13.6 MULTIVARIATE VOLATILITY MODELS

13.6 MULTIVARIATE VOLATILITY MODELS

So far we have focused on univariate density forecasting problems. However, some applications require predicting joint distributions of two or more variables. For example, investors may be interested in pricing options with multiple underlying assets or computing Value at Risk at the portfolio level for many risky assets. In such situations, the outcome will depend on the degree of dependency across multiple underlying variables. This section discusses different ways to model multivariate volatility, while the next section covers copula modeling.

First, consider extending the conditional location–scale models to the multivariate case by modeling the conditional mean and volatility of an $(n\times 1)$ vector of variables $y_{t+1}:$

$$y_{t+1}={\mu }_{t+1|t}(\theta )+H_{t+1|t}^{1/2}(\theta ){\eta }_{t+1},{\eta }_{t+1}\sim (0,I_n),$$

where $\theta$ is a vector of underlying parameters. Here $H_{t+1|t}^{1/2}$ is an $(n\times n)$ positivesemidefinite matrix with the property that $\mathrm{var}(y_{t+1}|Z_t)=H_{t+1|t}=H_{t+1|t}^{1/2}(H_{t+1|t}^{1/2}{)}^{\prime }.$

where $H_{t+1|t}^{1/2}$ can be obtained from the Cholesky decomposition of $H_{t+1|t}$ . The question is how to best model $H_{t+1|t}$ in the multivariate case where $n>1$ . Bauwens, Laurent, and Rombouts (2006) review the large literature on this issue and we shall simply highlight a few key models from their review.

The so-called BEKK(1,1,1) specification takes the form

$$H_{t+1|t}=C^{\prime }C+A^{\prime }{\epsilon }_t{\epsilon }_t^{\prime }A+G^{\prime }{H}_{t|t-1}G,\text{\hspace{1em}(13.49)}$$

where C, A, G are $(n\times n)$ matrices and C is upper triangular and ${\epsilon }_t=H_{t\mid t-1}^{1/2}(\theta ){\eta }_t$ This specification has $n(5n+1)/2$ parameters and can readily be generalized to allow for additional lags of ${\epsilon }_t$ and $H_{t\mid t-1}$

A parsimonious model can be obtained by assuming a factor structure in the conditional variance. For example, Lin (1992) studies a K -factor GARCH model of the form

$$H_{t+1|t}=C^{\prime }C+\sum _{k=1}^K{\lambda }_k{\lambda }_k^{\prime }({\alpha }_k^2{\omega }_k^{\prime }{\epsilon }_t{\epsilon }_t^{\prime }{\omega }_k+{\beta }_k^2{\omega }_k^{\prime }{H}_{t|t-1}{\omega }_k),\text{\hspace{1em}(13.50)}$$

where ${\omega }_k^{\prime }\iota =1$ serves as an identification restriction. Provided that $C^{\prime }C$ is positive definite, $H_{t\mid t-1}$ will be of full rank. This specification has $n(n+5)/2$ parameters.

Not only the variance parameters, but also the correlation parameters may vary over time. Early work such as Bollerslev (1990) assumed constant pairwise correlations and time-varying volatilities:

$$\begin{array}{l}{H}_{t+1|t}=D_{t+1|t}R{D}_{t+1|t},\\ \\ D_{t+1|t}=\mathrm{diag}(h_{11t+1|t}^{1/2},\dots ,h_{nnt+1|t}^{1/2}).\end{array}\text{\hspace{1em}(13.51)}$$

Here $h_{iit+1|t}$ can be obtained from a univariate GARCH model and the elements of the correlation matrix R are $R_{ij}={\rho }_{ij}$ with ${\rho }_{ii}=1$ , so that the $(i,j)$ element of $H_{t+1|t}$ takes the form ${\rho }_{ij}\sqrt{h_{iit+1|t}{h}_{jjt+1|t}}.$ In this model the conditional correlation is constant, but the conditional covariance will vary over time because of time-varying variances.

