*3.2. Methodology*

This study estimated the return and volatility transmissions using the Vector Autoregressive-Generalized Autoregressive Conditional Heteroskedasticity (VAR-AGARCH) model proposed by McAleer et al. (2009). Several studies have previously used the VAR-GARCH and VAR-AGARCH model to estimate spillover between di fferent asset classes (Arouri et al. 2011; Arouri et al. 2012; Jouini 2013; Yousaf and Hassan 2019). This model includes the Constant Conditional Correlation (CCC-GARCH) model of Bollerslev (1990) as a special case. The selection of the model was based on three reasons. First, the most commonly used multivariate models are the BEKK (Baba, Engle, Kraft, and Kroner) model and the DCC (dynamic conditional correlation) model. These models often su ffer from unreasonable parameter estimates and data convergence problems (Bouri 2015). The VAR-AGARCH model overcomes these problems regarding parameters and data convergence. Second, it incorporates asymmetry into the model. Third, this model can be used to calculate the optimal weights and hedge ratios.

Ling and McAleer (2003) propose the multivariate VAR-GARCH Model to estimate the return and volatility transmission between the di fferent series. For two series, the VAR-GARCH model has the following specifications for the conditional mean equation3:

$$R\_t = \mu + FR\_{t-1} + e\_t \text{ with } e\_t = D\_t^{1/2} \eta\_{t\prime} \tag{1}$$

in which *Rt* represents a 2 × 1 vector of daily returns<sup>4</sup> on the stocks *x* and *y* at time *t*, μ denotes a 2 × 1 vector of constants, *F* is a 2 × 2 matrix of parameters measuring the impacts of own lagged and cross mean transmissions between two series, *et* is the residual of the mean equation for the two stocks returns series at time t, η*t* indicates a 2 × 1 vector of independently and identically distributed random vectors, and *D*1/2 *t* = *diag* ( - *hx t* , *h y t* ), where *hx t* and *h y t* representing the conditional variances of the returns for stocks x and y, respectively, are given as

$$h\_t^{\mathbf{x}} = \mathbb{C}\_{\mathbf{x}} + a\_{11} \left(\mathbf{e}\_{t-1}^{\mathbf{x}}\right)^2 + a\_{21} \left(\mathbf{e}\_{t-1}^{\mathbf{y}}\right)^2 + b\_{11} h\_{t-1}^{\mathbf{x}} + b\_{21} h\_{t-1}^{\mathbf{y}} \tag{2}$$

$$h\_t^y = \mathbb{C}\_{\mathcal{Y}} + a\_{12} \left( e\_{t-1}^x \right)^2 + a\_{22} \left( e\_{t-1}^y \right)^2 + b\_{12} h\_{t-1}^x + b\_{22} h\_{t-1}^y. \tag{3}$$

Equations (2) and (3) reveal how shock and volatility are transmitted over time and across the markets under investigation. Furthermore, the conditional covariance between returns from two di fferent stock markets can be estimated as follows:

$$h\_t^{x,y} = p \times \sqrt{h\_t^x} \times \sqrt{h\_t^y}.\tag{4}$$

In the above equation, *h <sup>x</sup>*,*y t* refers to the conditional covariance between the returns of two stock markets (*<sup>x</sup>*, *y*) at time *t*. Moreover, *p* indicates the constant conditional correlation between the returns of two stock markets (*<sup>x</sup>*, *y*).

The VAR−GARCH model assumes that positive or negative shocks have the same impact on the conditional variance. To estimate the spillover between di fferent markets, we estimated spillover between two stock markets by using the VAR−AGARCH Model proposed by the McAleer et al. (2009).

<sup>3</sup> Several studies, for example, Hammoudeh et al. (2009), Arouri et al. (2011), and Dutta et al. (2018) have applied the VAR for the conditional mean equation. 

<sup>4</sup> *Stock Returnst* = *ln Stock Indext StockIndext*−<sup>1</sup>.

