Estimating Equations for a Class of Time-Irreversible Multi-Factor Models

26 Pages Posted: 31 Jan 2008

See all articles by Nina Boyarchenko

Nina Boyarchenko

Federal Reserve Bank of New York

Sergei Levendorskii

Calico Science Consulting

Date Written: January 30, 2008


The standard operator approach to the identification problem of diffusions and more general Markov processes relies on the variational principles for self-adjoint operators. If the process is not time reversible, equivalently, the infinitesimal operator of the process is not self-adjoint, these principles are not applicable. We develop the spectral decomposition for multi-factor time-irreversible OU processes, ATSMs and QTSMs, and use it to construct new estimating equations. Using these equations, we construct identification schemes for the leading eigenfunction, stationary distribution, gap between the leading eigenvalue and the real part of the rest of the spectrum, and more involved schemes for all parameters of the process. Finally, we use the variational principles for the self-adjoint operator associated to an appropriate quadratic form to construct the leading eigenfunction and a basis in the vector space of unobserved factors from observed yields in a QTSM, and provide the reduction of the identification problem of a QTSM to the identification problem of an OU model.

Keywords: estimating function, spectral decomposition, eigenfunctions, eigenvalues, time irreversible processes, quadratic term structure models, affine term structure models, Ornstein-Uhlenbeck process

JEL Classification: C22, G12

Suggested Citation

Boyarchenko, Nina and Levendorskii, Sergei Z., Estimating Equations for a Class of Time-Irreversible Multi-Factor Models (January 30, 2008). Available at SSRN: or

Nina Boyarchenko (Contact Author)

Federal Reserve Bank of New York ( email )

33 Liberty Street
New York, NY 10045
United States
212-720-7339 (Phone)
212-720-1582 (Fax)

Sergei Z. Levendorskii

Calico Science Consulting ( email )

Austin, TX
United States

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