Does the constraint matter? A note on the asymptotic properties of nonparametric instrumental variable estimation under monotonicity shape constraint

ABSTRACT

I show that constrained monotone instrumental variable estimators are asymptotically equivalent to their unconstrained counterparts whenever the true regression function is in the interior of the constrained set. In a simulation study, a sieve-based constrained estimator is shown to outperform the unconstrained one even in cases where both are asymptotically equivalent.

KEYWORDS:

• Shape constraints
• monotonicity
• nonparametric instrumental variable regression
• ill-posed inverse problem

Acknowledgements

I am very grateful to Enno Mammen for many helpful comments and discussions on this and several further related projects. I would also like to thank to Lena Boneva, Christoph Breunig and Martin Wahl for many helpful comments and ideas.

Disclosure statement

No potential conflict of interest was reported by the author.

Supplemental data

Supplemental data for this article can be accessed here.

Notes

¹ Imposing a shape constraint on the estimator can be of crucial importance for policy analysis. In particular, the unconstrained estimator can violate the constraint in small samples, even if the constraint is predicted by theory. See Blundell, Horowitz, and Parey (2012) for an example of imposing Slutsky restrictions on the demand for gasoline and Parmeter and Racine (2013) for an example of imposing Afriat conditions in production frontier analysis.

² See, for example, Engl, Hanke, and Neubauer (1996) and Carrasco, Florens, and Renault (2007) for a definition and discussion.

³ Due to the convexity of the constraint C, exists and is unique under mild conditions, see e.g. Engl, Hanke, and Neubauer (1996).

⁴ The online appendix (submitted also to the journal as a separate file) is available here: https://sites.google.com/site/petyobbonev/papers.

⁵ For further details of the data generation process, see the online appendix B.1.

⁶ The MISE-minimizing parameters are shown in the online appendix, Tables 1 and 2.