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Essays on testing conditional independence

Abstract

Conditional independence is of interest for testing unconfoundedness assumptions in causal inference, for selecting among semiparametric models, and for testing Granger noncausality, etc. This dissertation propose flexible tests for conditional independence, which are simple to implement yet powerful in the sense that they are consistent and achieve root n local power. In the literature, there are many tests available for the case in which the variables are categorical. But there are only a few nonparametric tests for the continuous case. On the other hand, in economics applications, it is common to condition on continuous variables. Chapter 1 provides a nonparametric test for continuous variables. The test statistic is a Wald type test based on an estimator of the topological "distance" between the restricted and unrestricted probability measures corresponding to conditional independence or its absence. The distance is evaluated using a family of Generically Comprehensively Revealing (GCR) functions indexed by a nuisance parameter vector. Although the test in chapter 1 is easy to calculate and has a tractable limiting null distribution, its consistency relies on the randomization of the choice of the nuisance parameters. In chapter 2, I obtain a Bierens type Integrated Conditional Moment test by integrating out the nuisance parameters. The test still achieves root n local power and its consistency does not rely on the randomization any more. Its limiting null distribution is a functional of a mean zero Gaussian process. I simulate the critical values by a conditional simulation approach. As an example of application, I test the key assumption of unconfoundedness in the context of estimating the returns to schooling. In applied microeconomics, many variables are categorical or binary. For example, in the returns-to-schooling example, the conditioning variables usually include a number of discrete variables such as sex, race, union or industry. However, in previous chapters I assume the conditioning variables to be continuous. In chapter 3, I extend the conditional independence tests to incorporate the case of mixed conditioning random variables, using the frequency approach and the smoothing approach.

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