  We describe our implementation of the multivariate Mat\'ern model for
multivariate spatial datasets, using Vecchia's approximation and a Fisher
scoring optimization algorithm. We consider various pararameterizations for the
multivariate Mat\'ern that have been proposed in the literature for ensuring
model validity, as well as an unconstrained model. A strength of our study is
that the code is tested on many real-world multivariate spatial datasets. We
use it to study the effect of ordering and conditioning in Vecchia's
approximation and the restrictions imposed by the various parameterizations. We
also consider a model in which co-located nuggets are correlated across
components and find that forcing this cross-component nugget correlation to be
zero can have a serious impact on the other model parameters, so we suggest
allowing cross-component correlation in co-located nugget terms.
