3.2.1. Model Specification

• Specify a conditional mean model for the stocks returns series (JSE.JO stock market of South Africa and its partners) which can be a constant or a VAR(p) model and the simplest one is VAR (1) is formed as follows:

  $${\mathit{z}}_t={\mathit{\varphi }}_0+{\mathit{\varphi }}_1{\mathit{z}}_{t-1}+{\mathit{a}}_t,$$

  (1)
  Where $z_t$ is a vector of JSE.JO stock market of South Africa and its four considered partners of 5-dimensional, and ${\varphi }_0$ is a 5-dimensional constant vector and ${\varphi }_1$ is a $5\times 5$ matrix and $a_t$ is a sequence of independent and identically distributed (iid) random vectors with mean zero and covariance matrix ${\sigma }_a$, which is positive-definite.
• Obtain the residuals of the conditional mean model from step 1 for five stock returns and use them to check the existence of the volatility effect. There are many tests for checking the volatility such as the Lagrange Multiplier test, the multivariate Portmanteau test and its robust version with 5% upper tail trimming, and rank-based test.
  The hypothesis (states no serial correlation in the series) that should be tested to check the existence of volatility effect in the residuals series is $H_0:{\mathit{\rho }}_1={\mathit{\rho }}_2=...={\mathit{\rho }}_m=0vs{H}_1:{\mathit{\rho }}_i\ne 0$, for some i satisfying, $1\le i\le m$ where m is a positive integer, and ${\rho }_i$ is the lag-i of cross-correlation matrix of ${\mathit{a}}_t^2$. The test statistic of the multivariate Portmanteau Test is defined as:

  $$Q_k^{*}\left(m\right)=T^2{\Sigma }_{i-1}^m\frac{1}{T-i}{\widehat{\mathit{b}}}_i({\widehat{\mathit{\rho }}}_0^{-1}\otimes {\widehat{\mathit{\rho }}}_0^{-1}){\widehat{\mathit{b}}}_i$$

  (2)

  where T is the sample size, $k=5$ is the dimension of ${\mathit{a}}_t$, and ${\widehat{\mathit{b}}}_i=vec\left({\rho }_i^{{}^{\prime }}\right)$ with ${\widehat{\rho }}_j$ being the lag-j of sample cross-correlation matrix of ${\mathit{a}}_t^2$. Under the null hypothesis that ${\mathit{a}}_t$ has no conditional heteroscedasticity, $Q_k^{*}\left(m\right)$ is asymtotically distributed as ${\chi }_{k^{2}m}^2$. For more information about the test statistic of the Lagrange Multiplier, the robust version of the Portmanteau test, and the rank-based test see [46] and references therein.
• Employ univariate volatility models, such as GARCH, EGARCH, and GJR-GARCH model, to each component residuals series from step 1. Check [47] for more details about univariate volatility models.
• Suppose the estimate of the volatility series ${\sigma }_{ii,t}$ is ${\widehat{h}}_{it}$, standardize the innovations of univariate volatility models via ${\widehat{\eta }}_{it}=\frac{{\widehat{a}}_{it}}{\sqrt{{\widehat{\sigma }}_{ii,t}}}$ and fit a DCC model as in equation (4) bellow to ${\widehat{\mathit{\eta }}}_t$.
  The DCC model of Engle (2002) is defined as in the following equations, where equation (3) represents the general form of the conditional mean model, equation (4) is residuals that univariate volatility models are applied, and equation (5) is the form of DCC model.

  $$\begin{array}{c}\hfill {\mathit{z}}_t={\mathit{\mu }}_t+{\mathit{a}}_t,\end{array}$$

  (3)

  $$\begin{array}{c}\hfill {\mathit{a}}_t={\mathit{H}}_t^{\frac{1}{2}}{\epsilon }_t,\end{array}$$

  (4)

  $$\begin{array}{c}\hfill {\mathit{H}}_t={\mathit{D}}_t{\mathit{R}}_t{\mathit{D}}_t,\end{array}$$

  (5)

  where

  $${\mathit{R}}_t=diag\{q_{11t}^{\frac{-1}{2}},...,q_{55t}^{\frac{-1}{2}}\}{\mathit{Q}}_{t}diag\{q_{11t}^{\frac{-1}{2}},...,q_{55t}^{\frac{-1}{2}}\},$$

  ${\mathit{\mu }}_t$ is a vector of expected value of ${\mathbf{z}}_t$, ${\epsilon }_t$ is a vector of identical independent distribution (i.i.d) of errors with $E\left({\epsilon }_t\right)=0$ and $E\left({\epsilon }_t{\epsilon }_t^{{}^{\prime }}\right)={\mathit{I}}_k$ and ${\mathit{D}}_t=diag\{h_{11t}^{\frac{1}{2}},...,h_{55t}^{\frac{1}{2}}\}$ is a diagonal matrix of square roots of the conditional covariance matrix from univariate models, and ${\mathit{Q}}_t=\left[q_{ij,t}\right]$ which forms the symmetric DCC model.

  $${\mathit{Q}}_t=(1-\alpha -\beta )\overline{\mathit{Q}}+\alpha {\mathit{\eta }}_{t-1}{\mathit{\eta }}_{t-1}^{{}^{\prime }}+\beta {\mathit{Q}}_{t-1},$$

  (6)
  $\alpha ,\beta >0$ are non-negative real numbers and satisfying $0<\alpha +\beta <1$ , and ${\mathit{\eta }}_{t-1}{\mathit{\eta }}_{t-1}^{{}^{\prime }}$ is the lagged function of the standardized residuals. $\overline{\mathit{Q}}$ is unconditional covariance matrix of ${\mathit{\eta }}_t$ and ${\mathit{Q}}_t$ is the unconditional variance between series, i and j.The conditional covariances are given by

  $$h_{ij,t}=\frac{q_{ii,t}\sqrt{h_{ii,t}{h}_{jj,t}}}{\sqrt{q_{ii,t}{q}_{jj,t}}},$$

  where

  $$h_{ii,t}={\omega }_i+{\alpha }_i{\epsilon }_{t-1}^2+{\beta }_i{h}_{ii,t-1},i=1,2,...,5.$$
  The ${\mathit{Q}}_t$ that forms the asymmetric version of the DCC model is as follows

  $${\mathit{Q}}_t=(1-\alpha -\beta )\overline{\mathit{Q}}-g\overline{N}+\alpha {\mathit{\eta }}_{t-1}{\mathit{\eta }}_{t-1}^{{}^{\prime }}+\beta {\mathit{Q}}_{t-1}+g{\mathit{n}}_{t-1}{\mathit{n}}_{t-1}^{{}^{\prime }}$$

  (7)

  where $\alpha +\beta +g\delta <1$, $\delta$ is the maximum eigenvalue of $\left[{\overline{\mathit{Q}}}^{\frac{-1}{2}}{N{\overline{\mathit{Q}}}^{\frac{-1}{2}}}^{\overline{}}\right]$, $\overline{N}=T^{-1}{\sum }_{t=1}^T{\mathit{n}}_t{\mathit{n}}_t^{{}^{\prime }}$, ${\mathit{n}}_t=\mathit{I}[{\mathit{\eta }}_t<0]\circ {\mathit{\eta }}_t$, $\mathit{I}[.]$ is a $k\times 1$ indicator function that takes the value 1 if the argument is true and 0 otherwise, and ∘ represents the elementwise operations notation.
• The performance of fitted models is checked and compared by information criteria AIC and BIC to select the best model.