Measuring extreme risk of sustainable financial system using GJR-GARCH model trading data-based

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Measuring extreme risk of sustainable financial system using GJR-GARCH model trading data-based Xiaomeng Maa,b, Ruixian Yangc, , Dong Zoud, Rui Liue ⁎

a

Management School, Shenzhen Polytechnic, Shenzhen 518055, China Post-Doctoral Scientific Research Workstation, China Merchants Bank, Shenzhen 518040, China c College of Information Management, Zhengzhou University, Zhengzhou 450001, China d School of Management, Huazhong University of Science and Technology, Wuhan 430074, China e Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China b

ARTICLE INFO

ABSTRACT

Keywords: Extreme risk spillover Value at Risk Safe haven Granger causality in risk Global financial crisis

This paper investigates the role of gold as a safe haven for stock markets and the US dollar by examining the extreme risk spillovers. The extreme risk is measured by Value at Risk (VaR), which is estimated by GJR-GARCH model based on skewed t distribution. Two test statistics of one-way and two-way Granger causality in risk are used to detect extreme risk spillovers. In general, the empirical results show that there are negative extreme risk spillovers between gold and stock markets and between gold and foreign exchange markets of US dollar, which indicate that gold can act as an effective safe haven against extreme stock and US dollar exchange rate movements. In addition, the global financial crisis can affect the safe haven role of gold.

1. Introduction The increasing trend of financial market integration and interdependence increases the likelihood of the risk transmission from one market or asset to other markets or assets (Liu, 2014; Zhou, 2013). During the turbulent periods, investors choose to rebalance their portfolios between the risky financial assets and safer assets to reduce the portfolios risk, and this ‘flight to quality’ phenomenon results in the price increase of safer assets (Tuysuz, 2013). After the global financial crisis, the global economy experiences the slow growth environment with increasing uncertainty, such as, the uncertainty of US monetary policy to raise interest rate, the slowdown of emerging markets and developing economies, and the slow recovery of advanced economies. The increasing uncertainty of global economy is also transmitted to the global financial market, which makes investors tend to reallocate the portfolios from high-risk markets to lowrisk markets. From the perspective of investors, risk managers and the financial media, gold is regularly considered as a hedge or a safe haven against financial markets (Ogiela, 2013). Therefore, understanding the relationship between gold and financial markets have important implications. The role of gold as a hedge or a safe haven has been examined in the literature by focusing on the relationship between gold and stock market movements, and between gold and currency depreciation. Most

⁎

previous studies have focused on the correlations or average dependences to examine the hedging ability of gold, and some studies have proposed different methods, such as, GARCH models, copula functions and threshold regression models, to study the role of gold as a safe haven. However, few studies have studied the safe haven role of gold by investigating extreme risk spillover effects. By focusing on the extreme risk spillover effects, this paper aims to investigate the potential role of gold as a safe haven against extreme stock and foreign exchange rate movements, specifically for the stock markets of USA, Japan and mainland China and foreign exchange market of the US dollar. In literature, it is difficult to model the extreme risk spillover effects when considering the complexity of the correlation and the nonlinearity of causality. In order to bridge this gap, we further develop the Granger causality in risk model proposed by Hong, Liu, and Wang, (2009) by using the Ant Colony Algorithm (ACA) to improve the calculation speed and efficiency of parameter estimation(Uthayakumar, Metawa, Shankar, & Lakshmanaprabu, 2018). In this paper, we only focus on the extreme risk spillover effects in financial market (Agarwal, Kumar, & Goel, 2019). In fact, the method we proposed can also be extended to other fields and can be used to analyze the interaction between different objects or business. For example, the method can be used to study the emotional contagion among the social network or in the social media (Kim & Hastak, 2018).

Corresponding author. E-mail address: [email protected] (R. Yang).

https://doi.org/10.1016/j.ijinfomgt.2018.12.013 Received 22 October 2018; Received in revised form 23 December 2018; Accepted 24 December 2018 0268-4012/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Ma, X., International Journal of Information Management, https://doi.org/10.1016/j.ijinfomgt.2018.12.013

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In this paper, we examine the extreme risk spillover effects among gold, stock and foreign exchange markets, where risk is originally measured by the left tail of the distribution, namely, the downside Value at Risk (VaR). In this paper, we also consider the extreme upside risk measured by upside VaR, which is introduced by Du and He (2015). Extreme risk spillover effects refer to comovements of large changes between two time series of markets or assets (Hong et al., 2009). For example, extreme price movements in one market will spill over to price movement in another market when markets are integrated and suffer from the same global shock, or due to “market contagion”. There are positive risk spillovers and negative risk spillovers. The former refer to extreme downside and upside comovements, the later refer to the extreme down-to-up and up-to-down comovements. In this paper, we focus mainly on the negative risk spillovers, which are important and useful to examine the role of gold as a safe haven. This paper will answer the following three questions. Firstly, if there exist extreme negative risk spillover effects gold and stock markets and between gold and the US dollar? Namely, is gold a safe haven for stock markets and the US dollar? Secondly, if there exist extreme negative risk spillover effects, are they unidirectional or bidirectional? Lastly, how the safe haven role of gold changes when considering the influence of global financial crisis? Our empirical data are daily closing prices of Shanghai Composite Index, Nikkei 225 Index, S&P 500 Index, London Gold and US Dollar Index during the period from January 1, 2005 to August 31, 2017. To analyze the influence of the global financial crisis, we consider two subsample periods, i.e. before and after the financial crisis. The empirical results show that there are extreme down-to-up risk spillover effects from the US dollar and stock markets to gold market over the whole study sample, which indicates that investors can choose gold as safe haven when facing extreme downside risk in stock markets and US dollar. However, when taking the influence of global financial crisis into consideration, the safe haven role of gold will change. In general, gold acts as a safe haven for US dollar before the global financial crisis, but is no longer a safe haven after the global financial crisis. Gold acts as a safe haven for the US and Japanese stock market before and after the global financial crisis. For the Chinese stock market, gold is not safe haven before the global financial crisis, but becomes a weak safe haven after the global financial crisis. The global financial crisis increase the ability of gold as a safe haven for the Japanese stock market. In addition, the down-to-up risk spillovers from the US stock market to gold market change to be up-to-down risk spillovers from the gold market to the US stock market. In addition, we also study the potential role of the US dollar as a safe haven for stock markets. The remainder of this article is organized as follows. Section 2 is the literature review. The methodology is described in Section 3. Section 4 presents the sample data and discusses the empirical results. The conclusion is presented in Section 5.

