Diagnostic checking in FARIMA models with uncorrelated but non-independent error terms

TL;DR

This paper develops modified portmanteau tests for FARIMA models assuming errors are uncorrelated but not independent, providing new asymptotic distributions and a self-normalization method, with simulations and real data applications.

Contribution

It introduces new asymptotic distributions for residual autocorrelations in weak FARIMA models and proposes a self-normalization test for model adequacy.

Findings

Modified portmanteau tests are effective for weak FARIMA models.

Asymptotic distributions are derived for residual autocorrelations.

Simulation studies support the theoretical results and practical application.

Abstract

This work considers the problem of modified portmanteau tests for testing the adequacy of FARIMA models under the assumption that the errors are uncorrelated but not necessarily independent (i.e. weak FARIMA). We first study the joint distribution of the least squares estimator and the noise empirical autocovariances. We then derive the asymp-totic distribution of residual empirical autocovariances and autocorrelations. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) modified portmanteau statistics for weak FARIMA models. We also propose another method based on a self-normalization approach to test the adequacy of FARIMA models. Finally some simulation studies are presented to corroborate our theoretical work. An application to the Standard \& Poor's 500 and Nikkei returns also illustrate the practical relevance of our theoretical results. AMS 2000 subject classifications: Primary 62M10, 62F03, 62F05; secondary 91B84, 62P05.