### Linear model

Following Chow et al. (2011), we construct a simple linear model to test the impact of the U.S. stock market on the Chinese stock market:

$$ {R}_{m,t}^c={\beta}_{m,0}+{\beta}_{m,1}{R}_{m,t-1}^c+{\beta}_{m,2}{R}_{t-1}^{s\&p}+{\varepsilon}_{m,t}, $$

(1)

$$ {R}_{a,t}^c={\beta}_{a,0}+{\beta}_{a,1}{R}_{a,t-1}^c+{\beta}_{a,2}{R}_{t-1}^{s\&p}+{\varepsilon}_{a,t}, $$

(2)

$$ {R}_{d,t}^c={\beta}_{d,0}+{\beta}_{d,1}{R}_{d,t-1}^c+{\beta}_{d,2}{R}_{t-1}^{s\&p}+{\varepsilon}_{d,t}, $$

(3)

where \( {R}_{m,t}^c \), \( {R}_{a,t}^c \), and \( {R}_{d,t}^c \) denote the semi-day logarithmic return in the morning, semi-day logarithmic return in the afternoon and daily logarithmic return of the CSI 300 Index (CSI300), respectively, and \( {R}_t^{s\&p} \) represents the daily return of the S&P 500 Index (SPX). *β*_{i, 2} measures the return spillovers from the U.S. stock market to the Chinese stock market. The null hypothesis is that the performance of the U.S. stock market on day *t*–1 (\( {R}_{t-1}^{s\&p} \)) cannot predict the return of the Chinese stock market on day *t*; in other words, for all *i = m*, *a*, *d*, *β*_{i, 2} = 0.

This linear model assumes that *ε*_{i, t} is a white noise process, neglecting heteroskedasticity. We construct more specific models based on different assumptions in Section 2.2.

### The multivariate GARCH model

To better measure the comovements between these two markets, we build a multivariate GARCH (MGARCH) model with DCC and asymmetric BEKK specifications in conditional variance functions. Before illustrating the conditional variance function, we use the function mentioned in Section 2.1 and build our conditional mean function, as follows:

$$ {R}_{i,t}={\mu}_i+{\Gamma}_i{R}_{i,t-1}+{\varepsilon}_{i,t},\kern0.4em i=m,a,d, $$

(4)

where \( {R}_{i,t}={\left({R}_{i,t}^c,{R}_t^{s\&p}\right)}^{\prime } \), \( {R}_{i,t}^c \) and \( {R}_t^{s\&p} \) are defined as above. In addition, we have

$$ {\mu}_i=\left(\begin{array}{c}{\mu}_{i,1}\\ {}{\mu}_{i,2}\end{array}\right),{\Gamma}_i=\left(\begin{array}{cc}{\gamma}_{i,11}& {\gamma}_{i,12}\\ {}{\gamma}_{i,21}& {\gamma}_{i,22}\end{array}\right),\kern0.5em {\varepsilon}_{i,t}=\left(\begin{array}{c}{\varepsilon}_{i,1t}\\ {}{\varepsilon}_{i,2t}\end{array}\right), $$

where *μ*_{i} is the constant vector, and *Γ*_{i} is the matrix of parameters that represent return spillover effects between the two markets. To be more specific, *γ*_{i, mn}, the diagonal element of *Γ*_{i}, measures the impact of its past returns, while *γ*_{i, mn}, the off-diagonal element of *Γ*_{i}, captures the influence of the past return of market *n* on the current return of market *m*. Thus, we should pay close attention to the significance of *γ*_{i, mn}. The variable *ε*_{i, t}, whose conditional covariance matrix is *H*_{i, t}, is the random error on day *t*. *H*_{i, t} is described by the DCC-GARCH and asymmetric BEKK-GARCH models.

Bollerslev et al. (1988) propose an MGARCH model, known as the General Vech GARCH model:

$$ vech\left({H}_t\right)= vech(C)+\sum \limits_{i=1}^q{A}_i vech\left({\varepsilon}_{t-i}{\varepsilon_{t-i}}^{\prime}\right)+\sum \limits_{i=1}^q{G}_i vech\left({H}_{t-i}\right), $$

where *vech* is the operator that stacks the lower triangular portion of a symmetric matrix into a vector.

However, the number of parameters to be estimated in this MGARCH is typically large. In addition, some restrictions are imposed on parameters to satisfy the positive definite property of the conditional variance matrix.

To solve these problems, many parametric formulations are introduced for the structure of the conditional variance-covariance matrices. Tse and Tsui (2002) propose the DCC-GARCH model, and Engle and Kroner (1995) introduce the BEKK model, which have been widely used, and both of these models effectively solve these problems.

#### DCC-GARCH model

Based on Tse and Tsui (2002), the DCC-GARCH model is applied for the conditional variance function. The mean function is

$$ {R}_{i,t}={\mu}_i+{\Gamma}_i{R}_{i,t-1}+{\varepsilon}_{i,t},\kern0.4em i=m,a,d, $$

and the variance function is

$$ {\displaystyle \begin{array}{c}{\upvarepsilon}_{i,t}\mid {\Omega}_{t-1}\sim D\left(0,{H}_{i,t}\right)\\ {}{H}_{i,t}={D}_{i,t}^{\prime }{R}_{i,t}{D}_{i,t}\end{array}}, $$

(5)

where \( {H}_{i,t}=\left(\begin{array}{cc}{h}_{i,11,t}& {h}_{i,12,t}\\ {}{h}_{i,21,t}& {h}_{i,22,t}\end{array}\right),{D}_{i,t}=\left(\begin{array}{cc}{h}_{i,11,t}& 0\\ {}0& {h}_{i,22,t}\end{array}\right),{R}_{i,t}=\left(\begin{array}{cc}1& {\rho}_{i,12,t}\\ {}{\rho}_{i,12,t}& 1\end{array}\right) \). *H*_{i, t}, *D*_{i, t} and *R*_{i, t} are the conditional covariance, variance, and correlation matrix of *ε*_{i, t}, respectively, and *Ω*_{t − 1} denotes the conditional information set at time *t* − 1.

