Diagnostic checking for conditional heteroscedasticity models

We suggest the score type tests for goodness-of-fit of conditional heteroscedasticity models in both univariate and multivariate time series. The tests can detect the alternatives converging to the null at a parametric rate. Weight functions are involved in the construction of the tests, which provides us with the flexibility to choose scores, especially under directional alternatives, for enhancing power performance. Furthermore, when the alternatives are not directional, we construct asymptotically distribution-free maximin tests for a large class of alternatives. A possibility to construct score-based omnibus tests is discussed when the alternative is saturated. The power performance is also investigated. A simulation study is carried out and a real data is analyzed.