This paper provides a polynomial factorization theorem that is then used to extend the characterization parts of the parametric representation theorems of Johansen (1992, Econometric Theory 8, 188–202) for vector autoregressive processes integrated of up to order 2. A characterization theorem is provided in the general case of an I(d) process. For the discussion of the complicated polynomial cointegration properties of such processes, the case of an I(3) process is considered as an example.