The cointegrated vector autoregressive model for I(2) variables is a non-linear parametric restriction on the linear I(2) regression model for variables of order I(0), I(1) and I(2). In this paper we discuss non-linear submodels given by smooth parametrizations. We give conditions on the parametrization which imply that the limit under local alternatives of the log likelihood ratio is quadratic, and show that the asymptotic distribution of the maximum likelihood estimator can be found by optimizing the limit function. This gives a reformulation of a condition by Boswijk (2000) and the reformulation is applied to show that some hypotheses on the cointegrating coefficients in the cointegrated I(2) model give asymptotic ¿² inference.