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Valeri Voev
Dynamic modelling of large-dimensional
covariance matrices
Abstract Modelling and forecasting the covariance of financial return series has
always been a challenge due to the so-called ‘curse of dimensionality’. This paper
proposes a methodology that is applicable in large-dimensional cases and is based
on a time series of realized covariance matrices. Some solutions are also presented
to the problem of non-positive definite forecasts. This methodology is then
compared to some traditional models on the basis of its forecasting performance
employing Diebold–Mariano tests. We show that our approach is better suited
to capture the dynamic features of volatilities and covolatilities compared to the
sample covariance based models.
1 Introduction
Modelling and forecasting the variances and covariances of asset returns is crucial
for financial management and portfolio selection and re-balancing. Recently, this
branch of the econometric literature has grown at a very fast pace. One of the simplest
methods used is the sample covariance matrix.Astylized fact, however, is that
there is a serial dependence in the second moments of returns. Thus, more sophisticated
models had to be developed which incorporate this property, as well as other
well-known features of financial return distributions such as leptokurtosis or the
so-called ‘leverage effect’. This led to the development of the univariate GARCH
processes and their extension—the multivariate GARCH (MGARCH) models (for
a comprehensive review see Bauwens et al. (2006)), which include also the modelling
of covariances. One of the most severe drawbacks of the MGARCH models,
however, is the difficulty of handling dimensions higher than 4 or 5 (or with very
restrictive assumptions). Another more practically oriented field of research deals
Valeri Voev (�)
University of Konstanz, CoFE, Konstanz, Germany
PBox D-124, University of Konstanz, 78457 Konstanz, Germany
E-mail: valeri.voev@uni-konstanz.de