Using the same model, Gürgün and Ünalmış (2014) extend the study of the hedge and safe haven properties of gold against stock markets to emerging and developing countries. The empirical results indicate that, gold acts a hedge and a safe haven in most of countries for domestic investors over the full sample period. However, gold acts as a safe haven in only a few markets for foreign investors. In addition, during times of extreme losses in equity markets, gold acts as a safe haven in an increasing number of countries for both domestic and foreign investors. Beckmann, Berger, and Czudaj, 2015; Beckmann, Czudaj, & Pilbeam, 2015) study the hedge and safe haven properties of gold by augmenting the model of Baur and Lucey (2010) to smooth transition regression with two extreme regimes. The results indicate that gold is both a hedge and a safe haven for stock markets. Gold can serve as a safe haven during the extreme financial markets. Using a VAR–GARCH framework, Arouri, Lahiani, and Nguyen, (2015) investigate the relationship between world gold prices and Chinese stock returns from the insight for hedging and diversification strategies. They find that gold can serve as a safe haven for Chinese stock market during the recent global financial crisis. Some studies focus on the nonlinear relationship between gold and stock markets (Choudhry, Hassan, & Shabi, 2015; Jain & Biswal, 2016). For example, Choudhry et al. (2015) apply the nonlinear Granger causality test to test the dynamic relationship between gold and stock markets during the global financial crisis. Their results show that gold is not a safe haven during the financial crisis period. However, gold can be used as a hedge in stable financial conditions. By using DCC-GARCH model and nonlinear causality tests, Jain and Biswal (2016) study the dynamic linkages among oil price, gold price, exchange rate, and stock market in India. The second strand of literature focuses on the relationship between gold and foreign exchange markets. Pukthuanthong and Roll (2011) find that currency appreciations or depreciations have effects on the price of gold. Some studies suggest that gold can act a hedge against the US dollar (Beckmann, Berger et al., 2015, 2015b; Capie, Mills, & Wood, 2005; Joy, 2011; Lin, Chen, & Yang, 2012; Reboredo & Rivera-Castro, 2014; Wang & Chueh, 2013). In an early study, Capie et al. (2005) examine the hedge property of gold against the US dollar. They find that the ability that gold serves as a dollar hedge is time-varying and highly depends on unpredictable political attitudes and events. Using a model of dynamic conditional correlations, Joy (2011) examines the hedge and safe haven property of gold against the US dollar. The results indicate that gold is a hedge against the US dollar but a poor safe haven, which are supported by the study of Reboredo and Rivera-Castro (2014). Wang and Chueh (2013) find that the value of the US dollar during the previous period negatively affects the gold prices during the following period, which indicate that gold can act as a hedge against the US dollar. By using GARCH-in-mean SVAR models, Beckmann, Czudaj et al. (2015) examine the link between gold prices and exchange rates of five different currencies. The results indicate that only the gold price denominated in the US dollar tends to increase after a depreciation of the US dollar. Different from the previous study, Lin et al. (2012) use wavelet analysis to examine the relationship between gold and the US dollar in short-term and longterm. Empirical results indicate that short-term negative correlations between gold and the US dollar are much higher and the financial crises accelerate the interdependence between gold and the US dollar. However, only a few studies suggest that gold can act as a safe haven against the US dollar. Using daily data from the US the UK markets, Ciner, Gurdgiev, and Lucey, (2013) find that gold plays the role of a monetary asset to act as a safe haven against the US dollar during extreme price movements. Using copulas, Reboredo (2013) examines the role of gold as a safe haven against the US dollar by focusing on the tail dependence between gold and USD exchange rates. The results indicate that gold can act as hedge against USD rate movements and can act as an effective safe haven against extreme USD rate movements. The last strand of literature focuses on the extreme risk spillover effects across different financial markets. In a seminal paper, Hong et al. (2009) introduce the concept of Granger causality in risk and propose a

2. Literature review This paper mainly adds to the following three strands of literature. The first strand focuses on the relationship between gold and stock markets to examine the role of gold as a hedge or a safe haven. In an earlier study, Baur and McDermott (2010) study the relationship between gold and stock markets by testing the safe haven effect. The results indicate that gold is a safe haven for most stock markets in developed countries, and can act as a strong safe haven during the peak of the recent financial crisis. However, the situation is quite different for emerging stock markets. Gold is not a safe haven for most of emerging stock markets and a weak safe haven for some emerging stock markets. Using the model proposed by Baur and McDermott (2010); Hood and Malik (2013) examine the role of gold, metals and Volatility Index (VIX) as a hedge and a safe haven for US stock market. They find that gold acts as a hedge and a weak safe haven, while VIX acts as a very strong hedge and a strong safe haven. Especially, in periods of extremely low or high volatility, gold does not act as a safe haven for the US stock market. 2

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class of kernel-based tests to examine extreme downside risk spillover between financial markets, where Value at Risk (VaR) is used to measure extreme market risk. The conventional definition of VaR refers to the downside risk, however in financial markets, the extreme upside price movement can also incur risk. In order to capture the upside risk, Du and He (2015) introduce the notion of upside VaR. Based on the study of Hong et al. (2009), several studies examine the extreme risk spillover effects in different markets, such as, in international TEIT markets (Zhou, 2013), in stock markets (Liu, 2014), in world gold markets, between crude oil and stock markets (Du & He, 2015) and among financial institutions (Wang, Xie, He, & Stanley, 2017). This paper contributes to the existing literature in two ways. First, we study the extreme risk spillover effects among gold, stock and foreign exchange markets, by using the Granger causality in risk, which provide a measure of extreme price comovement. This information is crucial in determining gold’s role as a safe haven. Most previous studies have proposed different methods including GARCH models, copula functions and threshold regression models to examine the role of gold as a safe haven. However, little attention has been paid to extreme risk spillover effects. To bridge this gap, we try to test the gold’s role as a safe haven by investigating extreme risk spillover effects. Second, the Granger causality in risk proposed by Hong et al. (2009) has been widely studied in different financial markets to investigate extreme risk spillover effects. In this paper, we broaden the research method to the study of extreme risk spillovers between gold and stock markets and between gold and the US dollar.

denoting bad news; otherwise dt 1 = 0 and denotes good news. The coefficient is used to measure the difference between the effects of good news and bad news on the conditional variance, namely, the leverage effects. As efficient estimation of VaR depends highly on the conditional distribution for the standardized error (Du & He, 2015), attention should be paid to the selection of the appropriate distribution. In this paper, we consider three distributions for the standardized error to estimate VaR: Student's t distribution, generalized error distribution (GED) and skewed t distribution.1 In this paper, the skewed t distribution proposed by Hansen is used. This distribution has two parameters to be estimated: the degrees of freedom parameter 2 < < and the skewness parameter 1 < < 1. The freedom degrees parameter controls the tail thickness and the skewness parameter controls the asymmetry. In fact, skewed t distribution covers several distribution. For example, when = 0, the skewed t distribution can be rewritten as standardized Student's t dis, the standardized normal distribution tribution. When = 0 and = can be obtained. Taking these characteristics of skewed t distribution into consideration, it has been widely used to describe financial returns series (Hong et al., 2009). From Eqs. (3)–(6), we can obtained the upside VaR and downside VaR of time seriesYt :

Vt (up) = µt + Vt (down) =

3.1. Extreme risk measurement In literature, Value at Risk (VaR) is a widely used measurement of extreme risk. The conventional definition of VaR refers to the downside risk, however in financial markets, the extreme upside price movement can also incur risk. Therefore, the notion of upside VaR is introduced by Du and He (2015) to capture the upside risk. In this paper, we use both the upside VaR and downside VaR, denoted by Vt (up) and Vt (down) , respectively. For a given time series of returns Yt , at the confidence level )%, the upside VaR and downside VaR are written as: of 100(1 (1)

P (Yt <

(2)

Vt (down)|It 1) = .

t

=

ht =

zt

p j=1

j Yt j

+ t,

0

+

m

2 i t i

+

2 t 1 dt 1

+

n j

j ht j .

m . d. s . (0, 1) withconditionalCDFF (.).

