to effectively estimate the time-varying risk spillover between crude oil and stock markets? Evidence from the expectile perspective, Energy Economics (2019), doi: https://doi.org/10.1016/j.eneco.2019.104562

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How to effectively estimate the time-varying risk spillover between crude oil and stock markets? Evidence from the expectile perspective Yue-Jun Zhang a,b*, Shu-Jiao Ma a,b a

Business School, Hunan University, Changsha 410082, China

b

Center for Resource and Environmental Management, Hunan University, Changsha 410082, China

*Corresponding

author. Tel./ fax : 86-731-88822899. E-mail: [email protected] (Yue-Jun Zhang).

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Highlights

 It tests the time-varying risk spillover between WTI and well-known stock markets  The EVaR approach based on CAR-ARCHE model is more adequate than QVaR

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approach

 The one-way spillover of WTI and stock markets is only significant during events

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 The two-way risk spillover of WTI and stock markets are significant for each time

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pronounced

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 The simultaneous risk spillover between WTI and stock markets are fairly

Abstract: With the integration and financialization of world economy, massive hot money has frequently flowed between crude oil and stock markets, and has brought

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significant extreme risks and their spillover. For this reason, this paper develops the ARCH-Expectile model with embedded Conditional AutoRegressive structure (namely

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CAR-ARCHE model) and expectile-based VaR (EVaR) approach, and investigates the time-varying risk spillover between WTI futures market and US, UK, Japanese and global stock markets, respectively. The results indicate that, for one thing, the EVaR approach based on CAR-ARCHE model is more adequate than the conventional quantile-based VaR (QVaR) approach based on GED-GARCH for WTI and stock markets, which is due to the evident advantages of expectile compared to quantile. For another, the unidirectional downside risk spillover effects from WTI to the four 1

stock markets and vice-versa are only remarkable during major events and present variations with jumps, but the bidirectional downside risk spillover effects between them are significant for each time point during the in-sample period, which indicate that the simultaneous risk spillover between WTI and stock markets are fairly pronounced. Keywords: CAR-ARCHE model; EVaR; prudence level; time-varying downside risk spillover effect

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JEL Classification: Q01; G15; O16

1. Introduction

As a non-renewable energy source, crude oil is an important strategic resource

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for world economic growth and has significant impact on the economy as well as

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financial markets. Over the past 10 years, due to the confluent influence of a series of risk factors (such as economic crisis, oil supply strategies by Organization of

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Petroleum Exporting Countries (OPEC) and US shale oil revolution), international crude oil prices have experienced sharp rises and falls (Wen et al., 2016, 2019), which

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indicates obvious extreme market risks. In fact, this pattern will persist in the long term (Zhang and Wei, 2011; Wang et al., 2016; Zhang and Li, 2019).

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Generally speaking, stock market is a barometer of economic and financial

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activities in a country or region. With the influence of various uncertainties such as economic and financial crises, stock prices fluctuate fiercely and extreme risks emerge in stock markets, which brings great potential losses for relevant participants. With the increasing breadth and depth of economic globalisation, investment or speculative funds frequently flow in or out of crude oil and stock

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markets, which strengthens the interaction between these two markets (Killian and Park, 2009; Fratzscher et al., 2013). Also, with the coming Internet finance and big data era, information can transmit quickly among financial and commodity markets. Moreover, some major emergencies, such as global financial crisis, extreme political events and military conflicts, often exert considerable influence on the confidence and asset allocation of investors, which may result in the changes of risk spillover

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among relevant markets. In addition, the volatility of crude oil market has an

essential influence on global economy and financial stability (Gong and Lin, 2017;

Gong and Lin, 2018a), while the price volatility of financial assets is closely related to

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risk (Gong and Lin, 2018b). For these reasons, the risk spillover between crude oil

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and stock markets has been increasingly evident, and it is possible to vary over time instead of being invariable (Basher and Sadorsky, 2006; Chai et al., 2011; Zhang et al.,

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2017).

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In addition, it should be noted that advanced stock markets in developed countries (such as USA, UK and Japan) dominate global stock market and have

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caused tremendous impacts on international crude oil market in recent years for several reasons (Zhang and Wei, 2011). On the one hand, since the beginning of the

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21st century, the macro-economy of these developed countries with advanced stock markets is inevitably, and more likely to be, affected by financial crisis, US shale oil revolution, Britain exiting from the EU and so on, which leads to significant changes in the supply and demand situation of crude oil. For example, according to BP (2018), global oil production only increased 0.6 million barrels per day in 2017, which was 3

lower than the historical average for the second consecutive year; by contrast, global oil consumption grew by 1.8%, namely 1.7 million barrels per day, which was larger than its ten-year annual average growth rate (1.2%) for the third consecutive year. On the other hand, the advanced stock markets of these developed countries often have high liquidity, then a large number of international speculative funds can freely enter and exit their stock markets. As a result, the links between crude oil and advanced

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stock markets have been getting increasingly close, while extreme market risks and their spillovers have been increasingly emerging.

To measure market risks, the Value at Risk (VaR) and conditional VaR (CVaR)

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approaches have become the essential tools within financial and commodity markets

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(Fan et al., 2008; Wang and Huang, 2016; Chen and Yang, 2017). However, both VaR and CVaR are representatives of the quantile-based VaR (QVaR) approach, which

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measure the risks caused by extreme values of asset distribution at the lower tail,

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and they are only related to the tail characteristic of asset returns but not involved with the entire distribution of returns, thus the measurement for tail loss is

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insufficient. Namely, they cannot examine the profit and loss information below a given quantile, and thus it is easy to ignore the huge loss of small probability and

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even extreme cases such as financial crisis (Santos et al., 2013; Zhang and Hong, 2015). Fortunately, expectile can completely describe the conditional distribution and it is more globally dependent than quantile under different distributions (Efron, 1991). Besides, expectile is not only related to the tail realisation of asset returns, but also related to their probability. Thus, compared to quantile, expectile proves more 4

sensitive to the size of extreme value of distribution (Kuan et al., 2009). As a result, this paper plays out the advantage of expectile and focuses on the downside expectile-based VaR (EVaR) of WTI crude oil market and US, UK, Japanese and global stock markets as well as their time-varying spillover effect, in order to effectively reveal the dynamic evolution of complex risk spillover across time. This paper mainly expands the existing research from three aspects as follows.

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First, through combining the Conditional Autoregressive Expectile (CARE) model and the line ARCH-Expectile (LARCHE) model, we put forward the ARCH-Expectile model with embedded CAR structure (namely CAR-ARCHE model) to analyse the

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contribution of market risk factors and measure the downside EVaRs of WTI and four

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stock markets. The new model here can not only deal with the heteroscedastic data, but also directly calculate the downside EVaRs. Second, based on the proposed

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model, we develop the EVaR approach for crude oil and stock markets returns, and

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find that the EVaR approach is advantageous over the traditional QVaR approach when used here, which not only proves more susceptible to the size of extreme loss,

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but also can avoid the assumption of asset return distribution. Finally, we construct the time-varying downside risk spillover statistics, and comprehensively explore the

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downside risk spillover between WTI and the four representative stock markets from both static and dynamic perspectives. The rest of this paper is organised as follows: Section 2 reviews relevant literature, Section 3 introduces econometric methods and data descriptions, Section 4 provides empirical results and analyses, and Section 5 concludes the paper. 5

2. Literature review Up to now, there have been abundant studies on risk spillover effect across financial or commodity markets. Here we mainly review the related studies from three aspects: the methods for risk measurement, the methods for risk spillover test and the link between financial asset prices and crude oil prices.

