Prediction Interval For Autoregressive Time Series Via Oracally Efficient Estimation Of Multi-Step Ahead Innovation Distribution Function

Kernel distribution estimator(KDE) is proposed for multi-step ahead prediction error distribution of autoregressive time series, based on prediction residuals. Under general assumptions, the KDE is proved to be oracally efficient as the infeasible KDE and the empirical cdf based on unobserved prediction errors. Quantile estimator is obtained from the oracally efficient KDE and prediction interval for multi-step ahead future observation is constructed using the estimated quantiles and shown to achieve asymptotically the nominal confidence levels. Simulation examples corroborate the asymptotic theory.