Meseguer (2006) Long-Range Forecasts of Mortality and Life Expectancy Using Bayesian Vector Autoregressions

Summary

Explores Bayesian vector autoregression (BVAR) models as an alternative to the Lee-Carter method for producing long-range probabilistic forecasts of age-specific U.S. mortality rates. The key problem is dimensionality: a VAR(1) for 21 age groups requires 462 coefficients, far exceeding the ~74 years of annual data (1928–2001). The Minnesota prior shrinks this toward age-specific random walks, with two cross-variable shrinkage specifications compared. Model selection via marginal likelihood favors the specification using sample-correlation–weighted cross-variable shrinkage; this model reduces point-forecast errors 2–22% vs. the near-zero cross-variable benchmark and produces wider, better-calibrated predictive intervals than the Lee-Carter method.

Key Claims

• Dimensionality problem in mortality VAR. With $m=21$ age groups and $p=1$ lag, an unrestricted VAR has $21^2 = 441$ autoregressive coefficients plus 21 constants. With $T \approx 74$ observations, estimation is infeasible without strong prior restrictions.
• Minnesota prior applied. Coefficients $a_{ijp}$ get independent normal priors: $N(1, \sigma^2_{ijp})$ for the first lag of the own variable (random-walk centering), $N(0, \sigma^2_{ijp})$ otherwise. Prior standard deviations (SDs): $\sigma_{ijp} = \lambda_1 / (p^{\lambda_3} \cdot \sigma_i) \times \lambda_2(i,j)$, where $\lambda_1$ governs overall tightness, $\lambda_3$ governs lag decay, and $\lambda_2(i,j)$ governs cross-variable relative weights.
• Two BVAR specifications compared.
  • BVAR(1)-I: $\lambda_2(i,j) = 0.001$ for $i \neq j$ — near-zero cross-variable shrinkage; effectively 21 near-independent random walks.
  • BVAR(1)-II: $\lambda_2(i,j) = 0.8 \times \Omega_{i,j}$ for $i \neq j$, where $\Omega_{i,j}$ is the sample correlation of $\Delta\log y_i$ and $\Delta\log y_j$ — adjacent age groups get higher cross-variable weight. This captures the strong correlation among neighboring age groups (correlations 0.7–0.97 for adjacent ages).
• Normal-diffuse prior and MCMC. A diffuse prior on the residual covariance $\Sigma$ is combined with a multivariate normal prior on the VAR coefficients. Unlike the Normal-Wishart conjugate, this yields no analytical posterior — Markov chain Monte Carlo (MCMC) is used for estimation.
• Model selection via marginal likelihood. Log marginal likelihood: BVAR(1)-I = 4,276.5 vs. BVAR(1)-II = 4,478.7. The difference strongly favors BVAR(1)-II, confirming that incorporating the correlation structure of age-specific mortality improves predictive fit.
• Forecast comparison vs. Lee-Carter. BVAR(1)-II reduces the sum of squared residuals (SSR) by 2–22% for most age groups (exception: ages 50–54). Both models have ~94–98% coverage of 90% credible intervals. Crucially, BVAR intervals widen with forecast horizon and age, while Lee-Carter intervals are too narrow for older ages — a known failure mode of the Lee-Carter approach which ignores parameter uncertainty.
• Long-range projections. Life expectancy at birth forecast to rise from ~78 years (2010) to ~83 years (2050) under BVAR(1)-II, with 90% interval width of 7.3 years by 2050 vs. 4.7 years for Lee-Carter — substantially greater uncertainty acknowledged.
• Precursor to Meseguer (2010). This 2006 report is an earlier, more concise version of the extended analysis in the 2010 paper, which adds additional model specifications, longer data, and Social Security trust-fund framing.

Concepts Introduced or Extended

• Minnesota Prior
• Bayesian VAR

Entities Mentioned

• (none with existing wiki pages)

Quotes

"The BVAR approach presented in this report seems promising in addressing some of the shortcomings of the Lee-Carter method."

My Take

A clear demonstration that the Minnesota prior solves the dimensionality problem in mortality VAR modeling and that incorporating sample-correlation information in the cross-variable shrinkage structure is beneficial. The comparison with Lee-Carter is methodologically important: BVAR naturally propagates parameter uncertainty into forecast intervals, while Lee-Carter's classical confidence intervals understate tail risk for older age groups. The paper is an internal Social Security Administration (SSA) working paper and less polished than the 2010 follow-up, but makes the key methodological points cleanly. The choice of normal-diffuse prior (vs. Normal-Wishart conjugate) adds MCMC cost without a clear payoff; the 2010 paper likely clarifies whether this choice was necessary.