Skip to main content
SpringerLink
Log in

High-Dimensional Volatility Matrix Estimation with Cross-Sectional Dependent and Heavy-Tailed Microstructural Noise

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

The estimates of the high-dimensional volatility matrix based on high-frequency data play a pivotal role in many financial applications. However, most existing studies have been built on the sub-Gaussian and cross-sectional independence assumptions of microstructure noise, which are typically violated in the financial markets. In this paper, the authors proposed a new robust volatility matrix estimator, with very mild assumptions on the cross-sectional dependence and tail behaviors of the noises, and demonstrated that it can achieve the optimal convergence rate n−1/4. Furthermore, the proposed model offered better explanatory and predictive powers by decomposing the estimator into low-rank and sparse components, using an appropriate regularization procedure. Simulation studies demonstrated that the proposed estimator outperforms its competitors under various dependence structures of microstructure noise. Additionally, an extensive analysis of the high-frequency data for stocks in the Shenzhen Stock Exchange of China demonstrated the practical effectiveness of the estimator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Markowitz H, Portfolio selection, Journal of Finance, 1952, 7(1): 77–91.

    Google Scholar 

  2. Markowitz H, Portfolio Selection: Efficent Diversification of Investments, Yale University Press, London, 1959.

    Google Scholar 

  3. Aït-Sahalia Y, Mykland P A, and Zhang L, How often to sample a continuous-time process in the presence of market microstructure noise, The Review of Financial Studies, 2005, 18(2): 351–416.

    Article  MATH  Google Scholar 

  4. Zhang L, Mykland P A, and Aït-Sahalia Y, A tale of two time scales: Determining integrated volatility with noisy high-frequency data, Journal of the American Statistical Association, 2005, 100(472): 1394–1411.

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhang L, Efficient estimation of stochastic volatility using noisy observations: A multi-scale approach, Bernoulli, 2006, 12(6): 1019–1043.

    Article  MathSciNet  MATH  Google Scholar 

  6. Christensen K, Kinnebrock S, and Podolskij M, Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data, Journal of Econometrics, 2010, 159(1): 116–133.

    Article  MathSciNet  MATH  Google Scholar 

  7. Wang Y and Zou J, Vast volatility matrix estimation for high-frequency financial data, The Annals of Statistics, 2010, 38(2): 943–978.

    Article  MathSciNet  MATH  Google Scholar 

  8. Tao M, Wang Y, Yao Q, et al., Large volatility matrix inference via combining low-frequency and high-frequency approaches, Journal of the American Statistical Association, 2011, 106(495): 1025–1040.

    Article  MathSciNet  MATH  Google Scholar 

  9. Tao M, Wang Y, and Chen X, Fast convergence rates in estimating large volatility matrices using high-frequency financial data, Econometric Theory, 2013, 29(4): 838–856.

    Article  MathSciNet  MATH  Google Scholar 

  10. Tao M, Wang Y, and Zhou HH, Optimal sparse volatility matrix estimation for high-dimensional Ito processes with measurement errors, The Annals of Statistics, 2013, 41(4): 1816–1864.

    Article  MathSciNet  MATH  Google Scholar 

  11. Hautsch N, Kyj L M, and Oomen RC, A blocking and regularization approach to high-dimensional realized covariance estimation, Journal of Applied Econometrics, 2012, 27(4): 625–645.

    Article  MathSciNet  Google Scholar 

  12. Kim D, Wang Y, and Zou J, Asymptotic theory for large volatility matrix estimation based on high-frequency financial data, Stochastic Processes and Their Applications, 2016, 126(11): 3527–3577.

    Article  MathSciNet  MATH  Google Scholar 

  13. Fan J, Fan Y, and Lü J, High dimensional covariance matrix estimation using a factor model, Journal of Econometrics, 2008, 147(1): 186–197.

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan J, Liao Y, and Mincheva M, High dimensional covariance matrix estimation in approximate factor models, The Annals of Statistics, 2011, 39(6): 3320–3356.

    Article  MathSciNet  MATH  Google Scholar 

  15. Fan J, Liao Y, and Mincheva M, Large covariance estimation by thresholding principal orthogonal complements, Journal of the Royal Statistical Society Series B, Statistical Methodology, 2013, 75(4): 603–680.

    Article  MathSciNet  MATH  Google Scholar 

  16. Fan J, Furger A, and Xiu D, Incorporating global industrial classification standard into portfolio allocation: A simple factor-based large covariance matrix estimator with high-frequency data, Journal of Business & Economic Statistics, 2016, 34(4): 489–503.

    Article  MathSciNet  Google Scholar 

  17. Aït-Sahalia Y and Xiu D, Using principal component analysis to estimate a high dimensional factor model with high-frequency data, Journal of Econometrics, 2017, 201(2): 384–399.

    Article  MathSciNet  MATH  Google Scholar 

  18. Fan J and Kim D, Robust high-dimensional volatility matrix estimation for high-frequency factor model, Journal of the American Statistical Association, 2018, 113(523): 1268–1283.

