A rigorous analysis is given of the asymptotic bias of the log maximum likelihood as an estimate of the expected log likelihood (the negative of the cross-entropy) of the maximum likelihood model, when an invertible, conditional, multivariate gaussian ARMA(p,q) model, with or without coefficient/innovations covariance constraints, is fit to stationary, possibly non-gaussian observations. It is assumed that these data either (i) arise from a model whose spectral density matrix coincides with that of a member of the class of models being fit, or (ii) do not conform to any ARMA model but do come from a process whose spectral density matrix can be well-approximated by invertible ARMA model spectral density matrix functions. For the gaussian sub-case of (i), the innovations covariance matrices of the models need not be parametrized separately from the coefficients, but otherwise a separate parametrization is assumed. The analysis shows that, for the purpose of comparing maximum likelihood models from different model classes, Akaike's AIC is asymptotically unbiased in case (i). In case (ii), its asymptotic bias is of the order of a number less than unity raised to the power max{p,q} and so is negligible if max{p,q} is not too small. These results extend and complete the somewhat heuristic analysis given by Ogata (1980) for exact or approximating univariate auto-regressive models.