COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1694 Principal Components and Long Run Implications of Multivariate Diffusions Xiaohong Chen, Lars Peter Hansen, and José Scheinkman April 2009 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. |
