# Probability and Statistics Seminar

### Tests for the Weights of the Global Minimum Variance Portfolio in a High-Dimensional Setting

In this talk tests for the weights of the global minimum variance portfolio (GMVP) in a high-dimensional setting are presented, namely when the number of assets $p$ depends on the sample size $n$ such that $p/n \to c$ in $(0,1)$, as $n$ tends to infinity. The introduced tests are based on the sample estimator and on a shrinkage estimator of the GMVP weights (cf. Bodnar et al. 2017). The asymptotic distributions of both test statistics under the null and alternative hypotheses are derived. Moreover, we provide a simulation study where the performance of the proposed tests is compared with each other and with an approach of Glombeck (2014). A good performance of the test based on the shrinkage estimator is observed even for values of $c$ close to $1$.

(joint work with Taras Bodnar, Solomiia Dmytriv and Nestor Parolya)

### References:

• Bodnar, T. and Schmid, W. (2008). A test for the weights of the global minimum variance portfolio in an elliptical model, Metrika, 67, 127-143.
• Bodnar, T., Parolya, N. and Schmid, W. (2017). Estimation of the minimum variance portfolio in high dimensions, European Journal of Operational Research, in press.
• Glombeck, K. (2014). Statistical inference for high-dimensional global minimum variance portfolios, Scandinavian Journal of Statistics, 41, 845-865.
• Okhrin, Y. and Schmid, W. (2006). Distributional properties of portfolio weights, Journal of Econometrics, 134, 235-256.
• Okhrin, Y. and Schmid, W. (2008). Estimation of optimal portfolio weights, International Journal of Theoretical and Applied Finance, 11, 249-276.