This paper tackles the problem of statistical inference in parametric models whose parameters are not identified. We focus on parametric models that are correctly specified, i.e. on sets of probability distributions to which we believe the true distribution belongs. This parametric model is a subset of all the possible probability distributions. Traditional inference approaches take as a starting point that the model is identified, i.e. that the distributions it generates are in a one-to-one mapping with the parameter set, assumed finite dimensional. As a consequence of this property, an identified parametric model usually forms a finite dimensional manifold embedded in the set of all possible probability distributions. This geometric structure of statistical models has given rise to a differential geometric approach to statistical inference pioneered in the work by Rao (1945), and which culminated in the books by Amari (1985), Amari, Barndorff-Nielsen, Kass, Lauritzen, and Rao (1987), Kass and Vos (1997), and Amari and Nagaoka (2000). Considering the problems of estimation and inference from a differential geometric viewpoint has lead to a number of important advances in statistics. In this paper, we show that the analytic tools provided by the differential-geometric theory can also be applied to the problem of inference in unidentified parametric models, problem which has occupied much of the recent work in econometrics.