Essays in hypothesis testing with instrumental variables

This dissertation develops methods for hypothesis testing in a nonparametric instrumental variables (IV) setting within a partial identification framework. In the first chapter I construct and derive the asymptotic distribution of a test statistic for the hypothesis that at least one element of the identified set satisfies a conjectured restriction. This procedure can be used to test for features of the model that may be identified even when the true model is not. This framework can also be employed to construct confidence regions for functionals of the elements of the identified set, such as consumer surplus and price elasticity of demand at a point. I apply this procedure to study Engel curves for gasoline and ethanol in Brazil and construct confidence regions for their level and slope at the sample average.; In the second chapter I construct an estimator for the coefficients in the true infinite series expansion of g(x) in the model Y = g(X)+epsilon. The estimation proceeds by transforming an instrument so that it is orthogonal to all elements in the expansion of g(x) except the term whose coefficient we wish to estimate. There exist multiple transformations that will work and for this reason we develop techniques that allow for the estimation of the whole set of valid transformations. I then show that by using this estimated set, it is possible to obtain a consistent estimator for a unique valid transformation. Such transformation is then employed to construct an estimator for the desired coefficient. Consistency and the asymptotic distribution of this estimator are established.; The third chapter (joint work with Azeem Shaikh) explores whether the empirical likelihood ratio test for testing moment restrictions is Hoeffding optimal. We show that without restricting the space of distributions empirical likelihood fails to be Hoeffding Optimal. We also make progress towards establishing optimality for a restricted set of distributions with a pointwise result that establishes the empirical likelihood ratio test can control the rate at which the probability of a Type I error vanishes.