*3.2. Methodology*

The econometric specification used in this study has two components. First, a vector autoregression (VAR) with one lag is used to model the returns.<sup>2</sup> This allows for autocorrelations and cross-autocorrelations in the returns. Second, a multivariate BEKK-GARCH model is used to model the time-varying variances and covariances developed by Engle and Kroner(1995).<sup>3</sup> BEKK-GARCH has the attractive characteristics that the conditional covariance matrices are positive definite (Chang et al. 2011). Several studies have used the BEKK-GARCH model to estimate the spillover between different asset classes; see, for example, Chang et al. (2011), Sadorsky (2012), Beirne et al. (2013), Chang et al. (2017), Cardona et al. (2017), and Sarwar et al. (2020). Moreover, we will estimate the optimal weights and hedge ratios using the BEKK-GARCH model.

<sup>2</sup> The number of lags is selected on the basis of the AIC and SIC criteria.

<sup>3</sup> We apply the BEKK-GARCH model on the valuable suggestion of a respected reviewer.

*JRFM* **2020**, *13*, 148

*t* This study aims to examine the return and volatility spillover between the stock markets, and thus we firstly focus on return spillover. For any pair of two series, the following are the specifications for the conditional mean equation:

$$R\_t = \mu + \mathcal{Q} \ R\_{t-1} + \varepsilon\_t \text{ with } \varepsilon\_t = H\_t^{1/2} \eta\_t. \tag{1}$$

*Rt* = (*Rxt* , *Ryt* ) is the vector of returns on the stock market indices x and y at time *t*, respectively; ∅ is the 2 × 2 matrix of parameters, measuring the impacts of own lagged and cross mean transmissions between two series; *et* = *ext* ,*eyt* is the vector of error terms of the conditional mean equations for the two series at time *t*; η*t* = η*xt* , η*yt* indicates a sequence of independently and identically distributed random errors; and *Ht* = *Hxt Hxyt Hxyt Hyt* denotes the conditional variance-covariance matrix of return seriesof*x*and*y*.Inaddition,*H*1/2 isthe2×2symmetricpositivedefinitematrix.

The full BEKK–GARCH, which imposes positive definiteness restrictions for *Ht*, is given by:

$$H\_l = \mathbb{C}'\mathbb{C} + A'e\_{t-1}e\_{t-1}'A + B'H\_{t-1}B\_t \tag{2}$$

where *A* and *B* are (*n* × *n*) coefficient matrices and *CC* is the decomposition of the intercept matrix. Each element (*i,j*)th in *Ht* depends on the corresponding (*i,j*)th element in (*et*−1*et*−<sup>1</sup>) and *Ht*−1. Accordingly, past shocks and volatility are allowed to directly spill over from a market to another, and they are captured by the coefficients of the *A* and *B* matrices. More specifically, the BEKK-GARCH matrices can be expanded as follows:

$$h\_t^x = \mathbb{C}\_x + a\_x^2 \Big(\boldsymbol{e}\_{t-1}^x\Big)^2 + 2a\_x a\_{yx} \boldsymbol{e}\_{t-1}^x \boldsymbol{e}\_{t-1}^y + a\_{yx}^2 \Big(\boldsymbol{e}\_{t-1}^y\Big)^2 + \beta\_x^2 h\_{t-1}^x + 2\beta\_{\overline{x}} \beta\_{\overline{y}x} h\_{t-1}^{xy} + \beta\_{\overline{y}x}^2 h\_{t-1}^y \tag{3}$$

$$h\_{l}^{xy} = \mathbb{C}\_{19} + a\_{3}\alpha\_{3}\big(\boldsymbol{\varepsilon}\_{l=1}^{x}\big)^{2} + \left(a\_{\mathcal{Y}}a\_{l\mathcal{Y}} + a\_{3}a\_{\mathcal{Y}}\right)\big\mathfrak{e}\_{l-1}^{x}\big\mathfrak{e}\_{l-1}^{y} + a\_{\mathcal{Y}}a\_{\mathcal{Y}}\big(\boldsymbol{\varepsilon}\_{l=1}^{y}\big)^{2} + \beta\_{l}\beta\_{\mathcal{Y}}h\_{l-1}^{x} + \left(\beta\_{\mathcal{Y}}\beta\_{\mathcal{Y}} + \beta\_{\mathcal{Y}}\beta\_{\mathcal{Y}}\right)\big\mathfrak{h}\_{l-1}^{xy} + \beta\_{\mathcal{Y}}\beta\_{\mathcal{Y}}h\_{l-1}^{y} \tag{4}$$

$$h\_t^y = \mathbb{C}\_{\mathcal{Y}} + a\_{\text{xy}}^2 \left(\boldsymbol{e}\_{t-1}^x\right)^2 + 2a\_{\text{xy}}a\_{\mathcal{Y}}\boldsymbol{e}\_{t-1}^x\boldsymbol{e}\_{t-1}^y + a\_{\mathcal{Y}}^2 \left(\boldsymbol{e}\_{t-1}^y\right)^2 + \beta\_{\text{xy}}^2 h\_{t-1}^x + 2\beta\_{\text{xy}}\beta\_{\mathcal{Y}}h\_{t-1}^{xy} + \beta\_{\mathcal{Y}}^2 h\_{t-1}^y \tag{5}$$

The BEKK-GARCH parameters are estimated by the maximum likelihood method using the BFGS algorithm. In addition to the return and volatility spillover, we also compute the optimal weights and hedge ratios for each pair of stocks.

The conditional variance and covariances are used for calculating the optimal portfolio weights and hedge ratios. This study follows Kroner and Ng (1998) in calculating the optimal portfolio weights of different pairs of stock markets:

$$w\_t^{xy} = \frac{h\_t^y - h\_t^{xy}}{h\_t^x - 2h\_t^{xy} + h\_t^y},\tag{6}$$

$$w\_t^{xy} = \begin{cases} 0, If \ w\_t^{xy} < 0\\ w\_t^{xy}, If \ 0 \le w\_t^{xy} \le 1\\ 1, If \ w\_t^{xy} > 1 \end{cases}$$

where *wxyt* is the weight of stock(*x*) in a \$1 stock(*x*)-stock(*y*) portfolio at time t; *hxyt* is the conditional covariance between the two stock markets; *hxt* and *hyt* are the conditional variance of stock(*x*) and stock(*y*), respectively; and 1 − *wxyt* is the weight of stock(*y*) in a \$1 stock(*x*)-stock(*y*) portfolio. As suggested by Kroner and Sultan (1993):

$$
\beta\_t^{xy} = \frac{h\_t^{xy}}{h\_t^y} \tag{7}
$$

where β*xyt* represents the hedge ratio. This shows that a short position in the stock (*y*) market can hedge a long position in stock (*x*).

### **4. Empirical Results and Implications**

In this section, we will discuss our empirical results and implications. We first discuss our preliminary analysis.
