Identification and well-posedness in a class of nonparametric problems

TL;DR

This paper clarifies and extends identification results for convolution-based nonparametric problems, emphasizing conditions for well-posedness and discussing implications for estimation and misspecification.

Contribution

It provides a unified approach to identification in various convolution equation problems and establishes conditions for well-posedness in generalized function spaces.

Findings

Identification can be achieved under more general assumptions.

Conditions for well-posedness are derived.

Illustrates issues of misspecification and estimation related to well-posedness.

Abstract

This is a companion note to Zinde-Walsh (2010), arXiv:1009.4217v1[MATH.ST], to clarify and extend results on identification in a number of problems that lead to a system of convolution equations. Examples include identification of the distribution of mismeasured variables, of a nonparametric regression function under Berkson type measurement error, some nonparametric panel data models, etc. The reason that identification in different problems can be considered in one approach is that they lead to the same system of convolution equations; moreover the solution can be given under more general assumptions than those usually considered, by examining these equations in spaces of generalized functions. An important issue that did not receive sufficient attention is that of well-posedness. This note gives conditions under which well-posedness obtains, an example that demonstrates that when well-posedness does not hold functions that are far apart can give rise to observable arbitrarily close functions and discusses misspecification and estimation from the stand-point of well-posedness.