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 %DIFDELCMD < {\it %%% realized covariance (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 limiting 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 Marc enko-Pastur type theorem for weighted sample covariance matrices, based on which we obtain a Marc enko-Pastur type theorem for RCV for a class %DIFDELCMD < \sC %%%  of diffusion processes. The results explicitly demonstrate how the time variability of the covolatility process affects the LSD of RCV. We further propose an alternative estimator, the %DIFDELCMD < {\it %%% time-variation adjusted realized covariance (TVARCV) matrix. We show that for processes in class %DIFDELCMD < \sC%%%   , the TVARCV possesses the desirable property that its LSD depends solely on that of the targeting ICV through the Marc enko-Pastur equation, and hence, in particular, the TVARCV can be used to recover the empirical spectral distribution of the ICV by using existing algorithms.
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 %DIFDELCMD < {\it %%% realized covariance (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 limiting 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 Marc enko--Pastur type theorem for weighted sample covariance matrices, based on which we obtain a Marc enko--Pastur type theorem for RCV for a class %DIFDELCMD < \sC %%% \mathcal{C of diffusion processes. The results explicitly demonstrate how the time variability of the covolatility process affects the LSD of RCV. We further propose an alternative estimator, the %DIFDELCMD < {\it %%% time-variation adjusted realized covariance (TVARCV) matrix. We show that for processes in class %DIFDELCMD < \sC%%% \mathcal C , the TVARCV possesses the desirable property that its LSD depends solely on that of the targeting ICV through the Marc enko--Pastur equation, and hence, in particular, the TVARCV can be used to recover the empirical spectral distribution of the ICV by using existing algorithms.
R
realized covariance
realized covariance
D
not only through the targeting ICV
D
,
R
but also on how the covolatility process varies in time
not only through the targeting ICV, but also on how the covolatility process varies in time
R
enko-Pastur
enko--Pastur
R
enko-Pastur
enko--Pastur
A
\mathcal{C
A
\mathcal
A
C
R
enko-Pastur
enko--Pastur