Low-rank Tensor Autoregressive Models for Mortality Modelling

Mortality patterns in closely related subpopulations often exhibit similarities, suggesting that mortality forecasts for individual subpopulations could be enhanced by borrowing strength from larger related groups. In this article, we focus on multipopulation mortality modelling, in which the data form a multiway mortality array comprising mortality rates of populations disaggregated by various sociodemographic attributes, such as gender, age, smoking/nonsmoking, and country or region. Each dimension of the array corresponds to one attribute. First, we propose a tensor autoregressive (TAR) model to efficiently model and forecast such multiway mortality arrays. Unlike existing vector autoregressive models, the TAR model preserves the multiway structure and more effectively incorporates patterns across groups and attributes. The proposed low-rank TAR models capture underlying low-dimensional tensor dynamics by utilizing the CANDECOMP/PARAFAC (CP) and Tucker decompositions. This yields a significant dimensionality reduction and a flexible model transformation. The CP decomposition addresses the overparameterization problem, while the Tucker decomposition enables demographic interpretations across multiple attributes. Finally, an empirical analysis using three-way mortality data (age, population, and gender) demonstrates that the proposed models achieve strong in-sample fit and satisfactory out-of-sample forecasting performance. Furthermore, we demonstrate that the proposed low-rank TAR models ensure coherence and nondivergence.