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 stimulating discussions and Jonathan Dennis for helpful
research assistance. Rahbek acknowledges continuing financial support from
the Danish Social Sciences Research Council. Kristensen received funding
from the Danish Research Agency and the Financial Markets Group, LSE,
during this research.