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

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 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 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 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 C, the TVARCV possesses 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.

Random matrix theory in statistics: A review

“…This model is also closely related to the separable covariance model (21). 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. Factor models have also been used extensively in finance and econometrics when the number of economic variables is high (Bai, 2003;Ng, 2002, 2007).…”

Performance analysis of minimum variance asset allocation with high frequency data

“…The test is to show the poor performance of RCV estimator and the error amplification caused by time variation effect. The samples y k are assumed to be the product of w Moreover, to describe the time variation, the entries of Θ k are assumed to approximate U-shape distribution during a day like in [14]. The diagonal entries of WN are assumed to be piecewise constants:…”

“…These problems motivate us to provide an estimator which can improve the accuracy of RCV estimator and also solve the problem of time variation effect in the time-varying high frequency GMVP problem. The TVARCV estimator in [14] is shown to be able to reduce time variation and is given as follows:…”

“…Another important characteristic of the high frequency data is the presence of "intraday" time variation [10,[14][15][16]. Earlier techniques, such as [10,15,16], used either autoregressive conditional heteroskedasticity (ARCH) or generalized ARCH model.…”

“…However the model restrains to a single asset or a small number of assets because of its complexity. The recent state of art estimator, time variation adjusted realized covariance (TVARCV) estimator in [14] reduces the intraday time variation. However, it still performs poorly in the condition when the sample size is of similar order to the number of assets.…”