The VAR AGARCH model incorporates asymmetry in the model as well. Specifically, instead of using Equations (2) and (3), the conditional variance of the VAR AGARCH model was defined as follows:

$$h\_t^{\mathbf{x}} = \mathbb{C}\_{\mathbf{x}} + a\_{11}A\left(\mathbf{e}\_{t-1}^{\mathbf{x}}\right)^2 + a\_{21}A\left(\mathbf{e}\_{t-1}^{\mathbf{y}}\right)^2 + b\_{11}h\_{t-1}^{\mathbf{x}} + b\_{21}h\_{t-1}^{\mathbf{y}} + a\_{11}B\left[\left(\mathbf{e}\_{t-1}^{\mathbf{x}}\right)\left(\left(\mathbf{e}\_{t-1}^{\mathbf{x}}\right) < 0\right)\right],\tag{5}$$

$$h\_t^y = \mathbb{C}\_{\mathcal{Y}} + a\_{12}A\left(e\_{t-1}^x\right)^2 + a\_{22}A\left(e\_{t-1}^y\right)^2 + b\_{12}h\_{t-1}^x + b\_{22}h\_{t-1}^y + a\_{22}B\left[\left(e\_{t-1}^y\right)\left(\left(e\_{t-1}^y\right) < 0\right)\right].\tag{6}$$

In the above equations, *A e<sup>x</sup> t*−1 2 and *B e<sup>x</sup> t*−1 ( *e<sup>x</sup> t*−1 < 0)) as well as *A e y t*−1 2 and *B e y t*−1 ( *e y t*−1 < 0)) reveal the relationships between a market's volatility and both positive and negative own lagged returns, respectively (Lin et al. 2014). Equations (5) and (6) show that the conditional variance of each market depends upon its past shock and past volatility, as well as the past shock and past volatility of other markets. In Equation (5), *e<sup>x</sup> t*−1 2 and *e y t*−1 2 explain how the past shocks of both x and y a ffect the current conditional volatility of *x*. Moreover, *hx t*−1 and *h y t*−1 measure how the past volatilities of both *x* and *y* affect the current conditional volatility of *x*. The parameters of the VAR-AGARCH model can be estimated by using the Quasi-Maximum Likelihood estimation (QMLE) and using the BFGS algorithm.<sup>5</sup>

The estimates of the VAR-AGARCH model can be used to calculate optimal portfolio weights. This study followed Kroner and Ng (1998) to calculate the optimal portfolio weights for the pairs of the stock market (*<sup>x</sup>*, *y*) as:

$$w\_{xy,t} = \frac{h\_{y,t} - h\_{xy,t}}{h\_{x,y} - 2h\_{xy,t} + h\_{y,t}}\tag{7}$$

$$w\_{xy,t} = \begin{cases} 0 & \text{if } W\_{xy,t} < 0\\ w\_{xy,t} & \text{if } 0 \le w\_{xy,t} \le 1\\ 1 & \text{if } w\_{xy,t} > 1 \end{cases},$$

where *wxy*,*<sup>t</sup>* is the weight of stock(*x*) in a \$1 stock(*x*)-stock(*y*) portfolio at time *t*, *hxy*,*<sup>t</sup>* is the conditional covariance between the two stock markets, *hx*,*<sup>t</sup>* and *hy*,*<sup>t</sup>* are the conditional variance of stock(*x*) and stock(*y*), respectively, and 1- *wxy*,*<sup>t</sup>* is the weight of stock(*y*) in a \$1 stock(*x*)-stock(*y*) portfolio.

It is also essential to estimate the risk-minimizing optimal hedge ratios for the portfolio of di fferent stocks. The estimates of the VAR-AGARCH model can also be used to calculate optimal hedge ratios. This study followed Kroner and Sultan (1993) to calculate the optimal hedge ratios as:

$$
\beta\_{xy,t} = \frac{h\_{xy,t}}{h\_{y,t}} \, ^\prime \tag{8}
$$

where β*xy*,*<sup>t</sup>* represents the hedge ratio. This shows that a short position in the stock (*y*) market can hedge a long position in the stock (*x*). Lastly, RATS 10.0 software is used for estimations.
