Dependencies and Volatility Spillovers among Chinese Stock and Crude Oil Future Markets: Evidence from Time-Varying Copula and BEKK-GARCH Models

Abstract

This paper investigates co-movements among the Chinese stock market, Shanghai International Energy Exchange (INE) crude oil futures and West Texas Intermediate (WTI) crude oil futures. We use Copula models to capture tail dependencies and employ the VAR-BEKK-GARCH model to examine the direction of volatility spillovers. We find that there are positively time-varying dependency relationships among the three markets. Compared with the corresponding upper-tail dependencies, the lower-tail dependencies were larger before the COVID-19 pandemic while relatively weaker after the breakout of the pandemic. Before the COVID-19 pandemic, there was only a statistically significant volatility spillover from WTI crude oil future market to the INE crude oil future market. After the breakout of the COVID-19 pandemic, there were statistically significant volatility spillovers in the two pairs of markets, namely, the WTI–INE and Chinese stock–WTI. However, we only find statistically significant evidence of unidirectional volatility spillover from the Chinese crude oil future market to the Chinese stock market during the pandemic.

1. Introduction

As the world’s biggest crude oil importer, China launched crude oil futures trading at the Shanghai International Energy Exchange (INE) on 26 March 2018. It is the first crude oil futures variety denominated in Chinese currency (RMB) on the Chinese mainland and open to overseas investors. As yet, it has been more than 4 years since the launch of INE crude oil futures.

The dependency on crude oil and financial markets has always been a hot topic amongst academicians, market participants and policymakers, in that the global economy may be influenced through various channels due to volatilities in oil prices. On the whole, earlier studies have shown interest in the nexus of crude oil and stock markets in developed countries. For instance, Aloui et al. (2013) investigated the conditional dependence between the Brent crude oil price and stock markets in Central and Eastern European countries. Tsai (2015); Ewing et al. (2018); Chen et al. (2019) and Huang and Mollick (2020) examined the response of U.S. stock returns to oil price shocks from different aspects. Bastianin et al. (2016) investigated the impacts of oil price shocks on stock market volatility in G7 countries. Bagirov and Mateus (2019) were interested in the relationship between stock markets, oil prices and firm performance in Europe. Recently, some scholars have turned their attention to emerging countries. Balcılar et al. (2017) investigated whether speculation in the oil market can drive investor herding in emerging stock markets. Bhatia and Basu (2020) examined the causality in quantiles between crude oil and stock markets in emerging economies. Boubaker and Raza (2017) analyzed mean and volatility spillovers between oil and the BRICS stock markets. Balcilar et al. (2019) investigated the quantile relationship between the oil and stock returns of emerging and frontier stock markets. Xiao et al. (2019) and Peng et al. (2020) studied the impact of global crude oil on the Chinese stock market. A strand of the literature examines oil market shocks on stock markets, dividing the sample into oil-exporting and oil-importing countries (e.g., Basher et al. 2018; Jiang and Yoon 2020). Besides this, there are few studies focusing on the connectedness of different crude oil futures (e.g., Yang et al. 2021; Li et al. 2022; Wei et al. 2022). The empirical methodologies most commonly used in the previous literature on the linkage of stock markets and crude oil markets, or the interactivity within different crude oil markets, include the TVP-VAR approach (Toparlı et al. 2019), the Copula-type approach (Aloui et al. 2013; Sukcharoen et al. 2014; Jammazi and Reboredo 2016), the wavelet-based approach (Martín-Barragán et al. 2015; Huang et al. 2016; Boubaker and Raza 2017; Jiang and Yoon 2020), the Markov-switching approach (Basher et al. 2018), the GFEVD approach (Li et al. 2022), the GARCH-type approach (Salisu and Oloko 2015; Alsalman 2016; Zhang and Wang 2019; Salisu and Gupta 2020; Abdulkarim et al. 2020), the nonlinear ARDL test (Al-hajj et al. 2018), the time series network model (An et al. 2018), the structural VAR approach (Gupta and Modise 2013), the nonparametric panel data model (Silvapulle et al. 2017), etc.

We investigate the interactivity of three selected financial markets, namely, the Chinese stock market, the Chinese crude oil future market and the global crude oil future market, taking an empirical sample spanning from 26 March 2018 to 26 September 2022. For this purpose, we have taken the CSI300 stock index (hereafter CHS) as a proxy of the Chinese stock market, the Shanghai International Energy Exchange crude oil futures (hereafter INE) as a proxy of the Chinese crude oil future market, and the West Texas Intermediate crude oil futures (hereinafter WTI) as a proxy of the global crude oil future market. An integrated methodology joining Copula-type models and the VAR-BEKK-GARCH model is used in our study. Firstly, we take the time-varying Symmetrized Joe-Clayton (SJC) Copula model proposed by Jondeau and Rockinger (2006) to capture the dynamic upper-tail and lower-tail dependencies of the three pairs of markets, namely, the Chinese stock market with the Chinese crude oil future market (CHS–INE), the Chinese stock market with the global crude oil future market (CHS–WTI), and the Chinese crude oil future market with the global crude oil future market (INE-WTI), respectively. Secondly, we employ a three-dimension C-vine SJC-Copula model to capture static tail dependencies among the three markets. Thirdly, a three-variable asymmetric VAR-BEKK-GARCH model is used to examine how the three markets spill volatilities over to each other, concentrating on the directions of spillovers. A Wald test is used to test the robustness of the volatility spillover effects in the last step. Besides this, considering the possible impact of the COVID-19 pandemic, we not only pay attention to the empirical results for the whole sample period but also divide the total sample into pre- and post-COVID-19 subsamples to examine shifts in the relationships among the three selected markets before and after the breakout of the COVID-19 pandemic.

Some interesting findings have been obtained in our study. First of all, a common feature of dynamic tail-dependence in the three pairs of markets, namely, CHS–INE, CHS–WTI and INE-WTI, has been found, according to the empirical results of the time-varying Symmetrized Joe-Clayton (SJC) Copula model. Both upper-tail dependence and lower-tail dependence are time-varying, and the upper-tail dependence is stronger than the lower-tail dependence most of the time. Secondly, using the three-dimension C-vine SJC-Copula model, we further found that the lower-tail dependencies were stronger than the corresponding upper-tail dependencies before the COVID-19, while weaker than the corresponding upper-tail dependencies after the COVID-19 pandemic breakout. Thirdly, the results of the multivariable asymmetric VAR-BEKK-GARCH model show that there are not only bidirectional volatility spillovers between WTI and INE, but also bidirectional volatility spillovers between WTI and CHS, while there is only statistical evidence of unidirectional volatility spillovers from INE to CHS. Fourthly, the volatility spillover effects during the post-COVID-19 era are obviously stronger than those before the COVID-19 pandemic. This implies that the WTI crude oil future market dominates the volatility spillovers, the emerging Chinese crude oil future market has a lower status, and the co-movement between the Chinese stock market and its crude oil future market is relatively weak.

