MULTIVARIATE-ARCH: BIAS EVALUATION FOR THE ML ESTIMATOR

Emma M. IGLESIAS (Cardiff University)
Garry D.A. PHILLIPS (Cardiff University)

At the present time, there exists an important and growing econometric literature that deals with the application of multivariate-ARCH models to a variety of economic and financial data. In a recent paper, Ling and McAleer (ET, 2002 forthcoming) have proved the asymptotic normality of the Quasi-Maximum Likelihood Estimator (QMLE) in a VARMA-GARCH model only requiring the existence of the second order moment of the unconditional errors, and a finite fourth-order moment of the conditional errors, which implies a considerable advance. However, the properties of the finite sample performance of this estimator have not yet been fully explored. In this paper, we provide analytical theoretical results concerning the important biases that can arise when the ML estimation method is employed in a simple bivariate structure under the assumption of conditional heteroscedasticity. We analyse two models: one proposed in Wong and Li (1997) (where the disturbances are dependent but uncorrelated) and another proposed by Liu and Polasek (1999, 2000) (where conditional correlation is allowed). We show the evolution of the biases in both models and how they are always larger than those of a univariate framework. While in the first model the authors reported very small biases in their simulation study, we prove theoretically that the biases can be very severe in some of the ML estimators (MLEs) of parameters if the difference between the intercepts in the two variance equations is relatively large. We address as well a constraint that should be included in the estimation of the second analysed model but which has so far been ignored. Following Lumsdaine (1995), we also examine theoretically some invariance properties in the biases of some of the MLEs of parameters in the conditional variance-covariance matrix. Finally, we propose as well a SUR (seemingly unrelated) specification in both models, that provides an alternative way to retrieve the information matrix.