| By: |
Xiaohong Chen (Cowles Foundation, Yale University);
Lars Peter Hansen (Dept. of Economics, University of Chicago);
Jose Scheinkman (Dept. of Economics, Princeton 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 |
| JEL: |
C12 C22 |
| Date: |
2009–04 |
| URL: |
http://d.repec.org/n?u=RePEc:cwl:cwldpp:1694&r=ets |