  This paper investigates multistep prediction errors for non-stationary
autoregressive processes with both model order and true parameters unknown. We
give asymptotic expressions for the multistep mean squared prediction errors
and accumulated prediction errors of two important methods, plug-in and direct
prediction. These expressions not only characterize how the prediction errors
are influenced by the model orders, prediction methods, values of parameters
and unit roots, but also inspire us to construct some new predictor selection
criteria that can ultimately choose the best combination of the model order and
prediction method with probability 1. Finally, simulation analysis confirms the
satisfactory finite sample performance of the newly proposed criteria.
