It is increasingly important in financial economics to estimate volatilities of asset returns. However most the available methods are not directly applicable when the number of assets involved is large, due to the lack of accuracy in estimating high dimensional matrices. Therefore it is pertinent to reduce the effective size of volatility matrices in order to produce adequate estimates and forecasts. Furthermore, since high-frequency financial data for different assets are typically not recorded at the same time points, conventional dimension-reduction techniques are not directly applicable. To overcome those difficulties we explore a novel approach that combines high-frequency volatility matrix estimation together with low-frequency dynamic models. The proposed methodology consists of three steps: (i) estimate daily realized co-volatility matrices directly based on high-frequency data, (ii) fit a matrix factor model to the estimated daily covolatility matrices, and (iii) fit a vector autoregressive (VAR) model to the estimated volatility factors. We establish the asymptotic theory for the proposed methodology in the framework that allows sample size, number of assets, and number of days go to infinity together. Our theory shows that the relevant eigenvalues and eigenvectors can be consistently estimated. We illustrate the methodology with the high-frequency price data on several hundreds of stocks traded in Shenzhen and Shanghai Stock Exchanges over a period of 177 days in 2003. Our approach pools together the strengths of modeling and estimation at both intradaily (high-frequency) and interdaily (low-frequency) levels.

Stochastic processes are often used to model complex scientific problems in fields ranging from biology and finance to engineering and physical science. This paper investigates rate-optimal estimation of the volatility matrix of a high-dimensional Itô process observed with measurement errors at discrete time points. The minimax rate of convergence is established for estimating sparse volatility matrices. By combining the multi-scale and threshold approaches we construct a volatility matrix estimator to achieve the optimal convergence rate. The minimax lower bound is derived by considering a subclass of Itô processes for which the minimax lower bound is obtained through a novel equivalent model of covariance matrix estimation for independent but nonidentically distributed observations and through a delicate construction of the least favorable parameters. In addition, a simulation study was conducted to test the finite sample performance of the optimal estimator, and the simulation results were found to support the established asymptotic theory.Continuous-time diffusion processes, or more generally, Itô processes, are frequently employed to model such complex dynamic systems. Data collected in the studies are treated as the processes observed at discrete time points with possible noise contamination. For example, the prices of financial assets are usually modeled by Itô processes, and the price data observed at high-frequencies are contaminated by market microstructure noise. In this paper we investigate estimation of the volatilities of the Itô processes based on noisy data.Several volatility estimation methods have been developed in the past several years. For estimating a univariate integrated volatility, popular estimators include two-scale realized volatility ], multi-scale realized volatility [Zhang (2006) and Fan and Wang (2007)], realized kernel volatility [Barndorff-Nielsen et al. (2008)] and preaveraging based realized volatility [Jacod et al. (2009)]. For estimating a bivariate integrated co-volatility, common methods are the previous-tick approach [Zhang (2011)], the refresh-time scheme and realized kernel volatility [Barndorff-Nielsen et al. (2011)], the generalized synchronization scheme [Aït-Sahalia, Fan and Xiu (2010)] and the pre-averaging approach [Christensen, Kinnebrock and Podolskij (2010)]. Optimal volatility and co-volatility estimation has been investigated in the parametric or nonparametric setting [Aït-Sahalia, Mykland and Zhang (2005), Bibinger and Reiß (2011), Gloter and Jacod (2001a, 2001b), Reiß (2011) and Xiu (2010)]. These works are for estimating scalar volatilities or volatility matrices of small size. Wang and Zou (2010) and Tao et al. (2011) studied the problem of estimating a large sparse volatility matrix based on noisy high-frequency financial data. Fan, Li and Yu (2012) employed a large volatility matrix estimator based on high-frequency data for portfolio allocation. The large volatility matrix estimation is a high-dimensional extension of the univariate case. It can be also...

Financial practices often need to estimate an integrated volatility matrix of a large number of assets using noisy high-frequency data. Many existing estimators of a volatility matrix of small dimensions become inconsistent when the size of the matrix is close to or larger than the sample size. This paper introduces a new type of large volatility matrix estimator based on nonsynchronized high-frequency data, allowing for the presence of microstructure noise. When both the number of assets and the sample size go to infinity, we show that our new estimator is consistent and achieves a fast convergence rate, where the rate is optimal with respect to the sample size. A simulation study is conducted to check the finite sample performance of the proposed estimator.