Tests for the weights of the global minimum variance portfolio in a high-dimensional setting

TL;DR

This paper develops and compares two statistical tests for the weights of the global minimum variance portfolio in high-dimensional settings where the number of assets is comparable to the sample size, including cases with singular covariance matrices.

Contribution

It introduces new tests based on sample and shrinkage estimators for GMVP weights in high-dimensional regimes, deriving their asymptotic distributions and evaluating their performance.

Findings

Shrinkage estimator-based test performs well near the boundary of high-dimensionality.

Proposed tests have accurate asymptotic distributions under null and alternative hypotheses.

Simulation results show the shrinkage-based test outperforms existing methods in power.

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 $\frac{p}{n}\to c \in (0,1)$ as $n$ tends to infinity. In the case of a singular covariance matrix with rank equal to $q$ we assume that $q/n\to \tilde{c}\in(0, 1)$ as $n\to\infty$. 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.