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

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Abstract

In this paper we derive the nite-sample distribution of the esti- mated 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 port- folio where the distribution of the test statistic is obtained under both the null and the 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 sam- ple size increase to in nity. The theoretical ndings are implemented in an empirical application dealing with the returns on the stocks included into the S&P 500 index.

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  • Bodnar, Taras & Mazur, Stepan & Podgórski, Krzysztof & Tyrcha, Joanna, 2018. "Tangency portfolio weights for singular covariance matrix in small and large dimensions: estimation and test theory," Working Papers 2018:1, Örebro University, School of Business.
    • Handle: RePEc:hhs:oruesi:2018_001
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    References listed on IDEAS

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    1. Jianqing Fan & Yuan Liao & Martina Mincheva, 2013. "Large covariance estimation by thresholding principal orthogonal complements," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(4), pages 603-680, September.
    2. Taras Bodnar & Stepan Mazur & Krzysztof Podgórski, 2017. "A test for the global minimum variance portfolio for small sample and singular covariance," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 101(3), pages 253-265, July.
    3. Bodnar, Taras & Reiß, Markus, 2016. "Exact and asymptotic tests on a factor model in low and large dimensions with applications," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 125-151.
    4. Bodnar, Taras & Mazur, Stepan & Podgórski, Krzysztof, 2016. "Singular inverse Wishart distribution and its application to portfolio theory," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 314-326.
    5. Bodnar, Taras & Okhrin, Yarema, 2008. "Properties of the singular, inverse and generalized inverse partitioned Wishart distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2389-2405, November.
    6. Jianqing Fan & Jingjin Zhang & Ke Yu, 2012. "Vast Portfolio Selection With Gross-Exposure Constraints," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(498), pages 592-606, June.
    7. Chamberlain, Gary, 1983. "Funds, Factors, and Diversification in Arbitrage Pricing Models," Econometrica, Econometric Society, vol. 51(5), pages 1305-1323, September.
    8. Taras Bodnar & Yarema Okhrin, 2011. "On the Product of Inverse Wishart and Normal Distributions with Applications to Discriminant Analysis and Portfolio Theory," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 38(2), pages 311-331, June.
    9. Fan, Jianqing & Fan, Yingying & Lv, Jinchi, 2008. "High dimensional covariance matrix estimation using a factor model," Journal of Econometrics, Elsevier, vol. 147(1), pages 186-197, November.
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    Cited by:

    1. Gulliksson, Mårten & Mazur, Stepan, 2019. "An Iterative Approach to Ill-Conditioned Optimal Portfolio Selection," Working Papers 2019:3, Örebro University, School of Business.
    2. Apostolos Chalkis & Emmanouil Christoforou & Ioannis Z. Emiris & Theodore Dalamagas, 2021. "Modeling asset allocations and a new portfolio performance score," Digital Finance, Springer, vol. 3(3), pages 333-371, December.

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    More about this item

    Keywords

    tangency portfolio; singular Wishart distribution; singular covariance matrix; high-dimensional asymptotics; hypothesis testing;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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