Asymmetric extreme risk spillovers between the Chinese stock market and index futures market: An MV-CAViaR based intraday CoVaR approach

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Asymmetric extreme risk spillovers between the Chinese stock market and index futures market: An MV-CAViaR based intraday CoVaR approach ⁎

Zhihong Jian , Shuai Wu, Zhican Zhu School of Economics, Huazhong University of Science and Technology, Wuhan, China

A R T IC LE I N F O

ABS TRA CT

JEL classification: C14 C32 G10 G1

This paper proposes a predictive CoVaR measure to analyze asynchronous risk spillovers between the Chinese stock and futures market. We jointly model the intraday CoVaR dynamics using an extended MV-CAViaR model. The results show the presence of asymmetric spillovers under different market states, different trading rules, and different confidence levels. Specifically, there exist significant downside spillovers and insignificant upside spillovers. Moreover, the futures (stock) market becomes dominant in risk transmission during bearish (bullish) market periods. Furthermore, high margin requirements would weaken the spillover effects of the futures market, but it would also strengthen the spillover effects of the stock market.

Keywords: Intraday CoVaR MV-CAViaR Asymmetric risk spillovers Spot stock and index futures markets

1. Introduction In recent years, the Chinese stock index futures market has developed rapidly and attracted much attention from both market participants and academic researchers. However, it remains unclear whether index futures trading is beneficial to the stability of the entire financial system, and the futures market is blamed for exacerbating the Chinese financial market turbulence in 2015–16, during which extreme events always occurred in the stock market and index futures market simultaneously or successively. In general, extreme price movements occurring in one market may impose an externality on the other related market, and an underestimation of the intrinsic relationship between the stock and futures market may result in severe consequences, such as substantial capital losses and even financial markets distress. Therefore, a thorough understanding of the extreme risk spillovers between the spot and futures market is critical for important financial market activities such as asset allocation, risk management and the implementation of regulatory policies. The interactions between index futures markets and stock markets have been widely discussed in the literature since the first introduction of index futures trading in the world. Early studies mainly focused on the lead-lag relationship (i.e., price discovery effect) between index futures and stock index returns, on the grounds that the futures market may incorporate information more efficiently than the stock market due to its inherent leverage, low transaction costs and lack of short sell restrictions (Tse, 1999; Koutmos and Tucker, 1996). However, the empirical results of the lead-lag effect are mixed and diverse. For example, Ghosh (1993), Kavussanos et al. (2008) and Theissen (2012) find that the futures market is the leader in price discovery. In contrast, Yang et al. (2012), Judge and Reancharoen (2014) and Chen and Gau (2009) show the leading role of stock markets. Recent papers pay much

⁎

Corresponding author at: School of Economics, Huazhong University of Science and Technology, 1037 Luoyu Road, Wuhan, Hubei, China. E-mail addresses: [email protected] (Z. Jian), [email protected] (S. Wu), [email protected] (Z. Zhu).

https://doi.org/10.1016/j.ememar.2018.06.001 Received 7 December 2017; Received in revised form 2 May 2018; Accepted 28 June 2018 1566-0141/ © 2018 Elsevier B.V. All rights reserved.

Please cite this article as: Jian, Z., Emerging Markets Review (2018), https://doi.org/10.1016/j.ememar.2018.06.001

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more attention to the volatility spillovers between the two markets. Drimbetas et al. (2007), Kasman and Kasman (2008) and Zhou et al. (2014) present evidence that the stock index futures market would stabilize the underlying stock market, but Zhong et al. (2004), Change et al. (1999) and Bae et al. (2004) blame the futures market as it is a source of instability for the stock market. The analyses of the lead-lag and volatility spillover effects provide insights into the relationship between the stock and futures markets, but they fail to disclose all the features of risk dependence. The extreme risk spillover effect is another important aspect that has not been fully analyzed in the extant literature. In particular, fewer studies have been conducted to analyze the intraday extreme risk spillover effect which is non-negligible for the Chinese emerging financial markets, because the price movements in these markets are more volatile and the history of index futures transactions is relatively short. Methodologically, various econometric models have been proposed to analyze the risk spillover effect between different financial markets. Some of them measure the return spillover effect, i.e. the interdependence of asset returns. For example, Angkinand et al. (2009) utilize a structural vector autoregressive (SVAR) framework to explore spillovers from the US financial crisis to many developed economies. Patton (2006) develops a conditional copulas method to examine the asymmetric and time-varying correlation between currency exchange rates. Diebold and Yilmaz (2009) construct a spillover index which is based on the variance decomposition analysis of the VAR model. Another portion of theoretical models focuses on the volatility spillover effect. For instance, Wagner and Szimayer (2004) construct a mean-reverting jump diffusion model for the implied volatility indices and analyze the volatility spillover effect using synchronous jump events. Baele (2005) develops a regime-switching spillover intensity model to better quantify the magnitude and time-varying nature of volatility spillovers. Asgharian and Nossman (2011) develop a stochastic volatility model with correlated jumps to simultaneously investigate the volatility spillover and jump spillover among markets. Given the time series of realized volatilities, Bubák et al. (2011) employ a vector heterogeneous autoregressive (VHAR) model to study both the short-term and long-term volatility transmissions. Meanwhile, the topic of extreme risk spillover and tail dependence has also been studied under different model frameworks. For example, Hong et al. (2009) introduce a new concept of Granger causality in VaR and propose a class of kernel-based tests to detect the extreme risk spillovers that occur with a time lag. Aboura and Wagner (2016) apply a bivariate EVT approach to characterize the extreme asymmetric volatility effect and find contemporaneous volatility-return tail dependence for market crashes. White et al. (2015) construct a multivariate, multi-quantile model framework to directly analyze spillovers in VaR between different financial assets. In this paper, our analysis of bidirectional and asynchronous extreme risk spillovers is based on the conditional Value-at-Risk measure (CoVaR), which is proposed by Adrian and Brunnermeier (2016) and has been widely used in the analyses of systemic risk (e.g., Castro and Ferrari, 2014; Girardi and Ergün, 2013; López-Espinosa et al., 2012). Specifically, CoVaRi∣j is defined as the VaR of market i conditional on the fact that market j is under some specific financial distress events, and the tail dependence is captured by ΔCoVaR, which stands for the difference between the CoVaR conditional on an extreme event and the CoVaR conditional on normal scenarios. In the existing literature, there are two types of CoVaR definitions and corresponding estimation methods. The first defines the conditional event of CoVaRi∣j as the return of market j being exactly at its VaR (Rj = VaRj). Under the assumption that VaR of market i is influenced by the contemporaneous return of market j, Adrian and Brunnermeier (2016) construct a quantile regression j model to estimate CoVaRi∣R =VaRj. However, this semiparametric model fails to consider bidirectional risk interactions between two assets simultaneously. The second defines the conditional event of CoVaRi∣j as the return of market j being at most at its VaR (Rj ≤ VaRj). Existing studies adopt copulas (e.g. Shahzad et al., 2018; Reboredo and Ugolini, 2015a, 2015b, 2015c; Mensi et al., 2017a, 2017b; Karimalis and Nomikos, 2014; Reboredo et al., 2016) or multivariate GARCH type models (e.g., Yu et al., 2018; j Girardi and Ergün, 2013) to estimate the joint distribution of multiple markets and then calculate the CoVaRi∣R ≤VaRj of interest. However, those parametric methods could only estimate the contemporaneous spillover effects and are unable to examine the lagged spillovers. To characterize the asynchronous extreme risk spillovers, the abovementioned definitions and models of CoVaR need to be revised. For this purpose, we propose a predictive CoVaR measure to examine the asynchronous extreme risk spillover effects between the Chinese stock market and index futures market. Moreover, we address the one-period CoVaR prediction1 under the MV-CAViaR model framework (White et al., 2015). Based on intraday high-frequency data, we extend the MV-CAViaR model to directly and jointly estimate the bidirectional spillover effects captured by intraday ΔCoVaR measures. In this paper, our definition and usage of CoVaR are different from Adrian and Brunnermeier (2016) in two aspects. First, we assume that the conditioning financial distress event refers to the return of market j being at most at its VaR (Rj ≤ VaRj) as opposed to being exactly at its VaR (Rj = VaRj), this modification allows us to consider more severe distress events and improve its consistency with respect to the dependence parameter (Girardi and Ergün, 2013; Mainik and Schaanning, 2014). Second, to further consider the potential lagged effect of extreme risk spillovers, we define CoVaR as the VaR conditional on a past extreme event and concentrate on the predictive power of extreme events. To estimate the asynchronous intraday spillover effects, we modify the MV-CAViaR model by additionally introducing a lagged dummy variable 1{Rj≤VaRj} into the VaR dynamics. The extended model enables us to jointly estimate the ΔCoVaR of each market. In contrast to the multivariate parametric technique, the advantage of our multivariate CoVaR framework lies in three aspects. First, our method imposes no distributional assumptions on asset returns, thus could avoid the potential problem of model misspecification.

