Open Access
December 2009 Nonlinear principal components and long-run implications of multivariate diffusions
Xiaohong Chen, Lars Peter Hansen, José Scheinkman
Ann. Statist. 37(6B): 4279-4312 (December 2009). DOI: 10.1214/09-AOS706


We investigate a method for extracting nonlinear principal components (NPCs). These NPCs maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and multivariate probability densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these NPCs. By exploiting the theory of continuous-time, reversible Markov diffusion processes, we give a different interpretation of these NPCs and the smoothness constraints. When the diffusion matrix is used to enforce smoothness, the NPCs maximize long-run variation relative to the overall variation subject to orthogonality constraints. Moreover, the NPCs behave as scalar autoregressions with heteroskedastic innovations; this supports semiparametric identification and estimation 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 nonreversible diffusion processes. Finally, we suggest a sieve method to estimate the NPCs from discretely-sampled data.


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Xiaohong Chen. Lars Peter Hansen. José Scheinkman. "Nonlinear principal components and long-run implications of multivariate diffusions." Ann. Statist. 37 (6B) 4279 - 4312, December 2009.


Published: December 2009
First available in Project Euclid: 23 October 2009

zbMATH: 1191.62107
MathSciNet: MR2572460
Digital Object Identifier: 10.1214/09-AOS706

Primary: 47D07 , 62H25
Secondary: 35P05

Keywords: conditional expectations operator , low-frequency data , multivariate diffusion , Nonlinear principal components , quadratic form

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 6B • December 2009
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