(8)

(9)

Zlt , up = 1(Ylt > Vlt (up)), l = 1, 2. Zlt , down = 1(Ylt <

Vlt (down), l = 1, 2.

(10)

where 1(•) is an indicator function and l represents the markets 1 or 2. When the market return exceeds VaR (larger than the upside VaR or less than the downside VaR) the risk indicator Zlt takes value of 1 and otherwise takes value of 0. For convenience, the test for Granger causality in down-to-up risk spillover from market 2 to market can be stated as: Hypothesis 1. There is no one-way Granger causality in down-to-up risk from market 2 to market 1

H0 : E (Z1t , up |I1(t

(4) i=1

ht Z .

In order to study the role as a safe haven, we focus on the negative extreme risk spillovers, namely, the down-to-up risk spillover and the up-to-down risk spillover. For two different markets {Y1t } and {Y2t } , the downside risk indicator and upside risk indicator functions are defined as:

(3)

ht z t ,

µt

3.2. Granger causality in risk

where It 1 = {Yt 1, Yt 2, ...} is the information set available at time t 1. Mathematically, the upside VaR and downside VaR are the right -quantile and left -quantile of the returns, respectively. In order to estimate VaR, this paper employ GJR-GARCH model which can capture the characteristics of financial returns, such as volatility clustering, fat tails, skewness and leverage effects (Du & He, 2015; Glosten, Jagannathan, & Runkle, 1993; Hung, Lee, & Liu, 2008). For a given time series of returns Yt , the AR-GJR-GARCH model takes the following form：

Yt = c +

(7)

where Z is the left-tailed critical value of the standardized innovation at level and Z1 is the upper -quantile . Following the study of Du and He (2015), the method proposed by Kupiec (1995) is used to test the accuracy of the VaR model for calculating extreme market risk. Under the null hypothesis, the test statistic LR follows 2 (1) distribution asymptotically. For a given significance level, if LR is larger than the critical value, the null hypothesis of correct specification of the VaR model should be rejected.

3. Methodology

P (Yt > Vt (up)|It 1) = ,

,

ht Z1

1), up )

= E (Z1t , up |(I1(t

1), up , I2(t 1), down )).

1), up)

E (Z1t , up |(I1(t

1), up, I2(t 1), down )).

Against

(5)

H1: E (Z1t , up |I1(t

(6)

where I1(t 1), up and I2(t 1), down are the information sets of upside and downside risk available at time t 1 for two time series respectively. If the alternative hypothesis H1 holds, we say that say that there is oneway Granger causality in down-to-up risk from market 2 to market 1.

where Eq. (3) and Eq. (5) are the conditional mean and variance equations, respectively. Let , then µt and ht are the conditional mean and variance of Yt given It 1, respectively. Eq. (4) defines the standardized residual z t and in Eq. (6), the standardized residual has a conditional distribution with mean zero and variance one. In the conditional variance equation, indicator dt 1 = 1 when t 1 < 0 ,

1 The results indicate that skewed t distribution is the most accurate, which is supported by Du and He (2015).

3

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That is, there is down-to-up risk spillover from market 2 to market 1, and an extreme downside risk in market 2 can be used to predict the upside risk in market 1 in the future. Similarly, the test for Granger causality in up-to-down risk spillover from market 2 to market 1 can be stated as:

4. Empirical results and discussions 4.1. Data In the paper, Shanghai Composite Index (SH), Nikkei 225 Index (N225), S&P 500 Index (SP500), London Gold Price (LGP) and US Dollar Index (USD) are chosen to represent the Chinese, US and Japanese stock markets, gold market and the foreign exchange market of the US dollar. We collect daily data from the Wind Database and our data cover the period from January 1, 2005 to August 31, 2017. Returns for different markets are obtained as the first difference in the natural logarithm of the daily closing price. Taking the non-synchronous trading problem into consideration, the daily returns of S&P 500, London Gold Price and US Dollar Index are lagged one day to match the current returns of Shanghai Composite Index and Nikkei 225 Index. Fig. 1 presents the daily returns during the period from January 1, 2005 to August 31, 2017. During the full sample period, the global financial market has witnessed several crisis events including the global finance crisis induced by the US subprime mortgage crisis, the European sovereign debt crisis and the stock market turbulence in China. Due to these crisis events, the global financial market becomes more volatile. Table 1 presents the descriptive statistics of returns. For the stock markets, as one of the emerging market, the Chinese stock market is more volatile with higher returns than the US and Japanese stock markets, which is supported by many empirical studies (Boubakri & Guillaumin, 2015; Zhou, Zhang, & Zhang, 2012). For the gold market, the mean of returns is relative high with lower volatility compared to the stock markets. For foreign exchange market of the US dollar, the mean and standard deviation of the returns are the lowest. For stock and gold markets, the skewness is negative, while the skewness for foreign exchange market is positive. The kurtosis for all markets are above 3, indicating that we cannot assume the return distribution to be normal. In addition, the Jarque–Bera test also rejects the assumption of normality. The Ljung–Box Q test statistic shows that, with the exception of gold market, serial autocorrelations exist in each set of returns. Therefore the AR model should be used to eliminate the serial autocorrelation effect in the returns.

Hypothesis 2. There is no one-way Granger causality in up-to-down risk from market 2 to market 1

H0 : E (Z1t , down |I1(t

1), down )

= E (Z1t , down |(I1(t

1), down , I2(t 1), up )).

1), down )

E (Z1t , down |(I1(t

1), down , I2(t 1), up )).

