Multistep Predictors for Unstable Autoregressive Processes

Abstract

Previous analysis on forecasting theory either assumes knowing the true parameters or assumes the stationarity of the series. Not much is known about the theory of forecasting for nosntationary models with estimated parameters. To fill this gap, this paper investigates multistep forecast errors for autoregressive (AR) processes with unit roots. We first give asymptotic expressions for the multistep versions of final prediction error (MFPE) and accumulated prediction error (MAPE) based on two different prediction/ estimation methods. It is shown that the term of order 1/n in MFPE and the term of order log n in MAPE share the same constant, where n is the sample size. This special feature enables us to construct a predictor selection criterion that can choose the best combination of the prediction model and prediction method with probability tending to 1. Simulation results are presented to illustrate this theoretical finding (joint work with Jin-Lung Lin).