IDEAS home Printed from https://ideas.repec.org/a/eee/csdana/v115y2017icp199-223.html
   My bibliography  Save this article

Numerical implementation of the QuEST function

Author

Listed:
  • Ledoit, Olivier
  • Wolf, Michael

Abstract

Certain estimation problems involving the covariance matrix in large dimensions are considered. Due to the breakdown of finite-dimensional asymptotic theory when the dimension is not negligible with respect to the sample size, it is necessary to resort to an alternative framework known as large-dimensional asymptotics. Recently, an estimator of the eigenvalues of the population covariance matrix has been proposed that is consistent according to a mean-squared criterion under large-dimensional asymptotics. It requires numerical inversion of a multivariate nonrandom function called the QuEST function. The numerical implementation of this QuEST function in practice is explained through a series of six successive steps. An algorithm is provided in order to compute the Jacobian of the QuEST function analytically, which is necessary for numerical inversion via a nonlinear optimizer. Monte Carlo simulations document the effectiveness of the code.

Suggested Citation

  • Ledoit, Olivier & Wolf, Michael, 2017. "Numerical implementation of the QuEST function," Computational Statistics & Data Analysis, Elsevier, vol. 115(C), pages 199-223.
  • Handle: RePEc:eee:csdana:v:115:y:2017:i:c:p:199-223
    DOI: 10.1016/j.csda.2017.06.004
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167947317301433
    Download Restriction: Full text for ScienceDirect subscribers only.