1 The multiple-period CoVaR prediction is often related to a time-scaling approach and thus could be more complex (see e.g., Kinateder and Wagner, 2014). We do not discuss this topic here and leave it for further studies.

2

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Second, our model allows researchers to directly quantify the extreme spillover effects between different markets, rather than recovering it indirectly via models of time-varying first and second moments. Finally, our method could further consider lagged spillover effects and provide early-warning indices for the prudential regulation policy. In our empirical analysis of extreme risk spillovers between the Chinese stock market and index futures market, we apply the proposed multivariate CoVaR model to study asymmetric spillover effects according to the following three considerations. First, the spillover effects during different market periods (bullish period, bearish period and normal period) may be significantly differed due to the change in investor sentiment and speculation activities. For example, the majority of investors in the Chinese financial markets are retail investors, and they are usually eager to buy when the market goes up and rush to sell when the market goes down. We select a sample period from April 16, 2015 to May 31, 2017, which contains the period of the recent market turbulence in China. We separately run the abovementioned CoVaR model using different subsample data and show that there exist bidirectional spillover effects between the two markets during normal periods, but the futures market has a unidirectional (and larger) spillover effect on the stock market during bearish periods, while the stock market has a unidirectional (and larger) spillover effect on the futures market during bullish periods. These results indicate that the information transmission mechanism between the two markets is bidirectional during normal periods, because a sudden price drop occurring in one market would increase the extreme risk of the other market. But this mechanism would be changed during extreme market periods, the stock market becomes more informative and influences the futures price movements when the market prices move upward, and the futures market becomes more informative and influences the stock price movements when the market prices move downward. One possible reason for this phenomenon is that informed investors prefer to trade in the stock market during bullish periods and trade in the futures market during bearish periods. Second, the characteristics of tail dependence may be altered due to changes in the margin requirement of index futures transactions. A tightened margin requirement puts restrictions on the market investors such as hedge funds (Adams et al., 2014), thus, it may influence asset price movements, market volatility (Wang, 2015) and even the risk spillover mechanism. We separately test the extreme risk spillover during the high margin requirement period (40%) and the low margin requirement period (20%). Our regression results indicate that the high margin requirement policy has various impacts on the extreme risk spillover mechanism. While it weakens the spillover effect from futures returns to index returns as expected, it also enhances the spillover effect from index returns to futures returns. A tightened margin requirement makes the futures market to be more likely affected by extreme price movements occurring in the stock market. These results have profound policy implications for regulators to set up a long-term effective market stabilization mechanism, because they reflect the benefits and costs of different trading rules from the perspective of extreme risk spillovers. After the increase in the margin ratio requirement of index futures transactions, the impact of extreme movements occurring in the futures market on the stock market decreases. The weakened futures shocks are beneficial to the stability of the stock market and meet the purpose of market regulators. However, the futures market becomes unstable and more vulnerable to the shocks in the stock market when the margin ratio requirement is high. The negative consequences of a high margin requirement may be related to the deterioration of market liquidity. Finally, the downside and upside risk spillovers could be markedly different due to several market factors such as asymmetric investor risk appetite, short-selling constraints and so on. In examining the asymmetries, we find consistent evidence of asymmetric downside and upside spillovers, i.e., a large and significant downside spillover effect and a small and insignificant upside spillover effect. This result indicates that among the Chinese stock market and index futures market, it is likely to observe extreme downward price movements simultaneously and observe extreme upward price movements separately. Therefore, market regulators should take into account the extreme downward price comovements between the spot and futures markets to develop efficient stabilization strategies. In summary, we draw comprehensive and instructive conclusions regarding the extreme risk spillover effects between the Chinese stock and futures market. Although spillovers are bidirectionally significant during a normal market period, changes in market state or margin requirement policy would distort and change the spillover mechanism. Therefore, our analysis not only has practical implications for market investors (particularly hedging investors) to reduce capital losses due to extreme price movements but also provides insightful instruction for market regulators to stabilize the entire financial system. Our contributions are twofold. First, we propose a predictive CoVaR measure to analyze the asynchronous spillover effects and construct a semiparametric multivariate CoVaR model to directly measure the intraday tail dependence between different markets. Based on the MV-CAViaR model proposed by White et al. (2015), this paper additionally considers the impacts of past extreme market events. In contrast to the existing parametric methods of CoVaR (e.g., copula, multivariate GARCH), our method imposes no assumption on the distributions and could directly measure the extreme spillover effects. Second, with regard to financial market interdependencies, our empirical analysis provides new evidence gleaned from asymmetric extreme risk linkages between the stock market and index futures market. To the best of our knowledge, it is the first to evaluate the asymmetric extreme risk spillovers between the spot stock and index futures market on the basis of intraday CoVaR measure. The remainder of this paper is organized as follows. Section 2 defines the CoVaR measure and presents the model framework. Sections 3 and 4 present the empirical data and regression results. Section 5 draws our conclusions. 2. Methodology Given the return Rti of an asset i at time t and the pre-specified confidence level q, we define the Value-at-Risk VaRq, 3

t

i

as the q-

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quantile of the return distribution2:

Pr(Rti ≤ VaRqi , t ) = q

(1)

For q < 0.5, VaR is typically lower than 0 and stands for the downside tail risk, a smaller value of VaR corresponds to greater downside extreme risk. For q > 0.5, VaR is typically larger than 0 and stands for the upside tail risk, a larger value of VaR indicates greater upside extreme risk. 2.1. Definition of CoVaR and ΔCoVaR We characterize the extreme risk spillover effects between stock index prices and index futures prices using the CoVaR measure because it provides information on the tail dependence of the joint distribution. Extended from the unconditional VaR to the conditional VaR, CoVaRq, ti∣j is defined as the VaR of asset i at time t for a given probability level q, conditional on some extreme event of asset j. In the original literature of CoVaR, Adrian and Brunnermeier (2016) define the extreme event as asset j being in financial distress, i.e., its return being exactly at its VaR:

Pr (Rti ≤ CoVaRqi , tj | Rt j = VaRqj, t ) = q

(2)

The definition of an extreme event is unpractical because for a continuous return distribution, the probability of the return being exactly equivalent to its VaRq is zero, which means that the extreme event would never be observed in a financial market. To consider more severe financial distress events and improve its consistency with respect to the dependence parameter, recent studies (e.g., Girardi and Ergün, 2013; Mainik and Schaanning, 2014; Reboredo et al., 2016) modify the definition of CoVaR and define the conditional distress event as the return of asset j is at most at its VaR3:

Pr (Rti ≤ CoVaRqi , tj | Rt j ≤ VaRqj, t ) = q

(3)

