A fast algorithm for the exact likelihood of stationary and partially nonstationary vector autoregressive-moving average processes

An expression for the likelihood function of a stationary vectorautoregressive-moving average process is developed. The expressionis very efficient numerically and applies to any stationarybut not necessarily invertible model. In particular, when themultivariate process is autoregressive, the exact likelihoodcan be evaluated with a small number of operations dependingon the order of the autoregressive operator and the processdimension, but not on the size of the observed series. The expressionalso provides an efficient method for the evaluation of theexact likelihood of a partially nonstationary vector autoregressive-movingaverage process, for which the determinant of the autoregressiveoperator has at least one unit root and the remaining rootsare outside the unit circle. This method does not require differencingthe series, so that complications caused by over-differencingthe series, such as noninvertibility and parameter identifiabilityproblems, are avoided. The results for autoregressive modelsare also applied to testing the stationarity and invertibilityof any autoregressive-moving average model with given parametervalues.