  We study the anisotropic nature of the Kerr nonlinear response in a
beta-barium borate (\beta-BaB2O4, BBO) nonlinear crystal. The focus is on
determining the relevant $\chi^{(3)}$ cubic tensor components that affect
interaction of type I cascaded second-harmonic generation. Various experiments
in the literature are analyzed and we correct the data from some of the
experiments for contributions from cascading as well as for updated material
parameters. We find that the Kerr nonlinear tensor component responsible for
self-phase modulation in cascading is considerably larger than what has been
used to date. We evaluate the impact of using such a cubic anisotropic response
in ultrafast cascading experiments.
