A data-driven score test of fit for testing the conditional distribution within the class of stationary GARCH(p,q) models is presented. In this paper extension of the complete results obtained by Inglot and Stawiarski in [7], as well as in Stawiarski [15] for the parsimonious GARCH(1,1) case is proposed. The null (composite) hypothesis subject to testing asserts that the innovations distribution, determining the GARCH conditional distribution, belongs to the specified parametric family. Generalized Error Distribution (called also Exponential Power) seems of special practical value. Applying the pioneer idea of Neyman [13] dating back to 1937, in combination with dimension selection device proposed by Ledwina [10] in 1994, lead to derivation uf the efficient score statistic and its data-driven version for this testing problem. In the case of GARCH(1,1) model both the asymptotic null distribution of the score statistic has been already established in [7] and [15], together with the asymptotics of the data-driven test statistic with appropriately regular estimators plugged in place of nuisance parameters. Main results are only stated herewith, while for detailed proofs inspection and power simulations, ample reference to these papers is provided. We show that the test derivation and asymptotic results carry over to stationary ARCH(q) models for any q 2 N. Moreover, thanks to ARCH(1) representation of the GARCH(p,q) model, the test can asymptotically encompass the full GARCH family, which as a final result provides the flexible testing tool in the GARCH(p, q) framework.

A data-driven score test of fit for testing the conditional distribution within the class of stationary GARCH(p,q) models is presented. In this paper extension of the complete results obtained by Inglot and Stawiarski in [7], as well as in Stawiarski [15] for the parsimonious GARCH(1,1) case is proposed. The null (composite) hypothesis subject to testing asserts that the innovations distribution, determining the GARCH conditional distribution, belongs to the specified parametric family. Generalized Error Distribution (called also Exponential Power) seems of special practical value. Applying the pioneer idea of Neyman [13] dating back to 1937, in combination with dimension selection device proposed by Ledwina [10] in 1994, lead to derivation uf the efficient score statistic and its data-driven version for this testing problem. In the case of GARCH(1,1) model both the asymptotic null distribution of the score statistic has been already established in [7] and [15], together with the asymptotics of the data-driven test statistic with appropriately regular estimators plugged in place of nuisance parameters. Main results are only stated herewith, while for detailed proofs inspection and power simulations, ample reference to these papers is provided. We show that the test derivation and asymptotic results carry over to stationary ARCH(q) models for any q 2 N. Moreover, thanks to ARCH(1) representation of the GARCH(p,q) model, the test can asymptotically encompass the full GARCH family, which as a final result provides the flexible testing tool in the GARCH(p, q) framework.