Automatic Spline Knot Selection in Modeling Mortgage Loan Default Using Shape Constrained Regression

In mortgage default modeling, many of the key variables, such as loan age, FICO score, Debt-to-Income Ratio (DTI), and Loan-to-House-Value Ratio (LTV), have nonlinear relationship with the target default/delinquency rates. To catch the nonlinearity in the response function, experienced modelers commonly segment the variables into brackets or apply a spline transformation with knots to the individual variables. In this paper, we introduce the Quantile-based Shape Constrained Maximum Likelihood Estimator (QSC-MLE) technique that features an automatic spline knot selection to a mortgage default model. QSC-MLE is an enhanced variant of SC-MLE (Chen & Samworth 2016) used in combination of a quantile-based knot set, which can effectively process large datasets, such as the mortgage loan data. In the model, the default event is defined as 120+ day delinquent in mort- gage payments. QSC-MLE requires generic shape information of the inputs, for example, monotonicity or convexity of the FICO score, DTI, and LTV to capture nonlinear effects. We show that the new default model considerably improves the accuracy of the out-of-sample prediction when compared with the standard logistic regression model or the Cox’s proportional hazards model without nonlinear splines. Moreover, the model conveniently generates component-wise spline functions, which facilitate the interpretation of the default rate response to the input variables