Portmanteau Tests for ARMA Models with Infinite Variance

TL;DR

This paper develops portmanteau tests for diagnosing ARMA models with infinite variance stable Paretian errors, addressing a gap in model checking for such heavy-tailed time series.

Contribution

It introduces portmanteau tests tailored for ARMA models with infinite variance, including a Monte Carlo approach due to slow convergence of asymptotic distributions.

Findings

Proposed tests effectively detect model inadequacy in heavy-tailed data.

Monte Carlo method improves test accuracy for practical series lengths.

Using traditional portmanteau tests can lead to incorrect conclusions with infinite variance data.

Abstract

Autoregressive and moving-average (ARMA) models with stable Paretian errors is one of the most studied models for time series with infinite variance. Estimation methods for these models have been studied by many researchers but the problem of diagnostic checking fitted models has not been addressed. In this paper, we develop portmanteau tests for checking randomness of a time series with infinite variance and as a diagnostic tool for checking model adequacy of fitted ARMA models. It is assumed that least-squares or an asymptotically equivalent estimation method, such as Gaussian maximum likelihood in the case of AR models, is used. And it is assumed that the distribution of the innovations is IID stable Paretian. It is seen via simulation that the proposed portmanteau tests do not converge well to the corresponding limiting distributions for practical series length so a Monte-Carlo test is suggested. Simulation experiments show that the proposed test procedure works effectively. Two illustrative applications to actual data are provided to demonstrate that an incorrect conclusion may result if the usual portmanteau test based on the finite variance assumption is used.