[Paper Review] Tests for the weights of the global minimum variance portfolio in a high-dimensional setting

Taras Bodnar, Solomiia Dmytriv+2 more|arXiv (Cornell University)|Oct 26, 2017

Random Matrices and Applications52 references30 citations

TL;DR

This paper proposes two high-dimensional statistical tests for the weights of the global minimum variance portfolio (GMVP), using both sample and shrinkage estimators of the weights. It derives asymptotic distributions under null and alternative hypotheses and demonstrates through simulations that the shrinkage-based test outperforms existing methods, especially when the dimension-to-sample ratio is near one, offering superior power and robustness in high-dimensional settings with singular covariance matrices.

ABSTRACT

In this study, we construct two tests for the weights of the global minimum variance portfolio (GMVP) in a high-dimensional setting, namely, when the number of assets $p$ depends on the sample size $n$ such that $\frac{p}{n} o c \in (0,1)$ as $n$ tends to infinity. In the case of a singular covariance matrix with rank equal to $q$ we assume that $q/n o ilde{c}\in(0, 1)$ as $n o\infty$. The considered tests are based on the sample estimator and on the shrinkage estimator of the GMVP weights. We derive the asymptotic distributions of the test statistics under the null and alternative hypotheses. Moreover, we provide a simulation study where the power functions and the receiver operating characteristic curves of the proposed tests are compared with other existing approaches. We observe that the test based on the shrinkage estimator performs well even for values of $c$ close to one.

Motivation & Objective

• Develop statistical tests for GMVP weights in high-dimensional settings where the number of assets p grows proportionally with sample size n (p/n → c ∈ (0,1)).
• Address the limitations of classical asymptotic methods when p and n grow simultaneously, particularly the poor performance of sample estimators under high dimensionality.
• Provide a novel application of shrinkage estimation in statistical test theory for portfolio weights, improving estimation and inference robustness.
• Extend testing procedures to the case of singular covariance matrices, where the rank q grows proportionally with n (q/n → c̃ ∈ (0,1)).
• Compare the empirical performance of proposed tests with existing approaches using power functions and ROC curves under varying high-dimensional scenarios.

Proposed method

• Derive asymptotic distributions of test statistics under both the null and alternative hypotheses for the GMVP weights using high-dimensional asymptotic theory.
• Propose a test based on the sample estimator of GMVP weights, extending prior work by Bodnar and Schmid (2008) to high-dimensional settings.
• Introduce a new test based on a shrinkage estimator of the GMVP weights, leveraging the optimal shrinkage approach from Bodnar et al. (2018) to reduce estimation risk.
• Account for singular covariance matrices by deriving new test statistics and their asymptotic distributions under the null and alternative hypotheses.
• Apply random matrix theory to analyze the behavior of eigenvalues and eigenvectors in high-dimensional settings, enabling consistent asymptotic approximations.
• Use simulation studies to evaluate and compare the empirical power and ROC curve performance of the proposed tests across different values of c and c̃.

Experimental results

Research questions

• RQ1 How do the asymptotic distributions of test statistics for GMVP weights behave under high-dimensional asymptotics (p/n → c ∈ (0,1))?
• RQ2 Does the use of a shrinkage estimator for GMVP weights lead to a more powerful test compared to the classical sample estimator in high-dimensional settings?
• RQ3 How do the proposed tests perform when the covariance matrix is singular (rank q < p) and q/n → c̃ ∈ (0,1)?
• RQ4 What is the relative performance of the shrinkage-based test compared to existing tests (e.g., Glombeck, 2014) in terms of empirical power and ROC curve characteristics?
• RQ5 To what extent does the Mahalanobis distance approximation accurately reflect the true power of the asymptotic test in moderate sample sizes (e.g., n = 500)?

Key findings

• The test based on the shrinkage estimator of the GMVP weights exhibits uniformly higher empirical power than the sample-based test and other existing methods, especially when c is close to one.
• The shrinkage-based test maintains strong performance even under high-dimensional asymptotics with c ≈ 1, where classical estimators fail due to increased sampling error.
• For singular covariance matrices, the proposed test based on the shrinkage estimator outperforms all other methods in terms of both power and ROC curve performance, except in rare cases where the type I error is inflated.
• The test of Bodnar, Mazur, and Podgorski (2016), which uses Bonferroni correction, shows poor ROC performance for moderate to large false positive rates due to its conservative critical values.
• The asymptotic test based on Mahalanobis distance provides a good approximation to the true power function even for moderate sample sizes (n = 500) and p up to 450, suggesting practical utility.
• The proposed shrinkage-based test is robust to changes in the diagonal of the covariance matrix (e.g., 20% and 50% changes), maintaining high power across different scenarios.