Extending a Logistic Approach to Risk Modeling through Semiparametric Mixing

The new proposal of the Basel Committee on banking regulation issued in January 2001 allows banks to use internal ratings systems to classify firms. Within this context, the main problem is to find a model that fits the data as well as possible, but one that also provides good prediction and explicative capabilities. In this paper, our aim is to compare two kinds of classification models applied to creditworthiness using weighted classification error as the performance function: the standard logistic model and a mixed logistic model, adopting, respectively, a parametric and a semiparametric approach. The main problem of the former is related to the assumption of an i.i.d. hypothesis, but it is often necessary to consider the possible presence of unobservable heterogeneity that characterizes microeconomic data. To better consider this phenomenon, we defined and applied a random effect logistic model, avoiding parametric assumptions upon the random effect distribution. This leads to a likelihood that is defined as the integral of the kernel density with respect to the mixing density, which has no analytical solution. This problem can be obviated by approximating the integral with a finite sum of kernel densities, each one characterized by a different set of model parameters. This discrete nature helps us in detecting non-overlapping clusters characterized by homogeneous values of insolvency risk, and in classifying firms to one of these clusters by means of estimated posterior probabilities of component membership.