Identification and well-posedness in nonparametric models with independence conditions

TL;DR

This paper analyzes nonparametric models with independence conditions, focusing on existence, identification, and well-posedness of solutions in generalized function spaces, broadening applicability beyond traditional density assumptions.

Contribution

It introduces a nonparametric framework using generalized functions to analyze models like measurement error and factor models, relaxing density requirements and establishing conditions for solutions and identification.

Findings

Conditions for existence of solutions are established.

Identification and partial identification criteria are provided.

Well-posedness conditions and implications for estimation are discussed.

Abstract

This paper provides a nonparametric analysis for several classes of models, with cases such as classical measurement error, regression with errors in variables, factor models and other models that may be represented in a form involving convolution equations. The focus here is on conditions for existence of solutions, nonparametric identification and well-posedness in the space of generalized functions (tempered distributions). This space provides advantages over working in function spaces by relaxing assumptions and extending the results to include a wider variety of models, for example by not requiring existence of density. Classes of (generalized) functions for which solutions exist are defined; identification conditions, partial identification and its implications are discussed. Conditions for well-posedness are given and the related issues of plug-in estimation and regularization are examined.