On the Inference About the Spectral Distribution of High-Dimensional Covariance Matrix Based on High-Frequency Noisy Observations

Abstract

In practice, observations are often contaminated by noise, making the resulting sample covariance matrix a signal-plus-noise sample covariance matrix. Aiming to make inferences about the spectral distribution of the population covariance matrix under such a situation, we establish an asymptotic relationship that describes how the limiting spectral distribution of (signal) sample covariance matrices depends on that of signalplus-noise-type sample covariance matrices. As an application, we consider inferences about the spectral distribution 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 a signal-plus-noise sample covariance matrix, and the aforementioned result, together with a (generalized) connection between the spectral distribution of signal sample covariance matrices and that of the population covariance matrix, enables us to propose a two-step procedure to consistently estimate the spectral distribution of ICV for a class of diffusion processes. An alternative approach is further proposed, which possesses several desirable properties: it is more robust, it eliminates the effects of microstructure noise, and the asymptotic relationship that enables consistent estimation of the spectral distribution of ICV is the standard Marčenko-Pastur equation. The performance of the two approaches is examined via simulation studies under both synchronous and asynchronous observation settings.

On Structure Testing for Component Covariance Matrices of a High Dimensional Mixture

“…A major difference here is that El Karoui's model makes a Gaussian assumption on z, whereas our model allows non-Gaussian distributions for z. Recently, Xia and Zheng (2014) proposed a similar model in the study of high dimensional integrated covolatility matrices. Their sample data can be modelled as x i = w i T p z i , which has the same form as the scale mixture (2), but w i in their model is a non-random function of the index i and .z i / is again a sequence of Gaussian vectors.…”

Limiting Spectral Distribution of High-dimensional Hayashi-Yoshida Estimator of Integrated Covariance Matrix

“…So a natural direction to extend this work is by adding microstructure noise to it. Significant insights can be obtained from 28 where the same was derived for realized (co)volatility matrix. In presence of noise the spectral distribution may deviate from the ideal situation in significant ways.…”

“…27 established the limiting spectral distribution of realized covariance matrix obtained from synchronized data. Recently an asymptotic relationship has been established between the limiting spectral distributions of the true sample covariance matrix and noisy sample covariance matrix 28 . 29 studied the estimation of integrated covariance matrix based on noisy high-frequency data with multiple transactions using random matrix theory.…”

Optimal covariance matrix estimation for high-dimensional noise in high-frequency data

“…At each time point t k , we only observe p i=1 I(t k ∈ G i ) components of Y t k . As a conventional assumption for analyzing noisy high-frequency data, U t k is taken to be independent of X t k ; see, among others, Aït-Sahalia, Fan and Xiu (2010), Xiu (2010), Liu and Tang (2014), Chang et al (2018), Xia and Zheng (2018), and the monograph Aït-Sahalia and Jacod (2014). Specifically, we make the following assumption:…”

“…In the literature on multivariate and high-dimensional high-frequency data analysis, existing studies mainly concern the estimations of the so-called realized covariance matrix. Specifically, the major objective is on the signal part, attempting to eliminate the impact from the noises; see, for example, Aït-Sahalia, Fan and Xiu (2010), Fan, Li and Yu (2012), Tao, Wang and Zhou (2013), Liu and Tang (2014) and Xia and Zheng (2018). However, it remains little explored on the high-dimensional covariance matrix for the noise in high-frequency data.…”