Instantaneous dependence among several asset returns is the main reason for the computational and statistical complexities in working with full multivariate GARCH models. Using the Cholesky decomposition of the covariance matrix of such returns, we introduce a broad class of multivariate models where univariate GARCH models are used for variances of individual assets and parsimonious models for the time-varying unit lower triangular matrices. This approach, while reducing the number of parameters and severity of the positive-definiteness constraint, has several advantages compared to the traditional orthogonal and related GARCH models. Its major drawback is the potential need for an a priori ordering or grouping of the stocks in a portfolio, which through a case study we show can be taken advantage of so far as reducing the forecast error of the volatilities and the dimension of the parameter space are concerned. Moreover, the Cholesky decomposition, unlike its 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 seven assets, and its forecasting power is compared with other existing models.
Autoregressive conditional heteroscedastic models Latent factor models Time-varying ARMA coefficients Cholesky decomposition Principal components Spectral decomposition Stochastic volatility models Maximum likelihood estimation
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Barndorff-Nielsen, O.E., Shephard, N.: Econometric analysis of realised covariation: high frequency based covariance, regression and correlation in financial economics. Econometrica 72, 885–925 (2004)