The subspace identification methods (SIMs), such as past outputs multivariable output-error state space (PO-MOESP), numerical algorithms for subspace state-space system identification, and canonic correlation analysis, build state-space models from a set of input-output data of a linear time invariant system. In these algorithms, the key step is the computation of the extended observability matrix, which is done via an orthogonal projection between spaces generated by the input-output data. However, the PO-MOESP algorithm, unlike the others subspace methods, uses a different space for computing such projection. This fact hinders the analysis and direct comparison with other algorithms. In this paper, we introduce a modified version of the PO-MOESP algorithm, which is conceptually simpler, requires fewer steps for computing the model parameters, and enables the comparison with other SIMs. Three numerical examples are provided in order to show the effectiveness of the proposed approach.