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Finding suitable risk measurement tool is the first and very important step in the process of risk measurement. Thanks to intuitive concept and simple calculation of VaR, it has gradually become one of the most essential tools for measuring market

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risks among numerous methods. However, because VaR does not satisfy the

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condition of risk additivity, and it is an inconsistent risk measurement method, so it is controversial in violation of asset diversification (Xie et al., 2014b). To overcome the

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above shortcomings of VaR, Rockafeller and Uryasev (2000) propose the CVaR

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approach. However, both VaR and CVaR are representatives of the QVaR approach, which cannot effectively utilize the overall information of distribution and are

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insensitive to extreme losses, while expectile can flexibly characterize the tail behaviours of a single asset. Therefore, in recent years, the expectile-based VaR

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(EVaR) has gradually become well known in the financial field (Xu et al., 2018). For example, based on the CARE model, Taylor (2008) calculates the EVaRs of stock indices. Kuan et al. (2009) extend the method of iterated weighted least squares (IWLS) provided by Newey and Powell (1987) to accommodate stationary and weakly-dependent data and present two types of CARE models to measure EVaR. 6

However, the CARE model does not take into account heteroscedasticity, which often appears in financial data, so it is not appropriate to be applied to financial data. Therefore, Xie et al. (2014a) come up with a linear ARCH-Expectile (LARCHE) model to extend the study of Kuan et al. (2009). In view of the coefficient estimates of the LARCHE model, not only the contribution to the risk factors of average return for China Minsheng Bank stock index can be analysed, but also the

is no test for the accuracy of their EVaR measurement.

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influence factors and magnitude of downside risk can be evaluated. However, there

When extreme market risks are concerned, the risk spillovers among different

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markets have attracted extensive attention in recent years. In particular, whether the

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historical information of extreme risks in one market is helpful to predict present or future extreme risk in another has been often investigated. If this is so, then we can

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say that the latter market is subjected to risk spillover from the former one. To this

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end, Hong (2001a) proposes a kind of kernel-based Granger causality test, which can be used in mean, variance and extreme risk spillovers and show good test efficacy.

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Then, the idea of Granger causality in risk is improved by Hong (2001b). Furthermore, on the basis of the improved method, the extreme risk spillovers between Chinese

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and major global futures markets (Hong et al., 2008), between exchange rates Euro/Dollar and Yen/Dollar (Hong et al., 2009), and among 84 publicly listed financial institutions (Wang et al., 2017) are specifically investigated, respectively. In addition, combining the information spillover with rolling window method, Lu and Hong (2012) empirically study the time-varying mean spillover between Shanghai Futures Market 7

and London Metal Exchange copper futures market. Actually, the method of Hong (2001b) is not only applicable to financial markets but also plays an important role in measuring extreme risk spillovers among different crude oil markets, or between crude oil and stock markets. For instance, using the Granger causality test in risk, the risk spillovers between WTI and Daqing crude oil markets (Pan and Zhang, 2007), between WTI and Brent crude oil markets (Fan et al., 2008), as well as between WTI

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market and S&P 500 index (Du and He, 2015), are thoroughly inspected, respectively. In particular, taking into account the time zone, Lu et al. (2014) add an instantaneous time-varying information spillover statistic into the approach presented by Lu and

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Hong (2012), and investigate the time-varying mean spillover among global crude oil

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markets. Overall, most of these studies explain the average risk spillover between markets during the entire period from a static perspective, and even if the time-

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varying characteristics are sometimes involved, they mostly examine time-varying

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mean or volatility spillovers, but often neglect the extreme risk spillover caused by various economic or financial related events, thus further fail to effectively depict the

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dynamic evolution of risk spillovers across time. Besides, the relationship between financial asset prices and crude oil prices has

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also been widely discussed by many scholars. For example, Killian and Park (2009) explore the impact of crude oil price changes on the volatility of US stock market based on the SVAR model, and find that the explanation extent of long-term changes in US stock returns from both supply and demand changes in global crude oil market reaches up to 22%. Fratzscher et al. (2013) use the multifactor model to unveil the 8

relationship among crude oil prices, US dollar exchange rate and asset prices. They find the bidirectional causality between crude oil prices and US dollar exchange rate since the early 2000s; by contrast, crude oil prices do not react to changes in these financial assets before 2001. Kim and Jung (2018) utilize multiple GARCH type models and Granger causality test to reveal the dependence among oil prices, exchange rates and the US interest rates, and find the unidirectional volatility spillovers from WTI to

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the US interest rates and to the exchange rates of oil importers. Bai and Koong (2018) use the diagonal BEKK and VAR models to detect the dynamic trilateral dependences among crude oil prices, stock market returns, and exchange rates in China and the

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United States.

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To sum up, although more and more scholars have fully appreciated the tight links between crude oil and financial markets, and paid attention to the risk overflow

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between these two types of markets, there are at least three aspects to be further

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explored. First, before testing risk spillover effect, it is necessary to measure the market risks, but most of existing studies are based on quantile to measure the risks.

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Even if there are some studies considering expectile, they basically lack direct interpretations for expectile and the accuracy test for the EVaR results. Second,

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previous studies on EVaR estimation often consider parametric autoregressive (such as CARE) model set-ups, which are not suitable for crude oil and stock return data with heteroscedasticity. Third, the changes of crude oil supply and demand, the adjustments of national macroeconomic policy as well as the changes of external market environment will affect crude oil and stock markets simultaneously, which 9

will further prompt the risk spillover between them to exhibit time-varying characteristics. However, most of the existing studies take a static perspective in examining the risk spillovers, which cannot effectively depict the dynamic evolution of risk spillovers across time. Therefore, considering the complex links between crude oil market and advanced stock market, this paper combines the CARE model with the LARCHE model

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based on expectile, and proposes the CAR-ARCHE model to evaluate the downside

risks of crude oil and stock markets. Then, it constructs the time-varying risk spillover statistics to explore the downside risk spillovers between WTI crude oil market and

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US, UK, Japanese and global stock markets, respectively, from both static and

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dynamic perspectives. This paper not only contributes to identifying future trends of crude oil market, advanced and global stock market indices, but also can provide

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information support for the scientific decision-making of investors, enterprises and

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governmental departments concerned.

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3. Methods and data description 3.1 Definitions of Expectile-based VaR (EVaR)

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Suppose Y represents an asset return, and its distribution function is FY . The

  quantile of FY is q( )  inf x  R : FY ( x)   for any given   (0,1) . Thus,

at the significance level  , the QVaR of Y is the negative value of the   quantile of

FY , namely QVaR( )  q( ) . From the definition of QVaR, it is easy to find that QVaR is unsusceptible to the size of extreme losses. The concept of expectile is first put forward by Aigner et al. (1976) and then 10

further developed by Newey and Powell (1987), who take into account asymmetric quadratic loss function similar to quantile as Eq. (1):

E   (Y   )  E    IY    (Y   )2   

(1)

where I{} denotes an indicator function, and   (0,1) denotes the relative cost of expected marginal loss, and indicates both the asymmetry degree of loss function and the prudence level. Minimising the quadratic loss function, we can obtain the

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  expectile of Y, namely  ( ) (Kuan et al., 2009). The superiority of the  

expectile is that its loss function is differentiable and thus can be directly derived.

Specifically, the first-order condition of minimising the quadratic loss function is 









    y   dF ( y )   y   dF ( y )  



y   dFY ( y )

 



Y



y   dFY ( y )

y   dFY ( y )



(2)

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

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shows that the expectile  ( ) satisfies Eq. (2):

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  y   dFY ( y )  (  1) y   dFY ( y )  0 , then the straightforward calculation

Y



Comparatively, q( ) is only related to the tail probability  , while  ( ) is

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decided by the tail realisation of Y and its probability  , so the latter proves more

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susceptible to the extreme values of distribution. Efron (1991) first applies the expectile to estimate the quantile-based VaR. The

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basic principle of this method is that the expectile with a given  is equivalent to a quantile with corresponding  (Yao and Tong, 1996; Abdous and Remillard, 1995). For any   (0,1) , let  ( ) be such that  ( )  q( ) . That is, the quantile of Y at the significance level  is equal to the expectile of Y at the prudence level  , and the relationship between  ( ) and q( ) is established as Eq. (3) (Yao and Tong, 11

1996):

 ( ) 

 q( )   2 E (Y )  2 

q ( )



q ( )



ydF ( y )

(3)

ydF ( y )  (1  2 ) q( )

3.2 Estimation of EVaR based on CAR-ARCHE model 3.2.1 CARE model The CARE model, which takes into consideration some stylised facts, is provided

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by Kuan et al. (2009) to measure EVaR of time series in financial markets. In view of the different effects of past positive and negative returns (namely

CARE model specifications as Eqs. (4) and (5):

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ytn  max  yt n ,0  , ytn  max   yt n ,0  ) on expectiles, we consider two kinds of

yt  c0    c1   yt 1  b1    yt1   d1    yt1   d n    ytn   et   2

2

 bn    ytn   2

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yt  c0    1   yt1  1   yt1    n   ytn  n   ytn  et  

(4)

(5)

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where n denotes the lag order, and et   denotes the error term. 3.2.2 CAR-ARCHE model

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Given the financial data with heteroscedasticity, we combine the CARE model presented by Kuan et al. (2009) with the LARCHE model provided by Xie et al.