The changes in the definition of financial distress events convert the estimation of CoVaR into the estimation of a joint cumulative distribution function. It is more feasible to estimate and backtest the revised CoVaR compared to the original CoVaR proposed by Adrian and Brunnermeier (2016). Nevertheless, both types of CoVaR merely measure the contemporaneous tail dependence of different asset returns. The studies of the lead-lag effect between the spot and futures markets show that different market may react to information shocks asynchronously, leading to different asset price movements. As a result, the lead-lag effect may be shown in both price discovery and risk linkage. Thus, it is necessary to take into account the asynchronous spillover effects. In our study, we further modify the definition of distress events and concentrate on the predictive power of extreme events occurring in the other market, our CoVaR measure is defined as follows:

Pr (Rti ≤ CoVaRqi , tj | Rt −j 1 ≤ VaRqj, t − 1) = q

(4)

≤ VaRq, t−1 is more reasonable because in real financial Compared to the CoVaR definition in Eq. (3), conditional on markets with imperfect information, when an extreme event occurring in one market is observed by investors (such as portfolio investors and hedging investors), it needs time for them to analyze the event and then take relevant actions in the other market. Similarly, we can measure (upside) CoVaR conditional on an extreme upward event occurring in the other market as follows: Rt−1j

Pr (Rti ≥ CoVaRqi , tj | Rt −j 1 ≥ VaRqj, t − 1) = 1 − q

j

(5)

The CoVaR measure alone cannot reveal the extreme risk spillover effect between different markets because it only reflects the risk level of one market under some specified conditions. We need to construct a benchmark extreme risk variable, and the difference between CoVaR and the benchmark (i.e., ΔCoVaR) could reveal the spillover effect of interest. Previous studies also provide several choices about the benchmark. For example, Reboredo et al. (2016) directly choose the unconditional VaR as a benchmark and use the KS bootstrapping test proposed by Abadie (2002) to compare the difference between CoVaR and VaR dynamics. Adrian and Brunnermeier (2016) choose the CoVaR conditional on a median state as the benchmark, i.e., j CoVaRqi∣R =VaR50j. In our analysis, we pay special attention to the occurrence of extreme events, so we define the benchmark j variable as the CoVaR conditional that the extreme event did not occur, i.e., CoVaRqi∣R > VaRqj. Thus, the extreme risk spillover effect of a distress event in asset j is given by i R j ≤ VaRqj

ΔCoVaR = CoVaRq, t

i R j > VaRqj

− CoVaRq, t

(6)

The value of ΔCoVaR reveals the change in VaR of asset i due to a distress event occurring in asset j. A larger absolute size of In general, VaR is defined as the maximum loss of a given portfolio with a probability level q, i.e., the 1 − q quantile of the loss function. Thus, the general version of VaR at the probability level q is equivalent to the negative value of the q-quantile of asset return. To keep our analysis briefly and straightforward, we follow Reboredo et al. (2016) and Girardi and Ergün (2013) and directly define VaR as the q-quantile of asset return, the difference in VaR definition does not affect our empirical results. 3 The CoVaR risk measurement approach has the same setting as bivariate extreme value statistics, which were initially applied in finance by Longin and Solnik (2001) and Marsh and Wagner (2000), because both of them depend on the bivariate distribution of return exceedances. 2

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ΔCoVaR indicates a larger extreme risk spillover effect from asset j to asset i. 2.2. Empirical model and estimation method Once we modify the definition of CoVaR, the abovementioned econometric methods are not suitable for estimating the CoVaR dynamics and asynchronous spillover effects. On the one hand, parametric methods such as copulas and bivariate GARCH type models could depict the features of joint distribution and estimate the CoVaR conditional on contemporaneous events Rtj ≤ VaRq, tj. However, the predictive CoVaR is conditional on past events Rt−1j ≤ VaRq, t−1j and cannot be derived from the joint distribution. On the other hand, semiparametric methods such as quantile regression and MV-CAViaR could estimate the CoVaR conditional on past events Rt−1j = VaRq, t−1j, because these models take into account the predictive power of lagged returns. However, semiparametric methods fail to estimate the predictive CoVaR conditional on Rt−1j ≤ VaRq, t−1j, which represents a more severe financial distress event and is more consistent with the dependence parameter compared to Rt−1j = VaRq, t−1j (see Girardi and Ergün, 2013; Mainik and Schaanning, 2014). To overcome these limitations of existing models, we extend the MV-CAViaR model by additionally considering the predictive power of distress events Rt−1i ≤ VaRq, t−1i and Rt−1j ≤ VaRq, t−1j. The modified model could directly characterize the bidirectional asynchronous spillover effects captured by ΔCoVaR. Engle and Manganelli (2004) assume that the quantile of a single return is autocorrelated and affected by past returns. Thus, they propose a conditional autoregressive quantile model (CAViaR model). To further consider the quantile interactions of different returns, White et al. (2015) construct a vector autoregressive (VAR) extension to the quantile models (MV-CAViaR model) as follows:

⎧ q1, t = α1,0 + α1,1 |R1, t − 1| + α1,2 |R2, t − 1| + α1,3q1, t − 1 + α1,4 q2, t − 1 ⎨ q2, t = α2,0 + α2,1 |R1, t − 1| + α2,2 |R2, t − 1| + α2,3q1, t − 1 + α2,4 q2, t − 1 ⎩

(7)

where qi, t stands for the conditional quantile of asset i at time t, and Ri, t is the return of asset i at time t. Eq. (7) assumes that for each quantile, it is not only affected by its past return and quantile values but also may be affected by the quantile and return of another asset due to spillover effects. However, the MV-CAViaR model merely considers the dependence of quantile variables and fails to consider the influence of conditional distress events. To further model the spillover effects captured by ΔCoVaR, we extend the MV-CAViaR model by introducing the dummy variable of distress events into quantile dynamics. The model specification is as follows:

⎧ q1, t = α1,0 + α1,1 |R1, t − 1| + α1,2 |R2, t − 1| + α1,3q1, t − 1 + α1,4 q2, t − 1 + α1,51(R1, t − 1 ≤ q1, t − 1) ⎪ + α1,61(R2, t − 1 ≤ q 2, t − 1) ⎨ q2, t = α2,0 + α2,1 |R1, t − 1| + α2,2 |R2, t − 1| + α2,3q1, t − 1 + α2,4 q2, t − 1 + α2,51(R1, t − 1 ≤ q1, t − 1) ⎪ + α2,61(R2, t − 1 ≤ q2, t − 1) ⎩

(8)

where I{·} is a dummy variable which equals 1 when an extreme event {·} occurs. Consequently, our model specifies that the quantile dynamic (i.e., the VaR dynamic) is not only influenced by past returns and lagged quantile values but also affected by extreme risk events occurring in each market. As we can see from the bivariate quantile model, extreme risk interactions may exist in various forms. For example, the coefficients α1, 2 and α2, 1 reveal the impact of one market's price movement on the other market's VaR. Moreover, α1, 4 and α2, 3 indicate the potential interactions between two VaR variables. Most importantly, we pay special attention to the coefficients α1, 6 and α2, 5, which stand for the difference between the VaR conditional on the occurrence of an extreme event and the VaR conditional on the extreme event did not occur. As a result, they could be considered as a measure of ΔCoVaR, i.e., the changes in extreme risk due to the spillovers of extreme events occurring in the other market. In our research, we estimate the intraday extreme risk spillovers using high frequency data, thus it is necessary to take into account the strong intraday trading pattern of market investors. Previous studies find that it may result in intraday periodic patterns of volatility (Engle and Sokalska, 2012) and trading volume (Bollerslev et al., 2016). We also detect significant intraday VaR pattern in the Chinese spot market and futures market. Our estimation procedures are as follows. STEP 1: Estimate the intraday VaR component. We model the VaR of each intraday interval k of asset i using univariate CAViaR model (Engle and Manganelli, 2004):

VaRti, k = β0 + β1VaRti− 1, k + β2 |Rti− 1, k |

(9)

i

Rt, ki

is the logarithmic return of asset i at the kth where VaRt, k stands for the VaR of asset i at the kth intraday interval of day t. intraday interval of day t. Given the estimated dynamic of VaRt, ki, we calculate the average VaR of each intraday interval k:

