Testing for ellipsoidal symmetry: A comparison study

Abstract

The focus of this paper is the methodology for testing ellipsoidal symmetry, which was recently proposed by Koltchinskii and Sakhanenko [Koltchinskii, V., Sakhanenko, L. 2000. Testing for ellipsoidal symmetry of a multivariate distribution. In: Giné, E., Mason, D., Wellner, J. (Eds.), High Dimensional Probability II. In: Progress in Probability, Birkhäuser, Boston, pp. 493–510]. It is a class of omnibus bootstrap tests that are affine invariant and consistent against any fixed alternative. First, we study their behavior under a sequence of local alternatives. Secondly, a finite sample comparison study of this new class of tests with other popular methods given by Beran, Manzotti et al., and Huffer et al. is carried out. We find that the new tests outperform other methods in preserving the level and have superior power for the most of the chosen alternatives. We also suggest a tool for identifying periods of financial instability and crises when these tests are applied to the distribution of the return rates of stock market indices. These tests can be used in place of tests for normality of asset return distributions since ellipsoidally symmetric distributions are the natural extensions of multivariate normal distributions, so that the capital asset pricing model holds.