Forecasting Vital Rates and Population with Bayesian Vector Autoregressions

Summary

Meseguer (2010) develops a Bayesian vector autoregression (BVAR) framework for jointly forecasting age-specific mortality and fertility rates over a 92-year horizon for 38 countries, applying Villani's (2008) mean-adjusted BVAR to demographic time series. Three models are compared: Lee-Carter (the standard demographic benchmark), a diffuse-prior BVAR, and an informative-prior BVAR. Lee-Carter dramatically understates long-run uncertainty — the actual 65+ female population is already outside its 90% credible interval by 2009. The informative BVAR closely matches the UN Medium Variant (Alt 2), while the diffuse BVAR overshoots. Population forecasts are obtained by plugging BVAR vital rate draws into the cohort-component projection identity.

Key Claims

• Lee-Carter understates forecast uncertainty severely at long horizons; the actual 65+ female population falls outside its 90% credible interval from 2009 onward despite a 1995 base year.
• The informative BVAR closely matches the UN Medium Variant (Alt 2) projection; the diffuse BVAR overshoots with total fertility rate (TFR) 2.13 vs. target 2.0 and total population 601.5M vs. Alt 2's 514.8M by 2087.
• Mortality is modeled with a 22-age-group BVAR for each gender (p=1 lag; $\Delta\log$ mortality; $\lambda_1=0.2$; prior AR mean = 0).
• Fertility is modeled with a 7-age-group BVAR (p=2 lags; log fertility; $\lambda_1=0.05$, tight; prior AR mean = 0.8 encoding persistence near replacement).
• The Minnesota $\lambda_2$ cross-variable tightness is made correlation-adaptive: $\lambda_2(i,j) = 0.8 \times \mathrm{Corr}[\Delta\log y_i, \Delta\log y_j]$, allowing closely related age groups to share information.
• Stationarity is enforced by rejecting any Gibbs draw for which the spectral radius of the companion matrix exceeds 1 — a simple indicator-function constraint: $I(\Phi) = 1$ iff $|\lambda_{\max}| < 1$, else 0.
• Population projections are obtained via the cohort-component identity applied to each BVAR posterior draw, giving a full distributional forecast for any population aggregate.

Concepts Introduced or Extended

• Steady State VAR — correlation-adaptive $\lambda_2$; stationarity rejection constraint; informative vs. diffuse BVAR comparison; demographic application
• Lee-Carter Model — Lee (1993) fertility extension; systematic underestimation of long-run uncertainty; BVAR comparison
• Cohort Component Method — demographic projection identity propagating vital rate forecasts to age-structured population
• Sims-Zha Prior — explicit $\lambda_1$–$\lambda_4$ Minnesota formulation with correlation-adapted $\lambda_2$

Entities Mentioned

• Javier Meseguer
• Mattias Villani

Quotes

"The advantage of the BVAR approach is that we can set prior beliefs about both the persistence of the vital rates and the steady-state level towards which they converge."

My Take

A valuable bridge between the Bayesian VAR econometrics literature (Villani 2008) and demographic projection practice (Lee-Carter). The correlation-adaptive $\lambda_2$ is a clever practical extension that relaxes the standard Minnesota assumption of uniform cross-variable shrinkage, exploiting the strong correlations between adjacent age groups in mortality and fertility schedules. The stationarity rejection constraint is simple but effective. The comparison with Lee-Carter is damning: the model that dominates academic demography and is used for official projections understates forecast uncertainty even over 10–15 year horizons. Limitations: migration is treated as exogenous rather than jointly modeled; the Normal-diffuse (non-conjugate) prior requires Gibbs sampling and prevents closed-form posterior inference; the paper focuses on aggregate population totals without detailed uncertainty attribution by component (mortality vs. fertility vs. migration).