Incorporating Model Uncertainty in the Construction of Bootstrap Prediction Intervals for Functional Time Series

Abstract

A sieve bootstrap method that incorporates model uncertainty for constructing pointwise or simultaneous prediction intervals of stationary functional time series is proposed. The bootstrap method exploits a general backward vector autoregressive representation of the time series of Fourier coefficients appearing in the well-established Karhunen-Loève expansion of the functional process. The bootstrap method generates, by running backward in time, functional bootstrap samples which adequately mimic the dependence structure of the underlying process and which all have the same conditionally fixed curves at the end of every functional bootstrap sample. The bootstrap prediction error distribution is then calculated as the difference between the model-free bootstrap generated future functional pseudo-observations and the functional forecasts obtained from a model used for prediction. In this way, the estimated prediction error distribution takes into account not only the innovation and estimation error associated with prediction, but also the possible error due to model uncertainty or misspecification. Through a simulation study, we demonstrate an excellent finite-sample performance of the proposed sieve bootstrap method.