COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1801

Examples of L²-Complete and Boundedly-Complete Distributions

Donald W.K. Andrews

May 2011

Completeness and bounded-completeness conditions are used increasingly in econometrics to obtain nonparametric identification in a variety of models from nonparametric instrumental variable regression to non-classical measurement error models. However, distributions that are known to be complete or boundedly complete are somewhat scarce.

In this paper, we consider an L²-completeness condition that lies between completeness and bounded completeness. We construct broad (nonparametric) classes of distributions that are L²-complete and boundedly complete. The distributions can have any marginal distributions and a wide range of strengths of dependence. Examples of L²-incomplete distributions also are provided.

Keywords: Bivariate distribution, Bounded completeness, Canonical correlation, Completeness, Identification, Measurement error, Nonparametric instrumental variable regression

JEL Classification: C14