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(2014a), and construct the CAR-ARCHE model as Eq. (6). This model can not only deal with the heteroscedastic data, but also analyse the contribution of risk factors, as well as directly evaluate the downside EVaR.

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yt  xt    et et   t t

(6) p

 t2   0   i et2i i 1

where  t is IID (independently identically distribution) with E  t   0, E  t2   1 , and p denotes the lag order in the ARCH term,  0  0,  i  0, i  1,2,



xt = 1, yt 1 ,  yt1  ,  yt1  , 2

2

,  ytn  ,  ytn  2

2



or xt = 1, yt1, yt1,

, p . When

, ytn , ytn  (for

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brevity, the former and the latter are denoted as the CAR1(n) and CAR2(n) structures, respectively), the CAR structure in the CARE model, which replaces the linear

structure in the LARCHE model, is embedded in the ARCHE model, and it is marked as

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the CARi-ARCHE(n, p) model, i  1,2 .

Based on the CAR-ARCHE model, it can be assumed that the expectile of asset

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return Y (namely EVaR) satisfies  y    xt   . According to the definition of

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expectile, we have  yt  t   yt  t  xt  t , and we can estimate the asymptotic least square (ALS) estimator for    by minimising E  

 y  x      , t

t

t

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denoted as ˆ   (Kuan et al., 2009)1. Then the estimator for EVaR of return Y , denoted as ˆ y    xtˆ   , can be obtained.

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3.2.3 Reliability test for EVaR

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In order to ensure the reliability of EVaRs, it is necessary to back-test whether the real extreme risks have been adequately estimated or not by the EVaR model. Consequently, we introduce the back-test method proposed by Kupiec (1995). The

1

Due to limited space, the specific estimation steps for parameter

   are omitted. However, detailed results

are available upon request. Moreover, according to Kuan et al. (2009) and Xie et al. (2014), the ALS estimator for

   meets consistency and asymptotic normality.

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core idea of this method can be generalized as follows: supposing that the significance level  , the sample size T, and the actual number of failures N, then the failure rate, denoted as f  N T , can be gained. Subsequently, under the null hypothesis f   , Kupiec (1995) builds the statistic of the proper likelihood ratio (LR) test as Eq. (7):

LR  2ln 1  f  

T N

f N 1   

T N

 N . 

(7)

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where LR ~  2 1 , and its critical values at the significance levels of 5% and 1% are 3.84 and 6.64, respectively. If the LR statistic is less than its critical values, we cannot

3.3 Time-varying downside risk spillover test

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3.3.1 Indicator function of EVaR

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reject the null hypothesis, which means the EVaR model established is adequate.

First of all, an indicator function of downside risk based on EVaR series is defined

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as Eq. (8):

Z m,t  I  ym,t  EVaR m,t  , m  1,2

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(8)

where ym ,t and EVaR m,t are the logarithmic returns and EVaR of market m at

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time t, respectively, and I () denotes an indicator function. If the actual loss outstrips EVaR, Z m,t =1 ; otherwise Z m,t =0 .

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For the sake of testing the unidirectional downside risk spillover from market 2





to market 1, we can give the null hypothesis H 0 : E Z1,t 1,t 1  E  Z1,t t 1  ,





whereas the alternative hypothesis H1 : E Z1,t 1,t 1  E  Z1,t t 1  , in which t 1  1,t 1 , 2,t 1 , while 1,t 1   y1,t 1 , y1,t 2 ,

2,t 1   y2,t 1 , y2,t 2 ,

, y1,1 and

, y2,1 represent the information sets available for markets 1 14

and 2 at time t  1, respectively. If H 0 holds, we can say market 2 does not have unidirectional downside risk spillover to market 1. It means when there exist extreme risks in market 2, it cannot be applied to predict the possible future extreme risks in market 1. 3.3.2 Selection of rolling sample size We determine the rolling sample size (or subsample size) W, and then form the

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subsample t  W  1, t  . The selection of W is essential for the test power of

statistics. Appropriate W can balance the contradiction of Type I or II errors and

signal timeliness when risk spillovers vary across time (Lu and Hong, 2012). At the

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types I and II error levels (  1 and  2 , respectively), W can be calculated using Eq.



W  2 z1 1 2  z1 2



2

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(9) (Belle, 2008): 2

(9)

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where z1 represents the critical value of standard normal distribution at the

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significance level of  , and  denotes the relative error. 3.3.3 Time-varying downside risk spillover statistics

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At the significance level of  , let EVaR m,t  EVaR m  m,t 1 ,   and show the time series of EVaRs in market m, which is acquired by equation: ˆ y    xtˆ   . To

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assess the dynamic Granger causality in downside risk during the subsample

t  W  1, t  , suppose

Z m,t  I  ym,t  EVaR m,t  for two series

y

1,t

, y2,t  , then the

lag-j subsample cross covariance function for Z1,t and Z 2,t is specified as:

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 1 W W  Z1,t  f1 Z 2,t  j  f 2 , 0  j  W  1 t 1 j  Ct  j,W    W W 1 Z1,t  j  f1 Z 2,t  f 2 , 1  W  j  0   t  1  j 







where f m  W

1





(10)



W

Z t 1

m ,t

. And the lag-j subsample cross correlation function for Z1,t

and Z 2,t is written as Eq. (11):



where Dm  f m 1  f m

D1D2 , j  0, 1, ,  W  1



(11)

denotes the sample variance of Z m ,t 2.

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t  j,W   Ct  j,W 

To detect the unidirectional and bidirectional time-varying Granger causalities in downside risk for these two markets, referring to Lu et al. (2014), we introduce the

  2   t  j,W   C1W  k    

2 D1W  k 

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 W 1  j H1,t W   W  k 2   j 1  M

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Daniell kernel function and corresponding test statistics are written as Eq. (12):

  H 2,t W   W  k 2  j M   t2  j,W   C2W  k    j 1W 

(12)

2 D2W  k 

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W 1

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where k () denotes a kernel function and assigns weights to various lags. As stated by Hong (2001a), the Daniell kernel function is the optimal and its domain is

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unbounded, so it can maximise the power of the test. M represents the lag order, and M  W . Centring factors ( C1W  k  and C2W  k  ) and scaling factors ( D1W  k 

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and D2W  k  ) can be specified as Eq. (13) (Hong, 2001a):

2

By the definition of f m , as W rolls forward with time t, f m may be zero, which further leads

Dm to be zero, and then  t would be meaningless because its denominator is zero, which further makes the statistics impossible to calculate. Thus, the value of risk spillover statistic at the corresponding time in this case is defined as the average of preceding W time points which could be computed (Lu and Hong, 2012). 16

W 1 j  C1W  k     1  W j 1 

 2 j k   M

 , 

W 1 j  j 1 4  j   D1W  k     1    1  k   W  W  M  j 1 

j  j   C2W  k     1   k 2  , D2W  k   W M  j 1W  W 1

y 

If

1,t

and

y  2,t

j  j 1 4  j    1   1  k   W  W  M  j 1W  W 1

(13)

are mutually independent in the subsample, then

according to Hong (2001a), for i  1,2 , Hi ,t W   N  0,1 , as W   , and the time-varying downside risk spillover tests are the one-sided tests. If H i ,t W  is less

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than the right-tailed critical value for a given significance level of N  0,1 , then the null hypothesis cannot be rejected, which implies no significant Granger causality in downside risk for these two markets at time t.

