On the estimation of integrated covariance matrices of high dimensional diffusion processes

By Xinghua Zheng, Yingying Li. 2010

61 citations recorded by the source corpus.

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 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 Mar\v{c}enko--Pastur type theorem for weighted sample covariance matrices, based on which we obtain a Mar\v{c}enko--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 class $\mathcal {C}$, the TVARCV possesses the desirable property that its LSD depends solely on that of the targeting ICV through the Mar\v{c}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.

Subfields: Cognitive Neuroscience, Finance, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/c1568cefa0da6d0377db1f195bb9d8788634177f

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Citations in the corpus

Principal Component Analysis of High-Frequency Data (2016, methodology): “Allowing for an increasing dimension yet more parametric structure, Zheng and Li (2011) established a Marcenko–Pastur type theorem for analyzing the spectral distribution of the integrated covariance matrix for a special class of diffusion processes.”
Time-Varying Periodicity in Intraday Volatility (2019, background): “[26] study the commonality in the intraday periodicity across many assets ([41] and [42] recognize the importance of intraday volatility variation for high-dimensional multivariate volatility estimation).”
OPTIMAL SPARSE VOLATILITY MATRIX ESTIMATION FOR HIGH-DIMENSIONAL ITÔ PROCESSES WITH MEASUREMENT ERRORS (2013, result): “This is in contrast to sub-optimal convergence rate results in the literature where a n − 1 / 6 convergence rate was obtained; see, for example, Fan, Li and Yu (2012), Wang and Zou (2010), Zhang, Mykland and A¨ıt-Sahalia (2005), and Zheng and Li (2011).”
Random matrix theory in statistics: A review (2014, methodology): “A related model has been used by Zheng and Li (2011) in the context of estimation of integrated covolatility matrix based on highdimensional, high-frequency financial data.”