The main contributions of our study can be summarized below. Firstly, differently from most of the previous literature, which focuses on the linkage of oil markets and stock markets in developed countries (e.g., Aloui et al. 2013; Tsai 2015; Ewing et al. 2018; Chen et al. 2019; Huang and Mollick 2020; Bastianin et al. 2016; Bagirov and Mateus 2019), we concentrate on the stock market of China, which belongs to an emerging economy and a developing country. Secondly, different from the strand of literature on dependencies among global oil markets and stock markets within developing countries (e.g., Balcılar et al. 2017; Boubaker and Raza 2017; Basher et al. 2018; Balcilar et al. 2019; Xiao et al. 2019; Bhatia and Basu 2020; Jiang and Yoon 2020; Peng et al. 2020), we include Chinese crude oil future (INE crude oil futures), which is an emerging area of crude oil futures, in our study. Thirdly, different from the previous literature focusing purely on the linkage of Chinese and global crude oil future markets (Yang et al. 2021; Li et al. 2022), this is the first study to simultaneously examine the dependencies among the three markets, namely, the Chinese stock market, Chinese crude oil futures and the global crude oil future market. Fourthly, different from the methodologies used in previous related literature (e.g., Aloui et al. 2013; Gupta and Modise 2013; Sukcharoen et al. 2014; Li et al. 2022; Martín-Barragán et al. 2015; Salisu and Oloko 2015; Alsalman 2016; Jammazi and Reboredo 2016; Huang et al. 2016; Boubaker and Raza 2017; Silvapulle et al. 2017; An et al. 2018; Al-hajj et al. 2018; Basher et al. 2018; Zhang and Wang 2019; Toparlı et al. 2019; Jiang and Yoon 2020; Salisu and Gupta 2020; Abdulkarim et al. 2020), we use an integrated methodology joining Copula-type models and the VAR-BEKK-GARCH model. Both time-varying and static types of Copula models are used to capture the extreme dependencies of the three markets, and the BEKK-GARCH model is not only used to investigate their volatility spillover effects, but also to recognize the direction of spillovers. The joining of the two kinds of methods can provide relatively rich evidence. Last but not the least, we compare the changes in the relationships among the three selected markets, before and after the COVID-19 pandemic. This is relevant to observe the status of Chinese crude oil futures in the world and their impact on the stock market. It also provides a glimpse of the impact of the COVID-19 pandemic on the co-movement of financial markets.

The remainder of our study is organized as follows: Section 2 introduces the methodology; Section 3 describes the data, descriptive statistics and basic statistical tests; Section 4 details the empirical results; and Section 5 concludes.

2. Empirical Methodology

In this paper, we first use the time-varying SJC-Copula model to investigate the dependences between Chinese stock and crude oil futures, where the AR (1)-GARCH (1,1) model is used to construct marginal distribution for each return series. We chose the SJC Copula model for its advantage of capturing both upper-tail and lower-tail dependencies between paired markets. The time-varying Copula model can be specified as follows.

The AR (1)-GARCH (1,1) model with t distribution (Jondeau and Rockinger 2006) is as follows:

$$r_{i,t}=c_0+c_1{r}_{i,t}{}_{-1}+e_{i,t}$$

(1)

$$e_{i,t}=\sqrt{h_{i,t}}·{\epsilon }_t$$

(2)

$$h_{i,t}={\omega }_i+{\phi }_1{e}_{i,t-1}^2+{\phi }_2{h}_{i,t-1}$$

(3)

where $r_{i,t}$ is the daily log return for asset $i$ at time $t$, $c_0$ and $c_1$ are parameters of the AR (1) model, ${\omega }_i$, ${\phi }_1$ and ${\phi }_2$ are parameters of the ARCH (1,1) model. ${\epsilon }_t\sim T\left(0,\gamma \right)$ and $\gamma$ are the parameters of t distribution.

The time-varying Symmetrized Joe-Clayton (SJC) Copula model (Patton 2006):

$$C_{SJC}\left(v_1,v_2|{\tau }^U,{\tau }^L\right)=\frac{1}{2}\left[C_{JC}\left(v_1,v_2|{\tau }^U,{\tau }^L\right)+C_{JC}\left(1-v_1,1-v_2|{\tau }^U,{\tau }^L\right)+v_1+v_2-1\right]$$

(4)

$$C_{JC}\left(v_1,v_2|{\tau }^U,{\tau }^L\right)=1-{\left(1-{\left\{{\left[1-{\left(1-v_1\right)}^a\right]}^{-b}+{\left[1-{\left(1-v_2\right)}^a\right]}^{-b}-1\right\}}^{-\frac{1}{a}}\right)}^{\frac{1}{b}}$$

(5)

$$a=\frac{1}{{\log }_2\left(2-{\tau }^U\right)}$$

(6)

$$b=-\frac{1}{{\log }_2\left(2-{\tau }^L\right)}$$

(7)

$${\tau }_t^U={\left(1+e^{-\left({\delta }^U+{\varphi }^U{\tau }_{t-1}^U+{\psi }^U\frac{1}{10}{\displaystyle {\sum }_{i=1}^{10}\left|v_{1,t-i}-v_{2,t-i}\right|}\right)}\right)}^{-1}$$

(8)

$${\tau }_t^L={\left(1+e^{-\left({\delta }^L+{\varphi }^L{\tau }_{t-1}^L+{\psi }^L\frac{1}{10}{\displaystyle {\sum }_{i=1}^{10}\left|v_{1,t-i}-v_{2,t-i}\right|}\right)}\right)}^{-1}$$

(9)

where ${\tau }_t^U$ and ${\tau }_t^L$ are the upper-tail dependence and lower-tail dependence between two assets, respectively. ${\delta }^U$, ${\varphi }^U$ and ${\psi }^U$ are the parameters of upper-tail dependence corresponding to Equation (8). ${\delta }^L$, ${\varphi }^L$ and ${\psi }^L$ are the parameters of lower-tail dependence.

Owing to the fact that the time-varying SJC Copula model has a dimensional constraint and can only be used to analyze the relationship between two markets, we next employed a three-dimensional vine Copula to incorporate three return series into one model for further examining their dependencies. The vine Copula model is extended into the SJC Copula, decomposing a multivariate Copula into a cascade of bivariate Copulas.

The C-vine Copula model (Joe et al. 2010):

$$f\left(v_1,v_2,v_3\right)={\displaystyle \prod _{i=1}^{3}f\left(v_i\right)}{\displaystyle \prod _{g=1}^2{\displaystyle \prod _{k=1}^{3-g}{c}_{g,g+k|1,\dots ,g-1}\left(F\left(v_g|v_1,\dots ,v_{g-1}\right),F\left(v_{k+g}|v_1,\dots ,v_{g-1}\right)\right)}}$$

(10)

where $f\left(v_1,v_2,v_3\right)$ is the joint density function, $i$ is the edge of tree $g$, $c_{g,g+k|1,\dots ,g-1}$ is the Copula density function and $F(.)$ is the marginal distribution function.

Copula models can only estimate the dependence between markets, and cannot recognize the direction of volatility spillovers, so we further employ the VAR-BEKK-GARCH model to examine cross-volatility spillover effects, concentrating on the direction of spillovers. The three-variable VAR (1)-BEKK-GARCH (1,1) model can be specified as below (Engle and Kroner 1995; Kroner and Ng 1998):

$$R_{{}^t}=\mu +{\mathsf{\Gamma }}_0{R}_t+{\mathsf{\Gamma }}_1{R}_t{}_{-1}+{\epsilon }_t$$

(11)

$$H_t=C^{\prime }C+A^{\prime }{\epsilon }_{t-1}{{\epsilon }^{\prime }}_{t-1}A+B^{\prime }{H}_{t-1}B+D^{\prime }{\xi }_{t-1}{{\xi }^{\prime }}_{t-1}D$$

(12)

where $R_{{}^t}$ is log returns matrix; $\mu$ is a 3 × 1 constant vector; ${\epsilon }_t$ is the N × 1 residual vector and ${\epsilon }_t\sim N\left(0,H_t\right)$; $H_t$ is the conditional variance–covariance matrix. $C$ is a 3 × 3 lower triangular constant matrix. $A$ is a 3 × 3 parameter matrix of conditional variance, where $a_{ij}$ captures the ARCH effect. $B$ is a 3 × 3 parameter matrix of residuals, where $b_{ij}$ captures the GARCH effect. $D$ is a 3 × 3 parameter matrix of leverage effects, where $d_{ij}$ captures the negative shock effect. If ${\epsilon }_t$ is negative, ${\xi }_t={\epsilon }_t$; otherwise, ${\xi }_t=0$.

