The exact likelihood function for a mixed autoregressive-moving average process

Box & Jenkins (1970) develop a strategy for the analysis and forecasting of time series, based on the construction, after suitable differencing, of autoregressive-moving average models. They present a model building procedure which works very well for most general purposes, and in addition derive the exact likelihood function for both pure autoregressive and pure moving average processes, in the latter case this being a new result. In this note it is shown that a similar approach can be employed to derive the exact likelihood function for the mixed process.