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Polyspectral Mean Estimation of General Nonlinear Processes

Dhrubajyoti Ghosh, Tucker McElroy, Soumendra Lahiri

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.

Polyspectral Mean Estimation of General Nonlinear Processes

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.

Paper Structure

This paper contains 21 sections, 6 theorems, 48 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and examples
  3. Asymptotic Results for Polyspectral Means
  4. Asymptotic mean and variance
  5. Asymptotic normality
  6. Testing of Linear Process Hypothesis Using Bispectrum
  7. Simulation
  8. AR(2) Process
  9. ARMA(2,1) Process
  10. Squared Hermite
  11. Simulation Results
  12. Power Curve for Linearity Test
  13. Real Data Analysis
  14. Sunspot Data Linearity Test
  15. GDP Trend Clustering
  16. ...and 6 more sections

Key Result

Proposition 1

Suppose that for some $k \geq 1$, $\{ X_t \}$ is a ${(k+1)}^{th}$ order stationary time series and suppose that Assumption A[r] holds for all $2\leq r\leq k+1$. Let $g$ be a weight function satisfying the symmetry condition (eq:sym-g) that has finite absolute integral over the k-torus. Then the poly The $o(1)$-remainder term is $O(T^{-1})$, provided where $g_T (\omega)= \sum_{\underline{\tilde{\l

Figures (7)

  • Figure 1: Differenced GDP (left panel) and raw GDP (right panel) for 136 countries, annual 1980-2020. There is a common nonlinear trend (right panel).
  • Figure 2: Heatmap for different g functions: (left panel) $g(\underline{\lambda}) = cos(2 \lambda)cos(3 \lambda)$, (center panel) $g(\underline{\lambda}) = \mathbb{I}(\lambda_1 \leq 0.2)\mathbb{I}(\lambda_2 \leq 0.2)$, (right panel) $g(\underline{\lambda}) = \mathbb{I}( 0.1 \leq \lambda_1^2 + \lambda_2 ^2 \leq 0.2)$.
  • Figure 3: Bispectrum of ARMA(2,1) process with parameters given in Section \ref{['sec:arma']}, left panel exhibiting the 3d-plot and the right panel showing the heatmap.
  • Figure 4: 3-d plot of the weight functions given in the text. Polyspectral means with different weight functions provide different features of the time series.
  • Figure 5: Power Curve of nonlinearity test for different choices of $\theta$ in the process (\ref{['eq:quad-proc']}). Power is increasing in $\theta$, which parameterizes deviation from linearity.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Theorem 1
  • Corollary 2
  • Theorem 2
  • proof
  • proof
  • proof
  • proof
  • ...and 1 more