3. Data and Preliminary Analysis

The Shanghai International Energy Exchange (INE) of China launched crude oil futures (hereinafter INE) on 26 March 2018. We obtained daily close prices of the Chinese INE crude oil futures from the Wind Database, which is one of the most commonly used databases by Chinese scholars. We collected daily close prices of the West Texas Intermediate crude oil futures (hereinafter WTI) from the U.S. Energy Information Administration website (https://www.eia.gov/, accessed on 27 September 2022). We took the CSI300 stock index as the proxy of the Chinese stock market (hereinafter CHS) and obtained daily close prices of the CSI300 index from the Wind Database. The period of our sample started from the first data available for the INE crude oil futures, namely, 26 March 2018 to 26 September 2022. We removed the missing values and an outlier from the original sample, consequently generating 1065 observations for price data and a corresponding 1064 observations for log return for each market1. Figure 1 depicts the evolution of the daily close price and log return for the three financial markets. Chinese stock, WTI and INE crude oil futures experienced a sharp price drop in December 2018 and March 2020, along with corresponding high volatilities in the log returns. Notably, the World Health Organization (WHO) announced the COVID-19 pandemic on 11 March 2020, which is marked with a green dotted line in Figure 1. It is obvious that there was a sharp decline in all three financial markets after the COVID-19 pandemic was announced.

Table 1. Descriptive statistics.

	Obs.	Mean	Midian	Min	Max	Std.	Skew.	Kurt.
Panel A: Total sample (27/03/2018~26/09/22)
CHS	1064	−0.001	0.002	−8.209	7.426	1.339	−0.267	6.366
INE	1064	0.033	0.076	−14.321	9.655	2.802	−0.454	5.743
WTI	1064	0.015	0.232	−60.168	31.963	4.081	−2.907	60.424
Panel B: Pre-COVID-19 (27/03/2018~10/03/2020)
CHS	466	0.011	−0.039	−8.209	5.778	1.398	−0.453	6.901
INE	466	−0.076	0.000	−10.615	8.269	2.320	−0.636	6.519
WTI	466	−0.139	0.082	−28.221	13.258	2.643	−2.701	32.217
Panel C: Post-COVID-19 (11/03/2020~26/09/22)
CHS	598	−0.010	0.024	−5.068	7.426	1.293	−0.086	5.735
INE	598	0.118	0.182	−14.321	9.655	3.125	−0.416	5.078
WTI	598	0.134	0.338	−60.168	31.963	4.917	−2.685	49.192

Notes: This table reports descriptive statistics for the return series of Chinese stock, INE crude oil futures and WTI oil crude oil futures. Panel A shows descriptive statistics for the total sample spanning from 27 March 2018 to 26 September 2022. Panel B shows descriptive statistics for the subsample spanning from 27 March 2018 to 10 March 2020, namely, the period before the COVID-19 pandemic was announced by the World Health Organization (WHO). Panel C reports descriptive statistics for the subsample spanning from 11 March 2020 to 26 September 2022, namely the period after the COVID-19 pandemic was announced by the WHO.

reports the descriptive statistics for the return series of the three financial markets, respectively. The total sample size of our study is 1064. To investigate whether there is a difference in the relationships among the three selected markets before and after the COVID-19 pandemic was announced on 11 March 2020, we split the total sample into two parts, namely, the pre-COVID-19 subsample (including 466 observations) and the post-COVID-19 subsample (including 598 observations). Panels A, B and C of Table 1 report the descriptive statistics for the total sample, and pre- and post-COVID-19 subsamples, respectively. The mean value of Chinese stock return (CHS) is positive and the mean values of two crude oil futures (INE and WTI) are negative in our total sample, shown in Panel A of Table 1. However, there is a difference between the pre- and post-COVID-19 subsamples. The mean value of CHS changed from positive to negative, while that of INE and WTI shifted from negative to positive, as reported in Panels B and C of Table 1. This implies that the main trend of Chinese stock return is decline, while the main trend of crude oil futures is one of increase. In sum, the skewness values of the three variables are all negative, and their kurtosis values are all positive.

Table 2. Basic statistical tests.

	JB	DF	ADF	PP	Vratio	LBQ(1)	LBQ(10)	ARCH
Panel A: Total sample (27/03/2018~26/09/22)
CHS	514.990 ***	−32.995 ***	−23.023 ***	−32.995 ***	−11.529 ***	9.406 ***	39.505 ***	9.368 ***
	(0.001)	(0.001)	(0.001)	(0.001)	(0.000)	(0.002)	(0.000)	(0.002)
INE	370.080 ***	−32.887 ***	−22.800 ***	−32.887 ***	−9.604 ***	98.318 ***	317.755 ***	49.128 ***
	(0.001)	(0.001)	(0.001)	(0.001)	(0.000)	(0.000)	(0.000)	(0.000)
WTI	147,690.000 ***	−33.789 ***	−25.346 ***	−33.789 ***	−3.113 ***	49.232 ***	246.997 ***	98.186 ***
	(0.001)	(0.001)	(0.001)	(0.001)	(0.002)	(0.000)	(0.000)	(0.000)
Panel B: Pre-COVID-19 (27/03/2018~10/03/2020)
CHS	311.390 ***	−22.414 ***	−16.100 ***	−22.414 ***	−7.216 ***	5.628 **	13.366	5.619 ***
	(0.001)	(0.001)	(0.001)	(0.001)	(0.000)	(0.018)	(0.204)	(0.018)
INE	271.860 ***	−21.517 ***	−15.377 ***	−21.517 ***	−5.325 ***	23.703 ***	24.504 ***	27.645 ***
	(0.001)	(0.001)	(0.001)	(0.001)	(0.000)	(0.000)	(0.006)	(0.000)
WTI	17,141.000 ***	−21.722 ***	−15.562 ***	−21.722 ***	−5.349 ***	25.607**	43.693 ***	23.503 ***
	(0.001)	(0.001)	(0.001)	(0.001)	(0.000)	(0.042)	(0.000)	(0.000)
Panel C: Post-COVID-19 (11/03/2020~26/09/22)
CHS	187.160 ***	−24.014 ***	−16.421 ***	−24.014 ***	−9.193 ***	2.986 *	44.517 ***	2.963 ***
	(0.001)	(0.001)	(0.001)	(0.001)	(0.000)	(0.084)	(0.000)	(0.085)
INE	124.810 ***	−25.310 ***	−17.587 ***	−25.310 ***	−8.005 ***	54.373 ***	165.251 ***	13.985 ***
	(0.001)	(0.001)	(0.001)	(0.001)	(0.000)	(0.000)	(0.000)	(0.000)
WTI	53,884.000 ***	−25.332 ***	−19.244 ***	−25.332 ***	−2.545 ***	13.093 ***	127.843 ***	55.213 ***
	(0.001)	(0.001)	(0.001)	(0.001)	(0.011)	(0.000)	(0.000)	(0.000)

Notes: This table reports basic statistical tests for the return series of Chinese stock, INE crude oil futures and WTI oil crude oil futures. Panel A shows basic statistical tests for the total sample spanning from 27 March 2018 to 26 September 2022. Panel B shows basic statistical tests for the subsample spanning from 27 March 2018 to 10 March 2020, namely, the period before the COVID-19 pandemic was announced by the World Health Organization (WHO). Panel C reports basic statistical tests for the subsample spanning from 11 March 2020 to 26 September 2022, namely the period after the COVID-19 pandemic was announced by the WHO. Jarque-Bera (JB) test is for testing normality. Dickey-Fuller (DF) test, Augmented Dickey-Fuller (ADF) test, Phillips-Perron (PP) test and Vratio test are for testing stationarity. Ljung-Box Q (LBQ) test is for testing autocorrelation in the residual series. ARCH test is for testing heteroscedasticity, with the null hypothesis of no residual heteroscedasticity. p-value is in parentheses. 0.000 means the p-value less than 0.001. *, ** and *** respectively indicate the significance level at 10%, 5% and 1%.

reports some basic statistical tests for the return series of the three financial markets, respectively. All return series passed the JB test at the 1% significance level, implying that neither of their distributions followed a normal distribution. According to the results of the DF, ADF, PP and V ratio tests, all the statistics significantly rejected the unit root null hypothesis at the 1% level, in favor of the stationary time series. Except for CHS in the pre-COVID-19 subsample, the Ljung–Box Q tests reject the null hypothesis of no autocorrelation, and the results of the ARCH tests for heteroscedasticity are all statistically significant for the total- and sub-samples. This indicates that GARCH-type models are suitable for our study.

