On the Identifiability of Additive Index Models

In this paper, we investigate the identifiability of the additive index model, also known as projection pursuit regression. Although a flexible regression tool, additive index models can be hard to interpret in practice due to a lack of identifiability. As noted by Horowitz (1998), “it is an open question whether there are identifying restrictions that yield useful forms”, in reference to additive index models that differ from the known structured nonparametric regression models such as additive models and single index models. We provide an affirmative answer to this question: the additive index model is identifiable as long as the projection indices are linearly independent and there is at most one quadratic ridge function. Furthermore, we show that when there are multiple quadratic ridge functions, the identifiability can still be ensured for the non-quadratic ridge functions and their corresponding projection indices, whereas it is not possible to identify the quadratic ridge functions. Such an identifiability result enables us to check if a more restrictive nonparametric model such as the additive model can be adopted as opposed to the more general additive index model.