Moment bounds for non-linear functionals of the periodogram

TL;DR

This paper establishes the validity of Edgeworth expansions for Fourier transforms of linear time series and applies this to analyze moments of non-linear periodogram functionals, including the mean square error of a long memory estimator.

Contribution

It extends the validity of Edgeworth expansions to non-Gaussian linear processes and analyzes the mean square error of the Geweke and Porter-Hudak estimator.

Findings

Edgeworth expansion validity for linear processes

Expression for the mean square error of the estimator

Estimator is rate optimal for long memory parameters

Abstract

In this paper, we prove the validity of the Edgeworth expansion of the Discrete Fourier transforms of some linear time series. This result is applied to approach moments of non linear functionals of the periodogram. As an illustration, we give an expression of the mean square error of the Geweke and Porter-Hudak estimator of the long memory parameter. We prove that this estimator is rate optimal, extending the result of Giraitis, Robinson, Samarov (1997) from Gaussian to linear processes.