Hahn and Christiansen 2016 — Mortality Projections for Non-converging Groups of Populations

Summary

Hahn and Christiansen (2016) propose the Bayesian Multi-Population Mortality Projection (BMPMP) model, a two-stage framework that couples a Cairns-Blake-Dowd (CBD) logit mortality model for ages 40–100 with a vector error correction model (VECM) for the CBD latent parameters, estimated via Bayesian Markov chain Monte Carlo (MCMC). The paper's central empirical finding is that European national mortality rates are not converging: discrepancies are stable or widening, making the convergence assumption embedded in the Li-Lee (2005) common-factor model empirically unjustified. BMPMP handles heterogeneous populations (e.g., Hungary, Czech Republic, Poland) that Li-Lee cannot accommodate and yields left-skewed posterior predictive distributions — more probability mass for extreme improvement than for deterioration — with skewness increasing in the cointegration rank $r$ .

Key Claims

Non-convergence as empirical norm: Mortality discrepancies between European countries remain stable or widen; the Li-Lee (2005) assumption of long-run convergence to a common reference population is not supported for heterogeneous groupings.
BMPMP two-stage structure: (1) CBD logit layer — $\text{logit}(q_{x,t}) = \kappa^0_t + \kappa^x_t(x - x_0) + \kappa^{x^2}_t(x - x_0)^2$ — applied at ages 40–100 for each population; (2) VECM on the stacked CBD parameter vectors $(\kappa^0, \kappa^x, \kappa^{x^2})$ across all populations, with cointegration rank $r$ and lag order $k$ as user-selected controls.
Reference pool $p^*$ : Populations are nested in a reference pool that guides long-run equilibrium without requiring convergence; this differs from Li-Lee's single dominating reference population.
Bayesian MCMC estimation: Gibbs sampler updates hyperparameters $H$ (VECM variance structure); Metropolis-Hastings updates latent CBD parameters $K$ ; posterior predictive distributions are obtained by forward simulation.
Cointegration rank r is critical: $r = 0$ (fully unrestricted vector autoregression, VAR) fails residual adequacy checks and produces diverging forecasts; $r = 5$ used in main Western Europe study. Higher $r$ produces increasingly left-skewed posteriors — more mass on extreme mortality improvement than deterioration.
Lag order k is unimportant: $k = 1$ and $k = 2$ produce nearly identical results; short-run autoregressive (AR) dynamics do not materially affect Western European mortality projections.
Western Europe case study (France, Germany, Italy, Spain, United Kingdom; ~1960–2010 Human Mortality Database (HMD) data): France-Germany and France-Italy pairs most correlated; Spain-UK near-zero correlation. Non-convergence confirmed by stable or widening pairwise mortality gaps.
Central Europe case study (Czech Republic, Hungary, Poland): Heterogeneous group (Hungary's post-1990 mortality trajectory differs sharply from Czech Republic/Poland); Li-Lee (2005) cannot accommodate this heterogeneity; BMPMP handles it via flexible cointegration structure.
CBD scope limitation: The CBD model is valid only for ages 40–100; the BMPMP framework inherits this restriction and does not cover child or young adult mortality.
Insurance application (corrgram): Correlogram (corrgram) of projected mortality improvements across populations provides a practical diagnostic for quantifying multi-population longevity basis risk in insurance portfolios.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"We find that the differences in mortality between the groups of populations do not converge in general, which justifies modeling multi-population mortality without the assumption of convergence."

"The cointegration rank r turns out to be a very sensitive model parameter: a fully unrestricted VAR ( $r = 0$ ) does not pass residual adequacy checks and cannot handle long-run dependencies between populations."

My Take

The paper's strongest contribution is empirical: it establishes non-convergence as the default state of European mortality differentials, which is an important corrective to the convergence-centric Li-Lee literature. The BMPMP model itself is well-motivated — the CBD+VECM combination is a natural extension of the Zhou et al. (2014) VECM approach to a genuinely parity-structured multi-population setting — but the Bayesian wrapper is heavy for what it achieves: the main output is predictive intervals rather than point forecasts, and the MCMC burden is nontrivial. The finding that lag order k is irrelevant but cointegration rank r is crucial is practically useful for practitioners calibrating the model. The CBD restriction to ages 40+ is a real limitation for life-table completeness but is honestly acknowledged. The corrgram idea for insurance applications is underexplored and could be developed further.