Denton Feaver and Spencer 2005 — Time Series Analysis and Stochastic Forecasting

Summary

Denton, Feaver, and Spencer (2005) explore alternative time series models and stochastic forecasting methods for Canadian mortality rates (1926–2000), explicitly positioning them as competitors to the Lee-Carter (LC) approach. They find that regardless of which stochastic method is used — nonparametric block bootstrap, partially parametric second-order autoregressive (AR(2)) + bootstrap, or fully parametric AR(2) + multivariate normal — the 50th percentile forecasts of life expectancy are nearly identical (within ~0.5 years at age 0 in 2050), and that the assumption of normality vs. the use of bootstrap residuals makes essentially no difference to interval width. The paper also investigates the separate effects of maximum life table age and long-run mortality slowdown on deterministic life expectancy forecasts.

Key Claims

• All four aggregate log-mortality indexes (three fixed-weight arithmetic, one Divisia) show near-linear downward trends over 1926–2000: linear trend fits account for >99% of total variation. This is consistent with the Lee-Carter first singular value series for Group of Seven (G7) countries.
• Disaggregated system models: (a) an AR(2) on log first-differences for each of 38 age-sex groups, estimated jointly by iterated seemingly unrelated regressions (SURE; system $R^2 = 0.9981$), and (b) a first-order quasi vector-autoregression (QVAR(1)) adding a lagged aggregate index to the own-lag. Both have negative autocorrelation in log first-differences (37/38 negative first-lag coefficients in AR(2)); statistical performance comparable; AR(2) slightly preferred for stochastic use.
• Three stochastic methods produce nearly identical central forecasts (Table 6): 50th percentile of mean life expectancy at age 0 in 2050: (a) nonparametric block bootstrap = 84.48 years, (b) AR(2)+bootstrap = 84.76 years, (c) AR(2)+multivariate normal = 84.74 years. These are 0.5–0.8 years below the Tuljapurkar et al. (2000) LC-based forecast for Canada (85.26), and 3.1–3.5 years above the official 1998 Canada Pension Plan central forecast (81.67).
• Parametric vs. bootstrap makes no material difference to interval width: methods (b) and (c) give nearly identical 90% probability intervals (e.g., 3.43 vs. 3.39 years at age 0 in 2050). Block bootstrap (method (a)) gives narrower intervals (2.51 years) because block length 25 constrains the effective sample; shorter block length converges to (b) and (c).
• Tuljapurkar et al. (2000) LC interval for Canada in 2050 = 2.78 years at age 0 — narrower than methods (b) and (c) and similar to (a).
• Maximum life table age (109 to 140 by 2100) has negligible effect on life expectancy forecasts (largest difference at age 0 in 2100 = 0.05 years); assumptions about long-run slowdown of mortality decline matter more (~5 years at age 0 between no-slowdown and zero-decline-by-2100 under deterministic forecasting).
• Correlation structure in AR(2) residuals: 703 cross-age correlations, mostly positive; only 44 (6.3%) negative; highest correlation 0.85 (male vs. female under-1); adjacent same-sex age groups mostly ≥ 0.4; high cross-sex correlations within age groups. Justifies SURE joint estimation on efficiency grounds.
• Calls for Bayesian combination of historical-data-based stochastic forecasts with subjective probability distributions over alternative long-run trend scenarios (citing Tuljapurkar and Boe 1999 as precedent for this approach in fertility).

Concepts Introduced or Extended

• Lee-Carter Model — method-robustness result: non-LC stochastic methods yield virtually identical 50th percentile mortality forecasts; normality vs. bootstrap residuals has negligible effect on interval width; LC itself not estimated but compared via Tuljapurkar et al. (2000) Canadian LC results.

Entities Mentioned

• Frank T. Denton
• Christine H. Feaver
• Byron G. Spencer
• Ronald Lee
• Shripad Tuljapurkar
• James W. Vaupel

Quotes

"We find that it makes little difference which method one uses — a nonparametric (bootstrap) method, a partially parametric method or a fully parametric method."

"Whether or not subjective probabilities are brought into play explicitly, stochastic demographic forecasting allows the possibility... of defining a loss function in a particular demographically based planning situation, a corresponding risk function, and then minimizing the latter as a guide to policy choice."

My Take

This is a thorough methods paper that provides strong robustness evidence for stochastic mortality forecasting: if non-LC methods give essentially the same central forecasts as LC, and bootstrap vs. normality barely changes the intervals, then the specific technical choices in stochastic mortality modeling are less consequential than is sometimes assumed. The real uncertainty is structural — whether historical trends will continue — not methodological. The paper's main limitation is its exclusive focus on Canadian data and its avoidance of out-of-sample validation (it doesn't test whether any method actually covers realized future mortality). The call for Bayesian mixture approaches to handle structural uncertainty is well-motivated and anticipates later work in this tradition.