As with nonparametric fitting of financial time series models to
the data, the neural network also provides an alternative to local
smoothing, as with LP method, in estimating default probabilities.
The logistic regression function
is nothing more than a function defined by a
neural network with only one neuron in a hidden layer, when the
logistic function
is chosen as a transfer function.
Through the combination of several neurons in one or more hidden
layers default probabilities can be estimated, as with the
nonparametric regression analysis, with flexibility. In order to
obtain estimates between 0 and 1, it is necessary to represent the
function
given by (18.1), for
example, with a function
over the interval [0,1]. We restrict
ourselves to one hidden layer with
neurons and choose
so that the default probability given by the neuron network
has the form
where
represents once
again the parameter vector built from
and
. To estimate the network weights from the
data we will not use the least squares method, which makes sense
for the regression model with normally distributed residuals, but
instead we will maximize the log-likelihood function
following the procedure
used in the logistic regression. By substituting in the estimator
we obtain an estimator for the default
probability
In order to obtain an especially simple model with fewer
parameters, Anders (1997) trivially modified the method
for the insolvency prognoses for small and middle sized firms and
assume a default probability of the form
which has obvious similarities to the
general partial linear model, besides the fact that here a part of
or all of the influential variables, i.e., the coordinates of
can appear in linear as well as in nonparametric portions. The
linear term
can be interpreted as the value of
an additional neuron whose transfer function is not the logistic
function
but the identity
. Estimating
the network from the application of a model selection technique
used to find the insolvency probability is surprisingly easy. It
contains in addition to a linear term only one single neuron
From the 6 input variables only 4 contribute to the
linear part (Age of the business, sales development, indicator for
limited liability, dummy variable for processed business), that
means the other two coefficients
are 0, and only 3
(Dummy variables for processed business and for trade, indicator
variable for educational degree of entrepreneur) contribute to the
sigmoid part, that means the corresponding weights
are 0.
With this simple model using a validation data set, which is not
used to estimate the parameters, a ratio of the correct
identifications of 83.3 % was obtained for the insolvencies and
of 63.3 % for the solvent firms.