High-Dimensional Covariance Matrices Under Dynamic Volatility Models : Asymptotics and Shrinkage Estimation

We study the estimation of the high-dimensional covariance matrix and its eigenvalues under dynamic volatility models. Data under such models have nonlinear dependency both cross-sectionally and temporally. We first investigate the empirical spectral distribution (ESD) of the sample covariance matrix under scalar BEKK models and establish conditions under which the limiting spectral distribution (LSD) is either the same as or different from the 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 Marˇcenko-Pastur law 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 unconditional covariance matrix