We assume that migration data are given for periods. This
data consist in
matrices of migration counts
for
each of type
. The generic
element
of the matrix
is the number
of migrations from
to
in period
. These matrices may be
computed from
data sets of migration events.
An obvious question in this context is whether the transition
probabilities can be assumed to be constant in time or not. A
first approach to analyze the time-stability of transition
probabilities is to compare the estimated transition
probabilities per period for periods with estimates from
pooled data.
The aggregated migration counts from periods are
Under the assumption of independence for the migration
events the vector of migration counts
starting from
is in each period
a
realization from a multinomial distributed random vector
The combined hypothesis of homogeneity
This approach creates two problems. Firstly, the two tests are
based on the assumption of independence. Secondly, the test
statistics are only asymptotically -distributed. This
means that sufficiently large sample sizes are required. A rule
of thumb given in the literature is
for all
and
which is hardly fulfilled in the context of
credit migrations.
The two -statistics in (4.13) and (4.14)
are of the Pearson type. Two other frequently used and
asymptotically equivalent statistics are the corresponding
-statistics of the Neyman type
Considering the strong assumptions on which these test procedures
are based on, one may prefer a simpler approach complementing the
point estimates
by estimated standard errors
The quantlet
XFGRatMig3.xpl
computes aggregated migration
counts, estimated transition probabilities and
-statistics. The call is
out = XFGRatMig3(c, rho, s),
where c is a
array of counts
for
periods and rho is a non-negative correlation
parameter. For
the independent case is
computed, compare Section 4.1.4. The last input
parameter s controls the accuracy of the computation,
see Section 4.1.4.
The result is assigned to the variable out, which is a list containing:
The quantlet
XFGRatMig4.xpl
(XFGRatMig4(etp, esd, etpagg))
graphs migration rates per period with estimated standard deviations and migration rates from pooled data. The
inputs are:
The following examples are based on transition matrices given by
Nickell et al. (2000, pp. 208, 213). The data set covers
long-term bonds rated by Moody's in the period 1970-1997. Instead
of the original matrices of type 8 9 we use condensed
matrices of type 3
4 by combining the original data in
the
basic rating categories A, B, C, and D, where D
stands for the category of defaulted credits.
The aggregated data for the full period from 1970 to 1997 are
In the following we use these matrices for illustrative purposes
as if data from periods are given. Figure
4.1 gives a graphical presentation for
rating
categories and
periods.
In order to illustrate the testing procedures presented in
Section 4.2.2 in the following the hypothesis is
tested that the data from the three periods came from the same
theoretical transition probabilities. Clearly, from the
construction of the three periods we may expect, that the test
rejects the null hypothesis. The three -statistics with
degrees of freedom for testing the equality of
the rows of the transition matrices have
-values 0.994,
, and 0.303. Thus, the null hypothesis must be clearly
rejected for the first two rows at any usual level of confidence
while the test for the last row suffers from the limited sample
size. Nevertheless, the
-statistic for the simultaneous
test of the equality of the transition matrices has
degrees of freedom and a
-value
.
Consequently, the null hypothesis must be rejected at any usual level of confidence.
A second example is given by comparing the matrix
based on the whole data with the matrix
based on the data of the normal phase of the business
cycle. In this case a test possibly may not indicate that
differences between
and
are
significant. Indeed, the
-statistics for testing the
equality of the rows of the transition matrices with 3 degrees of freedom have
-values
0.85, 0.82, and 0.02. The statistic of the simultaneous test with 9 degrees of freedom has a
-value of 0.69.