VaRki =

1 T

T

∑t=1

VaRti, k

(10)

where T is the number of trading days in our estimation sample. The periodic VaR component of each intraday interval is the relative size of its average VaR compared to the average VaR of the total intervals:

1 VaRFactorki = VaRki / ⎛ ⎝N

N

∑k =1 VaRki ⎞

(11)

⎠ 5

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where N is the number of intraday intervals of each trading day. We assume that for each equidistance intraday interval, the expected VaR should be the same and VaRFactorki should equals to 1 if the intraday periodic effect does not exist. A larger size of VaRFactorki indicates a larger absolute value of the average VaR at the kth intraday interval. STEP 2: White et al. (2015) propose a quasi-maximum likelihood (QML) method to estimate the parameters vector β of the MVCAViaR model, the optimization problem is as follows:

min β

1 M

M

n

∑ ⎧⎨∑ ρθ (Rit − VaRit (∙, β )) ⎫⎬ t=1

⎩ i=1

(12)

⎭

where ρθ(e) = e(θ − 1{e≤0}) is the standard check function. M is the total sample size, n is the number of assets, and 1{∙} is an indicator variable. Considering the intraday periodic effect of VaR, we modify the estimation problem as follows:

min β

1 M

M

n

∑ ⎧⎨∑ ρθ (Rit − VaRit (∙, β ) ∙VaRFactorti) ⎫⎬ t=1

⎩ i=1

(13)

⎭

We note that the quasi-maximum likelihood estimator does not impose any distributional assumption on the return distribution. Moreover, we specify the total VaR to be a multiplicative product of the periodic intraday component (VaRFactorti) and the stochastic intraday component (VaRit).4 3. Data 3.1. Preliminary analysis We empirically study the extreme risk spillover effects between the Chinese stock market and index futures market using a set of indices (CSI 300, CSI 500 and SSE 50) and their corresponding dominant contracts of index futures. Our high frequency data (fiveminute frequency) are derived from the Wind Database and cover the period from April 16, 2015 to May 31, 2017. Fig. 1 shows the overall market movement in our sample period using the daily data of CS I 300 index price and CSI 300 futures price.5 It can be seen that market prices move upward at the beginning (April 16, 2015 to June 12, 2015), followed by a market crash (June 13, 2015 to February 1, 2016). After the market-wide financial turbulence, the market prices become comparatively normal and stable (February 2, 2016 to May 31, 2017). The index and futures price movements are highly correlated at daily observation frequency. The market sentiment is remarkably different during different market periods, so it may lead to possible structural changes in extreme risk spillovers between the two markets. For example, during the bullish period, majority investors are optimistic about the market future and eager to buy rather than sell. However, during the bearish period, sudden collapses in market prices may frighten investors (particularly retail investors) and result in panic selling behaviors. Several studies (e.g. Du and He, 2015; Adams et al., 2014) also stress the necessity of taking into account the market state when analyzing risk spillovers. Table 1 gives descriptive statistics of the 5-minute frequency return series of stock indices and their corresponding dominant futures contracts. Different from daily price movements, intraday index and futures price movements may be inconsistent. For example, average futures returns become negative during the bearish period, while average index returns remain above zero. For both stock index and futures returns, the standard deviation during the stable period is lower than half of its value during the bullish and bearish period, which means that both markets became more stable after the recent financial turbulence in China. Generally, the kurtosis is greater than three, which indicates the existence of heavy tails of return distributions. Moreover, the Jarque-Bera tests also show that all the return series are non-normal, thus stress the importance of tail characteristics (e.g. extreme risk) of intraday asset return distributions. Finally, the market state also has an influence on the serial correlation relationship, both the stock index and index futures returns are significantly serial correlated during normal time, while there are no (or weak) serial correlations in futures returns during the bullish period. 3.2. Intraday periodic VaR effect Previous studies (e.g., Andersen and Bollerslev, 1997; Bollerslev et al., 2000) have shown that estimation and extraction of the intraday periodic component of return volatility are indispensable for meaningful intraday dynamic analysis. Therefore, when we conduct an analysis of the intraday VaR dynamic, which in essence is a quantile of the return distribution and may also be affected by the intraday periodic patterns, it is necessary to exclude the impact of intraday VaR periodic components. Following the assumption of the multiplicative form of volatility components in Engle and Sokalska (2012), we assume that the 4 Engle and Sokalska (2012) assume that the intraday periodic pattern exists only in the conditional volatility component and does not affect the standard error distribution. Following their assumptions, the total VaR is equal to the product of the standard deviation and the VaR of the standard error distribution. Therefore, the periodic pattern also has a multiplicative impact on VaR dynamics. 5 CSI 300 is a comprehensive market index that covers almost 60% of the market capitalization of both the Shanghai stock market and Shenzhen stock market. Two other stock indices (CSI 500 and SSE 50) share similar price movements.

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Fig. 1. Daily prices of the CSI 300 index and futures in China from April 16, 2015 to May 31, 2017. Dashed lines separate the entire sample into three subsamples, corresponding to the bullish market period (April 16, 2015 to June 12, 2015), bearish market period (June 13, 2015 to February 1, 2016) and stable market period (February 2, 2016 to May 31, 2017). Table 1 Descriptive statistics. CSI 300

CSI 500

SSE 50

Index

Futures

Index

Futures

Index

Futures

Bullish period Mean (1e−4) Std. dev. Skewness Kurtosis J-B LBQ

0.6435 0.0031 −1.1443 12.0917 1 47.04***

0.4514 0.0032 −1.7679 16.7010 1 25.72**

1.4394 0.0033 −1.2907 15.3308 1 90.25***

1.2433 0.0037 −1.1874 12.1411 1 28.96*

0.2077 0.0031 −0.7862 9.2635 1 40.04***

−0.1897 0.0035 −1.2770 12.7177 1 21.87

Bearish period Mean (1e−4) Std. dev. Skewness Kurtosis J-B LBQ

0.1575 0.0036 0.1435 9.8302 1 86.77***

−0.8170 0.0044 0.4092 11.7629 1 50.03***

0.0001 0.0041 −0.1874 10.3457 1 86.54***

−1.1831 0.0053 0.0532 9.3558 1 37.09*

0.2519 0.0035 0.6329 13.0710 1 75.54***

−0.7741 0.0039 0.5067 13.2335 1 76.10***

Stable period Mean (1e−4) Std. dev. Skewness Kurtosis J-B LBQ

0.2826 0.0012 −0.4189 13.9792 1 74.87***

0.0405 0.0014 −0.2609 12.6453 1 107.71***

0.1827 0.0016 −1.1535 21.7838 1 64.47***

0.0248 0.0019 −0.3523 12.4388 1 79.98***

0.3313 0.0011 0.3586 11.7758 1 98.12***

0.0946 0.0012 0.1385 11.3186 1 103.65***

Note: This table reports the descriptive statistics of 5-minute frequency intraday return series of three stock indices and their dominant futures contracts during the bullish market period (April 16, 2015 to June 12, 2015), bearish market period (June 13, 2015 to February 1, 2016) and stable market period (February 2, 2016 to May 31, 2017). J-B is the Jarque-Bera test for normality, with value 1 indicating the rejection of the null hypothesis of normal distribution at the 1% significance level. LBQ is the Ljung-Box Q test for serial correlation in returns. *** (**, *) stands for statistically significant at the 1 (5, 10) percent level.

