In this paper, two competing types of multistep predictors, i.e.,
plug-in and direct predictors, are considered in autoregressive (AR)
processes. When a working model AR(k) is used for the
h-step prediction with h > 1, the plug-in
predictor is obtained from repeatedly using the fitted (by least
squares) AR(k) model with an unknown future value replaced by
their own forecasts, and the direct predictor is obtained by estimating
the h-step prediction model's coefficients directly by
linear least squares. Under rather mild conditions, asymptotic
expressions for the mean-squared prediction errors (MSPEs) of these two
predictors are obtained in stationary cases. In addition, we also
extend these results to models with deterministic time trends. Based on
these expressions, performances of the plug-in and direct predictors
are compared. Finally, two examples are given to illustrate that some
stationary case results on these MSPEs can not be generalized to the
nonstationary case.The author is deeply
grateful to the co-editor Pentti Saikkonen and two referees for their
helpful suggestions and comments on a previous version of this
paper.