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

We study the estimation of high-dimensional covariance matrices and their empirical spectral distributions under dynamic volatility models. Data under such models have nonlinear dependency both cross-sectionally and temporally. We establish the condition under which the limiting spectral distribution (LSD) of the sample covariance matrix under scalar BEKK models is different from the i.i.d. case. We then propose a time-variation adjusted (TV-adj) sample covariance matrix and prove that its LSD follows the Marčenko–Pastur law. Based on the asymptotics of the TV-adj sample covariance matrix, we develop a consistent population spectrum estimator and an asymptotically optimal nonlinear shrinkage estimator of the unconditional covariance matrix.