Effect of Preliminary Unit Root Tests on Prediction Intervals for Gaussian Autoregressive Processes with Additive Outliers

The preliminary unit root test has been found to be a useful tool for improving the accuracy of a one-step-ahead predictor and prediction interval for the first-order autoregressive process when an autoregressive coefficient is close to one. This paper applies the aforementioned concepts of the preliminary unit root test in order to improve the efficiency of prediction intervals for the Gaussian autoregressive processes with additive outliers. The preliminary unit root tests considered are the augmented Dickey-Fuller (ADF) test and the Shin et al.’s (SSL) test. In addition, the analytic expressions of the coverage probability of prediction intervals are derived, and the structure of the coverage probability was proved to be independent from the mean of the process and the parameter of the innovation, it is a function of the autoregressive coefficients only. For the parameter estimation of processes we use the generalized M-estimates. The coverage probabilities and the widths of the standard prediction interval, the prediction interval following the ADF test, and the prediction interval following the SSL test are also compared via simulation studies. Simulation results have shown that the SSL test can help to improve the accuracy of the prediction intervals with additive outliers, especially when the sample size is large. The performance of the proposed prediction intervals is illustrated with an empirical application.