ON THE ESTIMATION OF INTEGRATED COVARIANCE MATRICES OF HIGH DIMENSIONAL DIFFUSION PROCESSES1

Abstract

We consider the estimation of integrated covariance (ICV) matrices of high dimensional diffusion processes based on high frequency observations. We start by studying the most commonly used estimator, the realized covari-ance (RCV) matrix. We show that in the high dimensional case when the dimension p and the observation frequency n grow in the same rate, the lim-iting spectral distribution (LSD) of RCV depends on the covolatility process not only through the targeting ICV, but also on how the covolatility process varies in time. We establish a Marčenko–Pastur type theorem for weighted sample covariance matrices, based on which we obtain a Marčenko–Pastur type theorem for RCV for a class C of diffusion processes. The results explic-itly demonstrate how the time variability of the covolatility process affects the LSD of RCV. We further propose an alternative estimator, the time-variation adjusted realized covariance (TVARCV) matrix. We show that for processes in class C, the TVARCV possesses the desirable property that its LSD de-pends solely on that of the targeting ICV through the Marčenko–Pastur equa-