Tests for sphericity in multivariate garch models

Tests for spherical symmetry of the innovation distribution are proposed in multivariate GARCH models. The new tests are of Kolmogorov--Smirnov and Cram\'er--von Mises--type and make use of the common geometry underlying the characteristic function of any spherically symmetric distribution. The asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions. It is shown that both the finite sample and the asymptotic null distribution depend on the unknown distribution of the Euclidean norm of the innovations. Therefore a conditional Monte Carlo procedure is used to actually carry out the tests. The validity of this resampling scheme is formally justified. Results on the behavior of the test in finite--samples are included, as well as an application on financial data.