Mixed Response Model in Credit Risk Modelling
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Statistical methods are motivated by the desire of learning from data to solve problems in the real world. The credit risk management area of the banking book in the financial industry is a field extensively applying statistical knowledge to solve the problems and continually innovating new statistical methods. In credit risk area, a fundamental assumption of the probability of default (PD) rates for a portfolio is that the PD rates are monotonic increasing as the borrower's creditworthiness worsen. However, since the banks' internal data are not big enough, the empirical realized PD rates often violate this assumption. For the same reason, the violation of the assumption for the PD transition matrix also happens often. These violations will cause a severe problem if we directly calibrate the risk models based this non-smoothed empirical observed PD rates. We propose a smoothing algorithm for the observed PD rates and PD transition matrix by using Constrained Maximum a Posteriori (CMAP) method to solve these problems. The results from the proposed smoothing method are validated by simulation and real default data showing that CMAP method can provide the smoothed and consistent PD rates. We also propose a new approach in this dissertation to estimate the correlated mixed response variable which is often found in credit risk area. The proposed approach simultaneously estimates the mixed response regression and estimates the correlation among the response variables. Moreover, we extend this methodology to the high dimensional mixed response regression models by using the pairwise composite likelihood method. The simulation results show that the proposed method can provide accurate coefficients and correlation for mixed response variables model.