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Parametric generalized spectrum for heavy-tailed time series

Yuichi Goto, Gaspard Bernard

Abstract

Recently, several spectra have emerged, designed to encapsulate the distributional characteristics of non-Gaussian stationary processes. This article introduces parametric families of generalized spectra based on the characteristic function, alongside inference procedures enabling $\sqrt{n}$-consistent estimation of the unknown parameters in a broad class of parametric models. These spectra capture non-linear dependencies without requiring that the underlying stochastic processes satisfy any moment assumptions. Crucially, this approach facilitates frequency domain analysis for heavy-tailed time series, including possibly non-causal Cauchy autoregressive models and discrete-stable integer-valued autoregressive models. To the best of our knowledge, the latter models have not been studied theoretically in the literature. By estimating parameters across both causal and non-causal parameter spaces, our method automatically identifies the causal or non-causal structure of Cauchy autoregressive models. Furthermore, our estimator does not depend on smoothing parameters since it is based on the integrated periodogram associated with the generalized spectrum. As applications, we develop goodness-of-fit tests, moving average unit-root tests, and tests for non-invertibility. We study the finite-sample performance of the proposed estimators and tests via Monte Carlo simulations, and apply the methodology to estimation and forecasting of a measles count dataset. We evaluate finite-sample performance using Monte Carlo simulations and illustrate the practical value of the procedure with an application to measles case-count estimation and forecasting.

Parametric generalized spectrum for heavy-tailed time series

Abstract

Recently, several spectra have emerged, designed to encapsulate the distributional characteristics of non-Gaussian stationary processes. This article introduces parametric families of generalized spectra based on the characteristic function, alongside inference procedures enabling -consistent estimation of the unknown parameters in a broad class of parametric models. These spectra capture non-linear dependencies without requiring that the underlying stochastic processes satisfy any moment assumptions. Crucially, this approach facilitates frequency domain analysis for heavy-tailed time series, including possibly non-causal Cauchy autoregressive models and discrete-stable integer-valued autoregressive models. To the best of our knowledge, the latter models have not been studied theoretically in the literature. By estimating parameters across both causal and non-causal parameter spaces, our method automatically identifies the causal or non-causal structure of Cauchy autoregressive models. Furthermore, our estimator does not depend on smoothing parameters since it is based on the integrated periodogram associated with the generalized spectrum. As applications, we develop goodness-of-fit tests, moving average unit-root tests, and tests for non-invertibility. We study the finite-sample performance of the proposed estimators and tests via Monte Carlo simulations, and apply the methodology to estimation and forecasting of a measles count dataset. We evaluate finite-sample performance using Monte Carlo simulations and illustrate the practical value of the procedure with an application to measles case-count estimation and forecasting.

Paper Structure

This paper contains 41 sections, 14 theorems, 165 equations, 12 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Generalized spectrum
  3. Parametric generalized spectrum
  4. Continuous-valued models
  5. Discrete-valued models
  6. Estimation of unknown parameter
  7. Goodness-of-fit test
  8. Hypothesis tests for unit-root and invertibility
  9. Two-sided hypothesis
  10. One-sided hypothesis
  11. Simulation studies
  12. Estimation
  13. Real-valued models
  14. Integer-valued models
  15. Hypothesis testing
  16. ...and 26 more sections

Key Result

Lemma 1

In the Cauchy MA(1) model, the vector of parameters ${\bm \theta}\coloneqq ( a, \delta)^\top$ is identifiable from the generalized spectra.

Figures (12)

  • Figure 1: Time series plot (left) and Hill plot (right) of the measles dataset.
  • Figure 2: Heatmaps of the real part (left panel) and the imaginary part (right panel) of the spectrum for the causal integer-valued AR(1) model in Example \ref{['IntegerAR(1)']}, with parameters $\delta = 2$, $p = 0.3$, $\alpha = 0.7$, and $\lambda = 0.785$.
  • Figure 3: Histograms of ${\hat{a}_n}$ and $\hat{\delta}_n$ (in green) with respect to limiting Gaussian distribution (in red) in the non-causal AR$(1)$ model with Cauchy innovations. $2000$ replication, $n=500$ and $(a, \delta)=(1.3, 2)$.
  • Figure 4: Histograms of ${\hat{p}_n}$ and $\hat{\delta}_n$ (in green) with respect to limiting Gaussian distribution (in red) in the INAR$(1)$ model with discrete stable innovations. $2000$ replication, $n=500$ and $(p, \delta)=(0.3, 2)$. The rate of selection of the proper $\alpha=0.7$ is a hundred percent.
  • Figure 5: One-step ahead predicted values of the measles data count time series (in red) and true value of the time series (in blue). The testing set size is $n_{\rm test}=246$.
  • ...and 7 more figures

Theorems & Definitions (30)

  • Example 1
  • Lemma 1
  • Example 2
  • Lemma 2
  • Lemma 3
  • Example 3
  • Lemma 4
  • Example 4
  • Lemma 5
  • Example 5
  • ...and 20 more