Asymptotics of the QMLE for Non-Linear ARCH Models

Asymptotic properties of the quasi-maximum likelihood estimator (QMLE) for non-linear ARCH(q) models -- including for example Asymmetric Power ARCH and log-ARCH -- are derived. Strong consistency is established under the assumptions that the ARCH process is geometrically ergodic, the conditional variance function has a finite log-moment, and finite second moment of the rescaled error. Asymptotic normality of the estimator is established under the additional assumption that certain ratios involving the conditional variance function are suitably bounded, and that the rescaled errors have little more than fourth moment. We verify our general conditions, including identification, for a wide range of leading specific ARCH models.