A Regression for Modeling Credit Defaults: The Generalized Extreme Value Model
Raffaella Calabrese, Silvia Osmetti
Department of Quantitative Methods, University of Milano-Bicocca; Statistical Sciences, Università Cattolica di Milano

In order to model credit defaults we propose a Generalized Linear Model (McCullagh and Neleder, 1989) whose link function is the quantile function of the Generalized Extreme Value (GEV) distribution (Kotz and Nadarajah, 2000). In particular, the dependent variable is binary and describes the rare event of a credit default. The goal of this paper is to overcome the drawbacks shown by the logistic regression in rare events. By using the logit link function the probability of rare events could be underestimated. Furthermore, the logit link is a symmetric function, not appropriated when the dependent variable is a rare event. We choose the GEV quantile function as skewed link function since we focus our attention on the tail of the response curve for the values close to 1. We define the proposed model GEV regression. After computing the likelihood and the score functions of the GEV regression model, we propose to estimate the parameters by the maximum likelihood method. Since the score functions do not have closed-form, an iterative algorithm is used. In order to compute the initial point estimates of the parameters we consider a particular case of the GEV regression, with the log-log link function, known in the literature (Agresti, 2002). Furthermore, we propose another particular case of the GEV regression whose link function is the quantile function of the Weibull random variable. Finally, we apply the GEV regression to empirical data on Italian Small and Medium Enterprizes in order to model the probability of default (PD) as a function of financial and economic variables (Altman et al., 1977), selected to cover the most relevant areas of firm activity (leverage, liquidity and profitability). The application shows that the GEV regression overcomes the PD underestimation of the logistic regression model and accommodates skewness. We highlight that PD estimates are important since they are required by the New Basel Capital Accord (Basel Committee on Banking Supervision, 2004) in the Internal Rating Based Approach. Moreover, our model shows an improvement in one-year forecasting accuracy.

Keywords: Generalized Linear Model; Generalized Extreme Value distribution; rare event; default probability

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