Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors

Abstract

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...

Low-rank diffusion matrix estimation for high-dimensional time-changed Lévy processes

“…For inference on the volatility matrix (in small dimensions) for continuous semi-martingales in a high-frequency regime we refer to [20,10] and references therein. Large sparse volatility matrix estimation for continuous Itô processes has been recently studied in [42,37,38].…”

Asymptotic theory for large volatility matrix estimation based on high-frequency financial data

“…Remark 13. As both p and n are allowed to go to infinity, if the micro-structure noises are sub-Gaussian, the drift and volatility are bounded, and the integrated volatility matrix is sparse, Tao et al [24] established the minimax convergence rate, π(p)[n −1/4 √ log p] 1−δ , and constructed a threshold MSRVM estimator to achieve the optimal convergence rate.…”

“…Cholesky decomposition implies that there is a lower triangular matrix, L, such that Σ(t) = L t L T t and dX t = µ t dt + L t dB t . As in the proof of Lemma 10 in [24], we can define a standard Brownian motion,…”

“…When there are a large number of assets in financial practices such as asset pricing, portfolio allocation, and risk management, the volatility estimators designed for estimating a small integrated volatility matrix perform very poorly, and in fact, they are inconsistent when both the number of assets and sample size go to infinity [27]. For the case of a large number of assets, we need to impose some sparse structure on the integrated volatility matrix and employ regularization such as thresholding to obtain consistent estimators of the large volatility matrix [23,24,25]. In particular, Tao et al [23,24] investigated convergence rates of multi-scale realized volatility matrix estimator in the asymptotic framework that allows both the number of assets and sample size to go to infinity, and showed that the estimator achieves optimal convergence rate with respect to the sample size.…”

“…For the case of a large number of assets, we need to impose some sparse structure on the integrated volatility matrix and employ regularization such as thresholding to obtain consistent estimators of the large volatility matrix [23,24,25]. In particular, Tao et al [23,24] investigated convergence rates of multi-scale realized volatility matrix estimator in the asymptotic framework that allows both the number of assets and sample size to go to infinity, and showed that the estimator achieves optimal convergence rate with respect to the sample size. This paper considers the kernel realized volatility (KRV) [4,5], the pre-averaging realized volatility (PRV) [13,18], and the multi-scale realized volatility (MSRV) [23,30] based on generalized sampling time scheme.…”