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Fig. 2. The periodic pattern of intraday VaR (5% significance level) dynamics of three stock indices and their dominant futures contracts. In each picture, the blue points stand for the calculated average VaR values (scaled by the overall average VaR) of each 5-minute interval, and the solid lines are the best Chebyshev polynomial fitted functions.

total VaR is the product of the periodic pattern component and the intraday VaR component. For the kth 5-minute intraday interval, we have the logarithmic return series in this interval for each trading day, Rt, k, t = 1, 2, …, T and calculate intraday VaR periodic components VaRFactork using Eqs. (9)–(11). Unlike the U.S. stock markets, which do not stop trading at lunchtime, the Chinese stock market and index futures market have a lunchtime break from 11:30 a.m. to 1:00 p.m.; thus, the intraday periodic pattern in Chinese financial markets may be different from the well-known U-shaped pattern. Fig. 2 plots the periodic pattern of intraday VaR dynamics and their Chebyshev polynomial fitted functions (up to 6 orders). Similar to the intraday volatility pattern in the U.S. markets, the VaR absolute values are quite large at the beginning of the trading day and become stable in the middle transaction periods. However, the intraday VaR pattern is different from previous results in two aspects. First, the values of VaR around the lunchtime break are also comparatively large. Second, during the last 30 min of each trading day, the VaR dynamic will move downward instead of moving upward. In a word, the market extreme risk is pronounced high at the beginning of a trading day and around the lunchtime break, but it moves downward in the final half hour of trading. 4. Empirical results 4.1. Asymmetric downside and upside extreme risk spillovers From the perspective of market investors, the downside extreme risk stands for the possible asset loss due to fast and great price downward movements, which are highly related to bad news announcements and of great concern to traders having long asset positions. However, the upside extreme risk stands for the possible asset loss due to extreme upward price movements, which are often related to good information shocks and have impacts on traders having short asset positions. Moreover, existing short-selling constraints in the Chinese stock market may distort the features of the downside and upside risk. In other words, the downside risk 8

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and upside risk are markedly different in terms of risk sources, market impacts and asset categories. Hence, downside and upside spillover effects between the spot market and futures market may be diverse and asymmetric (see, e.g., Mensi et al., 2017a, 2017b). In this section, we examine the asymmetries in downside and upside extreme risk spillovers using the extended MV-CAViaR model listed in Eq. (8).6 We select the 5% and 95% VaR as the proxy variable of downside risk and upside risk, and we separately use the return data7 of three pairs of stock indices (i.e., CSI 300, CSI 500 and SSE 50) and their corresponding index futures over the entire sample period (April 16, 2015 to May 31, 2017). Table 2 reports results for the downside extreme risk spillovers. First, the coefficients α1, 3 and α2, 4 are slightly smaller than 1 and are statistically significant at the 1% significance level, which indicates that the VaR series of each asset are highly autocorrelated. The coefficient α2, 3 in each group is statistically insignificant and α1, 4 is positively significant in the CSI 300 and SSE 50 group but insignificant in the CSI 500 group. These findings indicate a weak and unidirectional spillover effect in VaR values, i.e., a smaller size of VaRtFutures (a higher downside risk of the futures market) would lower the size of VaRt+1Index (a higher downside risk of the index market) but not vice versa. Second, the statistically negative coefficients α1, 1 and α2, 2 show that a large asset return will increase its extreme risk in the future, but the changes in one asset return may have mixed impacts on the extreme risk of another asset, as we can see from the results of α2, 1 and α1, 2. Finally, and most importantly, we focus on the impacts of extreme events (α1, 5 and α2, 6) and the extreme spillover effects captured by ΔCoVaR (α1, 6 and α2, 5). Our empirical results show that the extreme events occurring in index price movements would increase the extreme risk of the stock market because α1, 5 is significantly negative in each group, but the extreme events occurred in futures price movements do not have a similar impact on the futures market because α2, 6 is statistically insignificant in most groups. Furthermore, the coefficients α1, 6 (i.e., ΔCoVaRIndex∣Futures) and α2, 5 (i.e., ΔCoVaRFutures∣Index) in each group are significantly negative at the 1% significance level. Thus, an extreme event occurring in the index (futures) price would lead to a higher extreme risk of the futures (index) price. In a word, comparing the estimation results of six spillover variables (α2, 1, α1, 2, α2, 3, α1, 4, α2, 5, and α1, 6) in the VaR model, spillovers due to extreme events (captured by α2, 5 and α1, 6) are the most significant and consistent impacts among the three estimation groups, and the spillover effects due to market returns (captured by α2, 1and α1, 2) or VaR values (captured by α2, 3 and α1, 4) are mixed and inconsistent among different groups. The result shows that it is crucial and necessary to take into account the extreme events when modeling VaR dynamics. Table 3 reports the estimation results of upside VaR dynamics. Similar to the downside case, the VaR value of each asset return is significantly influenced by its past return and past VaR, which is shown by the significant coefficients α1, 1, α2, 2, α1, 3 and α1, 4. However, both α1, 5 and α1, 6 are insignificant in the CSI 300 and CSI 500 group, and their absolute sizes also decrease compared to the downside case (except α2, 6 in the SSE 50 group). Overall, the impact of upside events on upside VaR is smaller and less important compared to the impact of downside events on the downside VaR. As for the spillover effects, we find that α1, 5 and α2, 6 become insignificant in three groups, but they are statistically significant in each group of the downside VaR model (see Table 2). This evidence shows that upside and downside spillover effects between the spot market and futures market are asymmetric and distinct. Specifically, a downside event occurring in the index (futures) price would significantly contribute to the downside extreme risk of futures (index) returns, but an upside event occurring in the index (futures) price would not impose an impact on the upside extreme risk of futures (index) returns. The empirical results indicate that the stock market and index futures market in China are imperfectly related because only extreme downward price movements occurring in one market have impacts on the downside extreme risk of the other market. Take the CSI 500 as an example, if a downward extreme movement occurs in the futures price, the VaR value of the index return would decrease by 0.0163, indicating that the extreme downside risk of the index return would increase approximately 9.27%. However, the extreme upside risk interaction between the two markets is remarkably different, a sudden price boom occurring in one market would not influence the extreme upside risk of the other market significantly. For example, if an upward extreme movement occurs in the CSI 500 futures price, the VaR value of the index return would decrease by 0.0020, indicating that the extreme upside risk of the index return would slightly decrease by approximately 1.26%. Therefore, it is likely to observe extreme downward price movements simultaneously and observe extreme upward price movements separately. From the perspective of market regulation, policymakers should take into account the extreme downward price comovements between the spot and futures market to develop efficient stabilization strategies during periods of financial crisis. Fig. 3 displays the dynamics of both upside (5%) and downside (95%) VaR for each single asset. It is obvious that the time-varying features of the upside risk and downside risk are nearly symmetric, i.e., they are comparatively high during the bullish and bearish market period, and more stable during the normal market period. This empirical finding is also supported by the descriptive statistics listed in Table 4, because the absolute values of the mean and variance of the 5% VaR and 95% VaR are very close. Moreover, concerning the difference between the stock market and the futures market, we note that the index futures market is faced with a higher level of extreme risk compared to the stock market (larger absolute size of the mean and variance).

6

When modeling VaR dynamics at the 95% confidence level using the extended MV-CAViaR model, we change the definition of extreme events from Ri, t−1 ≤ qi, t−1 to Ri, t−1 ≥ qi, t−1, because it is more reasonable to take into account upper tail events instead of lower tail events when we model the VaR at the 95% confidence level. 7 To avoid the potential optimization bias due to the small size of high frequency returns, we rescale the return data by multiplying the ratio of the interval length of a trading day to the interval length of our sample. For example, when we use 5-minute frequency data, we multiply each return by 240/5 = 48, the size of rescaled returns is close to that of daily returns. 9

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Table 2 Downside spillovers in the stock market and futures market. Panel A. CSI 300 index and futures α1, 0 0.0006∗∗∗ (0.0001) α2, 0 0.0001 (0.0002)

α1, 1 −0.0896∗∗∗ (0.0107) α2, 1 −0.0694∗∗∗ (0.0140)

α1, 2 0.0228∗ (0.0136) α2, 2 −0.0793∗∗∗ (0.0160)

α1, 3 0.9352∗∗∗ (0.0074) α2, 3 −0.0185 (0.0115)

α1, 4 0.0229∗∗ (0.0097) α2, 4 0.9453∗∗∗ (0.0116)

α1, 5 −0.0203∗∗∗ (0.0022) α2, 5 −0.0119∗∗∗ (0.0026)