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In a word, as time t rolls forward, the time-varying risk spillover tests can be

described as the following procedures: ① employ the CAR-ARCHE model for each

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financial time series to produce an EVaR series; ② select the rolling sample size W,

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and compute each subsample cross correlations of the EVaR series; and ③ determine the kernel function k  z  and the lag order M. Then the time-varying

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downside risk spillover test can be conducted using Eq. (12)3. Two types of time-varying downside risk spillover statistics are calculated for

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each scrolling, and the average probability of rejecting the null hypothesis

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corresponding to a given significance level of  can be shown as Eq. (14): T

pi   I  pi ,t    T , i  1,2

(14)

t 1

where I () denotes an indicator function, pi ,t denotes the p-value of the i -th time-varying downside risk spillover statistic at time t, and T   T  W  1 . 3

By the definition of time-varying downside risk spillover statistics, they cannot be calculated in the first W  1

time points, so the real sample size of statistic is

T   T W 1. 17

Under the null hypothesis of no risk spillover between WTI and stock markets at every time, if the real sample size T  is long enough, then the distribution of statistic is close to its limiting distribution, and pi will be close to  . Thus, when the null hypothesis holds, if the closer pi converges to  , then the higher power the statistic has; however, when the null hypothesis does not hold, the larger pi means the bigger probability of rejecting the null hypothesis for the statistic.

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3.4 Data descriptions

As for stock markets, given the representativeness of advanced stock markets in

the world, especially the important influence of the countries where advanced stock

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markets are located in international crude oil market, we select the S&P 500 index in

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the United States, FTSE 100 index in the UK, as well as Nikkei 225 index in Japan on the basis of trading volumes, trading values and location of financial center (Baser

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and Sadorsky, 2006; Zhang and Wei, 2011), and we also use the MSCI global index

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produced by Morgan Stanley Capital International (MSCI) to reflect the dynamics of global stock market. Meanwhile, as for international crude oil market, we mainly use

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WTI crude oil futures prices, and we also employ Brent crude oil futures prices for robustness checks. All data are derived from the Wind database. The full sample

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period ranges from January 4, 2000 to March 29, 2019, and 4625 daily observations are gained, in which the in-sample period contains the observations from January 4, 2000 to December 31, 2014, which are used to estimate the EVaR model, while the out-of-sample period includes the remaining observations and is applied for the

18

robustness checks of the central results.

4

The returns of WTI and stock markets are

specified as yi ,t  100  ln  Pi ,t Pi ,t 1  , where Pi ,t denotes the closing price of market

i at the time t . Fig. 1 shows the trends of WTI and stock market returns during the full sample period. Note that all the returns display volatility clustering, and the volatility of WTI market returns is greater than that of other four stock market returns during some

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major event periods, which demonstrates that there is a greater possibility of extreme risk in WTI market.

Table 1 presents the descriptive statistics of WTI and stock market returns during

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the in-sample period. It can be found that the mean and volatility of WTI returns

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prove the largest, which can also be confirmed from Fig. 1. Meanwhile, the skewness is negative (i.e., left skewed) for WTI and stock returns, and their kurtoses are far

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greater than 3, so we can assert that they have leptokurtic distribution with fat tails.

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Also, the results of Jarque Bera (JB) statistics make it clear that all return series do not follow the standard normal distributions. Moreover, the Q statistics of Ljung Box

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(LB) demonstrate significant serial autocorrelations for all return series, and the ARCH tests reveal that they all have significant ARCH effect. In addition, all returns

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are statistically stationary based on the ADF and PP unit root tests. Figure 1 Table 1

4

The in-sample period, including fifteen years of trading activities, is long enough to estimate the EVaR model and can ensure the better in-sample fitting, while the out-of-sample period, including more than four years of trading activities, is also long enough to cover the upward and downward trends of oil and stock markets, which can be more suitable to verify the robustness of the model. 19

4. Empirical result analyses 4.1 Estimation of CAR-ARCHE model for WTI and stock market returns Based on the idea of expectile, the CAR-ARCHE model above is employed to estimate the downside EVaRs for WTI and US S&P 500, UK FTSE 100, Japanese Nikkei 225, as well as MCSI returns, respectively. 4.1.1 Determination of prudence level

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First of all, in view of the method provided in Section 3.1 and the relationship

between  and  in Eq. (3), we calculate the prudence levels i (i  1,2) of each return series under the given significance levels  i (see the third column in Table

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2). The non-parametric technique is applied to estimate the probability density

window width is chosen automatically.

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4.1.2 Selection of CAR-ARCHE model

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function for each return series, where the Gaussian kernel is selected and the

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Based on the CAR-ARCHE model in Section 3.2, we measure the downside risks of WTI and four stock market returns.

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Firstly, referring to the CARE model by Kuan et al. (2009), xt in Eq. (6) is made up of the CAR structure (namely the CAR1(n) and CAR2(n) structures, respectively) to

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examine the risk factors of WTI and four stock market returns. To this end, we determine the lag in each class of CAR structure. Specifically, let n  5 first, and then we test the significance of parameter estimates for each model. If b5 and d 5 in the CAR1(5) structure (or  5 and 5 in the CAR2(5) structure) are statistically insignificant, then the lag-5 variables can be rejected and we re-estimate the CAR1(4) 20

(or CAR2(4)) structure; otherwise, we choose the CAR1(5) (or CAR2(5)) structure. It should be noted that the past positive and negative market returns of lagged variable are both remained, provided at least one of their parameters is significant. Then, we can inspect whether the past positive and negative returns have asymmetric influence on expectile. The above procedures are repeated until the final CAR1 (or CAR2) structure is determined.

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Meanwhile, considering WTI and stock market returns are biased and may be

heteroscedastic, the ARCH model is employed. Consequently, we need to select the order of lag in the ARCH (p) model. Also, we use the ALS method to estimate the

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estimators is less than 10-12 (Kuan et al., 2009).

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coefficients in the CAR-ARCHE model until the distance between two adjacent

In the light of the truncated orders of autocorrelation and partial autocorrelation

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functions, together with the ARCH-LM test and the principle of minimum AIC value, appropriate models are singled out through many attempts under the given 

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corresponding to expectiles with different  for WTI and four stock markets,

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respectively. Table 2 lists their specific model forms and lag orders. Table 2

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4.1.3 Estimation of CAR-ARCHE model Based on the CAR-ARCHE model, we estimate the regression coefficients of WTI

and four stock market returns under the given  corresponding to expectiles with different  (see Table 3). We can find that: ① For WTI, Japanese and global stock markets, under different prudence levels, 21

the past positive return ( ytn ) and negative return ( ytn ) have significant negative effects on their current tail losses (namely EVaRs) in most cases, but the coefficients of ytn and ytn are different, and with the decrease of prudence levels, the impacts of past positive and negative returns on their EVaRs may increase, which shows that both the up and down of WTI prices and Japanese and global stock market indices will affect the size of EVaR, but these effects on expectile are

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asymmetric. ② For the US stock market, at the 2.51% and 0.40% prudence levels,

the autoregressive coefficients of autoregressive term ( yt 1 ) are not significant, but the coefficients of

y 

2  t n

and

y 

2  t n

are significantly negative and their size are

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different, which demonstrates that both the up and down of US stock index will also

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affect the size of EVaR. ③ For the UK stock market, under different prudence levels, the model coefficients are highly significant. At the prudence level of 2.61%, positive

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lag-1 return ( yt1 ) and negative lag-1 return ( yt1 ) have significant negative impacts

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on the current tail losses. At the prudence level of 0.40%, the coefficient of yt 1 is highly significant. This result shows that the volatility of UK stock index has a typical autocorrelation features; both the direction and magnitude of the impacts of

y 

2  t 1

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and

y 

2  t 1

on the current risk are different, which means that the up and down of

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UK stock index have reverse asymmetric effects on the current EVAR. From the results in Table 3, we can summarize that whether in WTI or stock

markets, high returns are often accompanied by high risks. Relevant investors should make much account of the rational allocation of assets and diversify their investments. 22