Against

H1: E (Z1t , down |I1(t

where I1(t 1), down and I2(t 1), up are the information sets of downside and upside risk available at time t 1 for two time series respectively. If the alternative hypothesis H1 holds, we say that say that there is one-way Granger causality in up-to-down risk from market 2 to market 1. That is, there is up-to-down risk spillover from market 2 to market 1, and an extreme upside risk in market 2 can be used to predict the downside risk in market 1 in the future. 3.3. Test statistics In order to test the Granger causality in risk between different markets, Hong et al. (2009) propose a class of kernel-based statistics. For two estimated series of risk indicators {Zˆ1t } and {Zˆ 2t } , Hong et al. (2009) define the sample cross-covariance function between these two risk indicators as follows:

Cˆ (j ) =

T

T ˆ1 )(Zˆ 2(t j ) (Zˆ1t t=1 +j 1 T ˆ1 )(Zˆ 2t (Zˆ1(t + j) t=1 j

1

T

where j is the lag order and ˆ l = T lation function between {Zˆ1t } and {Zˆ 2t } is：

ˆ (j ) =

Cˆ (j) , j = 0, ± 1, ..., ± (T Sˆ1 Sˆ2

ˆ2), 0

j

T

ˆ2), 1-T 1

T t 1

1

j<0

,

(11)

Zˆlt . The sample cross-corre-

1),

(12)

4.2. Estimation of VaR

2 ˆ1) is the sample variance of {Zˆlt } . where Sˆl = ˆ l (1 The test statistic for one-way Granger causality in risk from market 2 to market 1 is：

Q1 (M ) = T

T 1 j=1

k 2 (j M ) ˆ 2 (j )

C1T (M )

1

D1T (M ) 2 ,

In this part the upside and downside VaRs are estimated to examine the extreme risk spillovers across different markets. According to the descriptive statistics in Table 1, the daily returns may show the characteristics of volatility clustering, fat tails, skewness and non-normality. In order to capture volatility clustering, GARCH models are used. It is well known that financial returns may have leverage effects which means that the current volatility caused by the previous increase and decrease in the returns is asymmetric. Therefore, the GJR-GARCH models are used in this paper. To capture the characteristics of fat tails, skewness and non-normality, the Student's t distribution, generalized error distribution (GED) and skewed t distribution are used to estimate the upside and downside risk. After comparing the results of the three distributions, we find that skewed t distribution is the most accurate one. The estimated results of the GJR-GARCH model under skewed t distribution are presented in Table 2. First, for all the time series, the coefficients of lagged conditional variance, 1, are significant at the 1% level, indicating the presence of volatility clustering. Second, there is a significant leverage effect, captured by coefficient , in the US and Japanese stock markets, gold and foreign exchange market. There is no significant leverage effect in the Chinese stock market. For stock markets, the positive leverage effects indicate that bad news has stronger impact than good news, while the situation is different for gold and foreign exchange markets. Third, the asymmetric parameters, , are all blow 0 at the 5% significance level except for that of gold and foreign exchange markets, indicating that the time series is left skewed. Last, the freedom degrees are all above 4 at the 1% significance level,

(13)

The test statistic for two-way Granger causality in risk including instantaneous risk spillovers between two markets is:

Q2 (M ) = T

T 1 j =1 T

k 2 (j M ) ˆ 2 (j )

C2T (M )

1

D2T (M ) 2 ,

(14)

where k (•) is a kernel function which can takes several form. In this paper, we select the Daniell kernel k (x ) = sin ( x ) ( x ) by minimizing k 4 (z ) dz . M is the largest lag order, indicating how many lagged 0 information sets are considered for the analysis of risk spillover. C1T (•) , C2T (•) , D1T (•) and D2T (•) are the centering and standardized constants defined as: T 1 j =1

C1T (M ) = D1T (M ) = 2

T 1 j=1

C2T (M ) = D2T (M ) = 2[1 + ˆ 4 (0)]

(1

j T ) k 2 (j M ),

(1

j T )(1 T 1 |j| = 1

T 1 |j| = 1

(1

(1

(j + 1) T ) k 4 (j M ), |j| T ) k2 (j M ),

|j| T )(1

(|j| + 1) T ) k 4 (j M ). (15)

Under the null hypothesis, Q1 (M ) and Q2 (M ) follow an asymptotically standard normal distribution. If Q1 (M ) and Q2 (M ) are greater than the right-tailed critical value at a specified significance level, the null hypothesis is rejected. 4

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Fig. 1. Daily market returns. Table 1 Descriptive statistics.

SH SP500 N225 LGP USD

Table 2 Estimation results of GJR-GARCH model.

Mean

Std.dev.

Skewness

Kurtosis

J-B

LB-Q(20)

0.0223 0.0157 0.0149 0.0422 −0.0021

1.6979 1.2318 1.5938 1.2075 0.5358

−0.5281 −0.9425 −0.6058 −0.2743 0.2940

7.3195 17.0450 11.8466 8.9966 9.9370

2323.3515*** 23595.8027*** 9368.3259*** 4260.5994*** 5694.9196***

43.2981*** 69.5103*** 41.1254*** 24.6825 44.4682***

SH Mean equation c 0.0316 −0.0054 1 0.0155 2 0.0381** 3 Variance equation 0.0071 0 0.0562*** 1 0.9491*** 1

Note: This table provides the summary statistics of the returns in various markets. SH is the Shanghai Composite Index, SP500 is the S&P 500 Index, N225 is the Nikkei 225 Index, LGP is the London Gold Price and USD is the US Dollar Index. J–B is the Jarque–Bera test for the null hypothesis of Gaussian distribution and LB-Q(20) is the Ljung–Box Q test of serial correlation of up to 30 lags in the returns. * indicates statistical significance at the 10% level. ** indicates statistical significance at the 5% level and ***indicates statistical significance at the 1% level.

−0.0106 −0.0599*** 4.8920***

SP

N225

LGP

USD

0.0394*** −0.0777*** −0.0436 −0.0202

0.0332 −0.0507** 0.0136 −0.0053

0.0395** −0.0344 −0.0124 −0.0144

−0.0033 −0.0322* 0.0180 −0.0156

0.0202*** 0.0000 0.8854***

0.0615*** 0.0306** 0.8653***

0.0103** 0.0597*** 0.9481***

0.0007* 0.0391*** 0.9670***

0.1966*** −0.1558*** 5.7130***

0.1551*** −0.0898*** 7.3688***

−0.0252** −0.0472 5.0802***

−0.0165** −0.0155 7.5325***

Note: Parameters are estimated by the GJR-GARCH model with skewed t distribution. * indicates statistical significance at the 10% level. ** indicates statistical significance at the 5% level. *** indicates statistical significance at the 1% level.

which indicates fat tails of daily returns. Based on the estimation of the GJR-GARCH models, the upside and downside VaR are calculated at the 5% significance level for all returns. Table 3 presents the results of upside and downside VaR. We can see that the downside VaR is larger than the upside VaR for all markets. Among all markets, the Chinese stock market has the largest mean values for both downside and upside VaR. which is followed by the Japanese stock market, the gold market and the US stock market. The foreign exchange market has the lowest mean values. The standard deviations of upside and downside VaR for stock markets are higher than the gold and foreign exchange market. The results indicate that the Chinese stock market is more risky than the US and Japanese stock markets. We also calculate the failure rate to test the accuracy of the VaR model. The failure rate is the ratio of failure days and sample size (Du & He, 2015). As presented in Table 3, the failure rate varies from 0.0479 to 0.0553 and is around 5% which indicates that the GJR-GARCH model based on skewed t distribution can estimate the VaR efficiently. The estimation of VaR is sensitive to the standardized residual conditional distribution. Therefore, we also consider the estimation accuracy of different distributions, such as Student's t distribution, generalized error distribution. The results indicate that the skewed t distribution is most accurate compared to Student’s t distribution and generalized error distribution (see Appendix A).

