The logistic regression model for the estimate of the conditional
probability suffers under the same restrictions as the linear
regression model when estimating the general functions. In order
to avoid the dependence on the special parametric form of the
model and to gain more flexibility in the function estimation it
is suggested to estimate nonparametrically, for
example, with the LP-method given by (13.4) and
(13.7). In doing this, however, it is not guaranteed that
the function estimator will lie between 0 and 1. In order to
enforce this possible, as was done in the previous section, we
transform the value space of the estimated function to the
interval [0,1] using a given function
:
A combination of nonparametric and parametric applications offers
the possibility to use the flexibility of the nonparametric method
by credit rating, Müller and Rönz (2000). In doing so the
influential variables are not combined in a random vector
but are separated into two random vectors
. The coordinates of
represent
several chosen exclusive quantitative characteristics and eventual
hierarchical qualitative characteristics with sufficiently
accurate subdivided value spaces. All remaining characteristics,
especially the dichotomic and the dummy variables of unordered
qualitative characteristics, are combined in
. In order to
estimate the default probability we consider a generalized partial
linear model (GPLM = generalized partial linear model)
:
There are various algorithms for estimating and
for example the profile likelihood method from
Severini and Wong (1992) and Severini and Staniswallis (1994) or the
back-fitting method from Hastie and Tibshirani (1990). Essentially they
use the fact that for the known function
of the parameter
vector
can be estimated through maximization of the
log-likelihood function analog to the logistic regression
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The probability of default on the credit increases when the
customer is unemployed respectively when the customer had
repayment problems in the past. It decreases with the duration of
the credit. The dependence on the transformed credit levels and
ages are nonparametrically estimated. From the Figure
20.2 it is obvious to see that the estimated
function
is clearly non-linear with a maximum
by the average value of the credit level and age. The decrease in
the probability of default by high levels of credit can be
explained by the fact that the random sample contains only those
credits that have actually been given and that the processor was
essentially reluctant to give out large credits when the customer
appeared to be unreliable. This effect, which is caused by the
credit ratings from the past, occurs on a regular basis in credit
rating. Even if a systematic, model based method was not used,
that exclude the credit screening of extreme risks from the very
beginning and thus these ratings no longer appear in the data.
Thus it must be considered when interpreting and applying a model.