The conditional variance of each market in *D*_{i, t} follows the univariate GARCH(1, 1) process. In other words, we have

$$ {h}_{i, jj,t}={c}_j+{\alpha}_j{\varepsilon}_{i,j,t-1}+{\beta}_j{h}_{i, jj,t-1},\kern1em j=1,2, $$

(6)

and *R*_{i, t} depends on

$$ {R}_{i,t}=\left(1-{\theta}_1-{\theta}_2\right){R}_i+{\theta}_1{\Psi}_{i,t-1}+{\theta}_2{R}_{i,t-1}, $$

(7)

where *R*_{i} is a symmetric positive definite constant matrix, and *Ψ*_{i, t} is a matrix of *ε*_{i, t} whose elements represent weighted averages of residuals (see the concrete elements of *Ψ*_{i, t} in Tse and Tsui (2002)). This structure of *H*_{i, t} guarantees that *R*_{i, t} is positive definite.

The above DCC-GARCH model is estimated with a two-step method. Applying the DCC-GARCH model, we can calculate the time-varying conditional correlation between the returns of two markets, giving us an excellent opportunity to explore the characteristics of the comovements of the two markets over different periods of time (morning, afternoon and whole day). This approach allows us to not only test the significance of the return spillovers but also assess the strength of the return spillovers. Specifically, the higher the time-varying correlation is, the stronger the linkage between two markets’ returns, and vice visa.

However, since the conditional variance of each market in period *t* is determined by only its variance and error term in period *t*-1, there is no straightforward parameter that can be interpreted as the volatility spillover effects of the two markets. Therefore, we need to build a model to measure the volatility spillovers.

#### Asymmetric BEKK-GARCH model

Since the DCC-GARCH model cannot test the volatility spillover effects, we use the full (unrestricted) BEKK model for the asymmetric responses of the volatility to calculate the *p*-value of the parameters associated with relations in terms of volatility across markets. Our mean function is proposed as follows:

$$ {R}_{i,t}={\mu}_i+{\Gamma}_i{R}_{i,t-1}+{\varepsilon}_{i,t},\kern0.4em i=m,a,d. $$

The variance function is

$$ {\displaystyle \begin{array}{c}{\upvarepsilon}_{i,t}\mid {\Omega}_{t-1}\sim D\left(0,{H}_{i,t}\right)\\ {}{H}_{i,t}={C_i}^{\prime }{C}_i+{A_i}^{\prime }{\varepsilon}_{i,t-1}{\varepsilon_{i,t-1}}^{\prime }{A}_i+{B_i}^{\prime }{H}_{i,t-1}{B}_i+{D_i}^{\prime }{\xi}_{i,t-1}^{\prime }{\xi}_{i,t-1}{D}_i\end{array}}, $$

(8)

where \( {C}_i=\left(\begin{array}{cc}{c}_{i,11}& {c}_{i,12}\\ {}0& {c}_{i,22}\end{array}\right),{A}_i=\left(\begin{array}{cc}{a}_{i,11}& {a}_{i,12}\\ {}{a}_{i,21}& {a}_{i,22}\end{array}\right),{B}_i=\left(\begin{array}{cc}{b}_{i,11}& {b}_{i,12}\\ {}{b}_{i,21}& {b}_{i,22}\end{array}\right),{D}_i=\left(\begin{array}{cc}{d}_{i,11}& {d}_{i,12}\\ {}{d}_{i,21}& {d}_{i,22}\end{array}\right) \).Here, *ξ*_{i, t − 1} is defined as *ε*_{i, t − 1} if *ε*_{i, t − 1} is negative, and 0 otherwise, which shows the impact of negative shocks on the conditional volatility. One advantage of BEKK is that it provides methods for measuring the volatility spillover effects between two markets. According to Eq. (8), the conditional variance of each market is determined by lagged error terms, lagged conditional variance, and lagged shocks from bad news from the two markets. The diagonal parameters in matrices *A*_{i}, *B*_{i} and *D*_{i} measure the effects of each market’s past shocks, volatilities, and negative shocks on its current conditional variance, while the off-diagonal parameters in matrices *A*_{i}, *B*_{i} and *D*_{i} (*a*_{i, mn}, *b*_{i, mn} and *d*_{i, mn}) measure the impacts of past shocks, volatilities, and negative shocks of market *m* on the current conditional variance of market *n*. Therefore, if *a*_{i, 21} = *b*_{i, 21} = *d*_{i, 21} = 0, there is no volatility spillover from the U.S. stock market to the Chinese stock market. Similarly, if *a*_{i, 12} = *b*_{i, 12} = *d*_{i, 12} = 0, there is no volatility spillover effect from the Chinese stock market to the U.S. stock market. In addition, another advantage of the BEKK model is that *H*_{i, t} is positive definite if the diagonal elements of *C*_{i} are positive.