Prediction intervals for the Gaussian autoregressive processes following the unit root tests

Abstract

Recently, Diebold and Kilian (9), Niwitpong (21,22), and Niwitpong and Panichkitkosolkul (23) indicated that the preliminary unit root test was a useful tool for improving the accuracy of a one-step-ahead predictor and prediction interval for an AR(1) process. This paper extends these mentioned concepts to the prediction interval for the Gaussian autoregressive processes. We propose the methods to construct the simple prediction interval based on the residual model, PIa, and the prediction intervals following the unit root tests, PIfi. The unit root tests applied in this paper consisted of the augmented Dickey-Fuller test, the Phillips-Perron test, and the Elliott-Rothenberg-Stock test. In addition, an expression of the coverage probability is derived and we found that the structure of the coverage probability is independent from the parameter of a random error, but it is a function of the autoregressive parameters only. The coverage probability and average width of prediction intervals are compared through Monte Carlo simulation studies. Simulation results have shown that the proposed prediction intervals have minimum coverage probabilities 0.95 for almost all situations. Furthermore, the average widths of prediction intervals PIfi are shorter than that of a prediction interval PIa when the first-order autoregressive parameter value approaches one. A comparison of the proposed methods is also illustrated by using an empirical application.