  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
