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Efficient Estimation of Jump Diffusions and General Dynamic Models with a Continuum of Moment Conditions

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  • Eric Ghysels

    ()

  • Jean-Pierre Florens
  • Mikhail Chernov
  • Marine Carrasco

    ()

Abstract

A general estimation approach combining the attractive features of method of moments with the efficiency of ML is proposed. The moment conditions are computed via the characteristic function. The two major difficulties with the implementation is that one needs to use an infinite set of moment conditions leading to the singularity of the covariance matrix in the GMM context, and the optimal instrument yielding the ML efficiency was previously shown to depend on the unknown probability density function. We resolve the two problems simultaneously in the framework of C-GMM (GMM with a continuum of moment conditions). First, we prove asymptotic properties of the C-GMM estimator applied to dependent data and then provide a reformulation of the estimator that enhances its computational ease. Second, we propose to span the unknown optimal instrument by an infinite basis consisting of simple exponential functions. Since the estimation framework already relies on a continuum of moment conditions, adding a continuum of spanning functions does not pose any problems. As a result, we achieve ML efficiency when we use the values of conditional CF indexed by its argument as moment functions. We also introduce HAC-type estimators so that the estimation methods are not restricted to settings involving martingale difference sequences. Hence, our methods apply to Markovian and nail-Markovian dynamic models. Finally, a simulated method of moments type estimator is proposed to deal with the cases where the characteristic function does not have a closed-form expression. Extensive Monte-Carlo study based on the models typically used in term-structure literature favorably documents the performance of our methodology. L'estimation des processus de diffusion (affine ou à sauts) est problématique car l'expression de la vraisemblance n'est pas disponible. D'un autre côté, la fonction caractéristique de ces modèles est souvent connue. Cet article propose un estimateur du type méthode des moments généralisés (GMM) fondé sur la fonction caractéristique. Comme l'on dispose d'un continuum de conditions de moments, on utilise une méthode spécifique appelée C-GMM. On dérive les propriétés asymptotiques de l'estimateur et discute son implémentation en pratique. Dans le contexte d'un processus markovien, une condition de moment conditionnelle résulte de la fonction caractéristique conditionnelle. Une question importante est le choix de l'instrument optimal. On montre que, lorsque l'instrument est une fonction exponentielle, l'estimateur C-GMM est asymptotiquement aussi efficace que l'estimateur du maximum de vraisemblance. Il faut noter que la méthode C-GMM n'est pas limitée aux processus markoviens et s'applique à des modèles dynamiques très généraux. De plus, on propose une méthode des moments simulés qui permet de traiter le cas où l'expression de la fonction caractéristique n'est pas connue. Finalement, une étude de Monte Carlo sur des modèles fréquemment utilisés en finance montre que notre estimateur a de bonnes propriétés.

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 2003s-02.

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Date of creation: 01 Jan 2003
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Handle: RePEc:cir:cirwor:2003s-02

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Keywords: Asymptotic efficiency. Characteristic function. GMM. Diffusion processes. Simulated Method of Moments; Efficacité asymptotique. Fonction caractéristique. Méthode des moments généralisés. Méthode des moments simulés. Processus de diffusion;

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Citations

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Cited by:
  1. Ghysels, Eric & Tauchen, George, 2003. "Frontiers of financial econometrics and financial engineering," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 1-7.
  2. Julie Lyng Forman & Michael Sørensen, 2008. "The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 438-465.
  3. Kim, Myung Suk & Wang, Suojin, 2008. "Consistent estimation in regression models for the drift function in some continuous time models," Computational Statistics & Data Analysis, Elsevier, vol. 52(5), pages 2682-2691, January.
  4. GARCIA, René & RENAULT, Eric & VEREDAS, David, 2006. "Estimation of stable distributions by indirect inference," CORE Discussion Papers 2006112, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  5. Hao Zhou, 2003. "Itô Conditional Moment Generator and the Estimation of Short-Rate Processes," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 1(2), pages 250-271.
  6. Bakshi, Gurdip & Panayotov, George, 2010. "First-passage probability, jump models, and intra-horizon risk," Journal of Financial Economics, Elsevier, vol. 95(1), pages 20-40, January.
  7. Marine Carrasco, 2004. "Chi-square Tests for Parameter Stability," RCER Working Papers 508, University of Rochester - Center for Economic Research (RCER).
  8. Carrasco, Marine & Florens, Jean-Pierre, 2003. "On the Asymptotic Efficiency of GMM," IDEI Working Papers 173, Institut d'Économie Industrielle (IDEI), Toulouse.
  9. Qiang Dai & Kenneth Singleton, 2003. "Term Structure Dynamics in Theory and Reality," Review of Financial Studies, Society for Financial Studies, vol. 16(3), pages 631-678, July.
  10. Eric Ghysels & Pedro Santa-Clara & Rossen Valkanov, 2004. "The MIDAS Touch: Mixed Data Sampling Regression Models," CIRANO Working Papers 2004s-20, CIRANO.

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