Table 3 4.2 Estimation of EVaR model for WTI and stock market returns 4.2.1 Estimation of EVaRs based on the CAR-ARCHE models We calculate the downside EVaRs by means of the methods above at corresponding prudence levels for WTI and US, UK, Japanese and global stock markets, respectively, and relevant results are shown in Table 4. Some findings are

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identified as follows: Table 4

① Except for the UK and MSCI stock market returns at the 5% significance level,

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all other p-values are larger than corresponding significance levels. Hence, we can

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say that, at the 5% and 1% significance levels, the CAR-ARCHE models have adequately estimated the downside EVaRs of the five returns. ② No matter at the

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significance levels of 5% or 1%, the mean values of EVaRs of WTI prove far greater

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than those of four stock returns. Therefore, in comparison, it is necessary for WTI crude oil market participants to prepare more risk reserves. This coincides with the

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findings of He & Du (2015), who argue that WTI crude oil futures market has greater risks than the US stock market. In fact, as shown in Fig. 2, if the returns go down

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during the in-sample period, the downside risks of WTI returns tend to exceed those of the four stock returns. Figure 2 4.2.2 Comparison of EVaR and QVaR models Referring to Fan et al. (2008), we use the GED-GARCH models based on variance23

covariance method to estimate the downside QVaRs for WTI and four stock market returns, respectively. Some relevant results about the downside QVaRs for the five returns at the 5% and 1% significance levels are shown in Table 55. We can find that for US, UK and Japanese stock markets, at the 5% significance level, the null hypotheses are rejected, which means their estimated QVaRs are not adequate, but in other cases, the QVaR model can adequately estimate the QVaRs for the five

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returns. Table 5

Meanwhile, the comparison of results in Tables 4 and 5 shows that, except for

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MSCI returns at the 5% significance level, all other p-values of EVaR measurement

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are larger than those of QVaR measurement. Therefore, we can say that it proves more adequate and acceptable to use the EVaR approach based on CAR-ARCHE

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model than the QVaR approach based on GED-GARCH model overall. This is possible

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related to the fact that the dynamic characteristics of assets is not the same under different conditional distributions, resulting in the specification of CAR structure may

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vary with the prudence level  (Kuan et al., 2009). Therefore, the EVaR method used can flexibly describe the tail behaviours of a single asset.

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4.2.3 Forecasting performance analyses To judge the predictive ability of the EVaR model, we take the significance level

of 1% as an example, and use the EVaR model above to predict the EVaRs during the out-of-sample period. Furthermore, the comparison of results between predicted

5

For simplicity, we do not report the determination of GED quantiles and parameter estimation results. However, detailed results can be obtained from authors upon request. 24

values and corresponding historical returns shows that (see Fig. 3 with the example of WTI and Nikkei 225 returns), for WTI and S&P 500, FTSE 100, Nikkei 225, as well as MSCI returns, the proportions of returns not exceeding the EVaRs to all the historical returns during the out-of-sample period are 99.03%, 99.61%, 98.64%, 98.93%, and 99.12%, respectively, and all of them are very close to 99%. The corresponding LR values are 0.0067, 5.0474, 1.2216, 0.0512, and 0.1655, respectively, all less than the

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critical value 6.64. Thus, the results can be generally acceptable. In other words, when the EVaR model is used for WTI and stock returns, its predictive ability is expected to be credible.

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Figure 3

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4.3 Time-varying downside risk spillover between WTI and stock market returns After the EVaR series of WTI and four stock returns are gained, based on Eqs.

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(12) and (14), we further calculate the two types of time-varying downside risk

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spillover statistics H i ,t W  , i  1,2 , the corresponding p-values, and their corresponding average probability pi of rejecting the null hypothesis of no risk

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spillover between WTI and stock markets at every time during the in-sample period. Subsequently, the bidirectional and unidirectional time-varying downside risk

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spillovers between WTI and four stock returns are examined, respectively6. Table 6 gives the results of pi for WTI and stock returns, where the rolling

6

We also inspect the bidirectional and unidirectional time-varying downside risk spillovers between WTI and US, UK, Japanese as well as global stock markets for the in-sample period from a static perspective. The results show that the bidirectional downside risk spillovers between them are highly significant, but the unidirectional downside risk spillovers from WTI to the four stock markets and vice-versa are not significant. Details are available upon request. 25

sample W is equal to 64, 104 and 200, respectively7; while the lag order M is set as 10, 20, and 30, respectively. Since the combinations of W and M are too numerous, we only list the significance probability trends of time-varying downside risk spillover statistics at the 5% and 1% significance levels with W  104, M  20 (see Fig. 4)8. Combining the results in Table 6 and Fig. 4, several findings can be boiled down to as follows.

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Table 6 Figure 4

(1) During the in-sample period, there are significant bidirectional downside risk

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spillovers between WTI and US, UK, Japanese as well as global stock markets at each

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time point. On the one hand, as Table 6 reveals, at the 5% and 1% significance levels, with the changes of rolling sample W and lag order M, the average probability p2 is

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always equal to 1. On the other hand, it can also be demonstrated from the trends of

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significance probabilities of the bidirectional time-varying statistic H 2,t that corresponding p-values are close to zero at each time point (see Fig. 4). This result is

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similar to those of most existing research. In particular, Du and He (2015) obtain similar conclusions by studying the extreme risk spillover between WTI and S&P 500

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index using the data from September 1, 2004 to September 11, 2012, and find that there exists remarkable bidirectional downside risk spillover between them.

7

Based on Eq. (9), we set

  0.5 . If  1  0.05 ,  2  0.2 , then W  64 ; if  1   2  0.05 , then

W  104 (Lu et al., 2014). Moreover, the case of W  200 is also considered in the empirical studies for robustness check. 8 At identical significance levels, no matter how W and M are combined, the trends of significance probability of

time-varying downside risk spillover statistics for WTI and four stock returns are similar. Details are available upon request. 26

It is not difficult to understand the bidirectional risk spillover effects between WTI and four stock markets. WTI is one of the key benchmark crude oil futures markets around the world, while S&P 500, FTSE 100, Nikkei 225 and MSCI are the representatives of US, UK, Japanese and global stock markets, respectively. With the rapid integration of global economy, the interaction among various countries becomes increasingly close. Moreover, crude oil market has significant financial

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properties and stock market proves the macroeconomic barometer. Consequently, it is inevitable for the existence of risk spillovers between WTI and advanced stock markets (Wang and Liu, 2016; Zhang and Wang, 2019; Zhang and Li, 2019).

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(2) With the exception of major event periods, there are no significant

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unidirectional downside risk spillovers from WTI to US, UK, Japanese and global stock markets and vice-versa during the in-sample period. However, there are highly

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significant bidirectional downside risk spillover effects between them at each time

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point. So it can be found that there exist remarkable synchronising downside risk spillovers between WTI and four stock markets in this paper, that is, the interactions

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between them are very rapid and direct. Table 6 shows that no matter at the 5% or 1% significance levels, with the

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changes of W and M, the average probabilities p1 are all less than 0.01. It can be deduced that the unidirectional time-varying statistic H1,t is not sensitive to the changes of W and M. Furthermore, it can also be affirmed from the trends of significance probabilities of statistic H1,t in Fig. 4 that the unidirectional timevarying downside risk spillover statistics from WTI to the four stock markets and vice27

versa are not statistically significant for most of in-sample period, but they are remarkable and present a variation with a jump during the major event periods (e.g. US 911 terrorist attacks in 2001, global financial crisis in 2008, European debt crisis in 2011 and US shale oil revolution in 2014). For example, at the 5% significance level, in the middle of October, 2014, no matter from WTI to US, UK and Japanese stock markets or vice-versa, there exist significant unidirectional time-varying downside

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risk spillovers. These are may be related to the facts that in 2014, OPEC claimed no

reduction in crude oil production again and again, and the large-scale exploitation of shale oil occurred in the US, leading to severe shocks of crude oil prices, then

