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Spectral estimation for high-dimensional linear processes

Jamshid Namdari, Alexander Aue, Debashis Paul

TL;DR

The paper develops a spectral-estimation framework for high-dimensional linear processes under the key assumption of simultaneous diagonalizability of the coefficient matrices and innovation covariance. By analyzing weighted integrals of the sample periodogram through Stieltjes transforms, it derives a limiting spectral distribution (LSD) description and provides a procedure to estimate the joint spectral distribution of the coefficients as a discrete grid mixture, minimizing an $L^\kappa$ discrepancy. It further offers a rotationally equivariant estimator of the spectral density matrix in a common eigenbasis and proves consistency in the $L^2$ sense when the grid is known, complemented by a bootstrap-based model selection method. The methodology is validated via simulations and applied to S&P 500 data, uncovering dependence structures beyond simple factor models and illustrating practical relevance for high-dimensional time series analysis.

Abstract

We propose a novel estimation procedure for certain spectral distributions associated with a class of high dimensional linear time series. The processes under consideration are of the form $X_t = \sum_{\ell=0}^\infty \mathbf{A}_\ell Z_{t-\ell}$ with iid innovations $(Z_t)$. The key structural assumption is that the coefficient matrices and the variance of the innovations are simultaneously diagonalizable in a common orthonormal basis. We develop a strategy for estimating the joint spectral distribution of the coefficient matrices and the innovation variance by making use of the asymptotic behavior of the eigenvalues of appropriately weighted integrals of the sample periodogram. Throughout we work under the asymptotic regime $p,n \to \infty$, such that $p/n\to c \in (0,\infty)$, where $p$ is the dimension and $n$ is the sample size. Under this setting, we first establish a weak limit for the empirical distribution of eigenvalues of the aforementioned integrated sample periodograms. This result is proved using techniques from random matrix theory, in particular the characterization of weak convergence by means of the Stieltjes transform of relevant distributions. We utilize this result to develop an estimator of the joint spectral distribution of the coefficient matrices, by minimizing an $L^κ$ discrepancy measure, for $κ\geq 1$, between the empirical and limiting Stieltjes transforms of the integrated sample periodograms. This is accomplished by assuming that the joint spectral distribution is a discrete mixture of point masses. We also prove consistency of the estimator corresponding to the $L^2$ discrepancy measure. We illustrate the methodology through simulations and an application to stock price data from the S\&P 500 series.

Spectral estimation for high-dimensional linear processes

TL;DR

The paper develops a spectral-estimation framework for high-dimensional linear processes under the key assumption of simultaneous diagonalizability of the coefficient matrices and innovation covariance. By analyzing weighted integrals of the sample periodogram through Stieltjes transforms, it derives a limiting spectral distribution (LSD) description and provides a procedure to estimate the joint spectral distribution of the coefficients as a discrete grid mixture, minimizing an discrepancy. It further offers a rotationally equivariant estimator of the spectral density matrix in a common eigenbasis and proves consistency in the sense when the grid is known, complemented by a bootstrap-based model selection method. The methodology is validated via simulations and applied to S&P 500 data, uncovering dependence structures beyond simple factor models and illustrating practical relevance for high-dimensional time series analysis.

Abstract

We propose a novel estimation procedure for certain spectral distributions associated with a class of high dimensional linear time series. The processes under consideration are of the form with iid innovations . The key structural assumption is that the coefficient matrices and the variance of the innovations are simultaneously diagonalizable in a common orthonormal basis. We develop a strategy for estimating the joint spectral distribution of the coefficient matrices and the innovation variance by making use of the asymptotic behavior of the eigenvalues of appropriately weighted integrals of the sample periodogram. Throughout we work under the asymptotic regime , such that , where is the dimension and is the sample size. Under this setting, we first establish a weak limit for the empirical distribution of eigenvalues of the aforementioned integrated sample periodograms. This result is proved using techniques from random matrix theory, in particular the characterization of weak convergence by means of the Stieltjes transform of relevant distributions. We utilize this result to develop an estimator of the joint spectral distribution of the coefficient matrices, by minimizing an discrepancy measure, for , between the empirical and limiting Stieltjes transforms of the integrated sample periodograms. This is accomplished by assuming that the joint spectral distribution is a discrete mixture of point masses. We also prove consistency of the estimator corresponding to the discrepancy measure. We illustrate the methodology through simulations and an application to stock price data from the S\&P 500 series.

Paper Structure

This paper contains 22 sections, 5 theorems, 81 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Spectral distributions of weighted integrals of the sample periodogram
  3. Estimation procedure
  4. Spectral estimation procedure
  5. Estimation of the spectral density matrix
  6. Estimation procedure
  7. Consistency under the $L^2$ loss
  8. Sufficient condition for consistency
  9. Special cases
  10. Independent observations
  11. ARMA processes
  12. AR$(1)$ processes
  13. Model selection
  14. Simulation study
  15. Estimation of the spectral distribution
  16. ...and 7 more sections

Key Result

Theorem 2.1

Consider the linear process $(X_t\colon t \in \mathbb{Z})$ satisfying assumptions A.0--A.4, and suppose that $p/n \to c \in (0,\infty)$ as $p,n\to \infty$. Then, $F_g^{(n)}$ converges almost surely to a nonrandom probability distribution $F_g$ with Stieltjes transform $\mathcal{S}_g(z)$ determined b where $K_g\colon \mathbb{K}\times\mathbb{C}^+ \to \mathbb{C}^+$ is the unique solution to (eq:K) su

Figures (8)

  • Figure 1.1: Plot of Median and $90\%$ confidence band for spectral cdf of AR coefficient matrix corresponding to the case 2.1. Dash-Dot Red curve: median, Dashed Blue curve: $90\%$ confidence band, Black Solid curve: true spectral cdf
  • Figure 1.2: Plot of Median and $90\%$ confidence band for spectral cdf of MA coefficient matrix corresponding to the case 2.1. Dash-Dot Red curve: median, Dashed Blue curve: $90\%$ confidence band, Black Solid curve: true spectral cdf
  • Figure 1.3: Plot of Median and $90\%$ confidence band for spectral cdf of $\boldsymbol{\Sigma}$ corresponding to the case 2.1. GreenDash-Dot Red curve: median, Dashed Blue curve: $90\%$ confidence band, Black Solid curve: true spectral cdf
  • Figure 1.4: Plot of Median and $90\%$ confidence band for spectral cdf of $\boldsymbol{A_1}$ corresponding to the case 2.2. Dash-Dot Red curve: median, Dashed Blue curve: $90\%$ confidence band, Black Solid curve: true spectral cdf
  • Figure 1.5: Plot of Median and $90\%$ confidence band for spectral cdf of $\boldsymbol{A_2}$ corresponding to the case 2.2. Dash-Dot Red curve: median, Dashed Blue curve: $90\%$ confidence band, Black Solid curve: true spectral cdf
  • ...and 3 more figures

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 5.1
  • Proposition 5.1
  • Proposition 5.2
  • Proposition 5.3