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On the estimation of integrated covariance matrices of high dimensional diffusion processes. (English) Zbl 1246.62182

Summary: 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 covariance (RCV) matrix. We show that in the high dimensional case when the dimension \(p\) and the observation frequency \(n\) grow at 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 Marčenko-Pastur type theorem for weighted sample covariance matrices, based on which we obtain a Marčenko-Pastur type theorem for RCV 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. We further propose an alternative estimator, the time-variation adjusted realized covariance (TVARCV) matrix. We show that for processes in the class \(\mathcal C\), the TVARCV possesses have the desirable property that its LSD depends solely on that of the targeting ICV through the Marčenko-Pastur equation, and hence, in particular, the TVARCV can be used to recover the empirical spectral distribution of the ICV by using existing algorithms.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62H12 Estimation in multivariate analysis
62E20 Asymptotic distribution theory in statistics
62G30 Order statistics; empirical distribution functions
65C60 Computational problems in statistics (MSC2010)
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