Essays on inference in econometric models

This thesis contains three essays on inference in econometric models. Chapter 1 considers the question of bootstrap inference for Propensity Score Matching. Propensity Score Matching, where the propensity scores are estimated in a first step, is widely used for estimating treatment effects. In this context, the naive bootstrap is invalid (Abadie and Imbens, 2008). This chapter proposes a novel bootstrap procedure for this context, and demonstrates its consistency. Simulations and real data examples demonstrate the superior performance of the proposed method relative to using the asymptotic distribution for inference, especially when the degree of overlap in propensity scores is poor. General versions of the procedure can also be applied to other causal effect estimators such as inverse probability weighting and propensity score subclassification, potentially leading to higher order refinements for inference in such contexts. Chapter 2 tackles the question of inference in incomplete econometric models. In many economic and statistical applications, the observed data take the form of sets rather than points. Examples include bracket data in survey analysis, tumor growth and rock grain images in morphology analysis, and noisy measurements on the support function of a convex set in medical imaging and robotic vision. Additionally, nonparametric bounds on treatment effects under imperfect compliance can be expressed by means of random sets. This chapter develops a concept of nonparametric likelihood for random sets and its mean, known as the Aumann expectation, and proposes general inference methods by adapting the theory of empirical likelihood. Chapter 3 considers inference on the cumulative distribution function (CDF) in the classical measurement error model. It proposes both asymptotic and bootstrap based uniform confidence bands for the estimator of the CDF under measurement error. The proposed techniques can also be used to obtain confidence bands for quantiles, and perform various CDF-based tests such as goodness-offit tests for parametric models of densities, two sample homogeneity tests, and tests for stochastic dominance; all for the first time under measurement error