Recent work has focused on modeling time-varying correlations. For example, the multivariate dynamic conditional correlation GARCH model proposed by Engle and Sheppard (2001) and Engle (2002) takes the form $y_{t+1}\sim \mathrm{N}(0,H_{t+1|t})$

$$H_{t+1|t}\equiv D_{t+1|t}{R}_{t+1|t}{D}_{t+1|t},\text{\hspace{1em}(13.52)}$$

where $D_{t+1|t}$ is an $(n\times n)$ diagonal matrix of time-varying standard deviations with ith diagonal element $\sqrt{h_{iit+1|t}}$ , and $R_{t+1|t}$ is now a time-varying correlation matrix.

The diagonal elements of $D_{t+1|t}$ are assumed to be generated by univariate GARCH processes:

$$h_{iit+1|t}={\omega }_i+\sum _{j=1}^{q_i}{\alpha }_{ij}{\epsilon }_{it+1-j}^2+\sum _{p=1}^{p_i}{\beta }_{ip}{h}_{iit+1-p|t-p},\text{\hspace{1em}(13.53)}$$

whereas the dynamics in the correlations take the form

$$Q_{t+1|t}=\left(1-\sum _{j=1}^J{\alpha }_j-\sum _{k=1}^K{\beta }_k\right)\bar{Q}+\sum _{j=1}^J{\alpha }_j{\epsilon }_{t+1-j}{\epsilon }_{t+1-j}^{\prime }+\sum _{k=1}^K{\beta }_k{Q}_{t+1-k|t-k},\text{\hspace{1em}(13.54)}$$

$$R_{t+1|t}=(Q_{t+1|t}^{\ast }{)}^{-1}{Q}_{t+1|t}(Q_{t+1|t}^{\ast }{)}^{-1}.$$

Here $\bar{Q}$ is the unconditional covariance of the standardized residuals, the ${\alpha }_j$ terms measure the news impact of previous shocks, and the ${\beta }_k$ terms capture persistence in the correlations. Finally, $Q_{t+1|t}^{\ast }$ is a diagonal matrix of the form

$$\begin{array}{r}{Q}_{t+1|t}^{\ast }=\left(\begin{array}{cccccc}\sqrt{q_{11t+1|t}}& 0& 0& \cdots & 0& \\ 0& \sqrt{q_{22t+1|t}}& 0& \cdots & 0& \\ & & \ddots & & & ⋮\\ ⋮& & ⋮& \ddots & & ⋮\\ 0& 0& 0& \cdots & \sqrt{q_{nnt+1|t}}& \end{array}\right),\end{array}\text{\hspace{1em}(13.55)}$$

so that the individual elements of the dynamic conditional correlation matrix $R_{t+1|t}$ can be written as ${\rho }_{ijt+1|t}=q_{ijt+1|t}/\sqrt{q_{iit+1|t}{q}_{jjt+1|t}}.$

Estimation of these models proceeds by maximum likelihood methods. For example, the log-likelihood function of the dynamic conditional correlation model is

$$\begin{array}{l}\mathrm{LL}=-\frac{1}{2}\sum _{t=1}^T(n\log (2\pi )+\log (|H_{t\mid t-1}|)+y_t^{\prime }{H}_{t\mid t-1}^{-1}{y}_t)\\ =-\frac{1}{2}\sum _{t=1}^T(n\log (2\pi )+2\log (|D_{t\mid t-1}|)+\log (|R_{t\mid t-1}|)+{\epsilon }_t^{\prime }{R}_{t\mid t-1}^{-1}{\epsilon }_t),\end{array}$$

where ${\epsilon }_t\sim \mathrm{N}(0,R_{t|t-1})$ is the standardized residual. Engle and Sheppard advocate using two-stage QMLE estimation. In a first stage, univariate GARCH models are estimated for each residual series and the estimates are used to normalize the residuals by their standard deviation. In the second stage, the normalized residuals are used to estimate dynamic correlations.

A related literature has also considered multivariate quantile models. For example, White $\mathrm{Jr},$ Kim, and Manganelli (2008) use a multiquantile CAViaR specification to study conditional skewness dynamics in daily stock returns, while Cappiello et al. (2014) use time-varying conditional quantiles to study comovements in returns on Latin American equity markets.