α1, 6 −0.0091∗∗∗ (0.0020) α2, 6 −0.0022 (0.0025)

α1, 2 −0.0005 (0.0102) α2, 2 −0.0848∗∗∗ (0.0141)

α1, 3 0.9472∗∗∗ (0.0059) α2, 3 −0.0010 (0.0065)

α1, 4 0.0033 (0.0054) α2, 4 0.9559∗∗∗ (0.0076)

α1, 5 −0.0266∗∗∗ (0.0031) α2, 5 −0.0152∗∗∗ (0.0031)

α1, 6 −0.0163∗∗∗ (0.0026) α2, 6 −0.0329∗∗∗ (0.0033)

α1, 2 0.0259 (0.0169) α2, 2 −0.0792∗∗∗ (0.0256)

α1, 3 0.9218∗∗∗ (0.0102) α2, 3 −0.0221 (0.0159)

α1, 4 0.0213∗ (0.0113) α2, 4 0.9416∗∗∗ (0.0174)

α1, 5 −0.0223∗∗∗ (0.0020) α2, 5 −0.0130∗∗∗ (0.0024)

α1, 6 −0.0054∗∗∗ (0.0019) α2, 6 −0.0003 (0.0026)

Panel B. CSI 500 index and futures α1, 0 0.0007∗∗∗ (0.0002) α2, 0 0.0011∗∗∗ (0.0002)

α1, 1 −0.0908∗∗∗ (0.0131) α2, 1 0.0016 (0.0116)

Panel C. SSE 50 index and futures α1, 0 0.0002∗ (0.0001) α2, 0 0.0003 (0.0002)

α1, 1 −0.1209∗∗∗ (0.0162) α2, 1 −0.0880∗∗∗ (0.0226)

Note: This table reports estimates and standard errors (in brackets) of the bivariate 5%-VaR model (see Eq. (8)) for different stock index and index futures returns. For each panel, the variables in the first row are the coefficients of the stock index return's VaR function, and the variables in the second row are the coefficients of the index futures return's VaR function. *** (**, *) stands for statistically significant at the 1 (5, 10) percent level. Table 3 Upside spillovers in the stock market and futures market. Panel A. CSI 300 index and futures α1, 1 α1, 0 0.0015∗∗∗ 0.1530∗∗∗ (0.0003) (0.0185) α2, 0 α2, 1 0.0008∗∗∗ 0.0690∗∗∗ (0.0002) (0.0174) Panel B. CSI 500 index and futures α1, 0 α1, 1 0.0008∗∗∗ 0.1356∗∗∗ (0.0002) (0.0129) α2, 0 α2, 1 ∗∗∗ 0.0004 0.0418∗∗ (0.0002) (0.0162) Panel C. SSE 50 index and futures α1, 0 α1, 1 0.0006∗∗∗ 0.0890∗∗∗ (0.0002) (0.0135) α2, 0 α2, 1 0.0013∗ 0.0106 (0.0007) (0.0319)

α1, 2 0.0584∗∗∗ (0.0086) α2, 2 0.0995∗∗∗ (0.0148)

α1, 3 0.8901∗∗∗ (0.0136) α2, 3 −0.0490∗∗∗ (0.0132)

α1, 4 −0.0051 (0.0067) α2, 4 0.9604∗∗∗ (0.0089)

α1, 5 −0.0002 (0.0028) α2, 5 0.0031 (0.0024)

α1, 6 0.0032 (0.0026) α2, 6 0.0037 (0.0023)

α1, 2 0.0279∗∗∗ (0.0067) α2, 2 0.0985∗∗∗ (0.0153)

α1, 3 0.9108∗∗∗ (0.0090) α2, 3 −0.0245∗∗ (0.0107)

α1, 4 0.0007 (0.0050) α2, 4 0.9542∗∗∗ (0.0084)

α1, 5 −0.0009 (0.0018) α2, 5 0.0028 (0.0022)

α1, 6 −0.0020 (0.0018) α2, 6 −0.0019 (0.0024)

α1, 2 0.0563∗∗∗ (0.0184) α2, 2 0.2299∗∗∗ (0.0359)

α1, 3 0.9474∗∗∗ (0.0191) α2, 3 0.1849∗∗∗ (0.0318)

α1, 4 −0.0237 (0.0209) α2, 4 0.7075∗∗∗ (0.0373)

α1, 5 0.0105∗∗∗ (0.0025) α2, 5 −0.0108∗ (0.0056)

α1, 6 0.0022 (0.0025) α2, 6 0.0354∗∗∗ (0.0058)

Note: This table reports estimates and standard errors (in brackets) of the bivariate 95%-VaR model for different stock index and index futures returns. For each panel, the variables in the first row are the coefficients of the stock index return's VaR function, and the variables in the second row are the coefficients of the index futures return's VaR function. *** (**, *) stands for statistically significant at the 1 (5, 10) percent level.

Meanwhile, it is necessary to test the adequacy of the intraday MV-CAViaR model. Kupiec (1995) proposed an unconditional coverage test of the VaR model. Suppose the confidence level is α, the sample size is T, the number of hits (the return exceeds the VaR) is N, then the sample hit ratio N/T should be equivalent to α, and the unconditional coverage test statistic is

LRUC = 2 ln [(1 − N / T )T − N (N / T ) N ] − 2 ln [(1 − α )T − N α N ]

(14)

Under the null hypothesis of model correctly specified, LRUC asymptotically follows a χ2(1) distribution. 10

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Fig. 3. Downside and upside VaR dynamics at 5-minute observation frequency.

Table 4 Descriptive statistics of VaR estimations and performance evaluation. Mean

Std. dev.

Hit number

Hit rate

UC

CC

JI

Panel A. Downside risk (5% VaR) CSI 300 Index Futures CSI 500 Index Futures SSE 50 Index Futures

−0.1378 −0.1612 −0.1759 −0.2086 −0.1298 −0.1476

0.1086 0.1390 0.1298 0.1664 0.1089 0.1347

1277 1248 1179 1262 1240 1310

0.0516 0.0504 0.0476 0.0510 0.0501 0.0529

0.2628 0.7798 0.0810 0.4927 0.9628 0.0386

0.0008 0.8848 0.0008 0.6371 0.0069 0.0540

0.3362

Panel B. Upside risk (95% VaR) CSI 300 Index Futures CSI 500 Index Futures SSE 50 Index Futures

0.1424 0.1642 0.1592 0.2065 0.1380 0.1525

0.1155 0.1386 0.1242 0.1691 0.1139 0.1277

23,543 23,523 23,475 23,515 23,524 23,565

0.9505 0.9497 0.9478 0.9494 0.9498 0.9514

0.6955 0.8475 0.1139 0.6709 0.8704 0.2998

0.0001 0.4394 0.0004 0.8663 0.0179 0.0040

0.0246

0.0713 0.0265

0.1457 0.0124

Note: This table reports the summary statistics and backtests p-values of the VaR sequences estimated by the MV-CAViaR model. UC, CC and JI stand for the unconditional coverage test (Kupiec, 1995), conditional coverage (Christoffersen, 1998) test and joint independence test, respectively. A better fitted VaR model has a larger p-value. Bold values represent p-values below 1%.

