Bootstrap-based ARMA order selection

Abstract

Modelling the underlying stochastic process is one of the main goals in the study of many dynamic phenomena, such as signal processing, system identification and time series. The issue is often addressed within the framework of ARMA (Autoregressive Moving Average) paradigm, so that the related task of identification of the ‘true’ order is crucial. As it is well known, the effectiveness of such an approach may be seriously compromised by misspecification errors since they may affect model capabilities in capturing dynamic structures of the process. As a result, inference and empirical outcomes may be heavily misleading. Despite the big number of available approaches aimed at determining the order of an ARMA model, the issue is still open. In this paper, we bring the problem in the framework of bootstrap theory in conjunction with the information-based criterion of Akaike (AIC), and a new method for ARMA model selection will be presented. A theoretical justification for the proposed approach as well as an evaluation of its small sample performances, via simulation study, are given.

Keywords:

• AIC
• ARMA processes
• order selection
• SIEVE bootstrap

Notes

As pointed out by Chatfield 3, such an automated, computer-assisted approach may lead to a danger for the analyst in that he ‘will choose a model to fit the software rather than vice versa’.

Akaike 11 showed that K is an asymptotic approximation of the bias term. however, the validity of this approximation holds under the condition that at least one model, in the set of candidate models, can be a good approximation of the ‘true’ model.

In particular, we made use of the computer EULER (maintained by the Mathematical Department) and the supercomputer IBM-TERAGRID.