# Bootstrap Prediction Inference of Non-linear Autoregressive Models

arXiv: https://arxiv.org/abs/2306.04126 · HTML: https://mlsanity.com/paper/2306.04126/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2023.

## Built on by (most cited first, 5 of 5)

- [Scalable Subsampling Inference for Deep Neural Networks](https://mlsanity.com/paper/2405.08276/) (2024)
- [Uncertainty quantification in metric spaces](https://mlsanity.com/paper/2405.05110/) (2024)
- [Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces](https://mlsanity.com/paper/2507.15741/) (2025)
- [ProbRes: Volatility Learning for Probabilistic Time-Series Forecasting](https://mlsanity.com/paper/2606.02117/) (2026)
- [Prediction Inference of Time Series with Standard ReLU Deep Neural Networks](https://mlsanity.com/paper/2608.15362/) (2026)

## Abstract (arXiv)

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 L1 or L2 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.