This paper addresses the problem of identifying echelon canonical forms for a vector autoregressive moving average model with exogenous variables using finite algorithms. For given values of the Kronecker indices a method for estimating the structural parameters of a model using ordinary least squares calculations is presented. These procedures give rise, rather naturally, to a technique for the determination of the structural indices based on the use of conventional model selection criteria. A detailed analysis of the statistical properties of the estimation and identification procedures is given and some evidence on the practical significance of the results obtained is also provided. Modifications designed to improve the performance of the methods are presented. Some discussion of the practical significance of the results obtained is also provided.ARMAX model, consistency, echelon canonical form, efficiency, estimation, identification, Kronecker invariants, least squares, selection criterion, structure determination, subspace algorithm.