Large Time-Varying Covariance Matrices with Applications to Finance

Correlations among the asset returns are the main reason for the computational and statistical complexities of the full multivariate GARCH models. We rely on the variancecorrelation separation strategy and introduce a broad class of multivariate models in the spirit of Engle’s (2002) dynamic conditional correlation models, that is univariate GARCH models are used for variances of individual assets coupled with parsimonious parametric models either for the timevarying correlation matrices or the components of their spectral and Cholesky decompositions. Numerous examples of structured correlation matrices along with structured components of the Cholesky decomposition are provided. This approach, while reducing the number of correlation parameters and severity of the positive-definiteness constraint, leaves intact the interpretation and magnitudes of the coefficients of the univariate GARCH models as if there were no correlations. This property makes the approach more appealing than the existing GARCH models. Moreover, the Cholesky decompositions, unlike their competitors, decompose the normal likelihood function as a product of univariate normal likelihoods with independent parameters resulting in fast estimation algorithms. Gaussian maximum likelihood methods of estimation of the parameters are developed. The methodology is implemented for a real financial dataset with one hundred assets, and its forecasting power is compared with other existing models. Our preliminary numerical results show that the methodology can be applied to much larger portfolios of assets and it compares favorably with other models developed in quantitative finance.