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Principal components and the long run

Author

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  • Xiaohong Chen

    () (Institute for Fiscal Studies and Yale University)

  • Lars Peter Hansen

    (Institute for Fiscal Studies)

  • Jose A. Scheinkman

    (Institute for Fiscal Studies)

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.

Suggested Citation

  • Xiaohong Chen & Lars Peter Hansen & Jose A. Scheinkman, 2009. "Principal components and the long run," CeMMAP working papers CWP07/09, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:07/09
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    File URL: http://cemmap.ifs.org.uk/wps/cwp0709.pdf
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    References listed on IDEAS

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    1. Nour Meddahi, 2001. "An Eigenfunction Approach for Volatility Modeling," CIRANO Working Papers 2001s-70, CIRANO.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
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    Cited by:

    1. Chen, Xiaohong & Hansen, Lars Peter & Carrasco, Marine, 2010. "Nonlinearity and temporal dependence," Journal of Econometrics, Elsevier, vol. 155(2), pages 155-169, April.
    2. 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.
    3. Dennis Kristensen, 2007. "Nonparametric Estimation and Misspecification Testing of Diffusion Models," CREATES Research Papers 2007-01, Department of Economics and Business Economics, Aarhus University.
    4. Meddahi, N., 2001. "An Eigenfunction Approach for Volatility Modeling," Cahiers de recherche 2001-29, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    5. Lars Peter Hansen & José A. Scheinkman, 2009. "Long-Term Risk: An Operator Approach," Econometrica, Econometric Society, vol. 77(1), pages 177-234, January.
    6. Nour Meddahi, 2003. "ARMA representation of integrated and realized variances," Econometrics Journal, Royal Economic Society, vol. 6(2), pages 335-356, December.
    7. Meddahi, Nour & Renault, Eric, 2004. "Temporal aggregation of volatility models," Journal of Econometrics, Elsevier, vol. 119(2), pages 355-379, April.
    8. Bontemps, Christian & Meddahi, Nour, 2005. "Testing normality: a GMM approach," Journal of Econometrics, Elsevier, vol. 124(1), pages 149-186, January.
    9. Nour Meddahi, 2001. "A Theoretical Comparison Between Integrated andRealized Volatilities / A Theoretical Comparison Between Integrated and Realized Volatilities," CIRANO Working Papers 2001s-71, CIRANO.
    10. 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.
    11. Corradi, Valentina & Swanson, Norman R., 2005. "Bootstrap specification tests for diffusion processes," Journal of Econometrics, Elsevier, vol. 124(1), pages 117-148, January.
    12. Nour Meddahi, 2002. "ARMA Representation of Two-Factor Models," CIRANO Working Papers 2002s-92, CIRANO.
    13. Christian Gouriéroux & Eric Renault & Pascale Valery, 2007. "Diffusion Processes with Polynomial Eigenfunctions," Annals of Economics and Statistics, GENES, issue 85, pages 115-130.

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