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Principal components and the long run

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Author Info
Xiaohong Chen (Institute for Fiscal Studies and Yale)
Lars Peter Hansen
Jose A. Scheinkman

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Abstract

We investigate a method for extracting nonlinear principal components. These principal components maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these principal components. We also characterize the limiting behavior of the associated eigenvalues, the objects used to quantify the incremental importance of the principal components. By exploiting the theory of continuous-time, reversible Markov processes, we give a different interpretation of the principal components and the smoothness constraints. When the diffusion matrix is used to enforce smoothness, the principal components maximize long-run variation relative to the overall variation subject to orthogonality constraints. Moreover, the principal components behave as scalar autoregressions with heteroskedastic innovations. Finally, we explore implications for a more general class of stationary, multivariate diffusion processes.

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Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP07/09.

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Date of creation: May 2009
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Handle: RePEc:ifs:cemmap:07/09

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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Torben G. Andersen & Tim Bollerslev & Nour Meddahi, 2004. "Analytical Evaluation Of Volatility Forecasts," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 45(4), pages 1079-1110, November. [Downloadable!] (restricted)
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  2. Nour Meddahi, 2001. "An Eigenfunction Approach for Volatility Modeling," CIRANO Working Papers 2001s-70, CIRANO. [Downloadable!]
  3. Meddahi, N., 2001. "An Eigenfunction Approach for Volatility Modeling," Cahiers de recherche 2001-29, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  4. Hansen, Lars Peter & Alexandre Scheinkman, Jose & Touzi, Nizar, 1998. "Spectral methods for identifying scalar diffusions," Journal of Econometrics, Elsevier, vol. 86(1), pages 1-32, June. [Downloadable!] (restricted)
  5. Darolles, Serge & Florens, Jean-Pierre & Gourieroux, Christian, 2004. "Kernel-based nonlinear canonical analysis and time reversibility," Journal of Econometrics, Elsevier, vol. 119(2), pages 323-353, April. [Downloadable!] (restricted)
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  6. MEDDAHI, Nour, 2001. "An Eigenfunction Approach for Volatility Modeling," Cahiers de recherche 2001-29, Universite de Montreal, Departement de sciences economiques. [Downloadable!]
  7. Florens, Jean-Pierre & Renault, Eric & Touzi, Nizar, 1998. "Testing For Embeddability By Stationary Reversible Continuous-Time Markov Processes," Econometric Theory, Cambridge University Press, vol. 14(06), pages 744-769, December. [Downloadable!]
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(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Nour Meddahi, 2001. "A Theoretical Comparison Between Integrated andRealized Volatilities / A Theoretical Comparison Between Integrated and Realized Volatilities," CIRANO Working Papers 2001s-71, CIRANO. [Downloadable!]
  2. Nour Meddahi & Éric Renault, 2000. "Temporal Aggregation of Volatility Models," CIRANO Working Papers 2000s-22, CIRANO. [Downloadable!]
  3. BONTEMPS, Christian & MEDDAHI, Nour, 2002. "Testing Normality : A GMM Approach," Cahiers de recherche 2002-14, Universite de Montreal, Departement de sciences economiques. [Downloadable!]
  4. Nour Meddahi, 2002. "A theoretical comparison between integrated and realized volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(5), pages 479-508. [Downloadable!]
  5. DAROLLES, Serge & FLORENS, Jean-Pierre & RENAULT, Eric, 2003. "Non Parametric Instrumental Regression," IDEI Working Papers 228, Institut d'Économie Industrielle (IDEI), Toulouse. [Downloadable!]
    Other versions:
  6. Valentina Corradi & Norman R. Swanson, 2003. "Bootstrap Specification Tests for Diffusion Processes," Departmental Working Papers 200321, Rutgers University, Department of Economics. [Downloadable!]
    Other versions:
  7. Nour Meddahi, 2002. "ARMA Representation of Two-Factor Models," CIRANO Working Papers 2002s-92, CIRANO. [Downloadable!]
  8. Lars Peter Hansen & Jose Scheinkman, 2006. "Long Term Risk: An Operator Approach," NBER Working Papers 12650, National Bureau of Economic Research, Inc. [Downloadable!] (restricted)
    Other versions:
  9. Xiaohong Chen & Lars P. Hansen & Marine Carrasco, 2008. "Nonlinearity and Temporal Dependence," Cowles Foundation Discussion Papers 1652, Cowles Foundation, Yale University. [Downloadable!]
    Other versions:
  10. BONTEMPS, Christian & MEDDAHI, Nour, 2007. "Testing Distributional Assumptions: A GMM Approach," IDEI Working Papers 486, Institut d'Économie Industrielle (IDEI), Toulouse. [Downloadable!]
  11. Nour Meddahi, 2002. "ARMA Representation of Integrated and Realized Variances," CIRANO Working Papers 2002s-93, CIRANO. [Downloadable!]
  12. Dennis Kristensen, 2007. "Nonparametric Estimation and Misspecification Testing of Diffusion Models," CREATES Research Papers 2007-01, School of Economics and Management, University of Aarhus. [Downloadable!]
  13. Christian Bontemps & Nour Meddahi, 2002. "Testing Normality: A GMM Approach," CIRANO Working Papers 2002s-63, CIRANO. [Downloadable!]
  14. Torben G. Andersen & Tim Bollerslev & Nour Meddahi, 2002. "Analytic Evaluation of Volatility Forecasts," CIRANO Working Papers 2002s-90, CIRANO. [Downloadable!]
    Other versions:
  15. Nour Meddahi, 2001. "An Eigenfunction Approach for Volatility Modeling," CIRANO Working Papers 2001s-70, CIRANO. [Downloadable!]
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