Based the analysis above, GJR-GARCH models based on skewed t distribution will be used to estimate the series of downside VaR and upside VaR and then to investigate the extreme risk spillovers across different markets. 4.3. Empirical results of extreme risk spillovers In this section, we calculate the test statistics (Q1 (M ) and Q2 (M ) ) for one-way and two-way Granger causality in risk, respectively. We investigate the extreme risk spillover effects with the largest effective lag truncation orders 5, 10, 20, 30. Following Hong et al. (2009); Du and He (2015); Wang, Xie, Jiang, and Stanley, (2016) and Zhou (2013), we select the largest value of M to be 30. In order to study the safe have role of gold and the US dollar, the negative extreme risk spillovers are considered. The empirical results are presented in Table 4. We consider seven pairwise markets, including the extreme risk spillovers between stock and gold markets, between stock and foreign exchange markets and between gold and the foreign exchange markets. We first consider the extreme risk spillovers between Chinese stock 5

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Table 3 VaR estimation based on skewed t distribution.

SH SP500 N225 LGP USD

Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR

Mean

Std.dev.

Failure time

Failure rate

LR statistics

2.5041 2.4363 1.7495 1.6006 2.3777 2.2736 1.8251 1.8244 0.8349 0.8180

1.1081 1.0439 1.2463 1.1526 1.2402 1.1733 0.6184 0.5918 0.2690 0.2643

148 135 156 142 148 149 147 138 142 141

0.0525 0.0479 0.0553 0.0504 0.0525 0.0528 0.0521 0.0489 0.0504 0.0500

0.3602 0.2724 1.6261 0.0074 0.3602 0.4695 0.2652 0.0676 0.0074 0.0000

Note: the critical values of LR statistics at the 1%, 5% and 10% significance levels are 6.635, 3.841 and 2.706, respectively.

and gold markets. The one-way Granger causality in risk shows that there are lagged effects of down-to-up risk spillover from stock market to gold market. The extremely loss news in the Chinese stock market is transmitted to gold market gradually. The results indicate that gold can act as a safe haven for the Chinese stock market. For the extreme risk spillovers between the US stock and gold markets, two kinds of negative extreme risk spillover effects can be found, namely, the down-to-up risk spillovers and up-to-down risk spillovers. The statistic values show that there are strong down-up risk spillovers from the US stock market to gold market, indicating that extremely loss news in US stock market will be transmitted to gold market quickly and steady (see Fig. 2). For the updown risk spillovers, the extremely gain news in US market is transmitted to gold market gradually. In addition, there exists strong instantaneous downto-up risk spillovers between the US stock market and gold market. The results indicate that gold can act as a safe haven for the US stock market. There are also down-to-up and up-to-down risk spillover effects between the Japanese stock market and gold markets (see Figs. 3 and 4). For the one-way Granger causality in risk, there are extreme down-up risk spillovers from Japanese stock market to gold market, up-down risk spillovers from the Japanese stock market to gold market in the short term and lagged down-up effects from gold market to the Japanese stock market. For the two-way Granger causality in risk, there are weak instantaneous down-to-up risk spillovers and strong up-to-down risk

spillovers between the Japanese stock market and gold market. The results indicate that gold can act as a safe haven for the Japanese stock market. We also consider the extreme risk spillovers between gold and foreign exchange markets (see Figs. 5 and 6). For the one-way Granger causality in risk, there are extreme down-to-up risk spillovers from gold market to foreign exchange market and weak lagged extreme up-todown risk spillovers from gold market to foreign exchange market. There are weak extreme up-to-down risk spillovers from foreign exchange market to gold market. In addition, there exist strong extreme down-up risk spillovers from foreign exchange market to gold market. Taking the two-way Granger causality in risk into consideration, there are instantaneous extreme negative risk spillovers between foreign exchange market and gold market. The results indicate that gold can act as a safe haven for foreign exchange market of the US dollar. For the extreme risk spillovers between stock and foreign exchange markets, there are weak extreme down-up risk spillovers from the Chinese stock market to foreign exchange market in very short term. No negative extreme risk spillovers are found between the US stock market and foreign exchange market and between the Japanese stock market and foreign exchange market. The results indicate that the US dollar can act as a weak safe haven only for the Chinese stock market. From the analysis above, we can conclude that investors can choose

Table 4 Results of negative extreme risk spillovers. Spillover Direction

Down-up risk spillovers M=5

Stock and gold markets SH LGP −0.4329 SH LGP 0.3196 SH LGP −0.9317 5.2786*** SP500 LGP SP500 LGP 7.9265*** SP500 LGP −0.4613 N225 LGP 1.4410* N225 LGP 1.8194** N225 LGP 0.2187 Stock and foreign exchange markets 0.5126 SH USD SH USD 1.5823* −0.8573 SH USD SP500 USD 0.5179 SP500 USD −0.1185 0.8520 SP500 USD N225 USD 0.1172 0.9299 N225 USD N225 USD −0.7642 Gold and foreign exchange markets LGP USD 2.8203*** LGP USD 2.3256** LGP USD 1.6647*

Up-down risk spillovers

M = 10

M = 20

M = 30

M=5

M = 10

M = 20

M = 30

0.4542 1.2003 −0.5573 4.5480*** 7.0966*** −0.6644 1.2229 1.2943* 0.4361

1.1130 2.0757** −0.5009 3.8836*** 6.3818*** −0.8893 1.2308 1.3559* 0.3862

1.0109 1.9108** −0.4800 3.3303*** 5.7624*** −1.0528 0.8531 0.8523 0.3556

−0.4177 −0.9111 0.3203 −0.3837 −0.6968 0.1551 1.9106** 2.0236** 0.6796

−0.3851 −1.0835 0.5387 0.6317 0.8875 0.0071 1.6961** 1.4458* 0.9558

−0.6374 −1.0710 0.1694 1.2651 1.4322* 0.3593 2.3756*** 0.6397 2.7268***

−0.6155 −0.6139 −0.2567 1.4428* 1.5093* 0.5339 2.7239*** 0.3520 3.5091***

0.0551 0.6917 −0.6139 0.3918 −0.0010 0.5568 −0.1218 0.8696 −1.0417

−0.1487 −0.0384 −0.1719 0.1234 −0.0753 0.2525 0.1136 1.0446 −0.8836

−0.3125 −0.1401 −0.3019 −0.0745 −0.4145 0.3112 0.0856 0.9431 −0.8216

−0.7162 −0.9373 −0.0755 −0.8272 −0.4871 −0.6828 0.8201 0.2493 0.9094

−0.9992 −1.1059 −0.3073 −0.8696 −0.2651 −0.9649 0.8572 0.3712 0.8399

−1.2623 −1.1079 −0.6781 −0.5211 0.6052 −1.3425 0.6280 0.2699 0.6174

−1.2727 −0.7407 −1.0603 −0.5733 0.4779 −1.2887 0.4995 0.4466 0.2587

2.0033** 1.5921* 1.2422

1.3971* 1.2893* 0.6873

1.2863* 0.9269 0.8927

2.6208*** 0.7052 3.0081***

2.0369** 0.0454 2.8407***

1.5775* 0.6539 1.5815*

1.7007** 1.4162* 0.9938

Note: the right critical values of Q1 and Q2 statistics at the 1%, 5% and 10% significance levels are 2.3263, 1.6449 and 1.2816, respectively. 6

