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Strong consistency and asymptotic normality are established for the quasi-maximum likelihood estimator for a class of ARCH(q) models. The conditions are that the ARCH process is geometrically ergodic with a moment of arbitrarily small order. Furthermore for consistency, we assume that the second-order moment exists for the nondegenerate rescaled errors and, similarly, that the fourth-order moment exists for asymptotic normality to hold. Contrary to existing literature on (G)ARCH models the parameter space is not assumed to be compact; we only impose a lower bound for the constant term in our parameterization of the conditional variance. It is demonstrated that the general conditions are satisfied for a range of specific models.We are grateful to the editor and the referees for their very helpful and detailed suggestions, which, we believe, improved the paper substantially. We thank Søren T. Jensen for …