Multi-Step-Ahead Prediction Intervals for Nonparametric Autoregressions via Bootstrap: Consistency, Debiasing, and Pertinence

To address the difficult problem of the multi-step-ahead prediction of nonparametric autoregressions, we consider a forward bootstrap approach. Employing a local constant estimator, we can analyze a general type of nonparametric time-series model and show that the proposed point predictions are consistent with the true optimal predictor. We construct a quantile prediction interval that is asymptotically valid. Moreover, using a debiasing technique, we can asymptotically approximate the distribution of multi-step-ahead nonparametric estimation by the bootstrap. As a result, we can build bootstrap prediction intervals that are pertinent, i.e., can capture the model estimation variability, thus improving the standard quantile prediction intervals. Simulation studies are presented to illustrate the performance of our point predictions and pertinent prediction intervals for finite samples.