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Ill-conditioned problems, Fisher information and weak instruments

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  • Giovanni Forchini
  • Grant Hillier

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 Ѥistance 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.

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Bibliographic Info

Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP04/05.

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Length: 32 pp.
Date of creation: Apr 2005
Date of revision:
Handle: RePEc:ifs:cemmap:04/05

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Related research

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

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  1. Honore, Bo & Khan, Shakeeb & Powell, James L., 2002. "Quantile regression under random censoring," Journal of Econometrics, Elsevier, vol. 109(1), pages 67-105, July.
  2. Gourieroux Christian & Monfort Alain & Trognon A, 1981. "Pseudo maximum likelihood methods : theory," CEPREMAP Working Papers (Couverture Orange) 8129, CEPREMAP.
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  5. Newey, Whitney K. & McFadden, Daniel, 1986. "Large sample estimation and hypothesis testing," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 36, pages 2111-2245 Elsevier.
  6. Jeffrey M. Wooldridge, 1999. "Asymptotic Properties of Weighted M-Estimators for Variable Probability Samples," Econometrica, Econometric Society, vol. 67(6), pages 1385-1406, November.
  7. Jeffrey M. Wooldridge, 1987. "Specification Testing and Quasi-Maximum Likelihood Estimation," Working papers 479, Massachusetts Institute of Technology (MIT), Department of Economics.
  8. Jeffrey M. Wooldridge, 2002. "Inverse probability weighted M-estimators for sample selection, attrition and stratification," CeMMAP working papers CWP11/02, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  9. Tauchen, George, 1985. "Diagnostic testing and evaluation of maximum likelihood models," Journal of Econometrics, Elsevier, vol. 30(1-2), pages 415-443.
  10. Guido Imbens, 2000. "Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score," Econometric Society World Congress 2000 Contributed Papers 1166, Econometric Society.
  11. Newey, Whitney K, 1985. "Maximum Likelihood Specification Testing and Conditional Moment Tests," Econometrica, Econometric Society, vol. 53(5), pages 1047-70, September.
  12. Rajeev H. Dehejia & Sadek Wahba, 1998. "Causal Effects in Non-Experimental Studies: Re-Evaluating the Evaluation of Training Programs," NBER Working Papers 6586, National Bureau of Economic Research, Inc.
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