In our empirical analysis we found that convergence
of the Markov Chain designed in section 3
to the steady state might be slow. Thus some care
should be given both to the choice of the
starting value 
and to assessing convergence.
There has been an extensive discussion on the question whether to use
a long single chain or many short multiple chains when implementing
MCMC methods (see e.g. the comments following the papers of
Gelman and Rubin, 1992, and Geyer, 1992).
Finding it rather difficult to obtain convergence from a single chain,
we used two seperate chains during
the burn-in phase and monitored the algorithm simply by plotting
the sampled values of the model parameters
as a function of m for both chains.
If the chains were not
in equilibrium, we increased 
and continued the burn-in phase for
each of the chains.
We switched to a single chain
when convergence was obvious from the plots.
To obtain the final MCMC sample of length M
we continued with this single chain
for further M steps.
We found that the speed of convergence of the Markov
chain towards the steady state heavily depends on the number of
factors to be included in the model.
Whereas the burn-in phase is rather short
for the one-factor-model (
), it increases to 
for
the two-factor-model and is as long as 
for the three-factor-model.
For illustration purposes we include Figures 1 and 2
showing the
parameters of the second factor both for the two- and the three-factor-model.
Figure 1 about here
Figure 2 about here
For the first starting value we
use approximate Kalman filtering and
QML estimation. The filtered estimates
of the state variable 
,
and the QML estimator
served as starting value for one chain.
The second starting value
has been obtained by random choice. We sampled 
different
values for 
from
the cartesian product of univariate
intervals,
where these intervals were chosen large enough to cover a reasonable range
of parameter values and computed
the quasi likelihood function from the approximate normal model.
The parameter with the largest functional value of the
quasi likelihood function then served as starting value 
for the second chain.
Starting values 
for the state process
were obtained by approximate Kalman filtering conditional on
.
The filtered estimates of the
state variable ,
served as a starting value for the
state process.
The QML estimator
proved in general to be a good
starting value in the sense
that the corresponding chain reached the steady state quicker than
the chain starting at a randomly chosen value (see Figures 1
and 2),
except for the three-factor-model.
For this model
the chain starting at the QML estimator showed
hardly any
convergence to the steady state for the model parameters
of the third factor.
However, after replacing the QML starting value
by the randomly chosen starting value for the third factor only,
the steady state was finally reached after a considerable
burn-in phase.
To construct the final MCMC sample
we continued with the chain starting from the
QML estimator for the one- and the two-factor-model,
and starting from the QML estimator with the starting
value for the model parameters of the third factor
substituted by the randomly chosen starting value
for the three-factor-model.
The length of the final MCMC sample is chosen to be equal to
M = 9500 for the
one-factor-model and M=10000 for the two- and the three-factor-model.
We used Dickey-Fuller tests to test for stationarity of the final
MCMC samples. The unit-root hypothesis
was rejected for all parameters and factors at the 5% level, except for
the MCMC samples of
and 
.
We conclude this subsection with a discussion of the acceptance rates
of the Metropolis-Hastings algorithm. In Section 3
we suggested to implement MCMC methods for the CIR-model by means
of a Metropolis-Hastings step within Gibbs sampling, where sampling takes
place from an approximate proposal density and a rejection step is
incorporated in such a way that finally we sample from the correct
density. The larger the acceptance rates
for 
and 
for
- see formula (24) and (25) - the
better are the chosen proposal densities.
The empirical average acceptance rate
= 
for each model parameter , 
,
for the chain starting at the QML estimator is reported in Table 1.
The normal proposal density chosen for 
is extremely
good, leading to an average acceptance rate of 99.7 - 99.9%.
The normal proposal density chosen for 
is also fine, leading
to high average acceptance rates of 89.4 - 98.9%.
The proposal densities chosen for 
and 
which are
a normal and an inverse gamma density, respectively, are "worse" in the
sense that the average acceptance rates
range from 79.0 to 97.8% for and from 76.2 to 98.6% for
. These rates are smaller than for
and but still in an acceptable range.
Table 1 about here
The acceptance rates
= 
for the state variable 
are analyzed in
Table 2. For all factors and for all models the median
of
- where the median is taken over
t, 
- ranges from 96.8% to 99.6%
and proves to be quite high.
The
0.05-quantile of
ranges from 66.6% to 98.5% which means that for 95%
of the time points t, , the average acceptance rate
is larger than the given number.
Furthermore, for 92.5% - 100%
of the time points t, , the average acceptance rate
is larger than 0.9.
Finally, for the one- and the two-factor-model the minimum of all
average acceptance rates over t,
is larger than 62.7%.
To sum up, in most cases the normal proposal density derived from
the normal approximation of Chen and Scott (1995) is a perfect
proposal density for the state variable .
For the three-factor-model, however,
there are a few time points where this is not true.
For the first and the second factor we find one
and for the third factor even ten time points out of N = 361,
where is smaller than 0.3.
Table 2 about here