Spectral analysis of high-dimensional spot volatility matrix with applications

TL;DR

This paper extends spectral analysis techniques from classical covariance matrices to high-frequency data for estimating the spot volatility matrix in high dimensions, providing theoretical limits and practical tests.

Contribution

It develops the first-order spectral distribution and a CLT for linear spectral statistics of the spot volatility matrix using high-frequency data, with applications to hypothesis testing.

Findings

Established the limiting spectral distribution for high-dimensional spot volatility matrices.

Derived a central limit theorem for linear spectral statistics.

Validated the methods through simulation studies.

Abstract

In random matrix theory, the spectral distribution of the covariance matrix has been well studied under the large dimensional asymptotic regime when the dimensionality and the sample size tend to infinity at the same rate. However, most existing theories are built upon the assumption of independent and identically distributed samples, which may be violated in practice. For example, the observational data of continuous-time processes at discrete time points, namely, the high-frequency data. In this paper, we extend the classical spectral analysis for the covariance matrix in large dimensional random matrix to the spot volatility matrix by using the high-frequency data. We establish the first-order limiting spectral distribution and obtain a second-order result, that is, the central limit theorem for linear spectral statistics. Moreover, we apply the results to design some feasible tests for the spot volatility matrix, including the identity and sphericity tests. Simulation studies justify the finite sample performance of the test statistics and verify our established theory.