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Optimal Estimation under Nonstandard Conditions

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Abstract

We analyze optimality properties of maximum likelihood (ML) and other estimators when the problem does not necessarily fall within the locally asymptotically normal (LAN) class, therefore covering cases that are excluded from conventional LAN theory such as unit root nonstationary time series. The classical Hájek-Le Cam optimality theory is adapted to cover this situation. We show that the expectation of certain monotone "bowl-shaped" functions of the squared estimation error are minimized by the ML estimator in locally asymptotically quadratic situations, which often occur in nonstationary time series analysis when the LAN property fails. Moreover, we demonstrate a direct connection between the (Bayesian property of) asymptotic normality of the posterior and the classical optimality properties of ML estimators.

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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 1748.

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Length: 19 pages
Date of creation: Jan 2010
Date of revision:
Publication status: Published in Journal of Econometrics (August 2012), 169(2): 258-265
Handle: RePEc:cwl:cwldpp:1748

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Keywords: Bayesian asymptotics; Asymptotic normality; Local asymptotic normality; Locally asymptotic quadratic; Optimality property of MLE; Weak convergence;

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  1. Shiqing Ling & W. K. Li & Michael McAleer, 2001. "Estimation and Testing for Unit Root Processes with GARCH(1,1) Errors: Theory and Monte Carlo Evidence," ISER Discussion Paper 0544, Institute of Social and Economic Research, Osaka University.
  2. Keisuke Hirano & Jack R. Porter, 2003. "Asymptotic Efficiency in Parametric Structural Models with Parameter-Dependent Support," Econometrica, Econometric Society, vol. 71(5), pages 1307-1338, 09.
  3. Peter C. B. Phillips, 2012. "Folklore Theorems, Implicit Maps, and Indirect Inference," Econometrica, Econometric Society, vol. 80(1), pages 425-454, 01.
  4. Jeganathan, P., 1995. "Some Aspects of Asymptotic Theory with Applications to Time Series Models," Econometric Theory, Cambridge University Press, vol. 11(05), pages 818-887, October.
  5. Jae-Young Kim, 1998. "Large Sample Properties of Posterior Densities, Bayesian Information Criterion and the Likelihood Principle in Nonstationary Time Series Models," Econometrica, Econometric Society, vol. 66(2), pages 359-380, March.
  6. Peter C.B. Phillips & Tassos Magdalinos, 2004. "Limit Theory for Moderate Deviations from a Unit Root," Cowles Foundation Discussion Papers 1471, Cowles Foundation for Research in Economics, Yale University.
  7. Peter C.B. Phillips, 1987. "Partially Identified Econometric Models," Cowles Foundation Discussion Papers 845R, Cowles Foundation for Research in Economics, Yale University, revised Aug 1988.
  8. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
  9. Phillips, Peter C B, 1996. "Econometric Model Determination," Econometrica, Econometric Society, vol. 64(4), pages 763-812, July.
  10. Kleibergen, F.R. & Paap, R., 1998. "Priors, posteriors and Bayes factors for a Bayesian analysis of cointegration," Econometric Institute Research Papers EI 9821, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  11. Phillips, Peter C B & Ploberger, Werner, 1996. "An Asymptotic Theory of Bayesian Inference for Time Series," Econometrica, Econometric Society, vol. 64(2), pages 381-412, March.
  12. Jeganathan, P., 1991. "On the Asymptotic Behavior of Least-Squares Estimators in AR Time Series with Roots Near the Unit Circle," Econometric Theory, Cambridge University Press, vol. 7(03), pages 269-306, September.
  13. Vaart,A. W. van der, 2000. "Asymptotic Statistics," Cambridge Books, Cambridge University Press, number 9780521784504.
  14. Shiqing Ling & Michael McAleer, 2001. "On Adaptive Estimation in Nonstationary ARMA Models with GARCH Errors," ISER Discussion Paper 0548, Institute of Social and Economic Research, Osaka University.
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Cited by:
  1. Marc Hallin & Ramon van den Akker & Bas Werker, 2012. "Rank-Based Tests of the Cointegrating Rank in Semiparametric Error Correction Models," Working Papers ECARES ECARES 2012-042, ULB -- Universite Libre de Bruxelles.
  2. Elena Andreou & Bas J.M. Werker, 2014. "Residual-based Rank Specification Tests for AR-GARCH type models," University of Cyprus Working Papers in Economics 02-2014, University of Cyprus Department of Economics.

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