4. Major Findings and Discussion

4.1. Empirical Results of Copula Models

Table 3. Estimate results of marginal distribution based on the AR (1)-GARCH (1,1) model (total sample).

	CHS	WTI	INE
$c_0$	0.0168	0.2077 ***	0.1498 **
	(0.0420)	(0.0640)	(0.0660)
$c_1$	−0.0423	−0.0201	−0.0261
	(0.0280)	(0.0270)	(0.0330)
$\omega$	0.0622 ***	0.1793 ***	0.1793 ***
	(0.0240)	(0.0570)	(0.0690)
${\phi }_1$	0.0564 ***	0.1364 ***	0.1263 ***
	(0.0140)	(0.0310)	(0.0320)
${\phi }_2$	0.9086 ***	0.8635 ***	0.8635 ***
	(0.0190)	(0.0160)	(0.0300)
$\gamma$	5.8507 ***	3.6359 ***	5.0828 ***
	(1.1060)	(0.4340)	(0.7830)
AIC	3506.2846	4955.2396	4977.2417
BIC	3536.1034	4985.0583	5007.0604
LL	−1747.142	−2471.62	−2482.621

Notes: This table reports the estimated parameters of the AR (1)-GARCH (1,1) model for each return series of the total sample from 27 March 2018 to 26 September 2022. The AR (1)-GARCH (1,1) model is specified in Equations (1)–(3). CHS represents Chinese stock, taking the CSI300 stock index as the proxy. WTI represents the global WTI crude oil futures and INE represents Chinese INE crude oil futures. $c_0$ and $c_1$ are the parameters of the AR (1) model, $\omega$,${\phi }_1$ and ${\phi }_2$ are the parameters of the GARCH (1,1) model with t-distribution. $\gamma$ is the parameter of the t distribution. AIC and BIC correspond to the Akaike criteria and the Swcharz criteria. LL is the value of the log-likelihood function at the optimum. Standard errors are shown in parentheses. ** and ***, respectively indicate the significance level at 5% and 1%.

shows the estimated parameters of the AR (1)-GARCH (1,1) model with the total sample. The AR (1)-GARCH (1,1) model is formed from Equations (1)–(3). The purpose of constructing the AR (1)-GARCH (1,1) model for each return series is to obtain marginal distributions for the estimation of Copula models, transforming original return data into data following a uniform distribution within [0,1]. Most parameters are statistically significant, as shown in Table 3, which means that the non-linear model is suitable.

Table 4. Estimate results of marginal distribution based on the AR (1)-GARCH (1,1) model (pre-and post-COVID-19 subsamples).

Variables	Pre-COVID-19			Post-COVID-19
	(1)	(2)	(3)	(4)	(5)	(6)
	CHS	WTI	INE	CHS	WTI	INE
$c_0$	0.0239	0.1149	0.1148	0.0076	0.3242	0.1950 *
	(0.0530)	(0.0790)	(0.0860)	(0.0220)	(0.6370)	(0.1040)
$c_1$	−0.0480	−0.0523	0.0122	−0.0367	−0.0133	−0.0384
	(0.0420)	(0.0420)	(0.1940)	(0.0420)	(4.7210)	(0.0400)
$\omega$	0.0302	0.1953 *	0.1953 *	0.0886 **	0.1672	0.1672 **
	(0.0260)	(0.1020)	(0.1120)	(0.0410)	(0.9720)	(0.0850)
${\phi }_1$	0.0447 ***	0.0808 ***	0.0980	0.0651 ***	0.1651	0.1173 ***
	(0.0160)	(0.0270)	(0.0620)	(0.0220)	(0.3930)	(0.0360)
${\phi }_2$	0.9429 ***	0.9192 ***	0.8806 ***	0.8779 ***	0.8349 ***	0.8730 ***
	(0.0190)	(0.0210)	(0.0690)	(0.0340)	(0.1200)	(0.0260)
$\gamma$	4.7481 ***	2.8291 ***	3.7458 ***	7.6107 ***	4.5109 ***	6.9385 ***
	(1.1390)	(0.4030)	(0.7900)	(2.3540)	(1.6980)	(1.5910)
AIC	1568.1304	2000.8866	2011.5246	1949.3144	2951.0605	2972.8096
BIC	1592.9956	2025.7517	2036.3897	1975.6759	2977.422	2999.1712
LL	−778.065	−994.443	−999.762	−968.657	−1469.53	−1480.405

Notes: This table reports the estimated parameters of the AR (1)-GARCH (1,1) model for each return series of the sub-samples. The left three columns of this table show estimation results for the pre-COVID-19 subsample which spans from 27 March 2018 to 10 March 2020. While the right three columns of this table show estimation results for the post-COVID-19 subsample which spans from 11 March 2020 to 26 September 2022. The AR (1)-GARCH (1,1) model is specified in Equations (1)–(3). CHS represents Chinese stock, taking the Shenzhen–Shanghai 300 stock index as the proxy. WTI represents the global WTI crude oil futures and INE represents Chinese INE crude oil futures. $c_0$ and $c_1$ are the parameters of the AR (1) model, $\omega$,${\phi }_1$ and ${\phi }_2$ are the parameters of the GARCH (1,1) model with t-distribution. $\gamma$ is the parameter of the t distribution. AIC and BIC correspond to the Akaike criteria and the Swcharz criteria. LL is the value of the log-likelihood function at the optimum. Standard errors are shown in parentheses. *, ** and ***, respectively indicate the significance level at 10%, 5% and 1%.

shows the estimated parameters of the AR (1)-GARCH (1,1) model with the subsamples. The left three columns of this table show estimation results for the pre-COVID-19 subsample, while the right three columns of this table show estimation results for the post-COVID-19 subsample. Similar to the estimation results for the total sample, most parameters (especially $\omega$,${\phi }_1$ and ${\phi }_2$) are statistically significant at the 1% significant level, and the parameters of the t distribution ($\gamma$) are all significant at the 1% level. This means that the non-linear model is also suitable for subsamples.

Table 5. Estimation results of the time-varying SJC-Copula model for paired markets (total sample).

