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Identification and Estimation of 'Maximal' Affine Term Structure Models: An Application to Stochastic Volatility

62 Pages Posted: 28 Jun 2003  

Pierre Collin-Dufresne

Ecole Polytechnique Fédérale de Lausanne - Ecole Polytechnique Fédérale de Lausanne; National Bureau of Economic Research (NBER)

Robert S. Goldstein

University of Minnesota - Twin Cities - Carlson School of Management; National Bureau of Economic Research (NBER)

Christopher S. Jones

University of Southern California - Marshall School of Business - Finance and Business Economics Department

Date Written: May 14, 2003

Abstract

We propose a canonical representation for affine term structure models where the state vector is comprised of the first few Taylor-series components of the yield curve and their quadratic (co-)variations. With this representation: (i) the state variables have simple physical interpretations such as level, slope and curvature, (ii) their dynamics remain affine and tractable, (iii) the model is by construction 'maximal' (i.e., it is the most general model that is econometrically identifiable), and (iv) model-insensitive estimates of the state vector process implied from the term structure are readily available. (Furthermore, this representation may be useful for identifying the state variables in a squared-Gaussian framework where typically there is no one-to-one mapping between observable yields and latent state variables). We find that the 'unrestricted' A1(3) model of Dai and Singleton (2000) estimated by 'inverting' the yield curve for the state variables generates volatility estimates that are negatively correlated with the time series of volatility estimated using a standard GARCH approach. This occurs because the 'unrestricted' A1(3) model imposes the restriction that the volatility state variable is simultaneously a linear combination of yields (i.e., it impacts the cross-section of yields), and the quadratic variation of the spot rate process (i.e., it impacts the time-series of yields). We then investigate the A1(3) model which exhibits 'unspanned stochastic volatility' (USV). This model predicts that the cross section of bond prices is independent of the volatility state variable, and hence breaks the tension between the time-series and cross-sectional features of the term structure inherent in the unrestricted model. We find that explicitly imposing the USV constraint on affine models significantly improves the volatility estimates, while maintaining a good fit cross-sectionally.

Keywords: Term Structure of Interest rates, Affine Models

JEL Classification: G12, G13

Suggested Citation

Collin-Dufresne, Pierre and Goldstein, Robert S. and Jones, Christopher S., Identification and Estimation of 'Maximal' Affine Term Structure Models: An Application to Stochastic Volatility (May 14, 2003). Available at SSRN: https://ssrn.com/abstract=410420 or http://dx.doi.org/10.2139/ssrn.410420

Pierre Collin-Dufresne (Contact Author)

Ecole Polytechnique Fédérale de Lausanne - Ecole Polytechnique Fédérale de Lausanne ( email )

Quartier UNIL-Dorigny, Bâtiment Extranef, # 211
40, Bd du Pont-d'Arve
CH-1015 Lausanne, CH-6900
Switzerland

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Robert S. Goldstein

University of Minnesota - Twin Cities - Carlson School of Management ( email )

19th Avenue South
Minneapolis, MN 55455
United States
612-624-8581 (Phone)

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Christopher S. Jones

University of Southern California - Marshall School of Business - Finance and Business Economics Department ( email )

Marshall School of Business
Los Angeles, CA 90089
United States

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