Asymptotic distributions for quasi-efficient estimators in echelon VARMA models

Two linear estimators for stationary invertible vector autoregressive moving average (VARMA) models in echelon form – to achieve parameter unicity (identification) – with known Kronecker indices are studied. It is shown that both estimators are consistent and asymptotically normal with strong innovations. The first estimator is a generalized-least-squares (GLS) version of the two-step least-squares estimator studied in Dufour and Jouini (2005). The second is an asymptotically efficient estimator which is computationally much simpler than the Gaussian maximum-likelihood (ML) estimator which requires highly nonlinear optimization, and “efficient linear estimators” proposed earlier (Hannan and Kavalieris, Adv. App. Prob., 1984, Reinsel, Basu and Yap, J. Time Series Anal., 1992, and Poskitt and Salau, J. Time Series Anal., 1995). It stands for a new relatively simple three-step estimator based on a linear regression involving innovation estimates which take into account the truncation error of the first-stage long autoregression. The complex dynamic structure of associated residuals is then exploited to derive an efficient covariance matrix estimator of the VARMA innovations, which is of order T−1 more accurate than the one by the fourth-stage of Hannan and Kavalieris’ procedure. Finally, finite-sample simulation evidence shows that, overall, the asymptotically efficient estimator suggested outperforms its competitors in terms of bias and mean squared errors (MSE) for the models studied.