We present a matricial version of a generalized moment problem with degree constraint. We introduce a new metric on multivariable spectral densities induced by the family of their spectral factors which, in the scalar case, reduces to the Hellinger distance. We solve the corresponding constrained optimization problem via duality theory. A highly nontrivial existence theorem for the dual problem is established in the Byrnes-Lindquist spirit. A globally convergent  matricial Newton-type algorithm is provided for the numerical solution of the dual problem. This procedure is applied to multivariate spectral estimation. Simulations show that, in the case of short observation records, this method may provide a valid alternative to standard multivariable identification techniques such as MATLAB's PEM and MATLAB's N4SID.