To further examine the serial independence of the hit sequence, Christoffersen (1998) proposed a joint test of coverage and independence, i.e., the conditional coverage test. n11 n 01 LR cc = 2 ln [(1 − π01 )n00 π01 (1 − π11 )n10 π11 ] − 2 ln [(1 − α )T − N α N ]

(15)

where nij is the number of observations with value i followed by j, and π01 = n01/(n00 + n01); π11 = n11/(n10 + n11). Under the null hypothesis of model correctly specified, LRCC asymptotically follows a χ2(2) distribution. 11

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Furthermore, it is more reasonable to test the serial independence of hit sequences under a joint framework because the VaR sequences of an index and its futures are estimated jointly instead of separately. Therefore, we extend the independence test proposed by Christoffersen (1998) and construct a joint independence test as below.8 Let {ItS} and {ItF} be hit sequences of the stock index and its corresponding index futures, respectively. Define a joint hit sequence S F ⎧ 0, if It = 0, It = 0 ⎪ S St = 1, if It = 1, ItF = 0 or ItS = 0, ItF = 1 ⎨ ⎪ 2, if ItS = 1, ItF = 1 ⎩

where nij is the number of observations with value i followed by j, and πij = nij/(ni0 + ni1 + ni2) is the estimate value of Pr (St = j| St−1 = i). Similarly, ni = n0i + n1i + n2i is the number of observations with value i and πi = ni/(n0 + n1 + n2). The joint independence test statistic is n 00 n 01 n 02 n10 n11 n12 n20 n21 n22 LRJI = 2 ln (π00 π01 π02 π10 π11 π12 π20 π21 π22 ) − 2 ln (π0n0 π2n2 π2n2)

(16)

Under the null hypothesis of {St} is serially independent, LRJI asymptotically follows the χ2(4) distribution.9 Table 4 reports the unconditional coverage (UC) test and the conditional coverage (CC) test results of each single asset and the joint independent (JI) test results of each pair of index and futures. We find that the hit rates are very close to the pre-specified probability level of VaR, and most of the UC test statistics are insignificant at the 5% significance level (except the case of the 5% VaR of SSE 50 futures), indicating that our model passes the UC test in both stock and futures cases. Concerning the conditional coverage (CC) test results, the test statistics are insignificant at the 5% significance level in most of the futures cases (except the 95% VaR of SSE 50 futures), but they are significant at the 1% significance level in most of the index cases (except the 95% VaR of SSE 50 index). The mixed results of the conditional coverage test indicate that the hit sequences of index returns may be serially correlated, and a potential explanation is that the probability of observing intraday successive extreme events is higher compared to the probability of observing interday successive extreme events in the Chinese stock market. Moreover, the results of the joint independence (JI) test are similar: half of the index-futures groups are statistically significant at the 5% significance level, but they are comparatively better because none of them are statistically significant at the 1% significance level, indicating that the joint hit sequence does not have a serious serial correlation problem. 4.2. Asymmetric spillovers under different market states Fig. 3 plots the time-varying VaR dynamics of each market, the graphical evidence shows that downside and upside VaR values share a similar trend, and they are significantly high during the bullish and bearish period and relatively low during the following stable period. Adams et al. (2014) stress the importance of the market state on the direction, size, and duration of market risk spillovers. Moreover, Mensi et al. (2017a, 2017b) find evidence that the market dependence is dynamic and varies during bear and bull markets. Therefore, it is necessary to look into structural changes due to the variation of market states, and we separately examine the downside ΔCoVaR in each subsample. The estimation results of risk spillovers during each subsample period are reported in Table 5. To overcome the difficulty of presenting large tables with numbers, we only present the coefficients of extreme event variables. βI (βF) stands for the contribution of an extreme event occurred in the index (futures) price to its VaR dynamic. In each period, βI and βF are lower than zero and statistically significant, which indicates that when an extreme event occurs in a financial market, this market would be faced with a higher level of extreme risk (a larger absolute size of VaR) in the future. However, the bidirectional spillover effects captured by ΔCoVaR vary from period to period. During the stable market period, we find similar results compared to the entire sample, both ΔCoVaRI∣F and ΔCoVaRF∣I are significantly negative in the three groups, i.e., an extreme event occurring in the index (futures) return has significant predictive power for the extreme risk of the futures (index) return. If a sudden price drop is observed in the stock (futures) market, the VaR values of futures (stock) returns will decrease by 0.0049 to 0.0069 (0.0098 to 0.0207), indicating that the extreme risk of the futures (stock) market will increase by 2.4% to 4.28% (7.11% to 11.77%). However, the spillover effects during the bearish and bullish periods are significantly different from the previous findings. In the bearish period, ΔCoVaRI∣F remains statistically significant, and its size becomes larger comparing to the stable period. It appears that the downside extreme risk running from the futures market to the spot market becomes stronger, but the ΔCoVaRF∣I becomes insignificant in two of the three regression groups, the spillover effect from index prices to futures prices weakens and even disappears. In a word, when the entire market moves downward, the extreme events occurring in the futures market have a non-negligible impact on the spot market risk (the VaR values of the spot market would increase by 12.56% to 38.83%), but the extreme events occurring in the spot market would not influence the extreme risk of the futures market. When the market moves upward, big changes are taking place in the risk spillover mechanism between spot and futures returns. Although ΔCoVaRI∣F remains negative and significant in the CSI 500 group, it becomes positive and insignificant in the CSI 300 and SSE 50 group, the risk spillover effects from futures returns to index returns are weakened and even vanish, but ΔCoVaRF∣I becomes 8 The UC and CC test under a joint framework is also accessible, but they dependent on the true probability distribution of St which is unknown in the joint case, so we only consider the joint independence test. 9 See Section 4.2 in Christoffersen (1998) for more details.

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Table 5 Extreme risk spillovers under different market state. Panel A. Bearish period CSI 300 βI −0.0261∗∗∗ (0.007) ΔCoVaRF∣I −0.0238∗∗∗ (0.0078)

CSI 500 ΔCoVaRI∣F −0.0487∗∗∗ (0.0061) βF −0.0595∗∗∗ (0.0081)

SSE 50

βI −0.0379∗∗∗ (0.0115) ΔCoVaRF∣I −0.0016 (0.0085)

ΔCoVaRI∣F −0.0693∗∗∗ (0.0123) βF −0.0450∗∗∗ (0.0075)

βI −0.0264∗∗∗ (0.0067) ΔCoVaRF∣I 0.0070 (0.0087)

ΔCoVaRI∣F −0.0163∗∗ (0.0065) βF −0.0414∗∗∗ (0.0078)

Panel B. Bullish period CSI 300 βI −0.1672∗∗∗ (0.0451) ΔCoVaRF∣I −0.0342∗∗∗ (0.0118)

CSI 500 ΔCoVaRI∣F 0.0019 (0.0208) βF −0.0440∗∗∗ (0.0096)

SSE 50

βI −0.1443∗∗∗ (0.0256) ΔCoVaRF∣I −0.0597∗∗∗ (0.0171)

ΔCoVaRI∣F −0.0507∗∗∗ (0.0176) βF −0.0294∗∗ (0.0138)

βI −0.0553∗∗ (0.0259) ΔCoVaRF∣I −0.0154 (0.0101)

ΔCoVaRI∣F 0.0144 (0.0254) βF −0.0215∗ (0.0128)

Panel C. Stable period CSI 300 βI −0.0098∗∗∗ (0.0021) ΔCoVaRF∣I −0.0069∗∗∗ (0.0025)

CSI 500 ΔCoVaRI∣F −0.0098∗∗∗ (0.0022) βF −0.0117∗∗∗ (0.0035)

SSE 50

βI −0.0192∗∗∗ (0.0027) ΔCoVaRF∣I −0.0050∗ (0.0026)

ΔCoVaRI∣F −0.0207∗∗∗ (0.0024) βF −0.0228∗∗∗ (0.0030)

βI −0.0123∗∗∗ (0.0018) ΔCoVaRF∣I −0.0049∗∗ (0.0024)

ΔCoVaRI∣F −0.0108∗∗∗ (0.0018) βF −0.0088∗∗∗ (0.0030)

Note: This table reports the coefficients and standard errors (in brackets) of the dummy variable 1(rt ≤ VaRt) during the bullish market period (April 16, 2015 to June 12, 2015), bearish market period (June 13, 2015 to February 1, 2016) and stable market period (February 2, 2016 to May 31, 2017). βI (βF) stands for the contribution of an extreme event occurring in the index (futures) price to its VaR dynamic. ΔCoVaRI∣F and ΔCoVaRF∣I reveal the bidirectional spillovers between the stock and futures market. *** (**, *) stands for statistically significant at the 1 (5, 10) percent level.

