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On the edge eigenvalues of the precision matrices of nonstationary autoregressive processes

Junho Yang

Abstract

This paper investigates structural changes in the parameters of first-order autoregressive models by analyzing the edge eigenvalues of the precision matrices. Specifically, edge eigenvalues in the precision matrix are observed if and only if there is a structural change in the autoregressive coefficients. We show that these edge eigenvalues correspond to the zeros of a determinantal equation. Additionally, we propose a consistent estimator for detecting outliers within the panel time series framework, supported by numerical experiments.

On the edge eigenvalues of the precision matrices of nonstationary autoregressive processes

Abstract

This paper investigates structural changes in the parameters of first-order autoregressive models by analyzing the edge eigenvalues of the precision matrices. Specifically, edge eigenvalues in the precision matrix are observed if and only if there is a structural change in the autoregressive coefficients. We show that these edge eigenvalues correspond to the zeros of a determinantal equation. Additionally, we propose a consistent estimator for detecting outliers within the panel time series framework, supported by numerical experiments.

Paper Structure

This paper contains 21 sections, 21 theorems, 157 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Limiting spectral distribution of $\mu_{A_n}$
  3. Preliminaries
  4. LSDs of the null model and SCM and proof of Theorem \ref{['thm:1']}
  5. Outliers of the SCM
  6. Outlier behavior of the null model and proof of Theorem \ref{['lemma:outlier.null']}
  7. Outliers of the single SCM
  8. Outliers of the single interval SCM
  9. Outliers of the general SCM
  10. Parameter identification
  11. Outlier detection of a panel time series
  12. Numerical experiment
  13. Proof of Theorem \ref{['thm:spacing']}
  14. Concluding remarks and extensions
  15. Properties of $\mu_\rho$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $A_n$ be the precision matrix of an AR$(1)$ model, where the AR coefficients adhere to (eq:rho.model) (including the case $m=0$). Furthermore, we assume Then, we have $\mu_{A_n} \xrightarrow{D} \mu_{\rho}$ as $n \rightarrow \infty$, where $\xrightarrow{D}$ denotes weak convergence. Moreover, the LSD $\mu_{\rho}$ is compactly supported.

Figures (2)

  • Figure 1: Time series trajectories for the null model (left) and the SCM (right). Vertical dashed line indicates the time of the structural change in the SCM.
  • Figure 2: Empirical spectral distributions the precision matrices. Left: null model, right: the SCM. Crosses on the right panel indicate the outliers.

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Lemma 3.1
  • Theorem 3.1
  • Remark 3.1
  • Proposition 3.1
  • ...and 14 more