On the Univariate Representation of BEKK Models with Common Factors

First, we investigate the minimal order univariate representation of some well known n-dimensionalconditional volatility models. Even simple low order systems (e.g. a multivariate GARCH(0,1)) forthe joint behavior of several variables imply individual processes with a lot of persistence inthe form of high order lags. However, we show that in the presence of common GARCH factors,parsimonious univariate representations (e.g. GARCH(1,1)) can result from large multivariatemodels generating the conditional variances and conditional covariances/correlations. The trivialdiagonal model without any contagion effects in conditional volatilities gives rise to the sameconclusions though.Consequently, we then propose an approach to detect the presence of these commonalities inmultivariate GARCH process. The factor we extract is the volatility of a portfolio made up by theoriginal assets whose weights are determined by the reduced rank analysis.We compare the small sample performances of two strategies. First, extending Engle and Marcucci(2006), we use reduced rank regressions in a multivariate system for squared returns andcross-returns. Second we investigate a likelihood ratio approach, where under the null the matrixparameters of the BEKK have a reduced rank structure (Lin, 1992). It emerged that the latterapproach has quite good properties enabling us to discriminate between a system with seeminglyunrelated assets (e.g. a diagonal model) and a model with few common sources of volatility.