Asymptotic distributions for some quasi-efficient estimators in echelon-form VARMA models

We study two linear estimators for stationary invertible VARMA models in echelon form (for identification), with known Kronecker indices. Such linear estimators are much simpler to compute than Gaussian maximum likelihood (ML) estimators often proposed for such models, which are highly nonlinear. The first estimator is an improved two-step estimator which can be interpreted as a generalized least squares (GLS) of the two-step least-squares estimator considered in Dufour and Jouini (2005), for a more general model which allows for the presence of drift parameters. The second estimator is a new relatively simple three-step linear estimator which is asymptotically equivalent to ML, hence efficient, when the innovations of the process are Gaussian. The proposed asymptotically efficient estimator is based on using modified approximate residuals which better take into account the truncation error associated with the approximate long autoregression used in the first step of the method. We show that both estimators are consistent and asymptotically normal under the assumption that the innovations are a strong white noise, possibly non-Gaussian. Explicit formulae for the asymptotic covariance matrices are provided. The proposed estimators make it relatively easy to estimate the parameters of VARMA models in echelon form, and the distributional theory does not rely on a Gaussian assumption, like maximum likelihood or the estimators considered by Hannan and Kavalieris (1984b) and Reinsel, Basu and Yap (1992). We present simulation evidence which indicate that the proposed three-step estimator typically performs better in finite samples than the alternative multi-step linear estimators suggested by Hannan and Kavalieris (1984b), Reinsel et al. (1992), and Poskitt and Salau (1995).