Variables	(1)	(2)	(3)
	CHS–WTI	CHS–INE	WTI–INE
${\delta }^U$	0.5176	0.6417	1.1326
	(2.3130)	(1.5090)	(1.3250)
${\varphi }^U$	−9.3829	−9.9999 *	−4.5870
	(10.4410)	(5.4780)	(5.3770)
${\psi }^U$	−0.8586 ***	−0.8885 ***	−0.9962 ***
	(0.0720)	(0.0750)	(0.0030)
${\delta }^L$	0.4269	0.0467	−1.7080
	(1.6150)	(0.1460)	(3.5410)
${\varphi }^L$	−10.0000	−0.5488	−4.5674
	(6.5290)	(0.5870)	(13.5380)
${\psi }^L$	−0.3500	0.9471 ***	−0.9131 ***
	(0.3690)	(0.0270)	(0.1580)
AIC	−48.3245	−51.7326	−113.8154
BIC	−18.5058	−21.9139	−83.9967
LL	30.162	31.866	62.908

Notes: This table shows the estimated parameters of the time-varying SJC-Copula model for the three paired markets, namely, CHS–WTI, CHS–INE and WTI–INE, for the total sample from 27 March 2018 to 26 September 2022, respectively. CHS represents Chinese stock, taking the CSI300 stock index as the proxy. WTI represents the global WTI crude oil futures and INE represents Chinese INE crude oil futures. The time-varying SJC-Copula model is specified in Equations (4)–(7).${\delta }^U,{\varphi }^U\mathrm{and} {\psi }^U$ are the parameters of upper-tail dependence corresponding to Equation (8).${\delta }^L,{\varphi }^L\mathrm{and} {\psi }^L$ are the parameters of lower-tail dependence corresponding to Equation (9). AIC and BIC correspond to the Akaike criteria and the Swcharz criteria. LL is the value of the log-likelihood function at the optimum. Standard errors are shown in parentheses. * and ***, respectively indicate the significance level at 10% and 1%.

reports the estimated parameters of the time-varying SJC Copula model with the total sample from 27 March 2018 to 26 September 2022. This model is specified in Equations (4)–(9), capturing the dependency relationship between the paired markets. Three paired markets, namely, Chinese stock and global crude oil futures (CHS–WTI), Chinese stock and Chinese crude oil futures (CHS–INE), and Chinese crude oil futures and global crude oil futures (WTI–INE), are considered in our study. Columns (1), (2) and (3) of Table 5 show the estimated parameters of higher- and lower-tail dependence for the paired markets, CHS–WTI, CHS–INE and WTI–INE, respectively.

Table 6. Estimation results of the time-varying SJC-Copula model for paired markets (pre-and post-COVID-19 subsamples).

Variables	Pre-COVID-19			Post-COVID-19
	(1)	(2)	(3)	(4)	(5)	(6)
	CHS–WTI	CHS–INE	WTI–INE	CHS–WTI	CHS–INE	WTI–INE
${\delta }^U$	1.8660	1.5336	2.5006 *	−0.3407	−0.3766	−0.3456
	(1.7490)	(1.6650)	(1.4580)	(467,477)	(0.7440)	(0.1500)
${\varphi }^U$	−9.9999	−9.0109	−7.8347	1.2855	−6.9836 ***	1.2873 **
	(8.7390)	(8.7120)	(20.2180)	(1,799,548)	(2.1730)	(0.5480)
${\psi }^U$	0.1531	0.3153	−1.0211 ***	1.0138	−0.8828 ***	0.9463 ***
	(0.2180)	(0.6710)	(0.0620)	(917,194)	(0.0610)	(0.0150)
${\delta }^L$	−0.2929	−0.2866	−0.6202	−1.6809	−0.1558	1.4237
	(0.1570)	(1.3580)	(5.0930)	(16,498)	(0.1030)	(1.2910)
${\varphi }^L$	1.1888 *	0.7201 *	−1.8067	−0.6375	0.6579	−9.9999
	(0.6620)	(3.7060)	(26.8220)	(2502.30)	(0.4580)	(6.9960)
${\psi }^L$	0.9530 ***	0.9440 ***	−0.9237 ***	2.6262	1.0241 ***	0.6185 ***
	(0.0240)	(0.1790)	(0.1320)	(36,013.1)	(0.0460)	(0.1390)
AIC	−49.9773	−16.283	−70.2036	−19.0421	−37.0759	−54.1353
BIC	−25.1122	8.5821	−45.3385	7.3194	−10.7143	−27.7738
LL	30.989	14.142	41.102	15.521	24.538	33.068

Notes: This table shows the estimated parameters of the time-varying SJC-Copula model with the subsamples for the three paired markets, namely, CHS–WTI, CHS–INE and WTI–INE, respectively. CHS represents Chinese stock, taking the CSI300 stock index as the proxy. WTI represents the global WTI crude oil futures and INE represents Chinese INE crude oil futures. The left three columns of this table show the estimation results for the pre-COVID-19 subsample, which spans from 27 March 2018 to 10 March 2020, while the right three columns of this table show estimation results for the post-COVID-19 subsample, which spans from 11 March 2020 to 26 September 2022. The time-varying SJC-Copula model is specified in Equations (4)–(7).${\delta }^U, {\varphi }^U \mathrm{and} {\psi }^U$ are the parameters of upper-tail dependence corresponding to Equation (8).${\delta }^L,{\varphi }^L \mathrm{and} {\psi }^L$ are the parameters of lower-tail dependence corresponding to Equation (9). AIC and BIC correspond to the Akaike criteria and the Swcharz criteria. LL is the value of the log-likelihood function at the optimum. Standard errors are shown in parentheses. *, ** and ***, respectively indicate the significance level at 10%, 5% and 1%.

reports the estimated parameters of the time-varying SJC Copula models for subsamples. Columns (1), (2) and (3) of Table 6 show the estimated parameters of higher- and lower-tail dependence for the paired markets (CHS–WTI, CHS–INE and WTI–INE) in the pre-COVID-19 subsample spanning from 27 March 2018 to 10 March 2020, respectively. Columns (4), (5) and (6) of Table 6 show the estimated parameters of higher- and lower-tail dependence for the three paired markets in the post-COVID-19 subsample spanning from 11 March 2020 to 26 September 2022, respectively.

The corresponding empirical time-varying upper-tail dependence and lower-tail dependence are depicted in Figure 2, Figure 3 and Figure 4, respectively. As shown in Figure 2, Chinese stock has a positive dynamic correlation with INE crude oil futures. The lower-tail dependence between CHS and INE is relatively stable and around 0.2. The upper-tail dependence of CHS–INE is time-varying, sometimes higher and sometimes lower than lower-tail dependence. From Figure 3, we can see that Chinese stock has a closer dependence on WTI crude oil futures. The upper- and lower-tail dependence of CHS–WTI grows stronger or weaker in correlation with price fluctuations. Figure 4 shows that there is a stronger upper-tail dependence between INE and WTI. The upper-tail dependence of INE–WTI fluctuates from 0.3 to 0.5.

Table 7. Estimation results of the SJC vine Copula model.

		Upper-Tail	Lower-Tail
Panel A: Total sample (27/03/2018~26/09/2022)
CHS–WTI	Tree1	0.2113 ***	0.1422 ***
		(0.0580)	(0.0440)
CHS–INE	Tree2	0.1860 ***	0.1116 **
		(0.0550)	(0.0480)
WTI–INE|CH	Tree3	0.3128 ***	0.0588
		(0.0460)	(0.0440)
AIC	−199.2789
BIC	−169.4602
LL	105.639
Panel B: Pre-COVID-19 (27/03/2018~10/03/2020)
CH–WTI	Tree1	0.2712 ***	0.3493 ***
		(0.0770)	(0.0510)
CH–INE	Tree2	0.1925 ***	0.2048 ***
		(0.0730)	(0.0680)
WTI–INE|CH	Tree3	0.4127 ***	0.0789
		(0.0620)	(0.0870)
AIC	−131.1717
BIC	−106.3065
LL	71.586
Panel C: Post-COVID-19 (11/03/2020~26/09/2022)
CH–WTI	Tree1	0.1753 **	0.0042
		(0.0790)	(0.0140)
CH–INE	Tree2	0.1896 **	0.0437
		(0.0790)	(0.0540)
WTI–INE|CH	Tree3	0.2566 ***	0.0516
		(0.0630)	(0.0490)
AIC	−83.7971
BIC	−57.4355
LL	47.899

