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Spectral analysis of high-dimensional spot volatility matrix with applications

Qiang Liu, Yiming Liu, Zhi Liu, Wang Zhou

TL;DR

This paper develops a high-dimensional spectral theory for the spot volatility matrix $c_t$ using high-frequency data, establishing a first-order limiting spectral distribution via a Marčenko–Pastur-type relation and a second-order central limit theorem for linear spectral statistics. The results hold under infill asymptotics with $p\to\infty$ and $p/n$ tending to a positive limit, and accommodate time-varying, potentially random volatility. The authors translate these theoretical insights into feasible tests for identity and sphericity of $c_t$, providing asymptotically normal statistics and validating finite-sample performance through simulations. The work enables robust multivariate inference on the local co-variation structure of asset prices in high dimensions, with practical tests for structural hypotheses on the spot volatility matrix.

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.

Spectral analysis of high-dimensional spot volatility matrix with applications

TL;DR

This paper develops a high-dimensional spectral theory for the spot volatility matrix using high-frequency data, establishing a first-order limiting spectral distribution via a Marčenko–Pastur-type relation and a second-order central limit theorem for linear spectral statistics. The results hold under infill asymptotics with and tending to a positive limit, and accommodate time-varying, potentially random volatility. The authors translate these theoretical insights into feasible tests for identity and sphericity of , providing asymptotically normal statistics and validating finite-sample performance through simulations. The work enables robust multivariate inference on the local co-variation structure of asset prices in high dimensions, with practical tests for structural hypotheses on the spot volatility matrix.

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.

Paper Structure

This paper contains 10 sections, 5 theorems, 55 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Setup and assumptions
  3. Asymptotic results
  4. Empirical spectral distribution (ESD)
  5. Linear spectral statistics (LSS)
  6. Application
  7. Identity test
  8. Sphericity test
  9. Simulation studies
  10. Conclusion and future work

Key Result

Theorem 1

Under Assumption asu:dgp and the following conditions: Then, in probability, the ESD of $\widehat{c_{t}}^{n}$ converges in distribution to a probability distribution $F^{\bar{p},H_t}$, which is determined by $H_t$ in that its Stieltjes transform $m(z)$ as defined in equ:st is the only solution to Mar$\check{\text{c}}$enko-Pastur equation equ:esd with $H

Figures (4)

  • Figure 1: The empirical spectral distribution functions of realized spot volatility matrix at $t=0$ for the deterministic volatility model \ref{['det:vol']} (left) and the stochastic volatility model \ref{['sto:vol']} (right), with fixed $k_n=68$ and $p=34,68,102$ (Correspondingly, $\bar{p} = 0.5 ,1, 1.5$.). The red solid lines are corresponding theoretical limiting spectral distribution functions, namely the Mar$\check{\text{c}}$enko-Pastur law given by Theorem \ref{['spot_esd']}.
  • Figure 2: Q-Q plot for BJYZ-test (left) and LW-test (right) when $\bar{p} = 0.5$ (Namely, $p=34, k_n=68$), based on 1000 repetitions.
  • Figure 3: Q-Q plot for LW-test when $p = 68$ (left) and $p=102$ (right) with fixed $k_n=68$, corresponding to $\bar{p} = 1, 1.5$ respectively, based on 1000 repetitions.
  • Figure 4: Q-Q plot for J-test when $p = 34$ (left), $p = 68$ (middle) and $p=102$ (right) with fixed $k_n=68$, corresponding to $\bar{p} = 0.5, 1, 1.5$ respectively, based on 1000 repetitions.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5