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Numerical implementation of the QuEST function

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  • Olivier Ledoit
  • Michael Wolf

Abstract

This paper deals with certain estimation problems involving the covariance matrix in large dimensions. 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, Ledoit and Wolf (2015) have proposed an estimator of the eigenvalues of the population covariance matrix that is consistent according to a mean-square criterion under large-dimensional asymptotics. It requires numerical inversion of a multivariate nonrandom function which they call the QuEST function. The present paper explains how to numerically implement the QuEST function in practice through a series of six successive steps. It also provides an algorithm to compute the Jacobian analytically, which is necessary for numerical inversion by a nonlinear optimizer. Monte Carlo simulations document the effectiveness of the code.

Suggested Citation

  • Olivier Ledoit & Michael Wolf, 2016. "Numerical implementation of the QuEST function," ECON - Working Papers 215, Department of Economics - University of Zurich, revised Jan 2017.
    • Handle: RePEc:zur:econwp:215
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    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," Review of Financial Studies, Society for Financial Studies, vol. 30(12), pages 4349-4388.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    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. 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 1651-1684, August.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    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.
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    Citations

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    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. 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.
    3. Ledoit, Olivier & Wolf, Michael, 2021. "Shrinkage estimation of large covariance matrices: Keep it simple, statistician?," Journal of Multivariate Analysis, Elsevier, vol. 186(C).
    4. 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.
    5. 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.
    6. Moura, Guilherme V. & Santos, André A. P. & Ruiz Ortega, Esther, 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.
    7. Anatolyev, Stanislav & Pyrlik, Vladimir, 2022. "Copula shrinkage and portfolio allocation in ultra-high dimensions," Journal of Economic Dynamics and Control, Elsevier, vol. 143(C).
    8. 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.
    9. Bernardo Freitas Paulo da Costa & Silvana M. Pesenti & Rodrigo S. Targino, 2023. "Risk Budgeting Portfolios from Simulations," Papers 2302.01196, arXiv.org.
    10. 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.
    11. Olivier Ledoit & Michael Wolf, 2019. "Quadratic shrinkage for large covariance matrices," ECON - Working Papers 335, Department of Economics - University of Zurich, revised Dec 2020.
    12. 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.
    13. Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    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).

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    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

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