Tangency portfolio weights for singular covariance matrix in small and large dimensions: estimation and test theory. (English) Zbl 1418.91453
Summary: 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.
MSC:
| 91G10 | Portfolio theory |
| 62H10 | Multivariate distribution of statistics |
| 62H12 | Estimation in multivariate analysis |
| 91G70 | Statistical methods; risk measures |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
Keywords:
tangency portfolio; singular Wishart distribution; singular covariance matrix; high-dimensional asymptotics; hypothesis testingReferences:
| [1] | Bodnar, T.; Mazur, S.; Okhrin, Y., Bayesian estimation of the global minimum variance portfolio, European J. Oper. Res., 256, 292-307 (2017) · Zbl 1395.91395 |
| [2] | Bodnar, T.; Mazur, S.; Podgórski, K., Singular inverse Wishart distribution and its application to portfolio theory, J. Multivariate Anal., 143, 314-326 (2016) · Zbl 1328.62313 |
| [3] | Bodnar, T.; Mazur, S.; Podgórski, K., A test for the global minimum variance portfolio for small sample and singular covariance, AStA Adv. Stat. Anal., 101, 253-265 (2017) · Zbl 1443.62332 |
| [4] | Bodnar, T.; Okhrin, Y., Properties of the singular, inverse and generalized inverse partitioned Wishart distributions, J. Multivariate Anal., 99, 2389-2405 (2008) · Zbl 1151.62046 |
| [5] | Bodnar, T.; Okhrin, Y., On the product of inverse Wishart and normal distributions with applications to discriminant analysis and portfolio theory, Scand. J. Stat., 38, 311-331 (2011) · Zbl 1246.62132 |
| [6] | Bodnar, T.; Reiß, M., Exact and asymptotic tests on a factor model in low and large dimensions with applications, J. Multivariate Anal., 150, 125-151 (2016) · Zbl 1397.62209 |
| [7] | Chamberlain, G., Funds, factors, and diversification in arbitrage pricing models, Econometrica, 51, 1305-1323 (1983) · Zbl 0523.90018 |
| [8] | Fan, J.; Fan, Y.; Lv, J., High dimensional covariance matrix estimation using a factor model, J. Econometrics, 147, 186-197 (2008) · Zbl 1429.62185 |
| [9] | Fan, J.; Liao, Y.; Mincheva, M., Large covariance estimation by thresholding principal orthogonal complements, J. R. Stat. Soc. Ser. B Stat. Methodol., 75, 603-680 (2013) · Zbl 1411.62138 |
| [10] | Fan, J.; Zhang, J.; Yu, K., Vast portfolio selection with gross-exposure constraints, J. Amer. Statist. Assoc., 107, 592-606 (2012) · Zbl 1261.62091 |
| [11] | Givens, G. H.; Hoeting, J. A., Computational Statistics (2012), John Wiley & Sons |
| [12] | Gupta, A.; Nagar, D., Matrix Variate Distributions (2000), Chapman and Hall/CRC: Chapman and Hall/CRC Boca Raton · Zbl 0935.62064 |
| [13] | Gupta, A.; Varga, T.; Bodnar, T., Elliptically Contoured Models in Statistics and Portfolio Theory (2013), Springer · Zbl 1306.62028 |
| [14] | Markowitz, H., Mean-variance analysis in portfolio choice and capital markets, J. Finance, 7, 77-91 (1952) |
| [15] | Nadakuditi, R. R.; Eldeman, A., Sample eigenvalue based detection of high-dimensional signals in white noise using relatively few samples, IEEE Trans. Signal Process., 56, 2625-2638 (2008) · Zbl 1390.94567 |
| [16] | Pappas, D.; Kiriakopoulos, K.; Kaimakamis, G., Optimal portfolio selection with singular covariance matrix, Int. Math. Forum, 5, 2305-2318 (2010) · Zbl 1214.91105 |
| [17] | Pruzek, R., High dimensional covariance estimation: avoiding the ‘curse of dimensionality’, (Bozdogan, H.; etal., Proceedings of the First US/Japan Conference on the Frontiers of Statistical Modeling: An Informational Approach (1994), Springer: Springer Dordrecht) |
| [18] | Stein, C.; Efron, B.; Morris, C., Improving the usual estimator of a normal covariance matrixTechnical Report 37 (1972), Dept. of Statistics, Stanford University |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
