Estimation of the variance of the quasi-maximum likelihood estimator of weak VARMA models

This paper considers the problems of computing and estimating the asymptotic variance matrix of the least squares (LS) and/or the quasi-maximum likelihood (QML) estimators of vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. We firstly give expressions for the derivatives of the VARMA residuals in terms of the parameters of the models. Secondly we give an explicit expression of the asymptotic variance matrix of the QML/LS estimator, in terms of the VAR and MA polynomials, and of the second and fourth-order structure of the noise. We then deduce a consistent estimator of this asymptotic variance matrix. Modified versions of the Wald, Lagrange Multiplier and Likelihood Ratio tests are proposed for testing linear restrictions on the parameters. The theoretical results are illustrated by means Monte Carlo experiments.