We consider a goodness-of-fit test for certain parametrizations of conditionally heteroscedastic time series with unobserved components. Our test is quite general in that it can be employed to validate any given specification of arbitrary order and may even be invoked for testing not just GARCH models but also some related models such as autoregressive conditional duration models. The test statistic utilizes the characterization of Bierens (J Econom 20:105–134, 1982) and may be written down in a convenient closed-form expression. Consistency of the test is proved, and the asymptotic distribution of the test statistic under the null hypothesis is studied. Since this distribution depends on unknown quantities, two bootstrap resampling schemes are investigated and compared in order to approximate critical points and actually carry out the test. Finite-sample results are presented as well as applications of the proposed procedures to real data from the financial markets.

中文翻译：

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条件异方差模型的参数规范的拟合优度检验

我们考虑具有条件的异方差时间序列中某些参数未观测到的参量的拟合优度检验。我们的测试相当笼统，因为它可以用来验证任意给定的任意顺序规范，甚至可以调用它来测试GARCH模型和一些相关模型，例如自回归条件持续时间模型。检验统计量利用Bierens的特征（J Econom 20：105–134，1982），可以用方便的封闭形式表达出来。证明了检验的一致性，并研究了原假设下检验统计量的渐近分布。由于此分布取决于未知数量，因此对两种自举重采样方案进行了研究和比较，以逼近临界点并实际进行测试。