Nonparametric instrumental variable estimation under monotonicity

Abstract

The ill-posedness of the nonparametric instrumental variable (NPIV) model leads to estimators that may suffer from poor statistical performance. In this paper, we explore the possibility of imposing shape restrictions to improve the performance of the NPIV estimators. We assume that the function to be estimated is monotone and consider a sieve estimator that enforces this monotonicity constraint. We define a constrained measure of ill-posedness that is relevant for the constrained estimator and show that, under a monotone IV assumption and certain other mild regularity conditions, this measure is bounded uniformly over the dimension of the sieve space. This finding is in stark contrast to the wellknown result that the unconstrained sieve measure of ill-posedness that is relevant for the unconstrained estimator grows to infinity with the dimension of the sieve space. Based on this result, we derive a novel non-asymptotic error bound for the constrained estimator. The bound gives a set of data-generating processes for which the monotonicity constraint has a particularly strong regularization effect and considerably improves the performance of the estimator. The form of the bound implies that the regularization effect can be strong even in large samples and even if the function to be estimated is steep, particularly so if the NPIV model is severely ill-posed. Our simulation study confirms these findings and reveals the potential for large performance gains from imposing the monotonicity constraint. * First version: January 2014. This version: March 25, 2017. We thank Alex Belloni, Richard Blundell, Stéphane Bonhomme, Moshe Buchinsky, Matias Cattaneo, Xiaohong Chen, Victor Chernozhukov, Andrew Chesher, Joachim Freyberger, Jerry Hausman, Jinyong Hahn, Joel Horowitz, Dennis Kristensen, Simon Lee, Zhipeng Liao, Rosa Matzkin, Eric Mbakop, Matthew Kahn, Ulrich Müller, Whitney Newey, Markus Reiß, Andres Santos, Susanne Schennach, Azeem Shaikh, Vladimir Spokoiny, and three referees for useful comments and discussions. We are also thankful for excellent research assistance by Dongwoo Kim who implemented the NPIV estimators in STATA

Testing for the presence of measurement error in Stata

“…Finally, monotonicity of the relationship between the outcome and the explanatory variable is a weak assumption that is often directly implied by economic theory, for example, when the conditional mean E ( Y | X ∗ = x ∗ ) is a production, cost, or utility function. Examples can be found in Matzkin (1994); Olley and Pakes (1996); Cunha, Heckman, and Schennach (2010); Blundell, Horowitz, and Parey (2012, 2017); Kasy (2014); Wilhelm (2015); Hoderlein et al (2015); and Chetverikov and Wilhelm (2017), among many others.…”

Inference in nonparametric/semiparametric moment equality models with shape restrictions

“…Shape restrictions can greatly help inference in this model. Chetverikov and Wilhelm () showed that monotonicity can improve rates of convergence. If the model is partially identified, shape restrictions can also dramatically reduce the identified set.…”

“…Other recent work in this area includes Chetverikov (), Chetverikov and Wilhelm (), Fang and Seo (), Freyberger and Horowitz (), Freyberger and Reeves (), Gutknecht (), and Horowitz and Lee (). All of these papers focus on nonparametric regression or nonparametric IV regression models, except Fang and Seo () and Freyberger and Reeves (), which consider general setups but require point identification.…”

Testing for the presence of measurement error

“…when E P [µ(Y )|X * = x * ] is a production, cost, or utility function. Examples can be found in Matzkin (1994), Olley and Pakes (1996), Cunha, Heckman, and Schennach (2010), Parey (2012, 2016), Kasy (2014), Wilhelm (2015), Hoderlein, Holzmann, Kasy, and Meister (2016), Chetverikov and Wilhelm (2017), among many others.…”