Diagnostic Checking in Multivariate ARMA Models With Dependent Errors Using Normalized Residual Autocorrelations

TL;DR

This paper develops new diagnostic tools for multivariate ARMA models with dependent errors, providing their asymptotic distributions and demonstrating their effectiveness through simulations and real data application.

Contribution

It introduces self-normalized portmanteau statistics for VARMA models with dependent innovations, deriving their asymptotic distributions under weak assumptions.

Findings

New portmanteau statistics for VARMA models with dependent errors.

Asymptotic distributions differ from classical chi-squared approximations.

Effective in simulations and real-world financial data.

Abstract

In this paper we derive the asymptotic distribution of normalized residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We propose new portmanteau statistics for vector autoregressive moving-average (VARMA) models with uncorrelated but non-independent innovations by using a self-normalization approach. We establish the asymptotic distribution of the proposed statistics. This asymptotic distribution is quite different from the usual chi-squared approximation used under the independent and identically distributed assumption on the noise, or the weighted sum of independent chi-squared random variables obtained under nonindependent innovations. A set of Monte Carlo experiments and an application to the daily returns of the CAC40 is presented.