    Article  MathSciNet  MATH  Google Scholar 

  19. Fan J and Kim D, Structured volatility matrix estimation for non-synchronized high-frequency financial data, Journal of Econometrics, 2019, 209(1): 61–78.

    Article  MathSciNet  MATH  Google Scholar 

  20. Dai C, Lu K, and Xiu D, Knowing factors or factor loadings, or neither? Evaluating estimators of large covariance matrices with noisy and asynchronous data, Journal of Econometrics, 2019, 208(1): 43–79.

    Article  MathSciNet  MATH  Google Scholar 

  21. Bollerslev T, Meddahi N, and Nyawa S, high-dimensional multivariate realized volatility estimation, Journal of Econometrics, 2019, 212(1): 116–136.

    Article  MathSciNet  MATH  Google Scholar 

  22. Fan J, Qi L, and Xiu D, Quasi-maximum likelihood estimation of GARCH models with heavy-tailed likelihoods, Journal of Business & Economic Statistics, 2014, 32(2): 178–191.

    Article  MathSciNet  Google Scholar 

  23. Moussa A M, Kamdem J S, and Terraza M, Fuzzy value-at-risk and expected shortfall for portfolios with heavy-tailed returns, Economic Modelling, 2014, 39: 247–256.

    Article  Google Scholar 

  24. Massacci D, Tail risk dynamics in stock returns: Links to the macroeconomy and global markets connectedness, Management Science, 2017, 63(9): 3072–3089.

    Article  Google Scholar 

  25. Mao G and Zhang Z, Stochastic tail index model for high frequency financial data with Bayesian analysis, Journal of Econometrics, 2018, 205(2): 470–487.

    Article  MathSciNet  MATH  Google Scholar 

  26. Opschoor A, Janus P, Lucas A, et al., New HEAVY models for fat–tailed realized covariances and returns, Journal of Business & Economic Statistics, 2018, 36(4): 643–657.

    Article  MathSciNet  Google Scholar 

  27. Fan J, Wang W, and Zhong Y, Robust covariance estimation for approximate factor models, Journal of Econometrics, 2019, 208(1): 5–22.

    Article  MathSciNet  MATH  Google Scholar 

  28. Thiele S, Modeling the conditional distribution of financial returns with asymmetric tails, Journal of Applied Econometrics, 2020, 35(1): 46–60.

    Article  MathSciNet  Google Scholar 

  29. Kwessi E, Double penalized semi-parametric signed-rank regression with adaptive lasso, Journal of Systems Science and Complexity, 2021, 34(1): 381–401.

    Article  MathSciNet  MATH  Google Scholar 

  30. Hasbrouck J and Seppi D J, Common factors in prices, order flows, and liquidity, Journal of Financial Economics, 2001, 59(3): 383–411.

    Article  Google Scholar 

  31. Voev V and Lunde A, Integrated covariance estimation using high-frequency data in the presence of noise, Journal of Financial Econometrics, 2007, 5(1): 68–104.

    Article  Google Scholar 

  32. Fan J, Wang W, and Zhong Y, An ℓ∞ eigenvector perturbation bound and its application to robust covariance estimation, Journal of Machine Learning Research, 2018, 18(207): 1–42.

    Google Scholar 

  33. Aït-Sahalia Y, Fan J, and Xiu D, High-frequency covariance estimates with noisy and asyn-chronous financial data, Journal of the American Statistical Association, 2010, 105(492): 1504–1517.

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang L, Estimating covariation: Epps effect, microstructure noise, Journal of Econometrics, 2011, 160(1): 33–47.

    Article  MathSciNet  MATH  Google Scholar 

  35. Barndorff-Nielsen O E, Hansen P R, Lunde A, et al., Multivariate realised kernels: Consistent positive semi-definite estimators of the covariation of equity prices with noise and nonsynchronous trading, Journal of Econometrics, 2011, 162(2): 149–169.

    Article  MathSciNet  MATH  Google Scholar 

  36. Fan J, Li Y, and Yu K, Vast volatility matrix estimation using high-frequency data for portfolio selection, Journal of the American Statistical Association, 2012, 107(497): 412–428.

    Article  MathSciNet  MATH  Google Scholar 

  37. Fan J and Wang Y, Multi-scale jump and volatility analysis for high-frequency financial data, Journal of the American Statistical Association, 2007, 102(480): 1349–1362.

    Article  MathSciNet  MATH  Google Scholar 

  38. Barndorff-Nielsen O E, Hansen P R, Lunde A, et al., Designing realized kernels to measure the ex post variation of equity prices in the presence of noise, Econometrica, 2008, 76(6): 1481–1536.

    Article  MathSciNet  MATH  Google Scholar 

  39. Jacod J, Li Y, Mykland P A, et al., Microstructure noise in the continuous case: The pre-averaging approach, Stochastic Processes and Their Applications, 2009, 119(7): 2249–2276.

