The likelihood of multivariate GARCH models is ill-conditioned

Abstract

The likelihood of multivariate GARCH models is ill-conditioned because of two facts. First, financial time series often display high correlations, implying that an eigenvalue af the conditional covariances fluctuates near the zero boundary. Second, GARCH models explain conditional covariances in terms of a linear combination of delayed squared errors and their conditional expectation; this functional form implies that the likelihood function is almost flat in the neighborhood of the optimal estimates. Building on this analysis we propose a linear transformation of data which, not only stabilizes the likelihood computation, but also provides insight about the statistical properties of data. The use of this transfonnation is illustrated by modeling the short-run conditional correlations of four nominal exchange rates.