Notes: This table reports the estimation results of the C-vine Copula model for the three financial markets. CHS represents Chinese stock, taking the CSI300 stock index as the proxy. WTI represents the global WTI crude oil futures and INE represents Chinese INE crude oil futures. For each paired market, the static SJC-Copula model is used to capture the upper-tail dependence and the lower dependence between them. The sign|is a delimiter departing the conditional paired markets and the root node. Panels A, B and C of this table show estimation results for the total sample (27/03/2018~26/09/2022), the pre-COVID-19 subsample (27/03/2018~10/03/2020) and the post-COVID-19 subsample (11/03/2020~26/09/2022), respectively. Standard error is shown in parentheses. ** and ***, respectively indicate the significance level at 5% and 1%.

shows the estimation results of the static vine Copula model with the total sample, and the pre- and post-COVID-19 subsamples, respectively. As shown in Panel A of Table 7, we can see that the dependence between Chinese stock (CHS) and global crude oil futures (WTI) is stronger than the dependence between Chinese stock and Chinese crude oil futures (INE), in that the upper-tail and lower-tail of CHS–WTI are 0.2113 and 0.1442, while those for CHS–INE are 0.1860 and 0.1116, respectively. Taking Chinese stock as the root node, INE and WTI have the relatively strongest upper-tail dependency relationship (Upper-tail = 0.3128), which is greater than their lower-tail dependence (lower-tail = 0.0588). We can see some shifts before and after the COVID-19 pandemic. When we re-estimate the static SJC vine Copula model with the pre-COVID-19 subsample, it is found that the lower-tail dependence of CHS–WTI (lower-tail = 0.3493) is stronger than the upper-tail dependence (upper-tail = 0.2712) before COVID-19, as reported in Panel B of Table 7. Nevertheless, the lower-tail dependence of CHS–WTI (lower-tail = 0.0042) becomes weaker than their upper-tail dependence (upper-tail = 0.1753) after the COVID-19 pandemic, which is shown in Panel C of Table 7.

In terms of the tail dependence of CHS–INE, similar shifts are obtained between the pre- and post-COVID-19 subsamples reported in Panel B and C of Table 7. The dependence of CHS–WTI is stronger than that of CHS–INE before COVID-19, while it is weaker than that of CHS–INE after COVID-19. Besides this, our empirical results show that all the coefficients of upper-tail dependence are statistically significant at least the 1% significance level. Except for estimates for the post-COVID-19 subsample and the lower-tail dependence of INE-WTI, all the coefficients of lower-tail dependence are also significant at the 1% level. This highlights the robustness of our conclusions on the dependent relationships among Chinese stock, Chinese crude oil futures and global crude oil futures.

4.2. Empirical Results of the VAR-BEKK-GARCH Model

Since the Copula model can only examine the dependence of different financial markets and cannot recognize the direction of shock or volatility spillovers, we next constructed a three-variable asymmetric VAR (1)-BEKK-GARCH (1,1) model to examine volatility spillover effects among them, where the optimal lag order for the VAR model was selected by the BIC information criteria shown in

Table 8. Lag selection for VAR (p) models.

Lags	Total Sample		Pre-COVID-19		Post-COVID-19
	AIC	SBC/BIC	AIC	SBC/BIC	AIC	SBC/BIC
0	13.8762	13.8903	12.7082	12.7352	14.2138	14.2360
1	13.5934	13.649 *	12.3392 *	12.4468 *	13.9319	14.0208 *
2	13.5713	13.6697	12.3460	12.5338	13.9090	14.0640
3	13.5425	13.6829	12.3696	12.6369	13.8721 *	14.0930
4	13.5377 *	13.7200	12.3852	12.7315	13.8721	14.1586
5	13.5436	13.7677	12.4155	12.8402	13.8809	14.2326
6	13.5552	13.8210	12.4350	12.9377	13.8969	14.3135
7	13.5449	13.8523	12.4450	13.0249	13.8926	14.3738
8	13.5519	13.9008	12.4700	13.1266	13.8841	14.4295

Notes: This table shows results of the lag order selection for the VAR(p) model with the total sample (27/03/2018~26/09/2022), the pre-COVID-19 subsample (27/03/2018~10/03/2020) and the post-COVID-19 subsample (11/03/2020~26/09/2022), respectively. The best lag order is selected by the BIC (SBC) and the AIC information, according to the rule of majority. * denotes the optimal lag order selected by the information criterion in the column, which is the smallest value in the same column.

.

Table 9. Preliminary estimation results of the VAR (1)-BEKK-GARCH (1,1) model for the total sample.

Spillover Effect			CHS (i = 1)	INE (i = 2)	WTI (i = 3)
ARCH	Matrix A	a1i	0.1893 ***	0.0959	−0.2691 ***
			(0.0359)	(0.1041)	(0.0662)
		a2i	0.0224	0.2928 ***	0.0523
			(0.0163)	(0.0550)	(0.0449)
		a3i	−0.0350 **	−0.5758 ***	0.2212 ***
			(0.0146)	(0.0839)	(0.0647)
GARCH	Matrix B	b1i	0.9575 ***	0.0556	0.0797 **
			(0.0117)	(0.0771)	(0.0381)
		b2i	−0.0216 *	0.5302 ***	0.1651 **
			(0.0121)	(0.0591)	(0.0670)
		b3i	0.0024	0.0806 **	0.8853 ***
			(0.0040)	(0.0330)	(0.0220)
Leverage	Matrix D	d1i	0.2152 ***	0.3472 *	0.1749
			(0.0653)	(0.1814)	(0.1300)
		d2i	−0.0898 ***	−0.3240 ***	0.0424
			(0.0198)	(0.1014)	(0.0726)
		d3i	−0.0158	−0.3102 *	−0.3027 ***
			(0.0152)	(0.1745)	(0.0901)

Notes: This table reports preliminary estimation results of the asymmetric VAR (1)-BEKK-GARCH (1,1) model with t distribution for the total sample from 27 March 2018 to 26 September 2022. CHS represents Chinese stock, taking the CSI300 stock index as the proxy. WTI represents the global WTI crude oil futures and INE represents Chinese INE crude oil futures. The VAR (1)-BEKK-GARCH (1,1) is specified as Equations (11) and (12). A and B are 3 × 3 matrices of conditional variance and residual, respectively, which can measure shock transmission (ARCH effect) and volatility spillover (GARCH effect). D is the 3 × 3 matrix measuring leverage effects. The model is estimated by the method of BFGS. Standard errors are shown in parentheses. *, ** and ***, respectively indicate the significance level at 10%, 5% and 1%.

reports the empirical results of the VAR (1)-BEKK-GARCH (1,1) model with the total sample from 27 March 2018 to 26 September 2022. As shown in Table 9, the diagonal coefficients of matrix A (a₁₁, a₂₂ and a₃₃) and matrix B (b₁₁, b₂₂ and b₃₃) for all markets are significant. This means that the conditional volatilities of these markets are affected by their own lagged shock and volatility. Moreover, the GARCH coefficients (b₁₁ = 0.9575, b₂₂ = 0.5302, b₃₃ = 0.8853) are relatively larger than the corresponding ARCH coefficients (a₁₁ = 0.1893, a₂₂ = 0.2928, a₃₃ = 0.2212), respectively. This indicates that the three assets are more sensitive to their continuous lagged fluctuations. The off-diagonal coefficients of matrix A of Table 9 provide information about across-market shock transmissions. The coefficient of a₃₁ (a₃₁ = −0.0350) is statistically significant at the 5% level, suggesting that WTI has significant shock spillover effects on Chinese stock (CHS). The coefficient of a₁₃ (a₁₃ = −0.2691) is statistically significant at the 1% level, suggesting that CHS also has significant shock spillover effects on WTI. The coefficient of a₂₃ (a₂₃ = 0.0523) is not statistically significant but the coefficient of a₃₂ (a₃₂ = −0.5758) is significant at the 1% level. This means that there are unilateral shock spillovers from WTI to INE. However, there is no shock spillover between INE and CHS, in that both the coefficients of a₁₂ and a₂₁ are statistically insignificant.

