The Annals of Statistics

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

Xinghua Zheng and Yingying Li

Full-text: Open access

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 $\mathcal {C}$ 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 $\mathcal {C}$, 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.

Article information

Source
Ann. Statist., Volume 39, Number 6 (2011), 3121-3151.

Dates
First available in Project Euclid: 27 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aos/1327672848

Digital Object Identifier
doi:10.1214/11-AOS939

Mathematical Reviews number (MathSciNet)
MR3012403

Zentralblatt MATH identifier
1246.62182

Subjects
Primary: 62H12: Estimation
Secondary: 62G99: None of the above, but in this section 60F15: Strong theorems

Keywords
High dimension high frequency integrated covariance matrix Marčenko–Pastur equation weighted sample covariance matrix realized covariance matrix

Citation

Zheng, Xinghua; Li, Yingying. On the estimation of integrated covariance matrices of high dimensional diffusion processes. Ann. Statist. 39 (2011), no. 6, 3121--3151. doi:10.1214/11-AOS939. https://projecteuclid.org/euclid.aos/1327672848


Export citation

References

  • Admati, A. R. and Pfleiderer, P. (1988). A theory of intraday patterns: Volume and price variability. Rev. Financ. Stud. 1 3–40.
  • Andersen, T. G. and Bollerslev, T. (1997). Intraday periodicity and volatility persistence in financial markets. Journal of Empirical Finance 4 115–158.
  • Andersen, T. G. and Bollerslev, T. (1998). Deutsche mark–dollar volatility: Intraday activity patterns, macroeconomic announcements, and longer run dependencies. J. Finance 53 219–265.
  • Andersen, T. G., Bollerslev, T., Diebold, F. X. and Labys, P. (2001). The distribution of realized exchange rate volatility. J. Amer. Statist. Assoc. 96 42–55.
  • Bai, Z. D. (1999). Methodologies in spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611–677.
  • Bai, Z., Chen, J. and Yao, J. (2010). On estimation of the population spectral distribution from a high-dimensional sample covariance matrix. Aust. N. Z. J. Stat. 52 423–437.
  • Bai, Z., Liu, H. and Wong, W.-K. (2009). Enhancement of the applicability of Markowitz’s portfolio optimization by utilizing random matrix theory. Math. Finance 19 639–667.
  • Bai, Z. D. and Silverstein, J. W. (1998). No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices. Ann. Probab. 26 316–345.
  • Barndorff-Nielsen, O. E. and Shephard, N. (2004). Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics. Econometrica 72 885–925.
  • 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.
  • El Karoui, N. (2008). Spectrum estimation for large dimensional covariance matrices using random matrix theory. Ann. Statist. 36 2757–2790.
  • Fan, J., Li, Y. and Yu, K. (2011). Vast volatility matrix estimation using high frequency data for portfolio selection. J. Amer. Statist. Assoc. To appear.
  • Geronimo, J. S. and Hill, T. P. (2003). Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform. J. Approx. Theory 121 54–60.
  • Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis. Cambridge Univ. Press, Cambridge. Corrected reprint of the 1985 original.
  • Jacod, J. and Protter, P. (1998). Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probab. 26 267–307.
  • Marčenko, V. A. and Pastur, L. A. (1967). Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. (N.S.) 72 507–536.
  • Markowitz, H. (1952). Portfolio selection. J. Finance 7 77–91.
  • Markowitz, H. M. (1959). Portfolio Selection: Efficient Diversification of Investments. Cowles Foundation for Research in Economics at Yale University, Monograph 16. Wiley, New York.
  • Mestre, X. (2008). Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates. IEEE Trans. Inform. Theory 54 5113–5129.
  • Silverstein, J. W. (1995). Strong convergence of the empirical distribution of eigenvalues of large-dimensional random matrices. J. Multivariate Anal. 55 331–339.
  • Silverstein, J. W. and Bai, Z. D. (1995). On the empirical distribution of eigenvalues of a class of large-dimensional random matrices. J. Multivariate Anal. 54 175–192.
  • Silverstein, J. W. and Choi, S.-I. (1995). Analysis of the limiting spectral distribution of large-dimensional random matrices. J. Multivariate Anal. 54 295–309.
  • Tao, M., Wang, Y., Yao, Y. and Zou, J. (2011). Large volatility matrix inference via combining low-frequency and high-frequency approaches. J. Amer. Statist. Assoc. 106 1025–1040.
  • Wang, Y. and Zou, J. (2010). Vast volatility matrix estimation for high-frequency financial data. Ann. Statist. 38 943–978.
  • Yin, Y. Q. (1986). Limiting spectral distribution for a class of random matrices. J. Multivariate Anal. 20 50–68.
  • Zheng, X. and Li, Y. (2011). Supplement to “On the estimation of integrated covariance matrices of high dimensional diffusion processes.” DOI:10.1214/11-AOS939SUPP.

Supplemental materials

  • Supplementary material: Supplement to “On the estimation of integrated covariance matrices of high dimensional diffusion processes”. This material contains the proof of Proposition 4, a detailed explanation of the second statement in Remark 3, and the proofs of the various lemmas in Section 3.1 and Proposition 7.