Tangency portfolio weights for singular covariance matrix in small and large dimensions: Estimation and test theory

• Published in: Journal of statistical planning and inference Vol. 201; pp. 40 - 57
• Main Authors: Bodnar, Taras, Mazur, Stepan, Podgórski, Krzysztof, Tyrcha, Joanna
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-07-2019
• Subjects: High-dimensional asymptotics; Hypothesis testing; Matematik; Mathematics; Natural Sciences; Naturvetenskap; Probability Theory and Statistics; Sannolikhetsteori och statistik; Singular covariance matrix; Singular Wishart distribution; Tangency portfolio; Hypothesis testing; 91G10; High-dimensional asymptotics; Tangency portfolio; 62H10; Singular Wishart distribution; 62H12; Singular covariance matrix
• ISSN: 0378-3758; 1873-1171; 1873-1171
• DOI: 10.1016/j.jspi.2018.11.003

In this paper we derive the finite-sample distribution of the estimated weights of the tangency portfolio when both the population and the sample covariance matrices are singular. These results are used in the derivation of a statistical test on the weights of the tangency portfolio where the distribution of the test statistic is obtained under both the null and alternative hypotheses. Moreover, we establish the high-dimensional asymptotic distribution of the estimated weights of the tangency portfolio when both the portfolio dimension and the sample size increase to infinity. The theoretical findings are implemented in an empirical application dealing with the returns on the stocks included into the S&P 500 index.
•Finite-sample distribution of estimated tangency portfolio weights is derived.•Both population and the sample covariance matrices are assumed to be singular.•Exact and high-dimensional asymptotic tests are proposed.•Distribution of the test statistic is obtained under both hypotheses.•Theoretical findings are implemented in an empirical application.