This dissertation studied two main topics: inference methods for directionally differentiable functions with first order degeneracy and estimation methods for semiparametric instrumental variables (IV) models with partial identication. Chapter 1 presents a general asymptotic framework for conducting inference on directionally differentiable functions with a zero first order derivative. Chapter 2 applies the theory to develop improved inference methods for the rank of a matrix, which is important in estimation and inference IV models.

Chapter 3 studies robust and optimal estimation in a canonical semiparametric IV model --- partially linear IV model --- with nonparametric partial identification.