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Ill-conditioned problems, Fisher information and weak instruments

Author

Listed:
  • Giovanni Forchini

    (Institute for Fiscal Studies)

  • Grant Hillier

    () (Institute for Fiscal Studies and University of Southampton)

Abstract

The existence of a uniformly consistent estimator for a particular parameter is well-known to depend on the uniform continuity of the functional that defines the parameter in terms of the model. Recently, Pötscher (Econometrica, 70, pp 1035 - 1065) showed that estimator risk may be bounded below by a term that depends on the oscillation (osc) of the functional, thus making the connection between continuity and risk quite explicit. However, osc has no direct statistical interpretation. In this paper we slightly modify the definition of osc so that it reflects a (generalized) derivative (der) of the functional. We show that der can be directly related to the familiar statistical concepts of Fisher information and identification, and also to the condition numbers that are used to measure ?istance from an ill-posed problem' in other branches of applied mathematics. We begin the analysis assuming a fully parametric setting, but then generalize to the nonparametric case, where the inverse of the Fisher information matrix is replaced by the covariance matrix of the efficient influence function. The results are applied to a number of examples, including the structural equation model, spectral density estimation, and estimation of variance and precision.

Suggested Citation

  • Giovanni Forchini & Grant Hillier, 2005. "Ill-conditioned problems, Fisher information and weak instruments," CeMMAP working papers CWP04/05, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:04/05
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    File URL: http://cemmap.ifs.org.uk/wps/cwp0405.pdf
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    References listed on IDEAS

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    Keywords

    Continuity; Derivative; Divergence; Fisher Information; Ill-conditioned problem; Ill-posed problem; Interest-functional; Oscillation; Precision;

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