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

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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File URL: http://cowles.econ.yale.edu/P/cd/d12b/d1269.pdf
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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
Note: CFP 1044.
Contact details of provider: Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
Phone: (203) 432-3702
Fax: (203) 432-6167
Web page: http://cowles.econ.yale.edu/

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Order Information: Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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  1. 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.
  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. Robinson, Peter M, 1988. "The Stochastic Difference between Econometric Statistics," Econometrica, Econometric Society, vol. 56(3), pages 531-48, May.
  4. 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.
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