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 100β% 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.

Key Words:

• Forecasting
• Non-Gaussian time series
• Resampling