  The non-linear autoregressive (NLAR) model plays an important role in
modeling and predicting time series. One-step ahead prediction is
straightforward using the NLAR model, but the multi-step ahead prediction is
cumbersome. For instance, iterating the one-step ahead predictor is a
convenient strategy for linear autoregressive (LAR) models, but it is
suboptimal under NLAR. In this paper, we first propose a simulation and/or
bootstrap algorithm to construct optimal point predictors under an $L_1$ or
$L_2$ loss criterion. In addition, we construct bootstrap prediction intervals
in the multi-step ahead prediction problem; in particular, we develop an
asymptotically valid quantile prediction interval as well as a pertinent
prediction interval for future values. In order to correct the undercoverage of
prediction intervals with finite samples, we further employ predictive -- as
opposed to fitted -- residuals in the bootstrap process. Simulation studies are
also given to substantiate the finite sample performance of our methods.
