Projection properties of constrained nonparametric instrumental variable estimators

ABSTRACT

I show that a broad class of constrained nonparametric instrumental variable estimators are projections of the corresponding unconstrained estimators on the constrained set with respect to some norm. In addition, for estimators based on Tikhonov regularization, constrained estimation can be viewed as two-step projection: first project the data on the unconstrained set, and then project this projection on the constrained set. As an application, I use the projection property to establish sufficient conditions for the consistency of a sieve-based and a kernel-based constrained estimator.

KEYWORDS:

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

JEL CLASSIFICATION:

• C29
• C14

Acknowledgments

I would like to thank to Enno Mammen and Martin Wahl for many helpful comments and ideas.

Disclosure statement

No potential conflict of interest was reported by the author.

Supplementary material

Supplemental data for this article can be accessed here.

Notes

¹ See Blundell, Horowitz, and Parey (2012) for an example of imposing Slutsky restrictions on the demand for gasoline.

² These two frameworks are chosen because they are popular approaches in the nonparametric IV literature, see e.g. Blundell, Chen, and Kristensen (2007) and Chen and Pouzo (2012) for sieves and Darolles et al. (2011) for kernels.

³ See for example Engl, Hanke, and Neubauer (1996).

⁴ It is not in the scope of this paper to derive the implications of this assumption on the set $C$. The direct method in the calculus of variations provides a proof of existence when $C$ is at least weakly (sequentially) closed.

⁵ Strictly seen, $<.,.{>}_{\mathfrak{V}}$ should be indexed by the constant $\alpha$. I omit the index for notational simplicity.

⁶ For the precise definition of the estimator in Grasmair, Scherzer, and Vanhems (2013), the scaling $1/n$ of the first term of $<,{>}_V$ must be omitted.