A GENERALIZED ADDITIVE MODEL FOR BINARY RARE EVENTS DATA: AN APPLICATION TO CREDIT DEFAULTS

We aim at proposing a Generalized Additive Model (GAM) for binary rare events, i.e. binary dependent variable with a very small number of ones. GAM is an extension of the family of Generalize Linear Models (GLMs) by replacing the linear predictor with an additive one defined as the sum of arbitrary smooth functions. In the GLMs the relationship between the independent variable and the predictor is constrained to be linear. Instead the GAMs do not involve strong assumptions about this relationship, which is merely constrained to be smooth. We extend the Generalized Extreme Value (GEV) regression model proposed by Calabrese and Osmetti (2011) for binary rare events data. In particular, we suggest the Generalized Extreme Value Additive (GEVA) model by considering the quantile function of the generalized extreme value distribution as a link function in a GAM. In order 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. Since defaults are rare events, we apply the GEVA regression to empirical data on Italian Small and Medium Enterprises (SMEs) to model their default probabilities. We compare on these data the performance of the GEVA model with the one of the most used regression model for binary dependent variable, the logistic additive model. By reducing the sample frequencies of rare events (defaults), the predictive performance of the logistic additive regression model to identify the rare events becomes worse. On the contrary, the GEVA model overcomes the underestimation problem and its accuracy to identify the rare events improves by reducing the sample percentage of rare events. Finally, we show that the GEVA model is a robust model, unlike the logistic additive regression model.