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Principal Components and the Long Run

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  • Xiaohong Chen
  • Lars Peter Hansen
  • Jos´e A. Scheinkman

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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Bibliographic Info

Paper provided by UCLA Department of Economics in its series Levine's Bibliography with number 122247000000000997.

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Date of creation: 31 Dec 2005
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Handle: RePEc:cla:levrem:122247000000000997

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  1. Hansen, Lars Peter & Alexandre Scheinkman, Jose & Touzi, Nizar, 1998. "Spectral methods for identifying scalar diffusions," Journal of Econometrics, Elsevier, Elsevier, vol. 86(1), pages 1-32, June.
  2. Serge Darolles & Jean-Pierre Florens & Christian Gourieroux, 2000. "Kernel Based Nonlinear Canonical Analysis and Time Reversibility," Working Papers, Centre de Recherche en Economie et Statistique 2000-18, Centre de Recherche en Economie et Statistique.
  3. Bontemps, Christian & Meddahi, Nour, 2007. "Testing Distributional Assumptions: A GMM Approach," IDEI Working Papers, Institut d'Économie Industrielle (IDEI), Toulouse 486, Institut d'Économie Industrielle (IDEI), Toulouse.
  4. MEDDAHI, Nour, 2001. "An Eigenfunction Approach for Volatility Modeling," Cahiers de recherche, Universite de Montreal, Departement de sciences economiques 2001-29, Universite de Montreal, Departement de sciences economiques.
  5. Darolles, S. & Florens, J.-P. & Gourieroux, C., 1999. "Kernel Based Nonlinear Canonical Analysis," Papers, Toulouse - GREMAQ 99.514, Toulouse - GREMAQ.
  6. 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, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 45(4), pages 1079-1110, November.
  7. Florens, Jean-Pierre & Renault, Eric & Touzi, Nizar, 1998. "Testing For Embeddability By Stationary Reversible Continuous-Time Markov Processes," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 14(06), pages 744-769, December.
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