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massive international hot money frequently enter and exit between crude oil and

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stock markets, which results in market extreme risks to increase sharply. 4.4 Robustness test

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For the sake of robustness of the results above, we adopt Brent crude oil futures

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prices to replace WTI for related research above. Under the given significance levels of 5% and 1%, the parameter estimation results of CAR-ARCHE model for Brent

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returns are presented in Table 7. It can be viewed that the past positive ( ytn ) and negative returns ( ytn ) have significant negative and asymmetric effects on the

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current EVaR. The results of EVaR are listed in Table 8. According to the p-values, it can be argued that the EVaR approach based on CAR-ARCHE is very adequate and reliable. Besides, to test the prediction ability of the CAR-ARCHE model used, we take the significance level of 1% as an example, and use the CAR-ARCHE model above to 28

predict the EVaRs during the out-of-sample period. It can be found that, during the out-of-sample period, the proportion of returns not exceeding the downside EVaR to all the real returns for Brent market is 98.83%, which is very close to 99%. Meanwhile, the corresponding p-value is 0.5972, far greater than 0.01. In addition, the test results of time-varying downside risk spillovers between Brent and four stock markets are shown in Table 9 and Fig. 5. It can be found that during the in-sample

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period, the bidirectional downside risk spillovers between them are significant for each time point, but the unidirectional downside risk spillovers from Brent to the

four stock markets and vice-versa are only remarkable during major events, which

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are similar to those between WTI and four markets, so we can say the central results

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5. Conclusions and implications

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above are largely robust.

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In view of the complex links between crude oil futures market and stock market, we propose the EVaR approach based on the CAR-ARCHE model, and construct two

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kinds of time-varying downside risk spillover statistics to comprehensively explore the risk spillovers between WTI and US, UK, Japanese and global stock markets,

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respectively. Taking into account what has been discussed above, several key conclusions can be safely obtained as follows. (1) Both the past positive and negative returns of WTI, S&P 500, FTSE100, Nikkei 225, and MSCI have significant negative effects on their current EVaRs in most cases. However, the past positive and negative returns exert asymmetric effects on 29

expectiles. Specifically, the effect of past negative returns proves much greater than that of past positive returns under most circumstances. (2) For WTI and US, UK, Japanese and global stock markets, it is more adequate and approved to apply the EVaR approach based on CAR-ARCHE model than the QVaR approach based on GED-GARCH model. In fact, the former can flexibly describe the tail behaviours of a single asset, which helps market traders to measure market

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risks more accurately and address extreme risks more effectively.

(3) During the in-sample period, there exist significant bidirectional downside

risk spillovers between WTI and US, UK, Japanese and global stock markets at each

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time point. However, the unidirectional downside risk spillovers from WTI to the four

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stock markets and vice-versa are only remarkable during major events and present variations with jumps, which indicates that the simultaneous risk interactions

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between WTI and four stock markets are more pronounced, that is, their spillover is

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fairly direct, but the duration is not necessarily long-lasting. The conclusions above have evident applicability in practice. During major

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events, for one thing, the significant unidirectional downside risk spillovers from WTI to S&P 500, FTSE 100, Nikkei 225, and MSCI stock markets mean that when the bad

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news in WTI market leads to the decline of returns, the risk information in WTI market can be helpful to predict the risks in the four stock markets. For another, the remarkable unidirectional downside risk spillovers from S&P 500, FTSE 100, Nikkei 225, as well as MSCI stock markets to WTI indicate that once a huge collapse occurs in the four stock markets, it is highly likely to face huge margin calls right immediately 30

for long traders in WTI market. Moreover, relevant government departments especially in giant crude oil importing countries can utilise the historical risk information in these four stock markets to make appropriate decisions about purchasing and storing crude oil. Besides, some future research work can be carried out in the following aspects. For instance, we can consider improving the estimation method of parameters in the

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CAR-ARCHE model, such as using the quasi-likelihood method. Moreover, we can also try to extend the CAR structure to the GARCH type models, so as to seek an effective combination of the expectile and GARCH type models to avoid the long lag order of

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the ARCH term.

Acknowledgments

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The authors are grateful for the financial support from National Natural Science

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Foundation of China (no. 71774051), Changjiang Scholars Program of the Ministry of Education of China (no. Q2016154), National Program for Support of Top-notch

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Young Professionals (no. W02070325), and Hunan Youth Talent Program.

31

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evidence from stock, crude oil and natural gas markets. Energ. Econ. 68, 228– 239. Zhang, Y.J., Li, S.H., 2019. The impact of investor sentiment on crude oil market risks: evidence from the wavelet approach. Quant. Financ. 19(8), 1357-1371. Zhang, Y.J., Wang, J.L., 2019. Do high-frequency stock market data help forecast crude oil prices? evidence from the MIDAS models. Energ. Econ. 78, 192–201.

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36

Biographical notes

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Dr. Yue-Jun Zhang is a Professor at Business School, Hunan University, China, as well as the Director of Center for Resource and Environmental Management, Hunan University. He got his PhD degree in Management Science from Chinese Academy of Sciences in 2009. His research interests include energy economics and policy modeling. Up to now, Prof. Zhang has published more than 100 articles in peer-reviewed journals, such as Energy Economics, Energy Policy, Applied Energy, European Journal of Operational Research, Journal of Policy Modeling, Resources Policy, Computational Economics, Empirical Economics, Economic Modelling. Prof. Zhang was a visiting scholar at Energy Studies Institute of National University of Singapore, and Lawrence Berkeley National Laboratory, USA. Ms. Shu-Jiao Ma is a PhD candidate at Business School, Hunan University. Her research interests focus on crude oil pricing and risk management.

37

Figures

WTI

20 0

Nikkei 225

FTSE 100

S&P 500

-20 20 0 -20 20 0 -20 20 0 -20

ro of

MSCI

20 0

-20 2000/01/05

2003/12/03

2007/10/15

2011/08/04

2015/06/10

2019/03/29

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Fig. 1. Daily WTI and stock market returns during 2000-2019

re

0

lP

-5

-15

na

EVaRs

-10

-20

EVaR of EVaR of EVaR of EVaR of EVaR of

ur

-25

Jo

-30 2000/01/06

2002/03/19

2004/05/18

2006/07/04

2008/05/29

2010/08/12

WTI S&P 500 FTSE 100 Nikkei 225 MSCI

2012/11/08

2014/12/31

Fig. 2. Downside EVaRs of WTI and stock returns during 2000-2014 at the 1% significance level.

38

12

8 Return of WTI EVaR of WTI

Return of Nikkei 225 EVaR of Nikkei 225

4

Nikkei 225 return & forecasted EVaR

WTI return & forecasted EVaR

6

0

-6

-12

-18 2015/01/08

0

-4

-8

2015/11/12

2016/09/09

2017/07/20

2018/05/28

-12 2015/01/07

2019/03/29

2015/11/11

(a) WTI

2016/09/08

2017/07/19

2018/05/25

2019/03/29

(b) Nikkei 225

Jo

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at the 1% significance level.

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Fig.3. Historical WTI and Nikkei 225 returns and their EVaRs during 2015-2019

39

(b1)   0.01

-p

ro of

(a1)   0.05

(b2)   0.01

na

lP

re

(a2)   0.05

(b3)   0.01

Jo

ur

(a3)   0.05

(a4)   0.05

(b4)   0.01

Fig. 4. Significance probabilities of time-varying downside risk spillover statistics for WTI and four stock markets during 2000-2014 at the 5% and 1% significance levels 40

(b1)   0.01

-p

ro of

(a1)   0.05

(b2)   0.01

na

lP

re

(a2)   0.05

(b3)   0.01

Jo

ur

(a3)   0.05

(a4)   0.05

(b4)   0.01

Fig. 5. Significance probabilities of time-varying downside risk spillover statistics for Brent and four stock markets during 2000-2014 at the 5% and 1% significance levels 41 / 52

Tables Table 1 Summary statistics for WTI and stock market returns during the entire perioda WTI

S&P 500

FTSE100

Nikkei 225

MSCI

Mean

0.0185

0.0153

0.0019

0.0024

0.0092

Max

16.4097

10.9579

9.3843

13.2346

8.9031

Min

-19.6625

-13.7779

-10.3274

-12.7154

-15.8073

Std. Dev.

