On prediction intervals for conditionally heteroscedastic processes

Kabaila (1999) argues that the standard 1-alpha prediction intervals for a broad class of conditionally heteroscedastic processes are justified by their possession of what he calls the 'relevance property'. He considers both the case that the parameters of the process are known and that these parameters are unknown. We consider the former case and ask whether these prediction intervals can, alternatively, be deduced from the requirements of both (a) unconditional coverage probability 1 - alpha and (b) minimum unconditional expected length. We show that the answer to this question is no, by presenting a counterexample. This counterexample concerns the standard 95% one-step-ahead prediction interval in the context of a simple Markoviau bilinear process.