A model specification test for GARCH(1,1) processes

We provide a consistent specification test for GARCH(1,1) models based on a test statistic of Cramér-von Mises type. Since the limit distribution of the test statistic under the null hypothesis depends on unknown quantities in a complicated manner, we propose a model-based (semiparametric)bootstrap method to approximate critical values of the test and verify its asymptotic validity. Finally, we illuminate the finite sample behavior of the test by some simulations.