[Paper Review] On the inference about the spectra of high-dimensional covariance matrix based on noisy observations-with applications to integrated covolatility matrix inference in the presence of microstructure noise

Ningning Xia, Xinghua Zheng|arXiv (Cornell University)|Sep 7, 2014

Financial Risk and Volatility Modeling25 references3 citations

TL;DR

This paper develops a two-step spectral inference method for high-dimensional covariance matrices when observations are corrupted by noise, specifically applying it to integrated covariance matrix estimation in high-frequency finance with microstructure noise. It establishes an asymptotic link between the limiting spectral distribution (LSD) of noisy sample covariance matrices and the true underlying covariance matrix, enabling consistent spectral estimation via a modified pre-averaging approach and a noise-robust estimator governed by the Marčenko-Pastur equation.

ABSTRACT

In practice, observations are often contaminated by noise, making the resulting sample covariance matrix to be an information-plus-noise-type covariance matrix. Aiming to make inferences about the spectra of the underlying true covariance matrix under such a situation, we establish an asymptotic relationship that describes how the limiting spectral distribution of (true) sample covariance matrices depends on that of information-plus-noise-type sample covariance matrices. As an application, we consider the inference about the spectra of integrated covolatility (ICV) matrices of high-dimensional diffusion processes based on high-frequency data with microstructure noise. The (slightly modified) pre-averaging estimator is an information-plus-noise-type covariance matrix, and the aforementioned result, together with a (generalized) connection between the spectral distribution of true sample covariance matrices and that of the population covariance matrix, enables us to propose a two-step procedure to estimate the spectral distribution of ICV for a class of diffusion processes. An alternative estimator is further proposed, which possesses two desirable properties: it eliminates the impact of microstructure noise, and its limiting spectral distribution depends only on that of the ICV through the standard Marčenko-Pastur equation. Numerical studies demonstrate that our proposed methods can be used to estimate the spectra of the underlying covariance matrix based on noisy observations.

Motivation & Objective

• To address the challenge of inferring the spectral distribution of high-dimensional true covariance matrices when observations are contaminated by noise, particularly in high-frequency financial data.
• To develop a theoretical framework linking the limiting spectral distribution (LSD) of noisy (information-plus-noise) sample covariance matrices to the LSD of the underlying true covariance matrix.
• To apply this framework to the inference of the spectral distribution of integrated covariance matrices (ICV) in high-dimensional diffusion processes observed with microstructure noise.
• To propose a two-step procedure that enables consistent estimation of the ICV's spectral distribution despite noisy high-frequency data.
• To design an alternative estimator that eliminates microstructure noise effects and whose LSD depends solely on the ICV through the standard Marčenko-Pastur equation.

Proposed method

• Establishes an asymptotic relationship between the LSD of true sample covariance matrices and that of information-plus-noise-type sample covariance matrices under high-dimensional asymptotics.
• Applies a generalized connection between the LSD of sample covariance matrices and the population covariance matrix, leveraging the Marčenko-Pastur equation as a foundational tool.
• Uses the (slightly modified) pre-averaging estimator as a noise-robust estimator of the ICV matrix, which naturally forms an information-plus-noise-type covariance matrix.
• Proposes a two-step procedure: first, infer the LSD of the noisy pre-averaged estimator; second, map it back to the LSD of the true ICV using the derived asymptotic relationship.
• Introduces an alternative estimator that is asymptotically equivalent to a sample covariance matrix of i.i.d. standard normal vectors, ensuring its LSD depends only on the ICV via the standard Marčenko-Pastur equation.
• Employs moment conditions and strong mixing assumptions on noise, along with moment bounds and Hölder’s inequality, to control error terms and establish almost sure convergence of spectral moments.

Experimental results

Research questions

• RQ1 How does the limiting spectral distribution (LSD) of a sample covariance matrix based on noisy observations relate to the LSD of the true underlying covariance matrix in high-dimensional settings?
• RQ2 Can the spectral distribution of the integrated covariance matrix (ICV) be consistently estimated when high-frequency financial data are contaminated by microstructure noise?
• RQ3 What is the asymptotic behavior of the pre-averaged estimator in the presence of microstructure noise, and how can its LSD be linked to the true ICV’s LSD?
• RQ4 Can a noise-robust estimator be constructed such that its LSD depends only on the ICV through the standard Marčenko-Pastur equation?
• RQ5 What are the sufficient conditions on the noise and the underlying process to ensure almost sure convergence of spectral moments and the LSD of the estimators?

Key findings

• The limiting spectral distribution (LSD) of the noisy (information-plus-noise) sample covariance matrix is asymptotically linked to the LSD of the true covariance matrix via a derived functional relationship.
• The two-step procedure based on the modified pre-averaging estimator successfully estimates the LSD of the integrated covariance matrix (ICV) under high-dimensional asymptotics.
• The alternative estimator proposed is asymptotically equivalent to a sample covariance matrix of i.i.d. standard normal vectors, ensuring its LSD depends only on the ICV through the standard Marčenko-Pastur equation.
• The LSD of the pre-averaged estimator converges almost surely to a deterministic limit determined by the ICV and the noise structure, under the stated moment and mixing conditions.
• Numerical studies confirm that the proposed methods can accurately estimate the spectra of the underlying covariance matrix even when observations are corrupted by microstructure noise.
• The error terms in the spectral moment approximations vanish almost surely under the assumptions of bounded moments and strong mixing, ensuring consistency of the LSD estimation.