Ill-conditioned problems, Fisher information, and weak instruments
Forchini, Giovanni and Hillier, Grant (2005) Ill-conditioned problems, Fisher information, and weak instruments. London, GB, Department of Economics, University College London (CeMMAP Working Papers, (CWP04/05) CWP04/05).
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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, Potscher (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 ‘distance 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.
|Item Type:||Monograph (Working Paper)|
|Subjects:||H Social Sciences > HB Economic Theory
H Social Sciences > HA Statistics
|Divisions:||University Structure - Pre August 2011 > School of Social Sciences > Economics
|Date Deposited:||24 Mar 2010|
|Last Modified:||02 Mar 2012 11:43|
|Publisher:||Department of Economics, University College London|
|RDF:||RDF+N-Triples, RDF+N3, RDF+XML, Browse.|
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