The Annals of Applied Statistics

Robust dependence modeling for high-dimensional covariance matrices with financial applications

Zhe Zhu and Roy E. Welsch

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

A very important problem in finance is the construction of portfolios of assets that balance risk and reward in an optimal way. A critical issue in portfolio development is how to address data outliers that reflect very unusual, generally non-recurring, market conditions. Should we allow these to have a significant impact on our estimation and portfolio construction process or should they be considered separately as evidence of a regime shift and/or be used to adjust baseline results? In financial asset allocation, a fundamental step is often a mean-variance optimization problem that makes use of the location vector and dispersion matrix of the financial assets. In this paper, we introduce a new high- dimensional covariance estimator that is much less sensitive to outliers compared to its classical counterparts. We then apply this estimator to the active asset allocation application, and show that our proposed new estimator delivers better results compared to many existing asset allocation methods. An important bonus is that on our examples, the method has a smaller proportion of stock weights greater than 10% and, in many cases, a higher alpha. Covariance estimation is more challenging than mean estimation and only locally and not globally optimal solutions are available. Our proposed new robust covariance estimator uses a regular vine dependence structure and only pairwise robust partial correlation estimators. The resulting robust covariance estimator delivers high performance for identifying outliers for large high dimensional datasets, has a high breakdown point, and is positive definite. When the full vine structure is not available, we propose using a minimal spanning tree algorithm to replace missing vine structure.

Article information

Source
Ann. Appl. Stat., Volume 12, Number 2 (2018), 1228-1249.

Dates
Received: March 2014
Revised: July 2017
First available in Project Euclid: 28 July 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1532743492

Digital Object Identifier
doi:10.1214/17-AOAS1087

Keywords
Active asset allocation portfolio selection robust estimation high-dimensional dependence modeling covariance/correlation estimation regular vine

Citation

Zhu, Zhe; Welsch, Roy E. Robust dependence modeling for high-dimensional covariance matrices with financial applications. Ann. Appl. Stat. 12 (2018), no. 2, 1228--1249. doi:10.1214/17-AOAS1087. https://projecteuclid.org/euclid.aoas/1532743492


Export citation

References

  • Anderson, T. W. (1958). An Introduction to Multivariate Statistical Analysis. Wiley, New York; Chapman & Hall, London.
  • Bedford, T. and Cooke, R. M. (2002). Vines—A new graphical model for dependent random variables. Ann. Statist. 30 1031–1068.
  • Billor, N., Hadi, A. S. and Velleman, P. F. (2000). BACON: Blocked adaptive computationally efficient outlier nominators. Comput. Statist. Data Anal. 34 279–298.
  • DeMiguel, V., Garlappi, L. and Uppal, R. (2009). Optimal versus naïve diversification: How inefficient is the $1/N$ portfolio strategy? Rev. Financ. Stud. 22 1915–53.
  • Diestel, R. (2005). Graph Theory, 3rd ed. Graduate Texts in Mathematics 173. Springer, Berlin.
  • Dissmann, J., Brechmann, E. C., Czado, C. and Kurowicka, D. (2012). Selecting and estimating regular vine copulae and application to financial returns. Comput. Statist. Data Anal. 59 52–69.
  • Donoho, D. and Huber, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann (P. J. Bickel, K. A. Doksum and J. L. Hodges, eds.) 157–184. Wadsworth, Belmont, CA.
  • Fan, J., Fan, Y. and Lv, J. (2008). High dimensional covariance matrix estimation using a factor model. J. Econometrics 147 186–197.
  • Garcia-Alvarez, L. and Luger, R. (2011). Dynamic correlations, estimation risk, and portfolio management during the financial crisis. Working Paper, CEMFI, Madrid.
  • Kurowicka, D. and Cooke, R. M. (2006). Uncertainty Analysis with High Dimensional Dependence Modelling. Wiley, Chichester.
  • Kurowicka, D., Cooke, R. M. and Callies, U. (2006). Vines inference. Braz. J. Probab. Stat. 20 103–120.
  • Lopuhaä, H. P. (1989). On the relation between $S$-estimators and $M$-estimators of multivariate location and covariance. Ann. Statist. 17 1662–1683.
  • Lopuhaä, H. P. and Rousseeuw, P. J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Ann. Statist. 19 229–248.
  • Maechler, M. and Stahel, W. (2009). Robust scatter estimators—The barrow wheel benchmark. ICORS 2009, Parma.
  • Markowitz, H. M. (1952). Portfolio selection. J. Finance 7 77–91.
  • Maronna, R. A. and Zamar, R. H. (2002). Robust estimates of location and dispersion for high-dimensional data sets. Technometrics 44 307–317.
  • Rousseeuw, P. J. and Driessen, K. V. (1999). A fast algorithm for the minimum covariance determinant estimator. Technometrics 41 212–223.
  • Rousseeuw, P. J. and Leroy, A. M. (1987). Robust Regression and Outlier Detection. Wiley.
  • Rousseeuw, P. and Yohai, V. (1984). Robust regression by means of S-estimators. In Robust and Nonlinear Time Series Analysis (Heidelberg, 1983). Lect. Notes Stat. 26 256–272. Springer, New York.
  • Rudin, W. (1976). Principles of Mathematical Analysis, 3rd ed. McGraw-Hill Book Co., New York–Auckland–Düsseldorf.
  • Tyler, D. E. (1987). A distribution-free $M$-estimator of multivariate scatter. Ann. Statist. 15 234–251.
  • Yohai, V. J. (1987). High breakdown-point and high efficiency robust estimates for regression. Ann. Statist. 15 642–656.
  • Yule, G. U. and Kendall, M. G. (1965). An Introduction to the Theory of Statistics, 14th ed. Hafner Publishing Co., New York.