"Frontmatter". In: Analysis of Financial Time Series

376 MULTIVARIATE VOLATILITY MODELSrho(t)0.2 0.4 0.6 0.8• • •• •• •• •• • •• •• •• ••• • • • • • •••• • •••• ••• •• • • • • •••• ••••• • •• •••• •• • •••••• • • •• ••••• •••• •••• • • •• ••• • •••• • •••••• • •• •• • ••• •• • • •• • • • ••• • •• • ••• • • • ••• •• •• •• •••• •• ••• • •• •••• • • • ••• •• •• ••• •• •••••••• •• • •• • • •••• • • • • ••• •••• • • ••• •• • •• •••• • • • •••• •• ••• •• •• • •• ••• • •••••••••• • •• • • ••• •• •• •• •••• • • •• • •• • • • •• ••• •• ••• • •• •• • • •• • •• •• •1940 1960 1980 2000yearFigure 9.7. The fitted conditional correlation coefficient between monthly log returns of IBMstock and the S&P 500 index using the time-varying correlation GARCH(1, 1) model ofExample 9.2 with Cholesky decomposition. The horizontal line denotes the average 0.612of the estimated coefficients.Remark: In a recent manuscript, Tse and Tsui (1998) consider a multivariateGARCH model with time-varying correlations. For a k-dimensional returns, theseauthors assume that the conditional correlation matrix ρ t follows the modelρ t = (1 − θ 1 − θ 2 )ρ + θ 1 ρ t−1 + θ 2 ψ t−1 ,where θ 1 and θ 2 are scalar parameters, ρ is a k × k positive definite matrix with unitdiagonal elements, and ψ t−1 is the k × k sample correlation matrix using shocksfrom t − m,...,t − 1 for a prespecified m. Estimation of the two scalar parametersθ 1 and θ 2 requires special constraints to ensure positive definiteness of the correlationmatrix. This approach seems much more complicated than the two methodsconsidered in this chapter.9.3 HIGHER DIMENSIONAL VOLATILITY MODELSIn this section, we make use of the sequential nature of Cholesky decomposition tosuggest a strategy for building a high-dimensional volatility model. Again, write thevector return series as r t = µ t + a t . The mean equations for r t can be specified by