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Fig. 2. Down-to-up risk spillovers between the US stock and gold markets.

gold as a safe haven when facing extreme downside risk in stock and foreign exchange market. In addition, the US dollar can act as a weak safe haven only for the Chinese stock market.

crisis affects the role of gold as a safe haven from the perspective of down-to-up risk spillovers. First, extreme risk spillovers from the Chinese stock market to gold market are insignificant before global financial crisis, while there are weak lagged down-to-up risk spillover effects from Chinese stock market to gold market after the global financial crisis. There are strong lagged down-to-up risk spillovers from the US stock market to gold market before the global financial crisis. However the effects change to be lagged up-to-down risk spillovers from the gold market to the US stock market after the global financial crisis. The weak and lagged down-to-up risk spillovers from the Japanese stock market to gold market before global financial crisis change to be strong and steady after the global financial crisis. Second, the down-to-up risk spillovers from the Chinese stock market to foreign exchange market become significant after the financial crisis. The strong and steady up-to-down risk spillovers from the foreign exchange market to the US stock market and the instantaneous down-to-up risk spillovers become significant after the financial crisis. There are strong and steady down-to-up risk spillovers from the Japanese stock market to foreign exchange market before the global financial crisis. However the down-to-up risk spillovers become weaker

4.4. Subsample analysis of extreme risk spillovers It is well known that financial markets are highly affected by the crisis events. Therefore, possible structural change in the extreme risk spillovers should be considered. In order to study the influence of the global financial crisis, we study the extreme risk spillovers in two subsample periods. Following Du and He (2015), we choose the date September 15, 2008, when Lehman Brothers filed for Chapter 11 protection, as the break point of the recent global financial crisis. The first period is from January 1, 2005 to September 14, 2008, which is a normal period. The second period is from September 15, 2008 to September 11, 2012, which covers the global financial crisis and European sovereign debt crisis. For the sake of brevity, the descriptive statistics and estimation of GJR-GARCH models and VaR for the two subsample periods are presented in Appendix B. The results of extreme risk spillovers before and after the global financial crisis are presented in Tables 5 and 6, respectively. From the comparison of the subsample periods, we find that the global financial

Fig. 3. Down-to-up risk spillovers between the Japanese stock and gold markets. 7

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Fig. 4. Up-to-down risk spillovers between the Japanese stock and gold markets.

and will diminish gradually after the global financial crisis. In addition, the down-to-up risk spillovers from foreign exchange market to gold market change to be insignificant after the global financial crisis. From the analysis above, we can conclude that gold acts as a safe haven for US dollar before the global financial crisis, but is no longer a safe haven after the global financial crisis. Gold acts as a safe haven for the US and Japanese stock market before and after the global financial crisis. For the Chinese stock market, gold is not safe haven before the global financial crisis, but becomes a weak safe haven after the global financial crisis. The global financial crisis increase the ability of gold as a safe haven for the Japanese stock market. The US dollar acts an effective safe haven for the US and Japanese stock markets before the global financial crisis, and is a weak for the Japanese stock market and no long a safe haven for the US stock market after the global financial crisis. In addition, the US dollar is not a safe haven for the Chinese stock market before the global financial crisis, but becomes a safe haven after the global financial crisis.

5. Conclusion During the turmoil period of global financial market, gold is widely considered as a safer asset. In literature, different methods have been proposed to investigate the market risk. In this paper, we contribute to the literature by extending the studies of Hong et al. (2009) and Du and He (2015) to examine the interactions across gold, foreign exchange and stock markets from the perspective of extreme risk spillovers. First, the GJR-GARCH models based on skewed t distribution are used to estimate the time series of downside VaR and upside VaR for all markets. Then, two test statistics of oneway and two-way Granger causality in risk are constructed to test extreme risk spillover effects between gold and stock markets, between foreign exchange and stock markets and between gold and foreign exchange markets. The empirical results show that there are extreme down-to-up risk spillover effects from the US dollar and stock markets to gold market over the whole study sample, which indicates that investors can choose gold as safe haven when facing extreme downside risk in stock markets and US

Fig. 5. Down-to-Up risk spillovers between gold and foreign exchange markets. 8

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Fig. 6. Up-to-down risk spillovers between gold and foreign exchange markets. Table 5 Results of negative extreme risk spillovers before global financial crisis. Spillover Direction

Down-up risk spillover M=5

Stock and gold markets SH LGP −0.0372 SH LGP 0.9412 −0.9941 SH LGP SP500 LGP 0.0329 SP500 LGP 1.0134 SP500 LGP −0.9669 N225 LGP 1.2658 N225 LGP −0.7992 2.5893*** N225 LGP Stock and foreign exchange markets SH USD −0.3422 SH USD 0.4308 −0.9147 SH USD SP500 USD 5.5846*** SP500 USD −0.9774 8.8762*** SP500 USD N225 USD 1.9720** 2.9759*** N225 USD N225 USD −0.1879 Gold and foreign exchange markets LGP USD 2.3599*** LGP USD 3.7716*** −0.4317 LGP USD

Up-down risk spillover

M = 10

M = 20

M = 30

M=5

M = 10

M = 20

M = 30

−0.4914 0.4259 −1.1216 0.7404 2.1668** −1.1197 2.7101*** 1.2176 2.6150***

−0.4418 0.2616 −0.8863 1.1524 2.7821*** −1.1529 2.3881*** 1.2873* 2.0900**

−0.1881 0.3484 −0.6149 1.4892* 3.2576*** −1.1522 1.8827** 0.8482 1.8144**

−0.2538 −0.7694 0.4075 2.2654** 0.4492 2.7518*** 0.9627 1.5334* −0.1717

−0.2972 −0.6166 0.1927 1.8714** 0.3592 2.2852** 0.8235 1.2501 −0.0845

−0.3468 −0.6572 0.1624 2.9877*** 2.0300** 2.1940** 0.9487 0.9841 0.3590

−0.4467 −0.7860 0.1515 3.2692*** 2.1109** 2.5117*** 0.7137 0.5447 0.4664

−0.6973 0.2270 −1.2130 5.4274*** −0.8556 8.5372*** 2.8135*** 4.3495*** −0.3719

−0.5101 0.2285 −0.9495 4.4196*** −0.4178 6.6789*** 1.7061** 3.1744*** −0.7623

−0.2421 0.2761 −0.6189 3.9764*** −0.5616 6.2050*** 1.1125 2.5989*** −1.0271

−0.6603 −0.4662 −0.4668 1.6926** −0.6860 3.0778*** 0.0320 −0.0755 0.1218

−0.5140 −0.2249 −0.5031 0.6034 −0.9653 1.8185** 0.4015 0.1057 0.4642

0.1492 0.2937 −0.0852 −0.0853 −1.0860 0.9664 0.3770 0.1358 0.4003

0.3652 0.4658 0.0477 −0.2409 −1.0536 0.7157 0.1367 0.2732 −0.0756

1.8982** 2.2014** 0.4853

1.1435 1.3442* 0.2744

0.8595 0.8539 0.3639

0.8924 −0.9834 2.2498***

0.4933 −0.8614 1.5618**

−0.1511 −0.6937 0.4812

0.0171 −0.0444 0.0705

Note: the right critical values of Q1 and Q2 statistics at the 1%, 5% and 10% significance levels are 2.3263, 1.6449 and 1.2816, respectively.

