# Three essays on hypotheses testing involving inequality constraints

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The focus of this research is on hypotheses testing involving inequality constraints. In the first chapter of this dissertation, we propose Kolmogorov-Smirnov type tests for stochastic dominance relations between the potential outcomes of a binary treatment under the unconfoundedness assumption. Our stochastic dominance tests compare every point of the cumulative distribution functions (CDF), so they can fully utilize all information in the distributions. For first order stochastic dominance, the test statistic is defined as the supremum of the difference of two inverse-probability-weighting estimators for the CDFs of the potential outcomes. The critical values are approximated based on a simulation method. We show that our test has good size properties and is consistent in the sense that it can detect any violation of the null hypothesis asymptotically. First order stochastic dominance tests in the treated subpopulation, and higher order stochastic dominance tests in the whole population and among the treated are shown to share the same properties. The tests are applied to evaluate the effect of a job training program on incomes, and we find that job training has a positive effect on real earnings. Finally, we extend our tests to cases in which the unconfoundedness assumption does not hold. On the other hand, there has been a considerable amount of attention paid to testing inequality restrictions using Wald type tests. As noted by Wolak (1991), there are certain situations where it is difficult to obtain tests with correct size even asymptotically. These situations occur when the variance-covariance matrix of the functions in the constraints depends on the unknown parameters as would be the case in nonlinear models. This dependence on the unknown parameters makes it computationally difficult to find the least favorable configuration (LFC) which can be used to bound the size of the test. In the second chapter of this dissertation, we extend Hansen's (2005) superior predictive ability (SPA) test to testing hypotheses involving general inequality constraints in which the variance-covariance matrix can be dependent on the unknown parameters. For our test we are able to obtain correct size asymptotically plus test consistency without requiring knowledge of the LFC. Also the test can be applied to a wider class of problems than considered in Wolak (1991). In the last chapter, we construct new Kolmogorov-Smirnov tests for stochastic dominance of any pre-specified order without resorting to the LFC to improve the power of Barrett and Donald's (2003) tests. To do this, we first show that under the null hypothesis if the objects being compared at a given income level are not equal, then the objects at this given income level will have no effect on the null distribution. Second, we extend Hansen's (2005) recentering method to a continuum of inequality constraints and construct a recentering function that will converge to the underlying parameter function uniformly asymptotically under the null hypothesis. We treat the recentering function as a true underlying parameter function and add it to the simulated Brownian bridge processes to simulate the critical values. We show that our tests can control the size asymptotically and are consistent. We also show that by avoiding the LFC, our tests are less conservative and more powerful than Barrett and Donald's (2003). Monte Carlo simulations support our results. We also examine the performances of our tests in an empirical example.