The Annals of Statistics

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

Minjing Tao, Yazhen Wang, and Harrison H. Zhou

Full-text: Open access

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.

Article information

Source
Ann. Statist., Volume 41, Number 4 (2013), 1816-1864.

Dates
First available in Project Euclid: 5 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1378386240

Digital Object Identifier
doi:10.1214/13-AOS1128

Mathematical Reviews number (MathSciNet)
MR3127850

Zentralblatt MATH identifier
1281.62178

Subjects
Primary: 62G05: Estimation 62H12: Estimation
Secondary: 62M05: Markov processes: estimation

Keywords
Large matrix estimation measurement error minimax lower bound multi-scale optimal convergence rate sparsity subGaussian tail threshold volatility matrix estimator

Citation

Tao, Minjing; Wang, Yazhen; Zhou, Harrison H. Optimal sparse volatility matrix estimation for high-dimensional Itô processes with measurement errors. Ann. Statist. 41 (2013), no. 4, 1816--1864. doi:10.1214/13-AOS1128. https://projecteuclid.org/euclid.aos/1378386240


Export citation

References

  • Aït-Sahalia, Y., Fan, J. and Xiu, D. (2010). High-frequency covariance estimates with noisy and asynchronous financial data. J. Amer. Statist. Assoc. 105 1504–1517.
  • Aït-Sahalia, Y., Mykland, P. A. and Zhang, L. (2005). How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial Studies 18 351–416.
  • Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. and Shephard, N. (2008). Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica 76 1481–1536.
  • Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. and Shephard, N. (2011). Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. J. Econometrics 162 149–169.
  • Bibinger, M. and Reiß, M (2011). Spectral estimation of covolatility from noisy observations using local weights. Preprint, Humboldt-Universität zu Berlin.
  • Bickel, P. J. and Levina, E. (2008a). Regularized estimation of large covariance matrices. Ann. Statist. 36 199–227.
  • Bickel, P. J. and Levina, E. (2008b). Covariance regularization by thresholding. Ann. Statist. 36 2577–2604.
  • Cai, T. T., Zhang, C.-H. and Zhou, H. H. (2010). Optimal rates of convergence for covariance matrix estimation. Ann. Statist. 38 2118–2144.
  • Cai, T. and Zhou, H. (2012). Optimal rates of convergence for sparse covariance matrix estimation. Ann. Statist. 40 2389–2420.
  • Callen, J., Govindaraj, S. and Xu, L. (2000). Large time and small noise asymptotic results for mean reverting diffusion processes with applications. Econom. Theory 16 401–419.
  • Christensen, K., Kinnebrock, S. and Podolskij, M. (2010). Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data. J. Econometrics 159 116–133.
  • Curci, G. and Corsi, F. (2012). Discrete sine transform for multi-scale realized volatility measures. Quant. Finance 12 263–279.
  • El Karoui, N. (2008). Operator norm consistent estimation of large-dimensional sparse covariance matrices. Ann. Statist. 36 2717–2756.
  • Fan, J., Fan, Y. and Lv, J. (2008). High dimensional covariance matrix estimation using a factor model. J. Econometrics 147 186–197.
  • Fan, J., Li, Y. and Yu, K. (2012). Vast volatility matrix estimation using high-frequency data for portfolio selection. J. Amer. Statist. Assoc. 107 412–428.
  • Fan, J. and Wang, Y. (2007). Multi-scale jump and volatility analysis for high-frequency financial data. J. Amer. Statist. Assoc. 102 1349–1362.
  • Gloter, A. and Jacod, J. (2001a). Diffusions with measurement errors. I. Local asymptotic normality. ESAIM Probab. Stat. 5 225–242 (electronic).
  • Gloter, A. and Jacod, J. (2001b). Diffusions with measurement errors. II. Optimal estimators. ESAIM Probab. Stat. 5 243-260.
  • Huang, J. Z., Liu, N., Pourahmadi, M. and Liu, L. (2006). Covariance matrix selection and estimation via penalised normal likelihood. Biometrika 93 85–98.
  • Jacod, J., Li, Y., Mykland, P. A., Podolskij, M. and Vetter, M. (2009). Microstructure noise in the continuous case: The pre-averaging approach. Stochastic Process. Appl. 119 2249–2276.
  • Johnstone, I. M. and Lu, A. Y. (2009). On consistency and sparsity for principal components analysis in high dimensions. J. Amer. Statist. Assoc. 104 682–693.
  • Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • Lam, C. and Fan, J. (2009). Sparsistency and rates of convergence in large covariance matrix estimation. Ann. Statist. 37 4254–4278.
  • Ledoit, O. and Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. J. Multivariate Anal. 88 365–411.
  • Mueschke, N. J. and Andrews, M. J. (2006). Investigation of scalar measurement error in diffusion and mixing processes. Experiments in Fluids 40 165–175.
  • Munk, A. and Schmidt-Hieber, J. (2010). Lower bounds for volatility estimation in microstructure noise models. In Borrowing Strength: Theory Powering Applications—A Festschrift for Lawrence D. Brown. Inst. Math. Stat. Collect. 6 43–55. IMS, Beachwood, OH.
  • Reiß, M. (2011). Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Statist. 39 772–802.
  • Salkuyeh, D. K. (2006). Positive integer powers of the tridiagonal Toeplitz matrices. Int. Math. Forum 1 1061–1065.
  • Saulis, L. and Statulevičius, V. A. (1991). Limit Theorems for Large Deviations. Mathematics and Its Applications (Soviet Series) 73. Kluwer Academic, Dordrecht.
  • Tao, M., Wang, Y., Yao, Q. and Zou, J. (2011). Large volatility matrix inference via combining low-frequency and high-frequency approaches. J. Amer. Statist. Assoc. 106 1025–1040.
  • Thorin, G. O. (1948). Convexity theorems generalizing those of M. Riesz and Hadamard with some applications. Medd. Lunds Univ. Mat. Sem. 9 1–58.
  • Wang, Y. and Zou, J. (2010). Vast volatility matrix estimation for high-frequency financial data. Ann. Statist. 38 943–978.
  • Whitmore, G. A. (1995). Estimating degradation by a Wiener diffusion process subject to measurement error. Lifetime Data Anal. 1 307–319.
  • Wilkinson, J. H. (1988). The Algebraic Eigenvalue Problem. Oxford Univ. Press, New York.
  • Wu, W. B. and Pourahmadi, M. (2003). Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika 90 831–844.
  • Xiu, D. (2010). Quasi-maximum likelihood estimation of volatility with high frequency data. J. Econometrics 159 235–250.
  • Yuan, M. and Lin, Y. (2007). Model selection and estimation in the Gaussian graphical model. Biometrika 94 19–35.
  • Zhang, L. (2006). Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach. Bernoulli 12 1019–1043.
  • Zhang, L. (2011). Estimating covariation: Epps effect, microstructure noise. J. Econometrics 160 33–47.
  • Zhang, L., Mykland, P. A. and Aït-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. J. Amer. Statist. Assoc. 100 1394–1411.
  • Zheng, X. and Li, Y. (2011). On the estimation of integrated covariance matrices of high dimensional diffusion processes. Ann. Statist. 39 3121–3151.