Tests for the Weights of the Global Minimum Variance Portfolio in a High-Dimensional Setting
In this paper, 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 <inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula> depends on the sample size <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> such that <inline-formula><tex-math notation="LaTeX">$\frac{p}{n}\rightarrow c \in (0,1)$</tex-math></inline-formula> as <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> tends to infinity. In the case of a singular covariance matrix with rank equal to <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula> we assume that <inline-formula><tex-math notation="LaTeX">$q/n\rightarrow \tilde{c}\in (0, 1)$</tex-math></inline-formula> as <inline-formula><tex-math notation="LaTeX">$n\rightarrow \infty$</tex-math></inline-formula>. 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 <inline-formula><tex-math notation="LaTeX">$c$</tex-math></inline-formula> close to one.