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Principal Components and Long Run Implications of Multivariate Diffusions

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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 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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  • Xiaohong Chen & Lars Peter Hansen & Jose Scheinkman, 2009. "Principal Components and Long Run Implications of Multivariate Diffusions," Cowles Foundation Discussion Papers 1694, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:1694
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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d16/d1694.pdf
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    1. 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.
    2. 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.
    3. Serge Darolles & Jean-Pierre Florens & Christian Gourieroux, 2004. "Kernel-based nonlinear canonical analysis and time reversibility," Post-Print halshs-00678062, HAL.
    4. Meddahi, N., 2001. "An Eigenfunction Approach for Volatility Modeling," Cahiers de recherche 2001-29, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    5. 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.
    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, 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, vol. 14(6), pages 744-769, December.
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    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. Christian Bontemps & Nour Meddahi, 2012. "Testing distributional assumptions: A GMM aproach," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 27(6), pages 978-1012, September.
    3. Chorowski, Jakub & Trabs, Mathias, 2016. "Spectral estimation for diffusions with random sampling times," Stochastic Processes and their Applications, Elsevier, vol. 126(10), pages 2976-3008.
    4. Andersen, Torben G. & Bollerslev, Tim & Meddahi, Nour, 2011. "Realized volatility forecasting and market microstructure noise," Journal of Econometrics, Elsevier, vol. 160(1), pages 220-234, January.
    5. Liang Chen & Juan J. Dolado & Jesús Gonzalo, 2021. "Quantile Factor Models," Econometrica, Econometric Society, vol. 89(2), pages 875-910, March.
    6. Kristensen, Dennis, 2011. "Semi-nonparametric estimation and misspecification testing of diffusion models," Journal of Econometrics, Elsevier, vol. 164(2), pages 382-403, October.
    7. Bu, Ruijun & Hadri, Kaddour & Kristensen, Dennis, 2021. "Diffusion copulas: Identification and estimation," Journal of Econometrics, Elsevier, vol. 221(2), pages 616-643.
    8. Strauch, Claudia, 2015. "Sharp adaptive drift estimation for ergodic diffusions: The multivariate case," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2562-2602.
    9. Thibaut Lamadon & Elena Manresa & Stephane Bonhomme, 2016. "Discretizing Unobserved Heterogeneity," 2016 Meeting Papers 1536, Society for Economic Dynamics.
    10. Gorodnichenko, Yuriy & Ng, Serena, 2017. "Level and volatility factors in macroeconomic data," Journal of Monetary Economics, Elsevier, vol. 91(C), pages 52-68.
    11. Escanciano, Juan Carlos & Hoderlein, Stefan & Lewbel, Arthur & Linton, Oliver & Srisuma, Sorawoot, 2021. "Nonparametric Euler Equation Identification And Estimation," Econometric Theory, Cambridge University Press, vol. 37(5), pages 851-891, October.

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    More about this item

    Keywords

    Nonlinear principal components; Discrete spectrum; Eigenvalue decay rates; Multivariate diffusion; Quadratic form; Conditional expectations operator;
    All these keywords.

    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; Diffusion Processes

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