Moment Bounds and Multistep Prediction of Linear Processes

Abstract

A uniform moment bound of the inverse Fisher's information matrix of a general linear process is established in this article. This bound is applied to derive the moment convergence of the normalized least squares estimate of the underlying linear process, which also includes the long-memory case. Based on this bound, an asymptotic expression for the corresponding mean squared prediction error (MSPE) is obtained. An important and intriguing application considered in this article is to establish and compare the asymptotic expressions of the multistep MSPEs of the least squares predictors for ARMA, I(d) and ARFIMA models. These asymptotic expressions not only offer means to assess the multistep prediction errors, but also explicitly demonstrate how the multistep MSPE manifests with the model complexity and the dependent structure of the underlying process, thereby shedding light about multistep prediction for general linear processes. Numerical findings are also conducted to verify the theoretical results. ? Key words and phrases: ARFIMA model, Fisher's information matrix, least squares estimates, long-memory process, mean squared prediction error, uniform moment bounds.