Abstract
The problem of endogeneity in statistics and econometrics is often handled by introducing instrumental variables (IV) which fulfill the mean independence assumption, i.e. the unobservable is mean independent of the instruments. When full independence of IV’s and the unobservable is assumed, nonparametric IV regression models and nonparametric demand models lead to nonlinear integral equations with unknown integral kernels. We prove convergence rates for the mean integrated square error of the iteratively regularized Newton method applied to these problems. Compared to related results we derive stronger convergence results that rely on weaker nonlinearity restrictions. We demonstrate in numerical simulations for a nonparametric IV regression that the method produces better results than the standard model.
Highlights
Dependence of an unobservable error term and covariates is a frequent problem in statistical and econometrical modeling known as endogeneity
An efficient way to deal with endogeneity is to use instrumental variables (IV) in the estimation
The novelty of this paper is that we prove significantly faster convergence rates for the mean integrated squared error (MISE) rather than convergence in probability under a different set of assumptions
Summary
Dependence of an unobservable error term and covariates is a frequent problem in statistical and econometrical modeling known as endogeneity. An efficient way to deal with endogeneity is to use instrumental variables (IV) in the estimation These are additional variables which can assumed to be independent or mean independent of the unobservable. In this paper we describe and analyze a consistent estimator for this type of problem, when F is an operator between Hilbert spaces. Important monographs on this topic are Bakushinskiı and Kokurin (2004) and Kaltenbacher, Neubauer and Scherzer (2008) These contributions consider only problems with known operators and deterministic right hand side in equation (1). The use of IRGNM for nonparametric IV problems was proposed and analyzed by Dunker et al (2014). They derived rates for convergence in probability with a priori parameter choice using variational methods.
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