In credit risk analysis, survival models with fixed and time-varying covariates are widely used to predict a borrower's time-to-event. When the time-varying drivers are endogenous, modelling jointly the evolution of the survival time and the endogenous covariates is the most appropriate approach, also known as the joint model for longitudinal and survival data. In addition to the temporal component, credit risk models can be enhanced when including borrowers' geographical information by considering spatial clustering and its variation over time. We propose the Spatio-Temporal Joint Model (STJM) to capture spatial and temporal effects and their interaction. This Bayesian hierarchical joint model reckons the survival effect of unobserved heterogeneity among borrowers located in the same region at a particular time. To estimate the STJM model for large datasets, we consider the Integrated Nested Laplace Approximation (INLA) methodology. 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.

翻译：在信用风险分析中，具有固定和时变协变量的生存模型被广泛用于预测借款人的风险事件发生时间。当时变驱动因素为内生变量时，最合适的方法是联合建模生存时间与内生协变量的演变过程，即所谓的纵向与生存数据联合模型。除时间成分外，考虑借款人地理位置信息的空间聚类及其随时间变化特征可增强信用风险模型。我们提出时空联合模型（STJM）以捕捉空间效应、时间效应及其交互作用。该贝叶斯层次联合模型考量了特定时段内同一区域借款人之间未观测异质性的生存效应。针对大规模数据集下的STJM模型估计，我们采用整合嵌套拉普拉斯近似（INLA）方法。将STJM应用于包含57,258位美国抵押贷款借款人（超过250万条观测值）的大规模数据集，以预测全额提前还款时间。实证结果表明，纳入空间效应能持续提升联合模型性能，但额外加入时空交互效应的增益则不够明确。