On Completeness and Consistency in Nonparametric Instrumental Variable Models

Abstract

This paper provides a first test for the identification condition in a nonparametric instrumental variable model, known as completeness, by linking the outcome of the test to consistency of an estimator. In particular, I show that uniformly over all distributions for which the test rejects with probability bounded away from 0, an estimator of the structural function is consistent. This is the case for a large class of complete distributions as well as certain sequences of incomplete distributions. As a byproduct of this result, the paper makes two additional contributions. First, I present a definition of weak instruments in the nonparametric instrumental variable model, which is equivalent to the failure of a restricted version of completeness. Second, I show that the null hypothesis of weak instruments, and thus failure of a restricted version of completeness, is testable and I provide a test statistic and a bootstrap procedure to obtain the critical values. Finally, I demonstrate the finite sample properties of the tests and the estimator in Monte Carlo simulations.