  Flexible multivariate covariance models for spatial data are on demand. This
paper addresses the problem of parametric constraints for positive
semidefiniteness of the multivariate Mat{\'e}rn model. Much attention has been
given to the bivariate case, while highly multivariate cases have been explored
to a limited extent only. The existing conditions often imply severe
restrictions on the upper bounds for the collocated correlation coefficients,
which makes the multivariate Mat{\'e}rn model appealing for the case of weak
spatial cross-dependence only. We provide a collection of validity conditions
for the multivariate Mat{\'e}rn covariance model that allows for more flexible
parameterizations than those currently available. We also prove that, in
several cases, we can attain much higher upper bounds for the collocated
correlation coefficients in comparison with our competitors. We conclude with a
simple illustration on a trivariate geochemical dataset and show that our
enlarged parametric space allows for better fitting performance with respect to
our competitors.