dollar. However, when taking the influence of global financial crisis into consideration, the safe haven role of gold will change. In general, gold acts as a safe haven for US dollar before the global financial crisis, but is no longer a safe haven after the global financial crisis. Gold acts as a safe haven for the US and Japanese stock market before and after the global financial crisis. For the Chinese stock market, gold is not safe haven before the global financial crisis, but becomes a weak safe haven after the global financial crisis. The global financial crisis increase the ability of gold as a safe haven for the Japanese stock market. In addition, the down-to-up risk spillovers from the US stock market to gold market change to be up-to-down risk spillovers from the gold market to the US stock market. In addition, we also study the potential role of the US dollar as a safe haven for stock markets. The empirical results indicate that the US dollar acts only as a weak safe haven for the Chinese stock market over the full study period. The US dollar acts an effective safe haven for the US and Japanese stock markets before the global financial crisis, and is a weak for the Japanese stock market and no long a safe haven for the US stock market after the global financial crisis. In addition, the US dollar is not a

safe haven for the Chinese stock market before the global financial crisis, but becomes a safe haven after the global financial crisis. Our findings have important implications for risk managers and international investors when reallocate the portfolios between high-risk assets to low-risk assets during different market conditions. For further studies, more attention should be paid to construct portfolios to examine the effectiveness of the investment strategies implied by the extreme risk spillover effects. Competing interests The authors declare that they have no competing interests. Acknowledgements This research is supported by the projects of China Postdoctoral Science Foundation (No. 2018M643213, 2018M640830 and 2018M631939) and the National Social Science Foundation of China (NSSF 17CTQ030). 9

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Table 6 Results of negative extreme risk spillovers after global financial crisis. Spillover Direction

Down-up risk spillover M=5

Stock and gold markets SH LGP 1.0765 −0.1884 SH LGP SH LGP 1.7097** 0.2032 SP500 LGP SP500 LGP −0.9166 SP500 LGP 1.1941 N225 LGP 5.8817*** N225 LGP 4.3160*** 4.0011*** N225 LGP Stock and foreign exchange markets SH USD 2.1613** 3.7346*** SH USD SH USD −0.6780 0.3107 SP500 USD SP500 USD −0.4827 0.9225 SP500 USD N225 USD 0.7649 N225 USD 1.6489** −0.5672 N225 USD Gold and foreign exchange markets LGP USD −0.6246 LGP USD −0.9259 0.0428 LGP USD

Up-down risk spillover

M = 10

M = 20

M = 30

M=5

M = 10

M = 20

M = 30

1.5960* 0.6219 1.6341* 1.1983 −1.1914 2.8662*** 5.4897*** 2.7121*** 5.0489***

1.6712** 1.2555 1.1058 1.8741** −0.0464 2.6675*** 5.0519*** 2.8319*** 4.3092***

1.6037* 1.2866* 0.9780 1.7692** 0.1303 2.3381*** 4.5124*** 2.8242*** 3.5541***

−0.0551 −0.4268 0.3469 1.1553 1.1008 0.5336 −0.2599 0.2219 −0.5906

0.2472 −0.2480 0.5961 2.5946*** 3.8448*** −0.1752 0.0370 0.8900 −0.8389

0.1600 −0.4465 0.6727 2.9757*** 4.3806*** −0.1731 −0.2611 0.6564 −1.0255

0.0840 −0.5854 0.7072 2.6264*** 3.8021*** −0.0861 −0.0925 0.4815 −0.6115

0.9252 2.1318** −0.8234 0.3830 0.3117 0.2316 −0.1743 0.7936 −1.0402

0.8080 1.1485 −0.0065 0.0217 0.5041 −0.4736 0.1165 0.8779 −0.7134

0.7991 1.1034 0.0261 0.3555 0.9372 −0.4342 −0.0487 0.5321 −0.6012

0.2824 −0.2832 0.6833 −0.6453 −0.8408 −0.0714 −0.6197 −0.1043 −0.7721

0.1842 −0.3090 0.5680 0.3050 −1.0083 1.4397* −0.7067 −0.3610 −0.6385

−0.3076 −0.4577 0.0183 1.6874** −0.8649 3.2491*** −1.1300 −0.6130 −0.9852

−0.3818 −0.2285 −0.3160 1.2391 −1.2496 3.0007*** −0.9327 −0.5721 −0.7470

−0.8541 −1.1244 −0.0845

−0.8436 −1.3864 0.1918

−1.0123 −1.5413 0.1073

−0.6862 −0.0216 −0.9502

−1.2320 −0.6146 −1.1278

−1.4742 −0.9343 −1.1530

−1.5220 −0.9307 −1.2262

Note: the right critical values of Q1 and Q2 statistics at the 1%, 5% and 10% significance levels are 2.3263, 1.6449 and 1.2816, respectively.

Appendix A. VaR estimation based on different distributions See Tables A1 and A2.

Table A1 VaR estimation based on Student’s t distribution.

SH SP500 N225 LGP USD

Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR

Mean

Std.dev.

Failure time

Failure rate

LR statistics

3.1676 3.2786 2.0477 2.1832 2.6990 2.8067 2.2930 2.3991 0.9665 0.9628

1.4100 1.4017 1.4705 1.5228 1.4228 1.4359 0.7762 0.7749 0.3119 0.3105

87 54 110 44 104 52 76 66 97 72

0.0309 0.0191 0.0390 0.0156 0.0369 0.0184 0.0270 0.0234 0.0344 0.0255

25.0650 73.1394 7.7358 94.9883 11.1999 77.1828 37.6244 51.8785 16.1532 42.9806

Note: the critical values of LR statistics at the 1%, 5% and 10% significance levels are 6.635, 3.841 and 2.706, respectively.

Table A2 VaR estimation based on GED.

SH SP500 N225 LGP USD

Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR

Mean

Std.dev.

