Polyspectral Mean Estimation of General Nonlinear Processes

TL;DR

This paper develops the asymptotic theory for polyspectral mean estimators of nonlinear processes, introduces a new bispectral-based test for linearity, and demonstrates applications in time series analysis and clustering.

Contribution

It derives the exact asymptotic distribution of polyspectral mean estimators and introduces a novel bispectral mean-based test for linear process hypothesis.

Findings

Asymptotic distribution depends on weighting functions and higher-order spectra.

Bispectral means can test linearity in time series.

Applications include Sunspot data analysis and GDP clustering.

Abstract

Higher-order spectra (or polyspectra), defined as the Fourier Transform of a stationary process' autocumulants, are useful in the analysis of nonlinear and non Gaussian processes. Polyspectral means are weighted averages over Fourier frequencies of the polyspectra, and estimators can be constructed from analogous weighted averages of the higher-order periodogram (a statistic computed from the data sample's discrete Fourier Transform). We derive the asymptotic distribution of a class of polyspectral mean estimators, obtaining an exact expression for the limit distribution that depends on both the given weighting function as well as on higher-order spectra. Secondly, we use bispectral means to define a new test of the linear process hypothesis. Simulations document the finite sample properties of the asymptotic results. Two applications illustrate our results' utility: we test the linear process hypothesis for a Sunspot time series, and for the Gross Domestic Product we conduct a clustering exercise based on bispectral means with different weight functions.