The Annals of Statistics

Nonparametric methods for inference in the presence of instrumental variables

Peter Hall and Joel L. Horowitz

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We suggest two nonparametric approaches, based on kernel methods and orthogonal series to estimating regression functions in the presence of instrumental variables. For the first time in this class of problems, we derive optimal convergence rates, and show that they are attained by particular estimators. In the presence of instrumental variables the relation that identifies the regression function also defines an ill-posed inverse problem, the “difficulty” of which depends on eigenvalues of a certain integral operator which is determined by the joint density of endogenous and instrumental variables. We delineate the role played by problem difficulty in determining both the optimal convergence rate and the appropriate choice of smoothing parameter.

Article information

Ann. Statist. Volume 33, Number 6 (2005), 2904-2929.

First available: 17 February 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Bandwidth convergence rate eigenvalue endogenous variable exogenous variable kernel method linear operator nonparametric regression smoothing optimality


Hall, Peter; Horowitz, Joel L. Nonparametric methods for inference in the presence of instrumental variables. The Annals of Statistics 33 (2005), no. 6, 2904--2929. doi:10.1214/009053605000000714.

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