Principal Components and Long Run Implications of Multivariate Diffusions

Xiaohong Chen (), Lars Hansen and Jose Scheinkman

No 1694, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University

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.

Keywords: Nonlinear principal components; Discrete spectrum; Eigenvalue decay rates; Multivariate diffusion; Quadratic form; Conditional expectations operator (search for similar items in EconPapers)
JEL-codes: C12 C22 (search for similar items in EconPapers)
Pages: 51 pages
Date: 2009-04
New Economics Papers: this item is included in nep-ecm and nep-ets
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (21)

Published in Annals of Statistics (2009), 37(6B): 4279-4312

Downloads: (external link)
https://cowles.yale.edu/sites/default/files/files/pub/d16/d1694.pdf (application/pdf)
Our link check indicates that this URL is bad, the error code is: 404 Not Found

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:cwl:cwldpp:1694

Ordering information: This working paper can be ordered from
Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA
The price is None.

Access Statistics for this paper

More papers in Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University Yale University, Box 208281, New Haven, CT 06520-8281 USA. Contact information at EDIRC.
Bibliographic data for series maintained by Brittany Ladd ().