Whittle estimator for finite-variance non-Gaussian time series with long memory

Abstract

We consider time series Yt = G(Xt) where Xt is Gaussian with long memory and G is a polynomial. The series Yt may or may not have long memory. The spectral density gθ(cursive Greek chi) of Yt is parameterized by a vector θ and we want to estimate its true value θ0. We use a least-squares Whittle-type estimator θ̂N for θ0, based on observations Y1 , . . . ,YN. If Yt is Gaussian, then √N(θ̂N - θ0) converges to a Gaussian distribution. We show that for non-Gaussian time series Yt, this √N consistency of the Whittle estimator does not always hold and that the limit is not necessarily Gaussian. This can happen even if Yt has short memory.