Nonparametric methods for inference in the presence of instrumental variables

Peter Hall*, Joel L. Horowitz

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

200 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)2904-2929
Number of pages26
JournalAnnals of Statistics
Issue number6
StatePublished - Dec 2005


  • Bandwidth
  • Convergence rate
  • Eigenvalue
  • Endogenous variable
  • Exogenous variable
  • Kernel method
  • Linear operator
  • Nonparametric regression
  • Optimality
  • Smoothing

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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