Journal of Real Estate Finance and Economics, 27:2, 143±172, 2003
# 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
Multi-Factor Cox-Ingersoll-Ross Models of the Term
Structure: Estimates and Tests from a Kalman
School of Business, Rutgers University, Piscataway, NJ 08854, U.S.A.
Morgan Stanley & Co., London E14 4QA, U.K.
This paper presents a method for estimating multi-factor versions of the Cox-Ingersoll-Ross (1985b) model of the
term structure of interest rates. The ®xed parameters in one, two, and three factor models are estimated by
applying an approximate maximum likelihood estimator in a state-space model using data for the U.S. treasury
market. A nonlinear Kalman ®lter is used to estimate the unobservable factors. Multi-factor models are necessary
to characterize the changing shape of the yield curve over time, and the statistical tests support the case for two
and three factor models. A three factor model would be able to incorporate random variation in short term interest
rates, long term rates, and interest rate volatility.
Key Words: interest rates, term structure, Kalman ®lter
The Cox-Ingersoll-Ross (1985b) model is an equilibrium asset pricing model for the term
structure of interest rates. The model provides solutions for bond prices and a complete
characterization of the term structure which incorporates risk premiums and expectations
for future interest rates. The model is frequently presented as a one factor model, but in
Sections 6 and 7 of their paper, Cox, Ingersoll, and Ross, hereafter CIR, show how to
incorporate multiple factors and how to extend the model to value nominal bonds and
nominal claims. The model is important for several reasons: it provides a link between
intertemporal asset pricing theory and the term structure of interest rates, preserves the
requirement that interest rates remain nonnegative, and produces relatively simple closed
form solutions for bond prices. The model is also useful as a tool for valuing interest rate
In this paper, we estimate multi-factor versions of the CIR model by using a state space
model in which estimates of the unobservable state variables are generated by a Kalman
®lter. One, two, and three factor models are estimated, and several tests are performed to
determine whether these models can accurately capture the variability of the term structure