Bivariate arch models: Finite-sample properties of QML estimators and an application to an LM-type test

This paper provides two main new results: the first shows theoretically that large biases and variances can arise when the quasi-maximum likelihood (QML) estimation method is employed in a simple bivariate structure under the assumption of conditional heteroskedasticity; and the second demonstrates how these analytical theoretical results can be used to improve the finite-sample performance of a test for multivariate autoregressive conditional heteroskedastic (ARCH) effects, suggesting an alternative to a traditional Bartlett-type correction. We analyze two models: one proposed in Wong and Li (1997, Biometrika 84, 111-123) and another proposed by Engle and Kroner (1995, Econometric Theory 11, 122-150) and Liu and Polasek (1999, Modelling and Decisions in Economics; 2000, working paper, University of Basel). We prove theoretically that a relatively large difference between the intercepts in the two conditional variance equations, which leads to the two series having correspondingly different volatilities in the restricted case, may produce very large variances in some QML estimators in the first model and very severe biases in some QML estimators in the second. Later we use our bias expressions to propose an LM-type test of multivariate ARCH effects and show through simulations that small-sample improvements are possible, especially in relation to the size, when we bias correct the estimators and use the expected hessian version of the test.