A Goodness-of-fit test for GARCH innovation density

We prove asymptotic normality of a suitably standardized integrated square difference between a kernel type error density estimator based on residuals and the expected value of the error density estimator based on innovations in GARCH models. This result is similar to that of Bickel-Rosenblatt under i.i.d. set up. Consequently the goodness-of-fit test for the innovation density of the GARCH processes based on this statistic is asymptotically distribution free, unlike the tests based on the residual empirical process. A simulation study comparing the finite sample behavior of this test with Kolmogorov-Smirnov test and the test based on integrated square difference between the kernel density estimate and null density shows some superiority of the proposed test.