An algorithm for the exact likelihood of a stationary vector autoregressive-moving average model

The so-called innovations form of the likelihood function implied by a stationary vector autoregressive-moving average model is considered without directly using a state-space representation. Specifically, it is shown in detail how to compute the exact likelihood by an adaptation to the multivariate case of the innovations algorithm of Ansley (1979) for univariate models. Comparisons with other existing methods are also provided, showing that the algorithm described here is computationally more efficient than the fastest methods currently available in many cases of practical interest.