Moment bounds and mean squared prediction errors of long-memory time series

TL;DR

This paper derives moment bounds for the CSS estimator in ARFIMA models with unknown memory parameters and analyzes the asymptotic behavior of mean squared prediction errors, highlighting advantages over LS predictors in high integration scenarios.

Contribution

It provides the first uniform moment bounds for the inverse normalized objective function in ARFIMA models with unknown memory parameters and derives asymptotic MSPE expressions for the CSS predictor.

Findings

CSS predictor outperforms LS predictor in high integration cases.

Asymptotic MSPE expressions reveal the impact of model complexity and dependence.

Numerical results support theoretical findings.

Abstract

A moment bound for the normalized conditional-sum-of-squares (CSS) estimate of a general autoregressive fractionally integrated moving average (ARFIMA) model with an arbitrary unknown memory parameter is derived in this paper. To achieve this goal, a uniform moment bound for the inverse of the normalized objective function is established. An important application of these results is to establish asymptotic expressions for the one-step and multi-step mean squared prediction errors (MSPE) of the CSS predictor. These asymptotic expressions not only explicitly demonstrate how the multi-step MSPE of the CSS predictor manifests with the model complexity and the dependent structure, but also offer means to compare the performance of the CSS predictor with the least squares (LS) predictor for integrated autoregressive models. It turns out that the CSS predictor can gain substantial advantage over the LS predictor when the integration order is high. Numerical findings are also conducted to illustrate the theoretical results.