Abstract: This paper presents the conditions under which the static function of a Multi-Input Single-Output (MISO) Non-linear AutoRegressive with eXogenous inputs (NARX) polynomial model can be written in rational or polynomial form. A particularly useful situation is when the static function is characterized by a set of polynomial functions. The existence and uniqueness of this set is shown to depend on conditions which are given in three lemmas. Based on these lemmas, a three step procedure for MISO Hammerstein model identification is presented. The procedure is illustrated identifying a Hammerstein model using data from an oil platform.