Next, we move our attention to the off-diagonal coefficients in matrix B of Table 9, which provides information about across-market volatility spillovers. The coefficient of b₁₃ (b₁₃ = 0.0797) is significant at the 1% level but b₃₁ (b₃₁ = 0.0024) is not statistically significant, highlighting a unilateral volatility spillover from CHS to WTI. The coefficients of b₂₃ (b₂₃ = −0.0216) and b₂₃ (b₃₂ = 0.0806) are, respectively significant at the 10% and 5% levels, implying bidirectional volatility spillover effects between INE and WTI. The coefficient of b₂₁ (b₂₁ = −0.0216) is significant at the 10% level but b₁₂ (b₁₂ = 0.0556) is not statistically significant, highlighting unilateral volatility spillovers from INE to CHS. Lastly, we turn to the coefficients of matrix D in Table 9, which estimate the across-market asymmetric leverage effects. We find bidirectional leverage spillover effects between CHS and INE (d₁₂ and d₂₁ are statistically significant), implying negative information from either of the two markets can shock the other. Meanwhile, d₃₁ and d₃₁ are not statistically significant, suggesting that negative information in either CHS or WTI cannot influence the other significantly.

Next, we examine whether there is a difference in the spillover effects between the pre- and post-COVID-19 subsamples.

Table 10. Preliminary estimation results of the VAR (1)-BEKK-GARCH (1,1) model for the pre- and post- COVID-19 subsamples.

Spillover Effect			CHS (i = 1)	INE (i = 2)	WTI (i = 3)
Panel A: Pre-COVID-19 (27/03/2018~10/03/2020)
ARCH	Matrix A	a1i	0.0894	0.0304	−0.1640
			(0.1087)	(0.2132)	(0.1906)
		a2i	0.0265	0.3328 ***	0.1029
			(0.0363)	(0.0837)	(0.0790)
		a3i	0.0242	−0.4270 ***	0.3386 ***
			(0.0298)	(0.0960)	(0.0658)
GARCH	Matrix B	b1i	0.9086 ***	−0.0214	−0.0059
			(0.0426)	(0.1183)	(0.1369)
		b2i	0.0438	0.2845 **	0.1725
			(0.0435)	(0.1263)	(0.1183)
		b3i	−0.1217 *	0.6499 ***	0.2191
			(0.0705)	(0.0779)	(0.2303)
Leverage	Matrix D	d1i	0.5161 ***	0.5413 ***	0.1936
			(0.1035)	(0.1954)	(0.1956)
		d2i	−0.0968	0.0595	−0.3391 **
			(0.0602)	(0.1450)	(0.1401)
		d3i	−0.1405 ***	−0.8082 ***	−0.0419
			(0.0433)	(0.1392)	(0.1515)
Panel B: Post-COVID-19 (11/03/2020~26/09/2022)
ARCH	Matrix A	a1i	0.1272 ***	0.0921	−0.3348 ***
			(0.0481)	(0.1463)	(0.0816)
		a2i	0.0306	0.1752 **	0.2211 ***
			(0.0205)	(0.0850)	(0.0463)
		a3i	−0.0386 **	−0.6847 ***	−0.0711
			(0.0176)	(0.0778)	(0.0994)
GARCH	Matrix B	b1i	0.9428 ***	0.0706	0.0929 **
			(0.0197)	(0.1751)	(0.0452)
		b2i	−0.0265 *	0.3537 ***	−0.0314
			(0.0160)	(0.0990)	(0.1234)
		b3i	0.0070	0.1289 ***	0.9211 ***
			(0.0050)	(0.0456)	(0.0306)
Leverage	Matrix D	d1i	0.2875 ***	−0.0586	0.1451
			(0.0586)	(0.2313)	(0.1601)
		d2i	−0.1155 ***	−0.4684 ***	0.0323
			(0.0274)	(0.1040)	(0.0953)
		d3i	0.0242	0.2639	−0.4195 ***
			(0.0161)	(0.2132)	(0.0737)

Notes: This table reports preliminary estimation results of the asymmetric VAR (1)-BEKK-GARCH (1,1) model with t distribution for the pre- and post- COVID-19 subsamples. Panel A of this table shows estimation results for the pre-COVID-19 subsample which spans from 27 March 2018 to 10 March 2020. While Panel B shows estimation results for the post-COVID-19 subsample which spans from 11 March 2020 to 26 September 2022. CHS represents Chinese stock, taking the CSI300 stock index as the proxy. WTI represents the global WTI crude oil futures and INE represents Chinese INE crude oil futures. The VAR (1)-BEKK-GARCH (1,1) is specified as Equations (11) and (12). A and B are 3 × 3 matrices of conditional variance and residual, respectively, which can measure shock transmission (ARCH effect) and volatility spillover (GARCH effect). D is the 3 × 3 matrix measuring leverage effects. The model is estimated by the method of BFGS. Standard errors are shown in parentheses. *, ** and ***, respectively indicate the significance level at 10%, 5% and 1%.

reports the empirical results of the VAR (1)-BEKK-GARCH (1,1) model with the two subsamples. Similar to the total sample, most of the diagonal coefficients of matrix A (a₁₁, a₂₂ and a₃₃) and matrix B (b₁₁, b₂₂ and b₃₃) are statistically significant, as reported in Table 10, implying the impact of lagged shock and volatilities in all three of the financial markets, namely, Chinese stock, and Chinese and global crude oil futures. Let us concentrate on the off-diagonal coefficients, comparing the coefficients for the pre-COVID-19 subsample shown in Panel A of Table 10 with the corresponding ones for the post-COVID-19 subsample reported in Panel B of Table 10. In terms of the shock spillover estimated in matrix A, there is no statistically significant and bidirectional shock spillover between CHS and INE, before or after COVID-19, which is in line with the corresponding results of the total sample reported in Table 9. On the coefficients of a₁₃ and a₃₁, there is no statistically significant shock spillover between CHS and WTI before COVID-19, while the shock spillover between them is significantly bidirectional after the COVID-19 pandemic. From the coefficients of a₂₃ and a₃₂, the shock spillover between INE and WTI is unidirectional (from WTI to INE) before COVID-19 but bidirectional after COVID-19.

In terms of the volatility spillover estimated in matrix B, there is no statistically significant volatility spillover between CHS and INE before COVID-19, while INE spills volatilities over to CHS significantly after COVID-19. From the coefficients of b₁₃ and b₃₁, the volatility significantly spills over from WTI to CHS before COVID-19, but it spills over from CHS to WTI after COVID-19. Interestingly, the direction of volatility spillover between INE and WTI is from WTI to INE, both before and after COVID-19, which is seen from the coefficients of b₂₃ and b₃₂. In terms of the leverage effects shown in matrix D of Table 10, bad news in the CHS will have affected INE significantly before COVID-19, while the direction was reversed after COVID-19. Before COVID-19, negative information from WTI had a significant impact on CHS, but this was not statistically significant after COVID-19. The leverage effect between INE and WTI was only significantly bidirectional before COVID-19.

In order to test the robustness of the empirical results above, we continue to use Wald tests to examine the volatility spillovers. For a different purpose, we assume different null hypotheses, as shown in the left column of

Table 11. Wald test results.

