Bootstrap Prediction Inference of Non-linear Autoregressive Models

TL;DR

This paper develops bootstrap-based methods for optimal point prediction and valid multi-step prediction intervals in non-linear autoregressive models, addressing challenges in multi-step ahead forecasting.

Contribution

It introduces a bootstrap algorithm for constructing optimal predictors and asymptotically valid prediction intervals specifically for NLAR models, improving multi-step forecasting accuracy.

Findings

Bootstrap predictors outperform naive methods in simulations.

Constructed prediction intervals achieve nominal coverage asymptotically.

Finite sample performance is validated through simulation studies.

Abstract

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.