  We use information from higher order moments to achieve identification of
non-Gaussian structural vector autoregressive moving average (SVARMA) models,
possibly non-fundamental or non-causal, through a frequency domain criterion
based on a new representation of the higher order spectral density arrays of
vector linear processes. This allows to identify the location of the roots of
the determinantal lag matrix polynomials based on higher order cumulants
dynamics and to identify the rotation of the model errors leading to the
structural shocks up to sign and permutation. We describe sufficient conditions
for global and local parameter identification that rely on simple rank
assumptions on the linear dynamics and on finite order serial and component
independence conditions for the structural innovations. We generalize previous
univariate analysis to develop asymptotically normal and efficient estimates
exploiting second and non-Gaussian higher order dynamics given a particular
structural shocks ordering without assumptions on causality or invertibility.
Bootstrap approximations to finite sample distributions and the properties of
numerical methods are explored with real and simulated data.