Failure time

Failure rate

LR statistics

2.6535 2.6702 1.8523 1.7794 2.5468 2.5051 2.0165 2.0419 0.9019 0.8789

1.1290 1.1236 1.2168 1.2610 1.3157 1.3179 0.6918 0.6859 0.2754 0.2753

128 104 136 98 123 93 114 104 111 105

0.0454 0.0369 0.0482 0.0348 0.0436 0.0330 0.0404 0.0369 0.0394 0.0372

1.3002 11.1999 0.1888 15.3832 2.5231 19.4491 5.8072 11.1999 7.2257 10.5737

Note: the critical values of LR statistics at the 1%, 5% and 10% significance levels are 6.635, 3.841 and 2.706, respectively.

10

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Appendix B. Subsample analysis of extreme risk spillovers Table B1 presents the descriptive statistics of different markets. From a comparison of the two subsample periods, we find that except for Chinese stock market, market returns become more volatile after the financial crisis in 2008. In addition, except for gold market, markets has lower mean returns after the financial crisis. Tables B2–B5 present the estimation results of GJR-GARCH models and VaR for the two subsample periods. The mean values of downside and upside VaR in Tables B3 and B5 indicate that except for Chinese stock market, markets become more risky after the financial crisis. The failure rates and LR statistics indicate that the GJR-GARCH models based on skewed t distribution can estimate the VaR efficiently for the two periods.

Table B1 Descriptive statistics for subsample period. Mean

Std.dev.

Panel A: Before the crisis (January 1, 2005-September, 14, 2008) SH 0.0317 1.9844 SP500 0.0004 0.9530 N225 0.0058 1.3213 LGP 0.0751 1.2735 USD −0.0176 0.4605 Panel B: After the crisis (September, 15, 2008-September 11, 2012) SH 0.0009 1.6382 SP500 −0.0135 1.7865 N225 −0.0429 1.9714 LGP 0.0828 1.3626 USD −0.0013 0.6760

Skewness

Kurtosis

J-B

LB-Q(20)

−0.4490 −0.2256 −0.2317 −0.6013 0.0199

5.9248 5.0219 5.1010 5.2766 4.2567

325.6727*** 149.3103*** 161.0479*** 230.6511*** 55.0059***

29.5330* 33.6573** 18.3605 16.3550 15.8298

−0.0712 −0.8860 −0.7954 0.1881 0.4945

6.2982 11.5245 12.7546 9.3496 10.3282

402.3413*** 2798.5155*** 3606.1477*** 1493.6341*** 2018.6458***

24.3634 44.0620*** 33.9218** 33.8671** 33.7494**

Note: This table provides the summary statistics of the returns in various markets. SH is the Shanghai Composite Index, SP500 is the S&P 500 Index, N225 is the Nikkei 225 Index, LGP is the London Gold Price and USD is the US Dollar Index. J–B is the Jarque–Bera test for the null hypothesis of Gaussian distribution and LB-Q (20) is the Ljung–Box Q test of serial correlation of up to 30 lags in the returns. * indicates statistical significance at the 10% level. ** indicates statistical significance at the 5% level and ***indicates statistical significance at the 1% level.

Table B2 Estimation results of GJR-GARCH model before global financial crisis.

Mean equation c 1

2 3

Variance equation 0

1 1

SH

SP500

N225

LGP

USD

0.0829 −0.0094 0.0168 0.0924***

0.0474 −0.1119*** −0.0982*** −0.0327

0.0364 −0.0678 0.0250 −0.0060

0.0882 −0.0639 0.0167 −0.0001

−0.0285** −0.0389 0.0550* −0.0286

0.0465 0.0625*** 0.9231***

0.0052 0.0596*** 0.9375***

0.0387** 0.0344 0.8861***

0.0083 0.0778*** 0.9482***

0.0006 0.0197** 0.9734***

0.0167 −0.0905** 4.6288***

0.1132** −0.0855 10.676*

— −0.1264*** 7.4174***

−0.0521* −0.1164** 8.3890**

0.0095 −0.0408 11.579***

Note: Parameters are estimated by the GJR-GARCH model with skewed t distribution, while the parameters for SP500 are estimated by the GARCH model with skewed t distribution. * indicates statistical significance at the 10% level. ** indicates statistical significance at the 5% level. *** indicates statistical significance at the 1% level.

Table B3 VaR estimation based on skewed t distribution before global financial crisis.

SH SP500 N225 LGP USD

Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR

Mean

Std.dev.

Failure time

Failure rate

LR statistics

3.0551 2.9790 1.4667 1.4149 2.0808 2.0203 1.9971 1.9839 0.7739 0.6950

1.0591 0.9395 0.5362 0.5138 0.7670 0.7350 0.5941 0.5414 0.1304 0.1264

48 36 44 40 45 43 46 45 46 47

0.0575 0.0431 0.0527 0.0479 0.0539 0.0515 0.0551 0.0539 0.0551 0.0563

0.9415 0.8726 0.1255 0.0783 0.2600 0.0390 0.4415 0.2600 0.4415 0.6689

Note: the critical values of LR statistics at the 1%, 5% and 10% significance levels are 6.6350, 3.8410 and 2.7060, respectively. 11

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Table B4 Estimation results of GJR-GARCH model after global financial crisis.

Mean equation c 1

2 3

Variance equation 0

1 1

SH

SP500

N225

LGP

USD

−0.0277 −0.0310 0.0036 0.0220

0.0671* −0.0522 −0.0408 −0.0108

0.0059 −0.0445 0.0295 0.0046

0.1151*** −0.0136 −0.0044 −0.0580*

−0.0038 −0.0175 −0.0017 −0.0364

0.0138 0.0382** 0.9571***

0.0287*** 0.0000 0.8931***

0.0953*** 0.0163 0.8644***

0.0155 0.0754*** 0.9383***

0.0040 0.0251** 0.9630***

0.1781*** −0.1282** 5.9068***

−0.0022 −0.00815** 5.5635***

0.1518*** −0.1100** 11.3450***

−0.0393 −0.0344 4.7637***

0.0016 0.0215 7.1515

Note: Parameters are estimated by the GJR-GARCH model with skewed t distribution. * indicates statistical significance at the 10% level. ** indicates statistical significance at the 5% level. *** indicates statistical significance at the 1% level. Table B5 VaR estimation based on skewed t distribution after global financial crisis.

SH SP500 N225 LGP USD

Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR Downside VaR Upside VaR

Mean

Std.dev.

Failure time

Failure rate

LR statistics

2.5639 2.3348 2.4774 2.3500 2.7861 2.5772 1.9002 2.0568 1.0184 1.0288

0.8036 0.7478 1.7638 1.6534 1.6641 1.5388 0.6414 0.6244 0.2456 0.2473

43 42 53 46 43 44 49 42 44 44

0.0485 0.0474 0.0598 0.0519 0.0485 0.0497 0.0553 0.0474 0.0497 0.0497

0.0405 0.1278 1.6968 0.0679 0.0405 0.0021 0.5082 0.1278 0.0021 0.0021

Note: the critical values of LR statistics at the 1%, 5% and 10% significance levels are 6.6350, 3.8410 and 2.7060, respectively.

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