Nonparametric methods for inference in the presence of instrumental variables

Details Nonparametric methods for inference in the presence of instrumental variables Nonparametric methods for inference in the presence of instrumental variables

2006-03-06

arxiv.org/abs/math/0603130v1

Annals of Statistics 2005, Vol. 33, No. 6, 2904-2929

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053605000000714 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053605000000714

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.

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