Score test of fit for composite hypothesis in the GARCH(1,1) model

Exploiting the pioneer idea of Neyman [1937. Smooth test for goodness of fit. Skand. Aktuarietidskr. 20, 149-199] and the device proposed by Ledwina [1994. Data driven version of Neyman's smooth test of fit. J. Amer. Stat. Assoc. 89, 1000-1005], the efficient score statistic and its data-driven version are derived for this testing problem. The asymptotic null distribution of the score statistic is established. Replacing the nuisance parameters with their square-root consistent estimators results in the data-driven test statistic. It is proved that in that case the asymptotic behaviour of the test statistic remains unchanged under appropriate regularity conditions and under discretization of the estimators. Computer simulations of the critical value and the power performance of the test for several alternatives in the case of generalized error distribution family serving as a null distribution are also presented.