Multivariate Volatility Models

Most applications deal with portfolios where it is necessary to forecast the entire covariance matrix of asset returns.

Consider the univariate volatility model:

    \[ Y_{t} = \sigma_{t} Z_{t} \]

where Y_{t} are returns; \sigma_{t} is conditional volatility, and Z_{t} are random shocks.

EWMA

The multivariate form of EWMA is

    \[ \hat{\Sigma}_{t}=\lambda\hat{\Sigma}_{t-1}+\left(1-\lambda\right)y_{t-1}^{\prime}y_{t-1} \]

with an individual element given by

    \[ \hat{\sigma}_{t,ij}=\lambda\hat{\sigma}_{t-1,ij}+\left(1-\lambda\right)y_{t-1,i}y_{t-1,j}\quad i,j=1,\ldots,K \]

where \lambda = 0.94 as per RiskMetrics.

A sample R code for EWMA is

Orthogonal GARCH (OGARCH)

It is usually very hard to estimate multivariate GARCH models. In practice, alternative methodologies for obtaining the covariance matrix are needed.

The orthogonal approach transforms linearly the observed returns matrix into a set of portfolios with the key property that they are uncorrelated, implying we can forecast their volatilities separately. This makes use of principal components analysis (PCA).

Orthogonalising covariance

The first step is to transform the return matrix y^{\left\{T\times K\right\}} into uncorrelated portfolio u^{\left\{T\times K\right\}}. Denote \hat{R}^{\left\{K\times K\right\}} as the sample correlation of y^{\left\{T\times K\right\}}. We then calculate orthogonal matrix of eigenvectors of \hat{R}^{\left\{K\times K\right\}}, denoted by \Lambda^{\left\{K\times K\right\}}. Then u^{\left\{T\times K\right\}} is defined by:

    \[ u^{\left\{T\times K\right\}}=\Lambda^{\left\{K\times K\right\}} \times y^{\left\{T\times K\right\}}. \]

The rows of u^{\left\{T\times K\right\}} are uncorrelated with each other so we can run a univariate GARCH or a similar model on each row in u^{\left\{T\times K\right\}} separately to obtain its conditional variance forecast, denoted by D_{t}. We then obtain the forecast of the conditional covariance matrix of the returns by:

    \[ \hat{\Sigma}_{t}=\Lambda \hat{D}_{t} \Lambda^{\prime}. \]

This implies that the covariance terms can be ignored when modeling the covariance matrix of u, and the problem has been reduced to a series of univariate estimations.

Large-scale implementations

In the above example, all the principal components (PCs) were used to construct the conditional covariance matrix. However, it is possible to use just a few of the columns. The highest eigenvalue corresponds to the most important principle component—the one that explains most of the variation in the data.

Such approaches are in widespread use because it is possible to construct the conditional covariance matrix for a very large number of assets. In a highly correlated environment, just a few principal components are required to represent system variation to a very high degree of accuracy. This is much easier than forecasting all volatilities directly in one go.

PCA also facilitates building a covariance matrix for an entire financial institution by iteratively combining the covariance matrices of the various trading desks, simply by using one or perhaps two principal components. For example, one can create the covariance matrices of small caps and large caps separately and use the first principal component to combine them into the covariance matrix of all equities. This can then be combined with the covariance matrix for fixed income assets, etc.

Correlation Models

Constant conditional correlations (CCC)

Bollerslev (1990) proposes the constant conditional correlations (CCC) model where time-varying covariances are proportional to the conditional standard deviation. The conditional covariance matrix \hat{\Sigma}_{t} consists of two components that are estimated separately: sample correlations \hat{R} and the diagonal matrix of time-varying volatilities \hat{D}_{t}.

    \[ \hat{\Sigma}_{t} = \hat{D}_{t} \hat{R} \hat{D}_{t} \]

where

    \[ \hat{D}_{t}=\left(\begin{array}{ccc} \hat{\sigma}_{t,1} & 0 & 0\\ 0 & \ddots & 0\\ 0 & 0 & \hat{\sigma}_{t,K} \end{array}\right). \]

The volatility of each asset \hat{\sigma}_{t,k} follows a GARCH process or any of the univariate models discussed here.

This model guarantees the positive definiteness of \hat{\Sigma}_{t} if \hat{R} is positive definite.

Dynamic conditional correlations (DCC)

In particular, the assumption of correlations being constant over time is at odds with the vast amount of empirical evidence supporting nonlinear dependence. To correct this defect, Engle (2002) and Tse and Tsui (2002) propose the dynamic conditional correlations (DCC) model as an extension to the CCC model.

