Predicting bivariate binary rare events responses using generalised extreme value regression model and copula function

A new bivariate Generalised Linear Model (GLM) is proposed for binary rare events, i.e. binary dependent variables with a very small number of ones. In a bivariate GLM model we suggest the quantile function of the Generalised Extreme Value (GEV) distribution. In this way, the drawback of the underestimation of the probability of the rare event in GLM models is overcome. The dependence between the response variables is modelled by the copula function. We explore different copula functions that provide a rich and flexible class of structures to derive joint distributions for bivariate binary data. Finally, we apply the proposed model to estimate the joint probability of defaults of UK and Italian small and medium enterprises.