    File URL: https://libkey.io/10.1016/j.csda.2017.06.004?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Olivier Ledoit & Michael Wolf, 2017. "Nonlinear Shrinkage of the Covariance Matrix for Portfolio Selection: Markowitz Meets Goldilocks," The Review of Financial Studies, Society for Financial Studies, vol. 30(12), pages 4349-4388.
    3. Chen, J. & Delyon, B. & Yao, J.-F., 2011. "On a model selection problem from high-dimensional sample covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1388-1398, November.
    4. Tsubasa Ito & Tatsuya Kubokawa, 2015. "Linear Ridge Estimator of High-Dimensional Precision Matrix Using Random Matrix Theory ," CIRJE F-Series CIRJE-F-995, CIRJE, Faculty of Economics, University of Tokyo.
    5. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1683, August.
    6. Ledoit, Olivier & Wolf, Michael, 2015. "Spectrum estimation: A unified framework for covariance matrix estimation and PCA in large dimensions," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 360-384.
    7. Ledoit, Olivier & Wolf, Michael, 2003. "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 603-621, December.
    8. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
    9. Silverstein, J. W. & Choi, S. I., 1995. "Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 295-309, August.
    10. Schäfer Juliane & Strimmer Korbinian, 2005. "A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 4(1), pages 1-32, November.
    11. Hafner, Christian M. & Reznikova, Olga, 2012. "On the estimation of dynamic conditional correlation models," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3533-3545.
    12. Robert F. Engle & Olivier Ledoit & Michael Wolf, 2019. "Large Dynamic Covariance Matrices," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(2), pages 363-375, April.
    13. repec:bla:jfinan:v:58:y:2003:i:4:p:1651-1684 is not listed on IDEAS
    14. Olivier Ledoit & Michael Wolf, 2014. "Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets Goldilocks," ECON - Working Papers 137, Department of Economics - University of Zurich, revised Feb 2017.
    15. Silverstein, J. W. & Bai, Z. D., 1995. "On the Empirical Distribution of Eigenvalues of a Class of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 54(2), pages 175-192, August.
    16. Ningning Xia & Zhidong Bai, 2015. "Functional CLT of eigenvectors for large sample covariance matrices," Statistical Papers, Springer, vol. 56(1), pages 23-60, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Zhao Zhao & Olivier Ledoit & Hui Jiang, 2019. "Risk reduction and efficiency increase in large portfolios: leverage and shrinkage," ECON - Working Papers 328, Department of Economics - University of Zurich, revised Jan 2020.
    2. Carlos Trucíos & Mauricio Zevallos & Luiz K. Hotta & André A. P. Santos, 2019. "Covariance Prediction in Large Portfolio Allocation," Econometrics, MDPI, vol. 7(2), pages 1-24, May.
    3. Olivier Ledoit & Michael Wolf, 2019. "Quadratic shrinkage for large covariance matrices," ECON - Working Papers 335, Department of Economics - University of Zurich, revised Dec 2020.
    4. Maaz Mahadi & Tarig Ballal & Muhammad Moinuddin & Tareq Y. Al-Naffouri & Ubaid Al-Saggaf, 2022. "Portfolio Optimization Using a Consistent Vector-Based MSE Estimation Approach," Papers 2204.05611, arXiv.org.
    5. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    6. Seonghun Cho & Shota Katayama & Johan Lim & Young-Geun Choi, 2021. "Positive-definite modification of a covariance matrix by minimizing the matrix $$\ell_{\infty}$$ ℓ ∞ norm with applications to portfolio optimization," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 105(4), pages 601-627, December.
    7. Ledoit, Olivier & Wolf, Michael, 2021. "Shrinkage estimation of large covariance matrices: Keep it simple, statistician?," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    8. Olivier Ledoit & Michael Wolf, 2019. "Shrinkage estimation of large covariance matrices: keep it simple, statistician?," ECON - Working Papers 327, Department of Economics - University of Zurich, revised Jun 2021.
    9. Robert F. Engle & Olivier Ledoit & Michael Wolf, 2019. "Large Dynamic Covariance Matrices," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(2), pages 363-375, April.
    10. Mörstedt, Torsten & Lutz, Bernhard & Neumann, Dirk, 2024. "Cross validation based transfer learning for cross-sectional non-linear shrinkage: A data-driven approach in portfolio optimization," European Journal of Operational Research, Elsevier, vol. 318(2), pages 670-685.
    11. Moura, Guilherme V. & Santos, André A. P., 2019. "Comparing Forecasts of Extremely Large Conditional Covariance Matrices," DES - Working Papers. Statistics and Econometrics. WS 29291, Universidad Carlos III de Madrid. Departamento de Estadística.
    12. Anatolyev, Stanislav & Pyrlik, Vladimir, 2022. "Copula shrinkage and portfolio allocation in ultra-high dimensions," Journal of Economic Dynamics and Control, Elsevier, vol. 143(C).
    13. da Costa, B. Freitas Paulo & Pesenti, Silvana M. & Targino, Rodrigo S., 2023. "Risk budgeting portfolios from simulations," European Journal of Operational Research, Elsevier, vol. 311(3), pages 1040-1056.
    14. Moura, Guilherme V. & Santos, André A.P. & Ruiz, Esther, 2020. "Comparing high-dimensional conditional covariance matrices: Implications for portfolio selection," Journal of Banking & Finance, Elsevier, vol. 118(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Robert F. Engle & Olivier Ledoit & Michael Wolf, 2019. "Large Dynamic Covariance Matrices," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 37(2), pages 363-375, April.
    2. Olivier Ledoit & Michael Wolf, 2019. "The power of (non-)linear shrinking: a review and guide to covariance matrix estimation," ECON - Working Papers 323, Department of Economics - University of Zurich, revised Feb 2020.
    3. Olivier Ledoit & Michael Wolf, 2019. "Quadratic shrinkage for large covariance matrices," ECON - Working Papers 335, Department of Economics - University of Zurich, revised Dec 2020.
    4. Mörstedt, Torsten & Lutz, Bernhard & Neumann, Dirk, 2024. "Cross validation based transfer learning for cross-sectional non-linear shrinkage: A data-driven approach in portfolio optimization," European Journal of Operational Research, Elsevier, vol. 318(2), pages 670-685.
    5. Couillet, Romain & McKay, Matthew, 2014. "Large dimensional analysis and optimization of robust shrinkage covariance matrix estimators," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 99-120.
    6. Olivier Ledoit & Michael Wolf, 2017. "Analytical nonlinear shrinkage of large-dimensional covariance matrices," ECON - Working Papers 264, Department of Economics - University of Zurich, revised Nov 2018.
    7. Bodnar, Taras & Parolya, Nestor & Schmid, Wolfgang, 2018. "Estimation of the global minimum variance portfolio in high dimensions," European Journal of Operational Research, Elsevier, vol. 266(1), pages 371-390.
    8. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    9. Olivier Ledoit & Michael Wolf, 2019. "Shrinkage estimation of large covariance matrices: keep it simple, statistician?," ECON - Working Papers 327, Department of Economics - University of Zurich, revised Jun 2021.
    10. Gianluca De Nard & Olivier Ledoit & Michael Wolf, 2018. "Factor models for portfolio selection in large dimensions: the good, the better and the ugly," ECON - Working Papers 290, Department of Economics - University of Zurich, revised Dec 2018.
    11. Lam, Clifford, 2020. "High-dimensional covariance matrix estimation," LSE Research Online Documents on Economics 101667, London School of Economics and Political Science, LSE Library.
    12. Sven Husmann & Antoniya Shivarova & Rick Steinert, 2022. "Sparsity and stability for minimum-variance portfolios," Risk Management, Palgrave Macmillan, vol. 24(3), pages 214-235, September.
    13. Ledoit, Olivier & Wolf, Michael, 2021. "Shrinkage estimation of large covariance matrices: Keep it simple, statistician?," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    14. Olivier Ledoit & Michael Wolf, 2013. "Optimal estimation of a large-dimensional covariance matrix under Stein’s loss," ECON - Working Papers 122, Department of Economics - University of Zurich, revised Mar 2017.
    15. De Nard, Gianluca & Zhao, Zhao, 2023. "Using, taming or avoiding the factor zoo? A double-shrinkage estimator for covariance matrices," Journal of Empirical Finance, Elsevier, vol. 72(C), pages 23-35.
    16. Zhao Zhao & Olivier Ledoit & Hui Jiang, 2019. "Risk reduction and efficiency increase in large portfolios: leverage and shrinkage," ECON - Working Papers 328, Department of Economics - University of Zurich, revised Jan 2020.
    17. Sven Husmann & Antoniya Shivarova & Rick Steinert, 2021. "Cross-validated covariance estimators for high-dimensional minimum-variance portfolios," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 35(3), pages 309-352, September.
    18. Richard Luger, 2024. "Regularizing stock return covariance matrices via multiple testing of correlations," Papers 2407.09696, arXiv.org.
    19. Ding, Yi & Li, Yingying & Zheng, Xinghua, 2021. "High dimensional minimum variance portfolio estimation under statistical factor models," Journal of Econometrics, Elsevier, vol. 222(1), pages 502-515.
    20. Wen, Jun, 2018. "Estimation of two high-dimensional covariance matrices and the spectrum of their ratio," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 1-29.

    More about this item

    Keywords

    Large-dimensional asymptotics; Numerical optimization; Random Matrix Theory; Spectrum estimation;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • C87 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Econometric Software

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:csdana:v:115:y:2017:i:c:p:199-223. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/csda .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.