*3.1. Dynamic Conditional Correlation Model*

As mentioned in the previous section, we use the DCC-GARCH model developed by Engle and Sheppard (2001), Engle (2002) to examine the time-varying correlation coefficients in this study. Generally, the DCC(1,1)-GARCH(1,1) specification is enough to capture the characteristics of heteroscedasticity of stock and financial variables (See Bollerslev et al. (1992)). This model is estimated by applying log likelihood estimation procedures.

The estimation of dynamic correlation coefficients between the returns of two markets consists of three steps. Firstly, we have to filter the returns in order to obtain residual returns (See Engle and Sheppard (2001)). We employ the model specification proposed by Chiang et al. (2007) as follows:

$$r\_t = \gamma\_0 + \gamma\_1 r\_{t-1} + \gamma\_2 r\_{t-1}^{IIS} + \varepsilon\_t \tag{1}$$

An AR(1) process is used to account for the autocorrelation of stock returns. *rUS* is the U.S. stock index returns, used as the global factor.

<sup>5</sup> Problem raised by Forbes and Rigobon (2002) as discussed above.

Secondly, the parameters in the variance models are estimated using the residual returns (*<sup>ε</sup>t*) from the first step.

$$
\varepsilon\_t = D\_t \upsilon\_t \sim N(0, H\_t) \tag{2}
$$

$$
\upsilon\_t \sim N(0, R\_t) \tag{3}
$$

and

$$H\_t = D\_t R\_t D\_t \tag{4}$$

where:


The elements in *Dt* are obtained from the univariate GARCH(1,1) models with *hi*,*<sup>t</sup>* on the *i*th diagonal.

$$h\_{i,t} = \omega\_i + \alpha\_i \varepsilon\_{i,t-1}^2 + \beta\_i h\_{i,t-1} \tag{5}$$

for *i* = 1,...,k.

The correlation coefficients are then estimated. The correlation between stock index returns *i* and *j* at time *t* is defined as:

$$\rho\_{ij,t} = \frac{E\_{t-1}(\varepsilon\_{i,t}\varepsilon\_{j,t})}{\sqrt{E\_{t-1}(\varepsilon\_{i,t}^2)E\_{t-1}(\varepsilon\_{j,t}^2)}}\tag{6}$$

Substituting *<sup>ε</sup>i*,*<sup>t</sup>* = *hi*,*tυi*,*<sup>t</sup>* and *<sup>ε</sup>j*,*<sup>t</sup>* = *hj*,*t<sup>υ</sup>j*,*<sup>t</sup>* to the Equation (6), we will have:

$$\rho\_{ij,t} = \frac{E\_{t-1}(\sqrt{h\_{i,t}}\upsilon\_{i,t}\sqrt{h\_{j,t}}\upsilon\_{j,t})}{\sqrt{E\_{t-1}(h\_{i,t}\upsilon\_{i,t}^2)E\_{t-1}(h\_{j,t}\upsilon\_{j,t}^2)}} = \frac{E\_{t-1}(\upsilon\_{i,t}\upsilon\_{i,t})}{\sqrt{E\_{t-1}(\upsilon\_{i,t}^2)E\_{t-1}(\upsilon\_{j,t}^2)}} = E\_{t-1}(\upsilon\_{i,t}\upsilon\_{j,t}) \tag{7}$$

with *Et*−<sup>1</sup>(*υ*2*i*,*<sup>t</sup>*) = *Et*−<sup>1</sup>(*h*−<sup>1</sup> *i*,*t <sup>ε</sup>*2*i*,*<sup>t</sup>*) = *h*−<sup>1</sup> *i*,*t Et*−<sup>1</sup>(*ε*2*i*,*<sup>t</sup>*) = 1 and *Et*−<sup>1</sup>(*υ*2*j*,*<sup>t</sup>*) = *Et*−<sup>1</sup>(*h*−<sup>1</sup> *j*,*t <sup>ε</sup>*2*j*,*<sup>t</sup>*) = *h*−<sup>1</sup> *j*,*t Et*−<sup>1</sup>(*ε*2*j*,*<sup>t</sup>*) = 1. The conditional correlation is hence the covariance of standardized disturbances. Let *Qt* the time-varying covariance matrix of *υt* (*Qt* = *Et*−<sup>1</sup>(*<sup>υ</sup>tυt*)) then we have:

$$R\_t = (\operatorname{diag} \mathbf{Q}\_t)^{-1/2} \mathbf{Q}\_t (\operatorname{diag} \mathbf{Q}\_t)^{-1/2} \tag{8}$$

