Bootstrap-after-bootstrap prediction intervals for autoregressive models

The use of the Bonferroni prediction interval based on the bootstrap-after-bootstrap is proposed for autoregressive (AR) models. Monte Carlo simulations are conducted using a number of AR models including stationary, unit-root, and near-unit-root processes. The major finding is that the bootstrap-after-bootstrap provides a superior small-sample alternative to asymptotic and standard bootstrap prediction intervals. The latter are often too narrow, substantially underestimating future uncertainty, especially when the model has unit roots or near unit roots. Bootstrap-after-bootstrap prediction intervals are found to provide accurate and conservative assessment of future uncertainty under nearly all circumstances considered.