From general state-space to VARMAX models

We propose two new algorithms to go from any state-space model to an output equivalent and invertible Vector AutoRegressive Moving Average model with eXogenous regressors (VARMAX). As the literature shows how to do the inverse transformation, these results imply that both representations, statespace and VARMAX, are equally general and freely interchangeable. These algorithms are useful to solve three practical problems: (i) discussing the identifiability of a state-space model, (ii) performing its diagnostic checking, and (iii) calibrating its parameters so that it realizes, exactly or approximately, a given reduced-form VARMAX. These applications are illustrated by means of practical examples with real data.