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Continuously updated extremum estimators

Author

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  • PATILEA, Valentin

    (Institut de Statistique, Université catholique de Louvain, Louvain-la-Neuve, Belgium)

  • RENAULT, Eric

    (GREMAQ-IDEI, Université de Toulouse I and Institut Universitaire de France)

Abstract

An important class of structural econometric models (nonlinear rational expectations, option pricing, auction models, ...) characterize observable variables as highly nonlinear transforma- tions of some latent variables. These transformations are one-to-one, but they depend on the unknown distribution of the latent variables through the equilibrium of the game and/or the learning process. Therefore numerical complexity of the equilibrium definition generates sub- stantial obstacles for the direct implementation of maximum likelihood inference. This is a particular case of argmax estimators based on a untractable sample based criterion Q[exp.T][thêta,lambda(thêta)] contaminated by the occurences of [thêta] in a ‘nuisance function’ [lambda(thêta)] to which corresponds a simple criterion Q[exp.T][thêta,lambda(thêta.exp0)] with [thêta.exp.0] the true, unknown value of the parameter. The natural idea is to replace the unknown value [lambda(thêta.exp0)] by some ’good proxy’, say [lambda(thêta.exp1)], to maximize Q[exp.T][thêta,lambda(thêta.exp1)] with respect to [thêta] and to get a new, updated estimate which, in turn, can be used for approxi- mating [lambda(thêta.exp0)],... Such steps can be considered for a fixed sample size (iterative M-estimators), or each time new data arrive (recursive estimators). In this paper we present and analyze several iterative and recursive (Robbins-Monro type) estimation procedures of this kind which, at the end, are applied to the class of structural econometric models which motivated this work.

Suggested Citation

  • PATILEA, Valentin & RENAULT, Eric, 1997. "Continuously updated extremum estimators," CORE Discussion Papers 1997076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvco:1997076
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    References listed on IDEAS

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    1. René Garcia & Éric Renault, 1998. "Risk Aversion, Intertemporal Substitution, and Option Pricing," CIRANO Working Papers 98s-02, CIRANO.
    2. Yacine Aït-Sahalia & Andrew W. Lo, 1998. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," Journal of Finance, American Finance Association, vol. 53(2), pages 499-547, April.
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    4. Hardle, W. & Vieu, P., 1990. "Kernel regression smoothing of time series," CORE Discussion Papers 1990031, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    6. Broadie, Mark & Detemple, Jerome & Ghysels, Eric & Torres, Olivier, 2000. "Nonparametric estimation of American options' exercise boundaries and call prices," Journal of Economic Dynamics and Control, Elsevier, vol. 24(11-12), pages 1829-1857, October.
    7. Peter Bossaerts & Pierre Hillion, 1993. "A Test Of A General Equilibrium Stock Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 311-347.
    8. GARCIA, René & RENAULT, Éric, 1998. "Risk Aversion, Intertemporal Substitution, and Option Pricing," Cahiers de recherche 9801, Universite de Montreal, Departement de sciences economiques.
    9. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    10. P. BOSSAERTS & C. HAFNER & Wolfgang HÄRDLE, 1996. "Foreign Exchange Rates Have Surprising Volatility," SFB 373 Discussion Papers 1996,68, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
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    12. Aït-Sahalia, Yacine. & Bickel, Peter J. & Stoker, Thomas M., 1994. "Goodness-of-fit tests for regression using kernel methods," Working papers 3747-94., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    13. repec:crs:wpaper:9329 is not listed on IDEAS
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    Cited by:

    1. René Garcia & Eric Ghysels & Éric Renault, 2004. "The Econometrics of Option Pricing," CIRANO Working Papers 2004s-04, CIRANO.

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