statistically significant (except the SSE 50 group), and its absolute size increases prominently in each group. In the bullish period, the absolute size of ΔCoVaRF∣I (range from 0.0154 to 0.0342) is approximately four times the size in the stable period (range from 0.0049 to 0.0069). As a result, during the bullish market state, only the extreme events occurring in the spot market have spillover effects (larger influence compared to the stable period) on the other financial market. Overall, these empirical results indicate that the bidirectional information (extreme movements) transmission mechanism between the two markets is bidirectional during most of the time (normal period), a sudden price drop occurring in one market would increase the extreme risk of the other market. However, this mechanism would be significantly different during extreme market periods. When the market prices move upward, the stock market becomes more informative and has a unidirectional impact on futures price movements. Additionally, when the market prices move downward, the futures market becomes more informative and has a unidirectional impact on stock price movements. One possible reason for this phenomenon is that the informed investors prefer to trade in the stock market during bullish periods and trade in the futures market during bearish periods. 4.3. Further discussion: could a high margin requirement policy stabilize the market? During the Chinese financial market turbulence in 2015–2016, critics often blamed the stock index futures market that index futures trading would exacerbate price fluctuations and induce more extreme price movements in the stock market. Under such pressure, the market regulator continually increased the margin ratio requirement of index futures from the lowest 10% to the highest 40%. A high margin ratio policy has various influences on market transaction activities and price movements, such as suppressing the transaction leverage, reducing the market liquidity and so on, but it is still unclear whether this adjustment is beneficial to the overall market stability. We evaluate the policy effects of high margin requirement from the perspective of extreme risk spillovers. If the increase in margin ratio requirement could effectively reduce the risk spillovers between the index futures market and the stock market, ΔCoVaRI∣F or ΔCoVaRF∣I should decrease and less significant during the high margin period. To avoid the bias due to changes in market state, we consider only the stable sample period and divide it into to two subsamples, the high margin requirement subsample

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Table 6 Extreme risk spillovers under different margin requirements. Panel A. High margin requirement period CSI 300 βI −0.0161∗∗∗ (0.0022) ΔCoVaRF∣I −0.0108∗∗∗ (0.0032)

CSI 500 ΔCoVaRI∣F −0.0016 (0.0019) βF −0.0099∗∗∗ (0.0032)

SSE 50

βI −0.0233∗∗∗ (0.0027) ΔCoVaRF∣I −0.0057∗ (0.0030)

ΔCoVaRI∣F −0.0179∗∗∗ (0.0027) βF −0.0200∗∗∗ (0.0035)

βI −0.0105∗∗∗ (0.0020) ΔCoVaRF∣I −0.0065∗∗∗ (0.0025)

ΔCoVaRI∣F −0.0074∗∗∗ (0.0021) βF −0.0137∗∗∗ (0.0027)

Panel B. Low margin requirement period CSI 300 βI −0.0115∗∗∗ (0.0041) ΔCoVaRF∣I −0.0136∗∗∗ (0.0035)

CSI 500 ΔCoVaRI∣F −0.0026 (0.0038) βF −0.0039 (0.0040)

SSE 50

βI −0.0165∗∗∗ (0.0045) ΔCoVaRF∣I 0.0026 (0.0053)

ΔCoVaRI∣F −0.0219∗∗∗ (0.0056) βF −0.0252∗∗∗ (0.0056)

βI −0.0010 (0.0038) ΔCoVaRF∣I 0.0009 (0.0033)

ΔCoVaRI∣F −0.0110∗∗∗ (0.0031) βF −0.0081∗∗ (0.0041)

Note: This table reports the coefficients and standard errors (in brackets) of the dummy variable 1(rt ≤ VaRt), βI (βF) stands for the contribution of an extreme event occurred in index (futures) price to its VaR dynamic. ΔCoVaRI∣F and ΔCoVaRF∣I reveal the bidirectional spillovers between the stock and futures market. *** (**, *) stands for statistically significant at the 1 (5, 10) percent level.

period (40%, February 2, 2016 to February 17, 2017) and the low margin requirement subsample period (20%,10 February 18, 2017 to May 31, 2017). Table 6 reports the estimation results of extreme risk spillovers under different margin requirements. In the sample period of high margin requirement, both ΔCoVaRI∣F and ΔCoVaRF∣I are significantly negative (except ΔCoVaRI∣F in the CSI 300 group), which indicates that the extreme risk of the spot market and futures market are highly correlated even when the margin requirement is as high as 40%. However, this finding does not indicate that changes in the margin requirement have no impact on the risk spillover mechanism. After the loosening of margin requirements, the absolute size of ΔCoVaRI∣F increases remarkably in each group, e.g., for the CSI 300 group, it increases from 0.0016 to 0.0026. In contrast, the ΔCoVaRF∣I becomes insignificant and positive in two of the three groups. In summary, the adjustment of margin requirements in the futures market has a double-sided effect on extreme risk interactions between the two markets. On the one hand, a higher margin ratio requirement would weaken the risk spillover from the futures market to the spot market as expected. On the other hand, a tightened margin requirement would make the futures market become more prone to the extreme events occurring in the stock market. These results have profound policy implications for regulators to set up a long-term effective market stabilization mechanism, because they reflect the benefits and costs of different trading rules from the perspective of extreme risk spillovers. If the margin ratio requirement of futures transaction increase from 20% to 40%, the futures market becomes less informative and has a smaller influence on the stock market because the extreme risk spillovers from futures returns to stock returns decrease by 18.26% to 38.46%. The weakened futures market shocks are beneficial to the stability of the stock market and meet the purpose of market regulators. However, the futures market becomes unstable and more vulnerable to the shocks of the stock market when the margin ratio requirement is high, because the extreme risk spillovers from stock returns to futures returns become more significant compared to those in the low margin requirement case. Therefore, policymakers should consider both the positive and negative impacts of the high margin ratio requirement policy in order to establish reasonable market trading rules.

5. Conclusion Using high frequency data of stock index and index futures prices, we investigate the extreme risk spillover effects between the stock and futures market. Unlike previous CoVaR studies which merely study the contemporaneous correlations of extreme movements, we propose a predictive CoVaR measure and construct a multivariate semiparametric CoVaR model to disclose the bidirectional lagged tail risk interactions simultaneously. To characterize the time-varying and asymmetric feature of extreme risk interactions, we separately measure downside and upside risk spillovers by computing downside and upside ΔCoVaR. Moreover, we further look into potential changes of the spillover effect due to the change in market state (bullish market, bearish market, and normal market) and the change in trading rules (high margin requirement and low margin requirement). 10 For CSI 300 and SSE 50 index futures, the reduced margin requirement is 20%, but for CSI 500 index futures contracts, the reduced margin requirement is 30%.

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Our empirical results show that there exist pronounced bidirectional downside spillover effects between the Chinese spot market and futures market, i.e., extreme downside movements occurring in one market would increase the extreme risk of the other market, but we do not find significant upside spillover effects between the two markets, which could be explained by the fact that market investors are more vulnerable to the downside risk than the upside risk. Second, we find the risk spillover effects would be remarkably distorted due to the change in market state. When the overall market price moves upward, the spillover effects from index returns to futures returns would be enlarged and enhanced, but the spillover effects from futures returns to index returns would be weakened and even vanish. The spot market becomes important and dominant during the bullish market period. When the overall market price moves downward, the spillovers from futures returns to index returns are strengthened and the adverse spillover effects, which transmits from index returns to futures returns, become weaker and insignificant, indicating that the futures market becomes more influential to the spot market but not vice versa. Finally, we find that a tight margin requirement of the futures transaction would weaken the spillover effect from the futures market to the stock market as expected. However, it also makes the futures market becomes more vulnerable to the extreme movements occurring in the stock market. Our empirical model and findings have important implications. Academically, it is shown that the asymmetry and lagged effects are crucial for the tail risk relationship between the Chinese stock market and index futures market. A static and contemporaneous model for the spillover effects may lead to substantial bias and incorrect results. 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