On ill-posedness of nonparametric instrumental variable regression with convexity constraints

TL;DR

This paper demonstrates that imposing monotonicity or convexity constraints in nonparametric instrumental variable regression does not resolve the problem of ill-posedness, which persists even without regularization.

Contribution

It provides a theoretical analysis showing that convexity and monotonicity constraints do not improve the well-posedness of the nonparametric IV regression problem.

Findings

Adding convexity constraints does not restore well-posedness.

The problem remains locally ill-posed without regularization.

Constraints do not mitigate the inherent ill-posedness of the regression.

Abstract

This note shows that adding monotonicity or convexity constraints on the regression function does not restore well-posedness in nonparametric instrumental variable regression. The minimum distance problem without regularisation is still locally ill-posed.