Testing for conditional heteroskedasticity with misspecified alternative hypotheses

For the multiple linear regression model, this paper examines the asymptotic behaviour of some tests for conditional heteroskedasticity when an alternative hypothesis is misspecified. In the standard likelihood-based framework, a Lagrange multiplier statistic based on a correctly specified alternative is ranked above one based on an overspecified alternative. It is shown that this ranking does not hold for statistics that are robust to the form of conditional heterokurticity. It is also shown that the ranking obtained in the standard likelihood-based framework does not depend on a joint density being specified in its entirety; the same ranking can be obtained when a type of information matrix equality holds.