Pseudo-maximum likelihood estimation of ARCH(∞) models

Strong consistency and asymptotic normality of the Gaussian pseudo-maximum likelihood estimate of the parameters in a wide class of ARCH(∞) processes are established. We require the ARCH weights to decay at least hyperbolically, with a faster rate needed for the central limit theorem than for the law of large numbers. Various rates are illustrated in examples of particular parameteriza- tions in which our conditions are shown to be satisfied.