  When modeling sub-national mortality rates, we should consider three
features: (1) how to incorporate any possible correlation among sub-populations
to potentially improve forecast accuracy through multi-population joint
modeling; (2) how to reconcile sub-national mortality forecasts so that they
aggregate adequately across various levels of a group structure; (3) among the
forecast reconciliation methods, how to combine their forecasts to achieve
improved forecast accuracy. To address these issues, we introduce an extension
of grouped univariate functional time series method. We first consider a
multivariate functional time series method to jointly forecast multiple related
series. We then evaluate the impact and benefit of using forecast combinations
among the forecast reconciliation methods. Using the Japanese regional
age-specific mortality rates, we investigate one-step-ahead to 15-step-ahead
point and interval forecast accuracies of our proposed extension and make
recommendations.
