Joint model for longitudinal and spatio-temporal survival data

In credit risk analysis, survival models with fixed and time-varying covariates are commonly used to predict a borrower’s time-to-event. When time-varying covariates are endogenous, jointly modeling their evolution with the event time — known as the joint model for longitudinal and time-to-event data — provides a principled approach. In addition to temporal dynamics, incorporating borrowers’ geographical information can enhance predictive accuracy by capturing spatial clustering and its variation over time. We propose the Spatio-Temporal Joint Model (STJM), a Bayesian hierarchical model that accounts for spatial and temporal effects and their interaction. The STJM captures the impact of unobserved heterogeneity across regions, affecting borrowers residing in the same area at a given time. To ensure scalability to large datasets, we implement the model using the Integrated Nested Laplace Approximation (INLA) framework. We apply the STJM to predict the time to full prepayment on a large dataset of 57,258 US mortgage borrowers with more than 2.5 million observations. Empirical results indicate that including spatial effects consistently improves the performance of the joint model. However, the gains are less definitive when we additionally include spatio-temporal interactions.