A Note on the QMLE Limit Theory in the Non-stationary ARCH(1) Model

In this note we extend the standard results for the limit theory of the popular quasi-maximum likelihood estimator (QMLE) in the context of the non-stationary autoregressive conditional heteroskedastic ARCH(1) model by allowing the innovation process not to possess fourth moments. Depending on the value of the index of stability, we either derive a-stable weak limits with nonstandard rates or inconsistency and non-tightness. We obtain the limit theory by the derivation of a limit theorem for multiplicative "martingale" transforms with limit mixtures of alpha-stable distributions for any alpha is an element of(0, 2].