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 relationships with the target default rates. Experienced modelers generally apply a spline transformation with knots to the individual variables. In this article, we introduce the Quantile-based Shape Constrained Maximum Likelihood Estimator (QSC-MLE), which features an automatic spline knot selection in a mortgage default model. QSC-MLE is an enhanced variant of SC-MLE (Chen and Samworth 2016) used in combination with a quantile-based knots set, to effectively process large datasets. QSC-MLE requires generic shape information of the inputs, for example, the monotonicity or convexity of the FICO score, DTI, and LTV, to capture any nonlinear effects. We show that the new default model considerably improves the accuracy of the out-of-sample prediction in comparison with the logistic regression and the Cox proportional hazards model. Moreover, the model conveniently generates component-wise spline functions, which facilitates the interpretation of the default rate response to the input variables. Key Findings ▪ A mortgage default model using the Quantile-based Shape Constrained Maximum Likelihood Estimator (QSC-MLE), which features automatic spline knot selection. ▪ QSC-MLE constructs shape-constrained spline functions to capture nonlinear effects of model inputs. ▪ The new default model considerably improves the accuracy of the out-of-sample prediction.