2.4121

1.2208

1.1876

1.5239

1.0342

Skewness

-0.2186

-0.3861

-0.2360

-0.5239

-0.9596

Kurtosis

8.0824

14.1839

10.4033

10.4447

21.2498

JB test

5013.5737***

24213.4833***

10602.7983***

10889.8929***

64878.2796***

LB-Q(10)

50.3546 ***

43.0655 ***

88.7467 ***

16.1967 *

159.7030 ***

LB-Q(20)

71.1638***

59.1764***

114.3074 ***

35.2661**

186.2047 ***

ARCH(10)

71.2425 ***

157.1321 ***

134.9107 ***

157.8624***

113.4747 ***

ARCH(20)

40.6960 ***

98.0295 ***

74.2094***

81.3934***

61.8623 ***

ADF test

-70.5877 ***

-51.7140 ***

-32.5990***

-69.7994***

-32.0109 ***

PP test

-70.5776 ***

-72.5643 ***

-70.3842***

-69.9761***

-58.0834 ***

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** and *** indicate the significance at 10%, 5% and 1% levels, respectively.

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Return

42 / 52

Table 2 Model selection of WTI and stock market returns Return WTI

S&P 500

FTSE 100

Nikkei 225

θ

CARi-ARCHE(n, p) model

0.05

0.0235

CAR2-ARCHE(1, 1)

0.01

0.0035

CAR2-ARCHE(3, 1)

0.05

0.0251

CAR1-ARCHE(2, 1)

0.01

0.0040

CAR1-ARCHE(1, 3)

0.05

0.0261

CAR2-ARCHE(1, 1)

0.01

0.0040

CAR1-ARCHE(1, 1)

0.05

0.0231

CAR2-ARCHE(1, 1)

0.01

0.0028

CAR2-ARCHE(2, 4)

0.05

0.0269

CAR2-ARCHE(1, 1)

0.01

0.0029

CAR2-ARCHE(2, 2)

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MSCI

α

43 / 52

WTI

f

S&P 500

FTSE 100

θ1=0.0251

θ1=0.0261

oo

Table 3 Regression results on expectiles of WTI and stock market returns a

θ1=0.0235

ˆ ( )

S.E.

Variable

ˆ ( )

S.E.

Variable

Cons.

-2.9330***

0.1045

Cons.

-1.6460***

0.0429

Cons.

y

 t 1

-0.3168*** -0.5119***

0.0547 0.0515

yt 1

y y y y

0.0589

   

 2 t 1

 t 1

 t 2

2

2

2  t 2

-0.0570***

0.0140

-0.0574***

0.0111

-0.0926***

0.0106

-0.0822***

0.0085

Panel B: α2=0.01 θ2=0.0040

Variable

ˆ ( )

S.E.

Cons.

-3.8211***

0.1939

y

 t 1

yt2 y

 t 2

y

 t 3

y

 t 3

a Cons.

-0.2502***

0.0816

-0.4746***

0.0768

-0.1769**

0.0813

0.0574

y

y

 t 1

0.0810

-0.7416***

0.0772

θ1=0.0269

S.E.

Variable

ˆ ( )

S.E.

Variable

ˆ ( )

S.E.

-1.4250***

0.0521

Cons.

-2.0389***

0.0692

Cons.

-1.2129***

0.0444

0.0532

y

 t 1

0.0567

y

 t 1

-0.1417***

0.0536

y

 t 1

y

 t 1

-0.8246***

0.0472

-0.4402*** -0.5583***

0.0498

θ2=0.0040

-0.2865*** -0.4932***

0.0509

θ2=0.0028

θ2=0.0029

ˆ ( )

S.E.

Variable

ˆ ( )

S.E.

Variable

ˆ ( )

S.E.

Variable

ˆ ( )

S.E.

Cons.

-3.2848***

0.0638

Cons.

-3.1261***

0.0655

Cons.

-2.7616***

0.1285

Cons.

-1.4629***

0.0934

0.0842

y

 t 1

0.0900

y

 t 1

-0.1896*

0.1005

 t 1

-0.9541***

0.0806

y

 t 1

-1.2313***

0.0898

-0.5205***

0.0895

yt2

-2.4483***

0.0999

0.0809

 t 2

-0.4122***

0.0900

yt 1

y  y 

-0.0036

0.0752

 2 t 1

-0.0417**

0.0212

 2 t 1

-0.0864***

0.0169

yt 1

y  y 

0.6132***

 2 t 1

-0.3664***

0.0260

y

 2 t 1

0.0406*

0.0237

yt2

0.0775

-0.7516***

θ1=0.0231

Variable

Jo ur

y

 t 1

na l

θ2=0.0035

0.0493

 t 1

MSCI

Pr

y

 t 1

Nikkei 225

ˆ ( )

e-

Variable

pr

Panel A: α1=0.05

y

 t 2

-0.4831***

-0.7095***

denotes the constant term. *, ** and *** indicate the significance at 10%, 5% and 1% levels, respectively. S.E. means the standard errors.

44 / 52

y

Failure

Rate

time

failure

statistic

0.7617

189

5.26%

0.4986

0.4801

-5.8093

1.5782

36

1.00%

0.0002

0.9893

0.0251

-1.8902

0.6828

192

5.34%

0.8749

0.3496

0.01

0.0040

-3.3953

0.4385

36

1.00%

0.0001

0.9920

FTSE

0.05

0.0261

-1.8504

0.4714

211

5.87%

5.4481

0.0196

100

0.01

0.0040

-3.3669

0.9318

44

1.22%

1.7041

0.1918

Nikkei

0.05

0.0231

-2.4685

0.4927

195

5.43%

1.3359

0.2478

225

0.01

0.0028

-4.2306

1.2562

36

1.00%

0.0001

0.9906

0.05

0.0269

-1.5667

0.5816

206

5.73%

3.8771

0.0489

0.01

0.0029

-3.0432

1.7427

37

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Table 4 Summary of EVaRs for WTI and stock market returns a Return

α

θ

Mean

Std. Dev.

0.05

0.0235

-3.6358

0.01

0.0035

S&P

0.05

500

WTI

MSCI

of

1.03%

LR

0.0319

a The p-value denotes the minimum significance level of rejecting the null hypothesis (namely there is

P-value

0.8583

no significant

difference between the rate of failure f and the significance level α). The larger p-value, the higher adequacy of the

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EVaR (Qian and Zhang, 2012).

45 / 52

Table 5 Summary of QVaRs based on GED-GARCH models for WTI and stock market returnsa

WTI

S&P 500

FTSE 100

Nikkei 225

MSCI

α

Mean

Failure time

Rate of failure

LR statistic

P-value

0.05

-3.6297

194

5.40%

1.2027

0.2728

0.01

-5.6544

48

1.34%

3.7258

0.0536

0.05

-1.7961

216

6.02%

7.3351

0.0068

0.01

-2.8922

44

1.23%

1.7177

0.1900

0.05

-1.8034

218

6.07%

8.0692

0.0045

0.01

-2.9070

45

1.25%

2.1361

0.1439

0.05

-1.7651

239

6.66%

18.9604

0.0000

0.01

-2.7713

49

1.37%

4.3493

0.0370

0.05

-2.4233

187

5.20%

0.3082

0.5788

0.01

-3.8796

31

0.86%

0.7193

0.3964

ro of

Return

a The p-value denotes the minimum significance level of rejecting the null hypothesis (namely there is

no significant

difference between the rate of failure f and the significance level α). The larger p-value, the higher adequacy of the

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QVaR (Qian and Zhang, 2012).