    Article  MathSciNet  MATH  Google Scholar 

  40. Podolskij M and Vetter M, Estimation of volatility functionals in the simultaneous presence of microstructure noise and jumps, Bernoulli, 2009, 15(3): 634–658.

    Article  MathSciNet  MATH  Google Scholar 

  41. Xiu D, Quasi-maximum likelihood estimation of volatility with high frequency data, Journal of Econometrics, 2010, 159(1): 235–250.

    Article  MathSciNet  MATH  Google Scholar 

  42. Hayashi T and Yoshida N, On covariance estimation of non-synchronously observed diffusion processes, Bernoulli, 2005, 11(2): 359–379.

    Article  MathSciNet  MATH  Google Scholar 

  43. Kim D and Wang Y, Sparse PCA-based on high-dimensional Ito processes with measurement errors, Journal of ultivariate Analysis, 2016, 152: 172–189.

    Article  MathSciNet  MATH  Google Scholar 

  44. Candes E J, Li X, Ma Y, et al., Robust principal component analysis? Journal of the ACM, 2011, 58(3): 1–37.

    Article  MathSciNet  MATH  Google Scholar 

  45. Kong X B, On the number of common factors with high-frequency data, Biometrika, 2017, 104(2): 397–410.

    Article  MathSciNet  MATH  Google Scholar 

  46. Kong X B, On the systematic and idiosyncratic volatility with large panel high-frequency data, The Annals of Statistics, 2018, 46(3): 1077–1108.

    Article  MathSciNet  MATH  Google Scholar 

  47. Li Y and Zhu J, L1-norm quantile regression, Journal of Computational and Graphical Statistics, 2008, 17(1): 163–185.

    Article  MathSciNet  Google Scholar 

  48. Wu Y and Liu Y, Variable selection in quantile regression, Statistica Sinica, 2009, 19(2): 801–817.

    MathSciNet  MATH  Google Scholar 

  49. Zou H and Yuan M, Composite quantile regression and the oracle model selection theory, The Annals of Statistics, 2008, 36(3): 1108–1126.

    Article  MathSciNet  MATH  Google Scholar 

  50. Belloni A and Chernozhukov V, ℓ1-penalized quantile regression in high-dimensional sparse models, The Annals of Statistics, 2011, 39(1): 82–130.

    Article  MathSciNet  MATH  Google Scholar 

  51. Fan J, Fan Y, and Barut E, Adaptive robust variable selection, The Annals of Statistics, 2014, 42(1): 324–351.

    Article  MathSciNet  MATH  Google Scholar 

  52. Wang L, The L1 penalized LAD estimator for high dimensional linear regression, Journal of Multivariate Analysis, 2013, 120: 135–151.

    Article  MathSciNet  MATH  Google Scholar 

  53. Fan J, Li Q, and Wang Y, Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions, Journal of the Royal Statistical Society Series B, Statistical Methodology, 2017, 79(1): 247–265.

    Article  MathSciNet  MATH  Google Scholar 

  54. Lam C and Yao Q, Factor modeling for high-dimensional time series: Inference for the number of factors, The Annals of Statistics, 2012, 40(2): 694–726.

    Article  MathSciNet  MATH  Google Scholar 

  55. Bai J and Ng S, Determining the number of factors in approximate factor models, Econometrica, 2002, 70(1): 191–221.

    Article  MathSciNet  MATH  Google Scholar 

  56. Fan J, Liao Y, and Shi X, Risks of large portfolios, Journal of Econometrics, 2015, 186(2): 367–387.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center for Applied Statistics, School of Statistics, Institute of Probability and Statistics, Renmin University of China, Beijing, 100086, China

    Wanwan Liang, Ben Wu, Xinyan Fan & Bo Zhang

  2. Department of Statistics & Data Science, Southern University of Science and Technology, Shenzhen, 518055, China

    Bingyi Jing

Authors
  1. Wanwan Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ben Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xinyan Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Bingyi Jing
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Bo Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bo Zhang.

Ethics declarations

The authors declare no conflict of interest.

Additional information

The research of Bo Zhang is supported by the National Natural Science Foundation of China under Grant Nos. 72271232, 71873137 and the MOE Project of Key Research Institute of Humanities and Social Sciences under Grant No. 22JJD110001. The authors gratefully acknowledge the support of Public Computing Cloud, Renmin University of China.

Supplement Materials

To save some space in the paper, additional numerical results are reported in the supplementary materials (http://stat.ruc.edu.cn/docs/2023-08/008f7961788a44d1845448688fb3d92e.pdf).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liang, W., Wu, B., Fan, X. et al. High-Dimensional Volatility Matrix Estimation with Cross-Sectional Dependent and Heavy-Tailed Microstructural Noise. J Syst Sci Complex 36, 2125–2154 (2023). https://doi.org/10.1007/s11424-023-2080-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-023-2080-5

Keywords

Search

Navigation