Positive-definite modification of a covariance matrix by minimizing the matrix $$\ell_{\infty}$$ ℓ ∞ norm with applications to portfolio optimization

Seonghun Cho, Shota Katayama, Johan Lim () and Young-Geun Choi
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Seonghun Cho: Seoul National University
Shota Katayama: Keio University
Johan Lim: Seoul National University
Young-Geun Choi: Sookmyung Women’s University

AStA Advances in Statistical Analysis, 2021, vol. 105, issue 4, No 4, 627 pages

Abstract: Abstract The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness (PDness) is essential for the validity of the procedures. Recently, many regularized estimators have been proposed and shown to be consistent in estimating the true matrix and its support under various structural assumptions on the true covariance matrix. However, they are often not PD. In this paper, we propose a simple modification to make a regularized covariance matrix be PD while preserving its support and the convergence rate. We focus on the matrix $$\ell_{\infty }$$ ℓ ∞ norm error in covariance matrix estimation because it could allow us to bound the error in the downstream multivariate procedure relying on it. Our proposal in this paper is an extension of the fixed support positive-definite (FSPD) modification by Choi et al. (2019) from spectral and Frobenius norms to the matrix $$\ell_{\infty }$$ ℓ ∞ norm. Like the original FSPD, we consider a convex combination between the initial estimator (the regularized covariance matrix without PDness) and a given form of the diagonal matrix minimize the $$\ell_{\infty }$$ ℓ ∞ distance between the initial estimator and the convex combination, and find a closed-form expression for the modification. We apply the procedure to the minimum variance portfolio (MVP) optimization problem and show that the vector $$\ell_{\infty }$$ ℓ ∞ error in the estimation of the optimal portfolio weight is bounded by the matrix $$\ell _{\infty }$$ ℓ ∞ error of the plug-in covariance matrix estimator. We illustrate the MVP results with S&P 500 daily returns data from January 1978 to December 2014.

Keywords: High-dimensional covariance matrix; Linear shrinkage; Matrix $$\ell _{\infty }$$ ℓ ∞ norm; Minimum variance portfolio; Positive definiteness; Regularized covariance matrix estimator (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10182-021-00396-7

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