How do multivariate GARCH models capture volatility dynamics across assets?

Question

AcadiFi · Accepted Answer

Multivariate GARCH (MGARCH) models the time-varying covariance matrix Ht of a k-vector of returns. Several parameterizations exist, each balancing flexibility and estimability:

1. VEC model: vech(Ht) = c + A vech(et-1 et-1') + B vech(Ht-1). Most general but has k(k+1)(k(k+1)+1)/2 parameters—explosive with k.
2. BEKK model: Ht = C'C + A' et-1 et-1' A + B' Ht-1 B. Guarantees positive definiteness and has k^2(p+q) + k(k+1)/2 parameters.
3. CCC (Constant Conditional Correlation): Ht = Dt R Dt with Dt diagonal of univariate GARCH volatilities and R constant correlation matrix. Simple and tractable but restrictive.
4. DCC (Dynamic Conditional Correlation): allows R to evolve. Discussed separately.

Example 3-asset BEKK captures cross-asset volatility spillovers: an innovation in asset 1 can affect asset 2's future conditional variance. Applications include portfolio optimization with time-varying covariances, hedge ratio calculation for cross-asset hedges, and VaR models that account for correlation increases in stress.

Estimation uses quasi-maximum likelihood. Diagnostic checks include Ljung-Box on standardized residuals and multivariate ARCH tests. Challenges include high parameter count, convergence issues with large systems, and interpretation of cross-parameters. Practitioners often favor DCC or factor-based MGARCH for large portfolios because they scale better while capturing the stylized fact that correlations rise during market stress.

How do multivariate GARCH models capture volatility dynamics across assets?

Master Level II with our CFA Course

Related Questions

Practice Questions