46 / 52

f M=10

M=20

M=30

1

1

1

1

0.0014

0.0009

0.0009

0.0006

0.0003

0.0003

0.0003

0.0014

0.0009

0.0006

0.0009

0.0003

0.0003

0.0003

M=20

M=30

1

1

1

WTI⇏S&P 500

0.0020

0.0017

WTI⇍S&P 500

0.0017

0.0011

WTI⇎FTSE 100

1

1

WTI⇏FTSE 100

0.0014

0.0014

WTI⇍FTSE 100

0.0017

0.0017

WTI⇎Nikkei 225

1

WTI⇏Nikkei 225

0.0020

WTI⇍Nikkei 225

0.0020

WTI⇎MSCI

M=10

W=200

1

1

1

1

1

1

1

1

0.0011

0.0006

0.0009

0.0009

0.0003

0.0003

0.0003

0.0017

0.0006

0.0006

0.0006

0.0012

0.0009

0.0009

1

1

1

1

1

1

1

1

0.0017

0.0011

0.0011

0.0011

0.0009

0.0003

0

0

0.0017

0.0020

0.0009

0.0006

0.0003

0.0003

0.0003

0

1

1

1

1

1

1

1

1

1

0.0006

0.0008

0.0008

0.0003

0.0003

0.0003

0

0

0

0.0011

0.0014

0.0014

0.0003

0.0003

0.0006

0.0003

0.0003

0.0003

1

1

1

1

1

1

1

1

1

e-

pr

M=10

WTI⇎S&P 500

WTI⇏S&P 500

0.0006

0.0008

0.0008

0.0011

0.0009

0.0009

0.0015

0.0015

0.0015

WTI⇍S&P 500

0.0008

0.0006

0.0006

0.0014

0.0009

0.0009

0.0018

0.0021

0.0018

WTI⇎FTSE 100

1

1

1

1

1

1

1

1

1

WTI⇏FTSE 100

0.0008

0.0008

0.0008

0.0011

0.0011

0.0011

0.0006

0.0009

0.0006

WTI⇍FTSE 100

0.0003

0.0006

0.0006

0.0003

0.0006

0.0003

0

0.0003

0.0003

WTI⇎Nikkei 225

1

1

1

1

1

1

1

1

1

WTI⇏Nikkei 225

0.0006

0.0006

0.0006

0.0006

0.0006

0.0009

0

0.0003

0.0003

WTI⇍Nikkei 225

0

0.0003

0.0003

0.0003

0.0006

0.0006

0

0.0003

0.0003

WTI⇏MSCI WTI⇍MSCI

Jo ur

WTI⇎S&P 500

1%

M=30

W=104

Pr

5%

M=20

1

W=64

Null hypothesis

na l

Sig. level

oo

Table 6 Average probabilities of rejecting the null hypotheses between WTI and stock marketsa

47 / 52

1

WTI⇏MSCI

0.0008

0.0014

0.0014

WTI⇍MSCI

0.0003

0.0006

0.0006

1

f

1

1

1

1

1

1

0.0009

0.0011

0.0011

0.0009

0.0012

0.0012

0.0003

0.0003

0.0003

0.0006

0.0006

0.0006

oo

1

represents there is no bidirectional time-varying downside risk spillover effect between the two markets.⇏ and ⇍ represent there are no unidirectional time-varying downside risk spillover

Jo ur

na l

Pr

e-

effects from the left to the right and from the right to the left, respectively.

pr

a⇎

WTI⇎MSCI

48 / 52

Table 7 Estimated results of CAR-GARCH for Brent returns a Parameter

α1=0.05

α2=0.01

θ

0.0242

0.0032

CARi-ARCHE(n, p) model

CAR2-ARCHE(1, 1)

CAR2-ARCHE(2, 7)

-1.4250***

-4.8894***

(0.0521)

(0.1802)

-0.4402***

-0.1854**

(0.0532)

(0.0878)

-0.5583***

-0.1791**

(0.0498)

(0.0825)

Cons. yt1 yt1

-0.5268**

ro of

yt2

(0.0877)

-0.5091***

yt2 a

(0.0826)

Cons. denotes the constant term. *, ** and *** indicate the significance at 10%, 5% and 1% levels, respectively.

Jo

ur

na

lP

re

-p

Numbers in parentheses are standard errors of the corresponding estimated coefficients.

49

Table 8 Summary of EVaRs for Brent returns a

a

θ

Mean

Failure time

Rate of failure

LRstatistic

P-value

0.0242

-3.3850

186

5.18%

0.2300

0.6316

0.0032

-5.9987

36

1.00%

0.0001

0.9906

The p-value denotes the minimum significance level of rejecting the null hypothesis (namely there is no significant

difference between the rate of failure f and the significance level α). The larger p-value, the higher adequacy of the

Jo

ur

na

lP

re

-p

ro of

EVaR (Qian and Zhang, 2012).

50

f M=30

M=10

M=20

M=30

1

1

1

1

1

1

0.0014

0.0011

0.0009

0.0009

0.0006

0.0003

0.0003

0.0020

0.0011

0.0014

0.0011

0.0003

0

0

1

1

1

1

1

1

1

0.0017

0.0009

0.0011

0.0011

0.0012

0.0006

0.0006

0.0017

0.0017

0.0017

0.0014

0.0012

0.0012

0.0009

1

1

1

1

1

1

1

1

0.0017

0.0017

0.0009

0.0011

0.0014

0.0003

0.0003

0.0003

0.0017

0.0017

0.0006

0.0009

0.0003

0

0

0

1

1

1

1

1

1

1

1

1

0.0011

0.0011

0.0011

0.0003

0.0003

0.0003

0.0003

0.0003

0

0.0014

0.0017

0.0017

0.0009

0.0014

0.0014

0.0006

0.0006

0.0003

1

1

1

1

1

1

1

1

1

M=20

M=30

Brent⇎S&P 500

1

1

1

Brent⇏S&P 500

0.0011

0.0014

Brent⇍S&P 500

0.0017

0.0017

Brent⇎FTSE 100

1

1

Brent⇏FTSE 100

0.0017

0.0017

Brent⇍FTSE 100

0.0014

0.0020

Brent⇎Nikkei 225

1

Brent⇏Nikkei 225

0.0014

Brent⇍Nikkei 225

0.0017

Brent⇎MSCI

M=10

e-

M=10

W=200

Brent⇏S&P 500

0.0006

0.0006

0.0008

0.0009

0.0011

0.0011

0.0006

0.0009

0.0009

Brent⇍S&P 500

0.0006

0.0006

0.0006

0.0006

0.0006

0.0006

0.0009

0.0009

0.0009

Brent⇎FTSE 100

1

1

1

1

1

1

1

1

1

Brent⇏FTSE 100

0.0003

0.0006

0.0006

0.0006

0.0006

0.0009

0.0000

0.0003

0.0003

Brent⇍FTSE 100

0

0

0

0.0003

0

0

0

0

0

Brent⇎Nikkei 225

1

1

1

1

1

1

1

1

1

Brent⇏Nikkei 225

0.0003

0.0003

0.0003

0.0006

0.0006

0.0011

0

0.0000

0.0006

Brent⇍Nikkei 225

0

0

0.0003

0.0003

0.0003

0.0009

0.0003

0.0003

0.0006

Brent⇏MSCI Brent⇍MSCI

Jo ur

Brent⇎S&P 500

1%

oo

5%

M=20

W=104

Pr

level

W=64

Null hypothesis

na l

Sig.

pr

Table 9 Average probabilities of rejecting the null hypotheses between Brent and stock marketsa

51

1

1

1

Brent⇏MSCI

0.0006

0.0014

0.0014

Brent⇍MSCI

0.0003

0.0003

0

f

1

1

1

1

1

1

0.0006

0.0014

0.0014

0.0009

0.0015

0.0015

0.0003

0.0003

0.0003

0.0003

0.0003

0.0003

oo

Brent⇎MSCI

represents there is no bidirectional time-varying downside risk spillover effect between the two markets.⇏ and ⇍ represent there are no unidirectional timevarying downside risk spillover effects from the left to the right and from the right to the left, respectively.

Jo ur

na l

Pr

e-

pr

a⇎

52