Principal Components and Long Run Implications of Multivariate Diffusions
AbstractWe 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 multivariate 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 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; this supports semiparametric identification of a multivariate reversible diffusion process and tests of the overidentifying restrictions implied by such a process from low frequency data. We also explore implications for stationary, possibly non-reversible diffusion processes.
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Bibliographic InfoPaper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1694.
Length: 51 pages
Date of creation: Apr 2009
Date of revision:
Publication status: Published in Annals of Statistics (2009), 37(6B): 4279-4312
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA
Find related papers by JEL classification:
- C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
- C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models
This paper has been announced in the following NEP Reports:
- NEP-ALL-2009-05-09 (All new papers)
- NEP-ECM-2009-05-09 (Econometrics)
- NEP-ETS-2009-05-09 (Econometric Time Series)
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- repec:wop:humbsf:2002-57 is not listed on IDEAS
- Darolles, Serge & Florens, Jean-Pierre & Gouriéroux, Christian, 2004.
"Kernel Based Nonlinear Canonical Analysis and Time Reversibility,"
Open Access publications from University of Toulouse 1 Capitole
http://neeo.univ-tlse1.fr, University of Toulouse 1 Capitole.
- 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.
- Serge Darolles & Jean-Pierre Florens & Christian Gourieroux, 2000. "Kernel Based Nonlinear Canonical Analysis and Time Reversibility," Working Papers 2000-18, Centre de Recherche en Economie et Statistique.
- Meddahi, N., 2001.
"An Eigenfunction Approach for Volatility Modeling,"
Cahiers de recherche
2001-29, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
- 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.
- Darolles, S. & Florens, J.-P. & Gourieroux, C., 1999.
"Kernel Based Nonlinear Canonical Analysis,"
99.514, Toulouse - GREMAQ.
- Serge Darolles & Jean-Pierre Florens & Christian Gourieroux, 1998. "Kernel Based Nonlinear Canonical Analysis," Working Papers 98-55, Centre de Recherche en Economie et Statistique.
- Darolles, Serge & Florens, Jean-Pierre & Gouriéroux, Christian, 1999. "Kernel Based Nonlinear Canonical Analysis," IDEI Working Papers 83, Institut d'Économie Industrielle (IDEI), Toulouse, revised 2001.
- Torben G. Andersen & Tim Bollerslev & Nour Meddahi, 2002.
"Analytic Evaluation of Volatility Forecasts,"
CIRANO Working Papers
- 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.
- Gobet, Emmanuel & Hoffmann, Marc & Reiß, Markus, 2002. "Nonparametric estimation of scalar diffusions based on low frequency data is ill-posed," SFB 373 Discussion Papers 2002,57, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
- Chen, Xiaohong & Hansen, Lars Peter & Carrasco, Marine, 2008.
"Nonlinearity and Temporal Dependence,"
48, Yale University, Department of Economics.
- Xiaohong Chen & Lars P. Hansen & Marine Carrasco, 2009. "Nonlinearity and Temporal Dependence," Cowles Foundation Discussion Papers 1652R, Cowles Foundation for Research in Economics, Yale University.
- Xiaohong Chen & Lars P. Hansen & Marine Carrasco, 2009. "Nonlinearity and Temporal Dependence," CIRANO Working Papers 2009s-17, CIRANO.
- Xiaohong Chen & Lars P. Hansen & Marine Carrasco, 2008. "Nonlinearity and Temporal Dependence," Cowles Foundation Discussion Papers 1652, Cowles Foundation for Research in Economics, Yale University.
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