High-dimensional covariance matrices under dynamic volatility models: asymptotics and shrinkage estimation

TL;DR

This paper investigates the spectral properties of high-dimensional covariance matrices under dynamic volatility models, proposing a time-variation adjustment and new estimators with proven asymptotic properties.

Contribution

It introduces a time-variation adjusted covariance estimator and establishes its spectral distribution, improving covariance estimation in dynamic volatility settings.

Findings

The LSD of the TV-adj covariance follows the Marcenko-Pastur law.

The proposed estimators are consistent and asymptotically optimal.

Conditions for the LSD to match the i.i.d. case are identified.

Abstract

We study the estimation of the high-dimensional covariance matrix andits eigenvalues under dynamic volatility models. Data under such modelshave nonlinear dependency both cross-sectionally and temporally. We firstinvestigate the empirical spectral distribution (ESD) of the sample covariancematrix under scalar BEKK models and establish conditions under which thelimiting spectral distribution (LSD) is either the same as or different fromthe i.i.d. case. We then propose a time-variation adjusted (TV-adj) sample co-variance matrix and prove that its LSD follows the same Marcenko-Pasturlaw as the i.i.d. case. Based on the asymptotics of the TV-adj sample co-variance matrix, we develop a consistent population spectrum estimator and an asymptotically optimal nonlinear shrinkage estimator of the unconditionalcovariance matrix