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Equivalence of the Higher-order Asymptotic Efficiency of k-step and Extremum Statistics

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Abstract

It is well known that a one-step scoring estimator that starts from any N^{1/2}-consistent estimator has the same first-order asymptotic efficiency as the maximum likelihood estimator. This paper extends this result to k-step estimators and test statistics for k >= 1, higher-order asymptotic efficiency, and general extremum estimators and test statistics. The paper shows that a k-step estimator has the same higher-order asymptotic efficiency, to any given order, as the extremum estimator towards which it is stepping, provided (i) k is sufficiently large, (ii) some smoothness and moment conditions hold, and (iii) a condition on the initial estimator holds. For example, for the Newton-Raphson k-step estimator, we obtain asymptotic equivalence to integer order s provided 2^{k} >= s + 1. Thus, for k = 1, 2, and 3, one obtains asymptotic equivalence to first, third, and seventh orders respectively. This means that the maximum differences between the probabilities that the (N^{1/2}-normalized) k-step and extremum estimators lie in any convex set are o(1), o(N^{-3/2}), and o(N^{-3}) respectively.

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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1269.

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Length: 42 pages
Date of creation: Jul 2000
Date of revision:
Publication status: Published in Econometric Theory (2002), 18: 1040-1085
Handle: RePEc:cwl:cwldpp:1269

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Keywords: Asymptotics; Edgeworth expansion; extremum estimator; Gauss-Newton; higher-order efficiency; Newton-Raphson.;

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  1. Rothenberg, Thomas J., 1984. "Approximating the distributions of econometric estimators and test statistics," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 2, chapter 15, pages 881-935 Elsevier.
  2. Pfanzagl, J. & Wefelmeyer, W., 1978. "A third-order optimum property of the maximum likelihood estimator," Journal of Multivariate Analysis, Elsevier, vol. 8(1), pages 1-29, March.
  3. Hall, Peter & Horowitz, Joel L, 1996. "Bootstrap Critical Values for Tests Based on Generalized-Method-of-Moments Estimators," Econometrica, Econometric Society, vol. 64(4), pages 891-916, July.
  4. Robinson, Peter M, 1988. "The Stochastic Difference between Econometric Statistics," Econometrica, Econometric Society, vol. 56(3), pages 531-48, May.
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Cited by:
  1. Dennis Kristensen & Bernard Salanié, 2010. "Higher Order Improvements for Approximate Estimators," CAM Working Papers 2010-04, University of Copenhagen. Department of Economics. Centre for Applied Microeconometrics.
  2. Fan, Yanqin & Gentry, Matthew & Li, Tong, 2011. "A new class of asymptotically efficient estimators for moment condition models," Journal of Econometrics, Elsevier, vol. 162(2), pages 268-277, June.
  3. Hiroyuki Kasahara & Katsumi Shimotsu, 2006. "Nested Pseudo-likelihood Estimation and Bootstrap-based Inference for Structural Discrete Markov Decision Models," Working Papers 1063, Queen's University, Department of Economics.
  4. Rasmus Tangsgaard Varneskov, 2011. "Generalized Flat-Top Realized Kernel Estimation of Ex-Post Variation of Asset Prices Contaminated by Noise," CREATES Research Papers 2011-31, School of Economics and Management, University of Aarhus.
  5. Atsushi Inoue & Mototsugu Shintani, 2001. "Bootstrapping GMM Estimators for Time Series," Vanderbilt University Department of Economics Working Papers 0129, Vanderbilt University Department of Economics, revised Aug 2003.
  6. Politis, D N, 2009. "Higher-Order Accurate, Positive Semi-definite Estimation of Large-Sample Covariance and Spectral Density Matrices," University of California at San Diego, Economics Working Paper Series qt66w826hz, Department of Economics, UC San Diego.
  7. Stelios Arvanitis & Antonis Demos, 2014. "On the Validity of Edgeworth Expansions and Moment Approximations for Three Indirect Inference Estimators," DEOS Working Papers 1406, Athens University of Economics and Business.
  8. Antonis Demos & Stelios Arvanitis, 2010. "Stochastic Expansions and Moment Approximations for Three Indirect Estimators," DEOS Working Papers 1004, Athens University of Economics and Business.
  9. Antonis Demos & Stelios Arvanitis, 2010. "A New Class of Indirect Estimators and Bias Correction," DEOS Working Papers 1023, Athens University of Economics and Business.
  10. Kasahara, Hiroyuki & Shimotsu, Katsumi, 2008. "Pseudo-likelihood estimation and bootstrap inference for structural discrete Markov decision models," Journal of Econometrics, Elsevier, vol. 146(1), pages 92-106, September.
  11. Yixiao Sun & Peter C.B. Phillips, 2008. "Optimal Bandwidth Choice for Interval Estimation in GMM Regression," Cowles Foundation Discussion Papers 1661, Cowles Foundation for Research in Economics, Yale University.

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