Principal Components and Long Run Implications of Multivariate Diffusions

Xiaohong Chen (), Lars Peter Hansen and Jose Scheinkman ()
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Xiaohong Chen: Cowles Foundation, Yale University, http://cowles.econ.yale.edu/faculty/chen.htm

No 1694, Cowles Foundation Discussion Papers from Cowles Foundation, 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)
New Economics Papers: this item is included in nep-ecm and nep-ets
Date: Written
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Published in Annals of Statistics (2009), 37(6B): 4279-4312

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