*Statistical Modelling* **15 (3)** (2015), 233–255

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Two Cholesky-log-GARCH models for multivariate volatilities

Xanthi Pedeli

Department of Statistics,

Athens University of Economics & Business,

Athens, Greece

e-mail: xpedeli@aueb.gr

Konstantinos Fokianos

Department of Mathematics & Statistics,

University of Cyprus,

Cyprus

Mohsen Pourahmadi

Department of Statistics,

Texas A&M University,

College Station, USA

### Abstract:

Parsimonious estimation of high-dimensional covariance matrices is of fundamental importance in multivariate statistics. Typical examples occur in finance, where the instantaneous dependence among several asset returns should be taken into account. Multivariate GARCH processes have been established as a standard approach for modelling such data. However, the majority of GARCH-type models are either based on strong assumptions that may not be realistic or require restrictions that are often too hard to be satisfied in practice. We consider two alternative decompositions of time-varying covariance matrices Σ_t. The first is based on the modified Cholesky decomposition of the covariance matrices and second relies on the hyperspherical parametrization of the standard Cholesky factor of their correlation matrices R_t . Then, we combine each Cholesky factor with the log-GARCH models for the corresponding time–varying volatilities and use a quasi maximum likelihood approach to estimate the parameters. Using log-GARCH models is quite natural for achieving the positive definiteness of Σ_t and this is a novelty of this work. Application of the proposed methodologies to two real financial datasets reveals their usefulness in terms of parsimony, ease of implementation and stresses the choice of the appropriate models using familiar data-driven processes such as various forms of the exploratory data analysis and regression.

#### Keywords:

Cholesky decomposition; Covariance matrix; GARCH model; hyperspherical coordinates; volatility.
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