*Qt* in this equation is a *nxn* positive symmetric matrix. It is defined by:

$$Q\_t = (1 - \theta\_1 - \theta\_2)\ddot{Q} + \theta\_1 \nu\_{t-1} \nu\_{t-1}' + \theta\_2 Q\_{t-1} \tag{9}$$

where:


The conditional correlation coefficient, also the element of matrix *Rt*, is then:

$$\rho\_{\vec{i}\vec{j},t} = \frac{q\_{\vec{i}\vec{j},t}}{\sqrt{q\_{\vec{i}\vec{i},t}q\_{\vec{j}\vec{j},t}}} \tag{10}$$

$$\rho\_{ij} = \frac{(1 - \theta\_1 - \theta\_2)\bar{q}\_{ij} + \theta\_1 v\_{i, t-1} v\_{j, t-1} + \theta\_2 q\_{ij, t-1}}{\sqrt{\left[ (1 - \theta\_1 - \theta\_2)\overline{q}\_{ii} + \theta\_1 v\_{i, t-1}^2 + \theta\_2 q\_{ii, t-1} \right]} \sqrt{\left[ (1 - \theta\_1 - \theta\_2)\bar{q}\_{jj} + \theta\_1 v\_{j, t-1}^2 + \theta\_2 q\_{jj, t-1} \right]}} \tag{11}$$

As proposed by Engle (2002), the DCC model can be estimated by using a two-stage approach to maximize the log-likelihood. Let *θ* and *φ* be denoted the parameters respectively in matrices *D* and *R*, the log-likelihood function to determine the parameters in the Equations (1) and (5) can be written as follows:

$$\begin{split} L(\theta,\phi) &= -\frac{1}{2} \sum\_{t=1}^{T} (n\log(2\pi) + \log|H\_t| + \varepsilon\_t' H\_t^{-1} \varepsilon\_t) \\ &= -\frac{1}{2} \sum\_{t=1}^{T} (n\log(2\pi) + \log|D\_t R\_t D\_t| + \varepsilon\_t' D\_t^{-1} R\_t^{-1} D\_t^{-1} \varepsilon\_t) \\ &= -\frac{1}{2} \sum\_{t=1}^{T} (n\log(2\pi) + 2\log|D\_t| + \log|R\_t| + v\_t' R\_t^{-1} v\_t) \end{split} \tag{12}$$

where *υt* ∼ N(0,*Rt*) are the residuals standardized on the basis of their conditional standard deviations. Rewriting (12) gives:

$$\begin{split} L(\boldsymbol{\theta}, \boldsymbol{\phi}) &= -\frac{1}{2} \sum\_{t=1}^{T} (n \log(2\pi) + 2 \log|D\_t| + \varepsilon\_t' D\_t^{-2} \varepsilon\_t \\ &\quad + -\frac{1}{2} \sum\_{t=1}^{T} (\log|\boldsymbol{R}\_t| + \nu\_t' \boldsymbol{R}^{-1} \boldsymbol{\nu}\_t - \boldsymbol{\nu}\_t' \boldsymbol{\nu}\_t) \\ &= -L\_1(\boldsymbol{\theta}) + L\_2(\boldsymbol{\phi}) \end{split} \tag{13}$$

where:

$$L\_1(\theta) = -\frac{1}{2} \sum\_{t=1}^{T} \left( n \log(2\pi) + 2 \log|D\_t| + \varepsilon\_t' D\_t^{-2} \varepsilon\_t \right) \tag{14}$$

$$L\_2(\phi) = -\frac{1}{2} \sum\_{t=1}^{T} (\log|\mathcal{R}\_l| + \nu\_t^{\prime} \mathcal{R}^{-1}\_{\ \ t} \upsilon\_t - \upsilon\_t^{\prime} \upsilon\_t) \tag{15}$$

*<sup>L</sup>*1(*θ*) is log-likelihood function of variances and *<sup>L</sup>*2(*φ*) is that of correlations. In the first stage, the parameters of variances in *L*1 are determined by maximizing *<sup>L</sup>*1(*θ*). In the second stage, given the estimated parameters in the first stage, the likelihood function *<sup>L</sup>*2(*φ*) is maximized to estimate the correlation parameters in *<sup>L</sup>*2(*φ*).
