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

摘要

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 nonstationary 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 α-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 α-stable distributions for any a e (0,2].