Different with CCC model, the correlation matrix is time dependent within the DCC framework as

    \[ \hat{R}_{t} = \hat{Q}_{t}^{\prime} \hat{Q}_{t} \]

where \hat{Q}_{t} is a symmetric positive definite autoregressive matrix and is given by

    \[ \hat{Q}_{t}=\left(1-\zeta-\xi\right)\bar{Q}+\zeta Y_{t-1}^{\prime}Y_{t-1}+\xi\hat{Q}_{t-1} \]

where \bar{Q} is the K\times K unconditional covariance matrix of Y; \zeta,\xi > 0 and \zeta + \xi <1 to ensure positive definiteness and stationarity, respectively.

  • Pros: it can be estimated in two steps: one for parameters determining univariate volatilities and another for parameters determining the correlations.
  • Cons: parameters \zeta and \xi are constants implying that the conditional correlations of all assets are driven by the same underlying dynamics — often an unrealistic assumption.

When we compare the correlations estimated by the above three models: EWMA, OGARCH and DCC, we will find the correlation forecasts for EWMA seem to be most volatile. Both DCC and OGARCH models have more stable correlations with the OGARCH having the lowest fluctuations but the highest average correlations. The large swings in EWMA correlations might be an overreaction.

Multiariate Extensions of GARCH

It is conceptually straightforward to develop multivariate extensions of the univariate GARCH-type models — such as multivariate GARCH (MVGARCH). Unfortunately, it is more difficult in practice because the most obvious model extensions result in the number of parameters exploding as the number of assets increases.

The BEKK model

There are a number of alternative MVGARCH models available, but the BEKK model, proposed by Engle and Kroner (1995), is probably the most widely used. The matrix of conditional covariances

The general BEKK \left(L_{1} ,L_{2} ,K \right) model is given by

    \[ \Sigma_{t}=\Omega\Omega^{\prime}+\sum_{k=1}^{K}\sum_{i=1}^{L_{1}}A_{i,k}^{\prime}Y_{t-i}^{\prime}Y_{t-i}A_{i,k}+\sum_{k=1}^{K}\sum_{j=1}^{L_{2}}B_{j,k}^{\prime}\Sigma_{t-j}B_{j,k} \]

The number of parameters in the BEKK(1,1,2) model is K(5K+1)/2, i.e. 11 in 2-asset case.

    \begin{eqnarray*} \Sigma_{t} & = & \left(\begin{array}{cc} \sigma_{t,11} & \sigma_{t,12}\\ \sigma_{t,12} & \sigma_{t,22} \end{array}\right)\\  & = & \underbrace{\left(\begin{array}{cc} \omega_{11} & 0\\ \omega_{21} & \omega_{22} \end{array}\right)}_{\Omega}\underbrace{\left(\begin{array}{cc} \omega_{11} & 0\\ \omega_{21} & \omega_{22} \end{array}\right)^{\prime}}_{\Omega^{\prime}}+\underbrace{\left(\begin{array}{cc} \alpha_{11} & \alpha_{12}\\ \alpha_{21} & \alpha_{22} \end{array}\right)^{\prime}}_{A^{\prime}}\underbrace{\left(\begin{array}{cc} Y_{t-1,1}^{2} & Y_{t-1,1}Y_{t-1,2}\\ Y_{t-1,2}Y_{t-1,1} & Y_{t-1,2}^{2} \end{array}\right)}_{Y_{t-1}^{\prime}Y_{t-1}}\underbrace{\left(\begin{array}{cc} \alpha_{11} & \alpha_{12}\\ \alpha_{21} & \alpha_{22} \end{array}\right)}_{A}\\  &  & +\underbrace{\left(\begin{array}{cc} \beta_{11} & \beta_{12}\\ \beta_{21} & \beta_{22} \end{array}\right)^{\prime}}_{B^{\prime}}\underbrace{\left(\begin{array}{cc} \sigma_{t-1,11} & \sigma_{t-1,12}\\ \sigma_{t-1,21} & \sigma_{t-1,22} \end{array}\right)}_{\Sigma_{t-1}}\underbrace{\left(\begin{array}{cc} \beta_{11} & \beta_{12}\\ \beta_{21} & \beta_{22} \end{array}\right)}_{B} \end{eqnarray*}

where \omega, \alpha and \beta are coefficients. We can find the simple idea behind the BEKK and DCC models are similar that the volatilities/correlations are dependent on their past realisations and the shocks from squared financial asset returns.

  • Cons: too many parameters. This implies those parameters may be hard to interpret. Furthermore, many parameters are often found to be statistically insignificant, which suggests the model may be overparameterized.

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