This thesis consists of three chapters, each contributing to a different aspect of econometrics inference using uniform asymptotics. The three chapters are respectively entitled Comparing Nonnested Moment Inequality Models, Size Distortion and Modification of Classical Vuong Tests, and Inference Based on Conditional Moment Inequalities.

In Comparing Nonnested Moment Inequality Models, I construct tests to statistically compare two nonnested partially-identified moment inequality models. The tests are the first that make the statistical comparison of such models possible. They evaluate the relative fits of the models according to the familiar likelihood criterion, building on the pioneering work by Vuong (1989) and Kitamura (2000). Using the uniform asymptotic theory, I show that the tests have correct asymptotic size and are consistent against all fixed alternatives. The uniform size control is important in this context because the pointwise asymptotic distribution of the log likelihood ratio statistic changes abruptly for continuous changes of the null distribution. Such abrupt changes deviate greatly from the finite sample behavior of the statistic. Uniform asymptotic approximation of the finite sample distribution of the statistic is needed to generate a valid critical value. The derivation of the uniform asymptotic approximation is made difficult by both partial identification and the parameter-on-boundary problem caused by the inequalities. I partially solve the problem by deriving quadratic bounds for the criterion functions using the Kuhn-Tucker conditions.

A complete uniform approximation for the moment inequality models is not tractable for the reasons mentioned above. But it is possible for point-identified models that do not involve inequality constraints. I study such models in the second chapter.

In Size Distortion and Modification of Classical Vuong Tests, I derive a uniform asymptotic approximation for the distribution of the classical Vuong test statistic. The new approximation reveals a serious size-distortion of the classical Vuong tests. Algebraic manipulation of the new approximation reveals that the distortion is caused by a O (1/ n) bias in the log-likelihood ratio statistic. I correct the bias and propose a modified test that has correct asymptotic size. The modified test is a one-step test, is easy to implement, and has good finite sample power property. I also extend the size-distortion analysis and the modification to the comparison of moment based models using the generalized empirical likelihood criteria. The findings in this chapter contribute to the rigorous comparison of models in applied research.

Inference Based on Conditional Moment Inequalities is jointly authored with Donald Andrews. In this chapter, we construct confidence sets for the parameters defined by partially-identified conditional moment inequality models. Conditional moment inequality models in practice are far more common than unconditional moment models (with finite moment conditions). They are also more difficult to deal with because one conditional moment implies an uncountable number of unconditional moments. Under appropriate choices of instrument functions, conditional moments are transformed in our approach into equivalent unconditional ones without information loss. We show that our confidence sets that rely on specially designed data-dependent critical values have uniformly correct asymptotic coverage probability. The uniformity result is obtained using a novel approach that allows the nuisance parameters to be infinite dimensional. In the new approach, instead of establishing an asymptotic distribution for the statistic considered, we provide a uniform approximation to its distribution for a given sample size n by the distribution of a functional of a Gaussian process that also depends on n. In spite of the infinite-dimensional character of the testing problem, we show that the tests have power against some n^-1/2 local alternatives.