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.

翻译：非线性自回归（NLAR）模型在时间序列建模与预测中发挥着重要作用。使用NLAR模型进行单步向前预测较为直接，但多步向前预测则较为复杂。例如，对线性自回归（LAR）模型而言，迭代单步向前预测器是一种便捷策略，但在NLAR框架下该方法并非最优。本文首先提出一种模拟和/或引导算法，用于构建基于$L_1$或$L_2$损失准则的最优点预测器。此外，我们针对多步向前预测问题构建了引导预测区间；具体而言，我们开发了渐近有效的分位数预测区间以及适用于未来值的相关预测区间。为修正有限样本下预测区间覆盖不足的问题，我们在引导过程中进一步采用预测残差（而非拟合残差）。最后通过仿真研究验证了所提方法在有限样本下的表现。