Null	(1)	(2)	(3)
	Total Sample	Pre-COVID-19	Post-COVID-19
	F-Statistics	F-Statistics	F-Statistics
no asymmetric volatility spillover effects (aij = 0, bij = 0, i ≠ j)	12.6640 ***	11.6208 ***	19.3840 ***
	(0.0000)	(0.0000)	(0.0000)
no asymmetric volatility spillovers from CHS (a12 = 0, a13 = 0, b12 = 0, b13 = 0)	5.3317 ***	0.2459	4.7553 ***
	(0.0003)	(0.9123)	(0.0008)
no volatility spillovers from CHS to INE (a12 = 0, b12 = 0)	0.7700	0.0286	0.4219
	(0.4630)	(0.9718)	(0.6558)
no volatility spillovers from CHS to WTI (a13 = 0, b13 = 0)	8.3547 ***	0.3966	9.3721 ***
	(0.0002)	(0.6726)	(0.0001)
no asymmetric volatility spillovers from INE (a21 = 0, a23 = 0, b21 = 0, b23 = 0)	4.8812 ***	1.0263	17.3912 ***
	(0.0006)	(0.3920)	(0.0000)
no volatility spillovers from INE to CHS (a21 = 0, b21 = 0)	1.9688	0.9125	2.9326 *
	(0.1396)	(0.4015)	(0.0533)
no volatility spillovers from INE to WTI (a23 = 0, b23 = 0)	7.2731 ***	1.9779	32.0453 ***
	(0.0007)	(0.1384)	(0.0000)
no asymmetric volatility spillovers from WTI (a31 = 0, a32 = 0, b31 = 0, b32 = 0)	13.3231 ***	28.3947 ***	24.2876 ***
	(0.0000)	(0.0000)	(0.0000)
no volatility spillovers from WTI to CHS (a31 = 0, b31 = 0)	3.4349 **	1.6650	4.1909 **
	(0.0322)	(0.1892)	(0.0151)
no volatility spillovers from WTI to INE (a32 = 0, b32 = 0)	24.9127 ***	44.1057 ***	46.0429 ***
	(0.0000)	(0.0000)	(0.0000)

Notes: This table shows the results of the Wald test for volatility spillovers among Chinese stock, WTI crude oil futures and INE crude oil futures. CHS represents Chinese stock market, taking the CSI300 stock index as the proxy. WTI represents the global WTI crude oil futures and INE represents Chinese INE crude oil futures. Columns (1), (2) and (3) of this table show the Wald test results for the total sample (27/03/2018~26/09/2022), the pre-COVID-19 subsample (27/03/2018~10/03/2020) and the post-COVID-19 subsample (11/03/2020~26/09/2022), respectively. The F-statistics are reported in the column and p-values are reported in parentheses. 0.0000 means a p-value less than 0.0001. *, ** and ***, respectively indicate the significance level at 10%, 5% and 1%.

. The Wald test results reject the null hypothesis of no asymmetric volatility spillover effects, in favor of volatility spillovers among these three financial markets. The right columns (1), (2) and (3) of Table 11 display the Wald test results for the total sample, and the pre- and post- COVID-19 subsamples, respectively. For the total sample, only the null hypothesis of no asymmetric spillovers from INE to CHS and the null hypothesis of no asymmetric spillovers from CHS to INE cannot be rejected significantly, suggesting that there is no volatility spillover between them. All the other Wald test results for the total sample are statistically significant at the 10% or 5% level, suggesting bidirectional volatility spillovers between the two paired markets, namely, CHS–WTI and INE–WTI.

Before the COVID-19 pandemic, the Wald test results for the null hypothesis of no asymmetric spillovers from WTI and the null hypothesis of no asymmetric spillovers from WTI to INE were the only expectations; the outlier is that they were statistically significant at the 1% level, while other hypotheses cannot be rejected. This indicates that there was only a significant volatility spillover from WTI to INE. After the COVID-19 pandemic, most of the Wald test results are statistically significant. Specifically, the volatility spillover effects between WTI and CHS are significantly bidirectional after COVID-19. The volatility spillover effects between WTI and INE are also significantly bidirectional after COVID-19, and WTI dominates the volatility spillover effects. This is because the F-statistic for the hypothesis of no asymmetric spillovers from WTI to INE is 46.0429, while that for the hypothesis of no asymmetric spillovers from INE to WTI is 32.0453, which can be seen in column (3) of Table 11.

As regards the volatility spillover from crude oil futures to the Chinese stock market, the Wald test results show that the spillover effect of WTI to CHS is stronger than the spillover effect of INE to CHS, by comparing the size of F-statistics. Besides, the Wald test for the null hypothesis of no volatility spillovers from INE to CHS is statistically significant at a 10% level which can be seen in column (3) of Table 11, highlighting significant unidirectional spillovers from INE to CHS after the breakout of COVID-19 pandemic. As a whole, the Wald test results are in line with the analysis of the coefficients of the BEKK-GARCH model above, making our results more robust.

5. Conclusions

We combined Copula-type models and the VAR-BEKK-GARCH model to investigate dependencies and volatility spillovers among three selected markets, namely, the Chinese stock market, the Chinese INE crude oil future market and the global WTI crude oil future market, taking a sample spanning from 26 March 2018 to 26 September 2022. Based on the AR (1)-GARCH (1,1) model for constructing marginal distribution for each return series, we firstly used the bivariable time-varying SJC-Copula model to capture dynamic lower- and upper-tail dependencies between the three pairs of markets (namely, Chinese stock–WTI, Chinese stock–INE and WTI–INE). Then, incorporating the three variables into a higher-dimension C-vine Copula model, we examined their static dependency relationships with the whole sample period, and the pre- and post-COVID-19 subsamples, respectively. We finally employed a three-variable VAR-BEKK-GARCH model to estimate the direction of cross-market volatility spillovers among them.

There were some interesting empirical results obtained from our study. Firstly, the correlations between Chinese stock and crude oil futures were positive and time-varying. The lower-tail dependence coefficients of the Chinese stock and crude oil future markets were larger than their upper-tail dependencies before the COVID-19 pandemic. However, after the breakout of the COVID-19 pandemic, the upper-tail dependence coefficients between Chinese stock and crude oil future markets were larger than their lower-tail dependencies. INE crude oil futures highly depend on the global WTI crude oil futures in the upper-tail part. Further, with respect to volatility spillover effects, we found that the Chinese stock market and WTI crude oil future market showed bidirectional volatility spillovers, and the volatility spillovers between the INE and WTI crude oil future markets were also bidirectional in the total sample, while the volatility spillovers between the Chinese stock market and Chinese crude oil futures were statistically unidirectional from INE to CHS. Before the COVID-19 pandemic, there was only a statistically significant volatility spillover from the WTI crude oil future market to the INE crude oil future market. After the breakout of the COVID-19 pandemic, there were statistically significant volatility spillovers in two pairs of markets, namely, WTI–INE and Chinese stock–WTI. The empirical finding that the global WTI crude oil future market was dominant in the volatility spillovers is in line with the findings of Yang et al. (2021) and Li et al. (2022). However, we only found statistically significant evidence of unidirectional volatility spillovers from the Chinese crude oil future market to the Chinese stock market after the COVID-19 breakout.

Our study offers some innovations and contributions. For example, it is the first study to simultaneously examine the dependencies among three markets, namely, the Chinese stock market, Chinese crude oil futures and the global crude oil future market. Differently from the methodologies used in the previous related literature, we used an integrated methodology joining Copula-type models and the VAR-BEKK-GARCH model. Further, we not only examined the dependencies and volatility spillovers among the three markets with the total sample but also explored shifts in the relationships before and after the breakout of the COVID-19 pandemic. These findings are meaningful in the context of observing the status of Chinese crude oil futures in the world and their impacts on the stock market. They also provide a glimpse of the impact of the COVID-19 pandemic on the co-movement of financial markets. However, there is still some space for improvement in future studies. For instance, we can further investigate the role of WTI and INE in reducing the downside risk and constructing optimal portfolios, using methodologies similar to those of Ali et al. (2021, 2022). Since we did not find evidence of significantly bidirectional volatility spillovers between the Chinese crude oil future market and the Chinese stock market in this study, we could try to re-examine the linkage when more INE data are available in the future, or we could further investigate the dependencies between INE and other financial markets in China.