Generalized Extreme Value Regression: an Application to Defaults in Credit Risk Analysis

We aim at proposing a Generalized Additive Model (GAM) for Small and Medium Enterprises (SMEs). The Generalized Extreme Value regression model (Calabrese and Osmetti, 2011) is extended by replacing the linear predictor with an additive one, defined as the sum of arbitrary smooth functions. In order to focus the attention on the tail of the response curve for values close to one, we consider the quantile function of the generalized extreme value distribution as a link function in a GAM. Thus we propose the Generalized Extreme Value Additive (GEVA) model. To estimate the smooth functions, the local scoring algorithm (Hastie and Tibshirani, 1986) is applied. In credit risk analysis a pivotal topic is the default probability estimation for SMEs. For this reason, we apply the GEVA regression to empirical data on Italian Small and Medium Enterprises (SMEs). On this dataset we compare the performance of the GEVA model with the one of the logistic additive model. The main advantage of the GEVA model is its excellent performance to identify defaults for low default portfolio. Thanks to this characteristic, the drawback of the logistic (additive) regression model in underestimating the default probability (King and Zeng, 2001) is overcome. Finally, the GEVA model is a robust model, unlike the logistic (additive) regression model, if the sample percentage of defaults is different from that in the out-of-sample analysis.