The
Cox-Ingersoll-Ross-model (Cox et al., 1985)
frequently is presented as a one-factor-model, but
already Cox et al. (1985) show how
to incorporate multiple factors.
The nominal
instantaneous interest rate is assumed to be the sum of K state
variables (factors) 
:
where the state variables are assumed to be independently
generated by a square root process:
where 
is a Wiener process.
is the long-term mean. is pulled towards
at a rate governed by the speed of adjustment
coefficient 
.
Based on Cox et al. (1985), Chen and Scott (1995)
derive the solution for the nominal price
at time t for a pure discount bond with face value 1 maturing at time
t+T as follows:
where
with 
.
Each state variable is associated with
a parameter 
which is negatively
related to the risk premium.
The yield to maturity at time t of a
pure discount bond which matures at time t+T is defined as:
which is a linear function of the state variables
.
Let 
=
be the 
-dimensional
vector of yields observed at time t, where is the number of
observed yields which need not be the same at each date. Let 
be the associated times to maturity.
To estimate the unobservable state variables from yields observed at
discrete time intervals, Chen and Scott (1995)
and - independently - Geyer and Pichler (1995)
suggest to use a state
space formulation of the CIR-model.
However, none of the above cited studies
uses the exact
state space formulation. The exact state space formulation
with state variable 
and observation vector 
is given by the following model assumptions:
is a Markov process with
and
. 
is called the prior
and 
is called the transition density.
are conditionally independent given and
is independent of 
given 
with
. 
is called the observation density.
For the CIR-model the exact
transition densities are known
to be the
product of K non-central 
-densities (Cox et al., 1985; Chen and
Scott, 1993):
where 
is the modified Bessel function of the first
kind of order 
.
The observation density
is based on the linear
relationship (5) between yields and the state variable
. The distribution of observed yields given the state variable
is
derived from the following measurement equation:
where 
is a -dimensional vector and 
is a 
matrix. Both quantities are derived from (2) - (5):
is a -dimensional random vector reflecting
pricing errors caused by market imperfections.
Geyer and Pichler (1995) assume that the errors of each maturity
have the same variance 
, whereas
Chen and Scott (1995), Duan and Simonato (1995) and Lund (1994) assume different
variances 
for different
maturities.
The following choice combines
individual variances for n maturities which are identical for each t,
and a common variance for the remaining ones:
The observation density is then the product of
normal densities
To complete the state space formulation the prior has to be
chosen.
We assume that 
are independent apriori:
For the distribution of the individual components 
one may either assume a
vague normal prior 
for each state variable with 
being large,
or assume - like Chen and Scott (1995) - that
each state variable
is
distributed according to the stationary gamma distribution
(see Cox et al., 1985):
