We consider the estimation of integrated covariance  matrices of high dimensional diffusion processes by using high frequency data . We start by studying the most commonly used estimator, the realized covariance matrix {\it  (RCV)  . We show that in the high dimensional case when the dimension p and the observation frequency n grow in the same rate, the limiting empirical spectral distribution of RCV depends on the covolatility processes not only through the underlying integrated covariance matrixSigma , but also on how the covolatility processes vary in time. In particular, for two high dimensional diffusion processes with the same integrated covariance matrix, the empirical spectral distributions of their RCVs can be very different. Hence in terms of making inference about the spectrum of the integrated covariance matrix, the RCV is in generalnot a good proxy to rely on in the high dimensional caseenko-Pastur type theorem for weighted sample covariance matrices, based on which we further establish a Mar\v{c}enko-Pastur type theorem for RCV matrices for a class \mathcal{C} of diffusion processes. The results explicitly demonstrate how the time-variability of the covolatility process affects the LSD of RCV matrix} . We then propose an alternative estimator, the {\it time-variation adjusted realized covariance matrix (TVARCV) , for a class of diffusion processes . We show that the limiting empirical spectral distribution of our proposed estimator TVARCV does depend solely on that of Sigma through a Marcenko-Pastur equation, and hence the TVARCV can be used to recover the empirical spectral distribution of Sigma by inverting the Marcenko-Pastur equation, which can then be applied to further applications such as portfolio allocation, risk management, etc. .
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 {\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 the RCV matrix depends on the covolatility process not only through the targeting ICV matrix , but also on how the covolatility process varies in time. We establish a Mar\v{cenko-Pastur type theorem for weighted sample covariance matrices, based on which we further establish a Mar\v{c}enko-Pastur type theorem for RCV matrices for a class \mathcal{C} of diffusion processes. The results explicitly demonstrate how the time-variability of the covolatility process affects the LSD of RCV matrix} . We then propose an alternative estimator, the {\it time-variation adjusted realized covariance (TVARCV) matrix . We show that for diffusion processes in class \mathcal{C solely on that of the targeting ICV matrix through a Mar\v{c .
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(ICV)
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by using high frequency data
based on high frequency observations
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realized covariance matrix
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realized covariance
A
matrix
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empirical spectral distribution of RCV
spectral distribution (LSD) of the RCV matrix
R
processes
process
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underlying integrated covariance matrixSigma
targeting ICV matrix
R
processes vary
process varies
D
In particular, for two high dimensional diffusion processes with the same integrated covariance matrix, the empirical spectral distributions of their RCVs can be very different. Hence in terms of making inference about the spectrum of the integrated covariance matrix, the RCV is in general
D
not
R
a good proxy to rely on in the high dimensional case
We establish a Mar\v{c
D
matrix
R
, for a class of diffusion processes
matrix
R
the limiting empirical spectral distribution of our proposed estimator TVARCV does depend
for diffusion processes in class \mathcal{C
R
Sigma through a Marcenko-Pastur equation, and hence the TVARCV can be used to recover the empirical spectral distribution of Sigma by inverting the Marcenko-Pastur equation, which can then be applied to further applications such as portfolio allocation, risk management, etc.
the targeting ICV matrix through a Mar\v{c