Bootstrap Prediction Intervals for Autoregression

Abstract

The nonparametric bootstrap is applied to the problem of prediction in autoregression. Let { Y t : t = 0, ±1, ±2, …} be a stationary autoregressive process of known order p [AR( p )]. Given a realization of the series up to time t , ( y 1 , y 2 , …., y t ), a 100 β % prediction interval for Y t+k is desired. Standard forecasting techniques, which assume that the error sequence of the process { Y t } is Gaussian, rely on the fact that the conditional distribution of Y t+k , given the data, is Gaussian as well. As a nonparametric alternative, the bootstrap provides an estimate of the conditional distribution of Y t+k . The method is similar to other applications of the bootstrap for linear models, because the residuals are resampled. The proposed methodology represents a different approach, since an alternative representation for AR( p ) series is used, allowing for bootstrap replicates generated backward in time. It follows that the resulting replicates all have the same conditionally fixed values at the end of every series. A simulation that compares the proposed technique with the standard technique for low-order Gaussian and non-Gaussian autoregressive models demonstrates the potential of the bootstrap technique.