Review of Quantitative Finance and Accounting, 13 (1999): 111±135
# 1999 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Estimating and Testing Exponential-Af®ne Term
Structure Models by Kalman Filter
Hong Kong University of Science and Technology,
Clear Water Bay, Hong Kong, E-mail: firstname.lastname@example.org
Abstract. This paper proposes a uni®ed state-space formulation for parameter estimation of exponential-af®ne
term structure models. The proposed method uses an approximate linear Kalman ®lter which only requires
specifying the conditional mean and variance of the system in an approximate sense. The method allows for
measurement errors in the observed yields to maturity, and can simultaneously deal with many yields on bonds
with different maturities. An empirical analysis of two special cases of this general class of model is carried out:
the Gaussian case (Vasicek 1977) and the non-Gaussian case (Cox Ingersoll and Ross 1985 and Chen and Scott
1992). Our test results indicate a strong rejection of these two cases. A Monte Carlo study indicates that the
procedure is reliable for moderate sample sizes.
Key words: term structure, Kalman ®lter, exponential-af®ne, state-space model, quasi-maximum likelihood
JEL Classi®cation: C22.
The term structure of interest rates describes the relationship between the yield on a
default-free debt security and its maturity. Given the high correlation among bond yields
of different maturities, many theoretical models attempt to use a small number of factors to
explain these joint movements. The typical macro-econometric approach is to specify a
time-series model for the short-term interest rate and then employ the expectation
hypothesis to derive a structural time series model for bond yields of different maturities.
Examples of this approach, such as Hamilton (1987) and Hall, Anderson and Granger
(1992), are abundant in the literature.
A different modelling approach, popular in the ®nance literature and pioneered by
Vasicek (1977) and Dothan (1978), starts out by assuming a diffusion process for the
instantaneous spot interest rate. Arbitrage arguments are then used to facilitate the
derivation of a bond pricing formula. According to these models, the bond price is a
function of the unobserved instantaneous spot interest rate and the model's parameters. A
more general approach is to assume a set of unobserved state variables and proceed to
derive the bond price as a function of these state variables by arbitrage and/or equilibrium
arguments. Cox, Ingersoll and Ross (1985) (hereafter CIR), Richard (1978), Longstaff and
Schwartz (1992) and Chen and Scott (1992) are some examples. Recently, Duf®e and Kan
(1996) have provided a characterization for the class of exponential-af®ne term structure