AStA Advances in Statistical Analysis

, Volume 101, Issue 3, pp 253–265 | Cite as

A test for the global minimum variance portfolio for small sample and singular covariance

Original Paper


Recently, a test dealing with the linear hypothesis for the global minimum variance portfolio weights was obtained under the assumption of non-singular covariance matrix. However, the problem of potential multicollinearity and correlations of assets constitutes a limitation of the classical portfolio theory. Therefore, there is an interest in developing theory in the presence of singularities in the covariance matrix. In this paper, we extend the test by analyzing the portfolio weights in the small sample case with a singular population covariance matrix. The results are illustrated using actual stock returns and a discussion of practical relevance of the model is presented.


Global minimum variance portfolio Singular Wishart distribution Singular covariance matrix Small sample problem 

Mathematics Subject Classification

91G10 62H12 



The authors are thankful to Prof. Yarema Okhrin and two anonymous referees for careful reading of the paper and for their suggestions which have improved earlier versions. Taras Bodnar appreciates the financial support of the German Science Foundation (DFG) via the Projects BO 3521/3-1 and SCHM 859/13-1 “Bayesian estimation of the multi-period optimal portfolio weights and risk measure”. Stepan Mazur gratefully acknowledges financial support from the research project “Ambit fields: Probabilistic properties and statistical inference” funded by Villum Fonden. Krzysztof Podgórski appreciates the financial support of the Swedish Research Council Grant Dnr: 2013-5180 and Riksbankens Jubileumsfond Grant Dnr: P13-1024:1.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Taras Bodnar
    • 1
  • Stepan Mazur
    • 2
  • Krzysztof Podgórski
    • 3
  1. 1.Department of MathematicsStockholm UniversityStockholmSweden
  2. 2.Department of MathematicsAarhus UniversityAarhusDenmark
  3. 3.Department of StatisticsLund UniversityLundSweden

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