  Multivariate space-time data are increasingly available in various scientific
disciplines. When analyzing these data, one of the key issues is to describe
the multivariate space-time dependencies. Under the Gaussian framework, one
needs to propose relevant models for multivariate space-time covariance
functions, i.e. matrix-valued mappings with the additional requirement of
non-negative definiteness. We propose a flexible parametric class of
cross-covariance functions for multivariate space-time Gaussian random fields.
Space-time components belong to the (univariate) Gneiting class of space-time
covariance functions, with Mat\'ern or Cauchy covariance functions in the
spatial margins. The smoothness and scale parameters can be different for each
variable. We provide sufficient conditions for positive definiteness. A
simulation study shows that the parameters of this model can be efficiently
estimated using weighted pairwise likelihood, which belongs to the class of
composite likelihood methods. We then illustrate the model on a French dataset
of weather variables.
