Multivariate arch modelsfinite sample properties of ml estimators and an application to an lm-type test

Abstract

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. However, the properties of the estimation procedures that are used have not yet been fully explored. This paper provides two main new results: the first concerns the large biases and variances that can arise when the ML estimation method is employed in a simple bivariate structure under the assumption of conditional heteroscedasticity; and the second examines how to use these analytical theoretical results to improve the size and the power of a test for multivariate ARCH effects. We analyse two models: one proposed in Wong and Li (1997) (where the disturbances are dependent but uncorrelated) and another proposed by Engle and Kroner (1995) and Liu and Polasek (1999, 2000) (where conditional correlation is allowed through a diagonal representation). We prove theoretically that a relatively large difference between the intercepts in the two conditional variance equations produces, in the first model, very large variances in some of the ML estimators and, in the second, very severe biases in some of the ML estimators of the parameters. Later we use our bias expressions to propose an LM type test of multivariate ARCH effects, showing that the size and the power of the test improve when we allow for bias correction in the estimators, and that the best recommendation in practical applications is always to use the expected hessian version of the LM. We address as well some constraints that should be included in the estimation of the models but which have so far been ignored. Finally, we present a SUR (seemingly unrelated) specification in both models, that provides an alternative way to retrieve the information matrix. We also extend Lumsdaine (1995) results in multivariate framework.