Spectral norm bounds for high-dimensional realized covariance matrices and application to weak factor models

TL;DR

This paper derives sharp spectral norm bounds for high-dimensional realized covariance matrices of Itô semimartingales with jumps, and applies these bounds to estimate the number of relevant factors in latent factor models from high-frequency data.

Contribution

It introduces new matrix martingale inequalities using non-commutative $L^p$ spaces and applies them to high-dimensional covariance estimation and factor number determination.

Findings

Established sharp spectral norm bounds for realized covariance matrices.

Applied bounds to estimate the number of relevant factors in high-frequency financial data.

Extended matrix martingale inequalities to handle infinite activity jumps.

Abstract

Motivated by statistical analysis of latent factor models for high-frequency financial data, we develop sharp upper bounds for the spectral norm of the realized covariance matrix of a high-dimensional It\^o semimartingale with possibly infinite activity jumps. For this purpose, we develop Burkholder-Gundy type inequalities for matrix martingales with the help of the theory of non-commutative $L^p$ spaces. The obtained bounds are applied to estimating the number of (relevant) common factors in a continuous-time latent factor model from high-frequency data in the presence of weak factors.