Carter and Prskawetz 2001 — Examining Structural Shifts in Mortality Using the Lee-Carter Method

Summary

Carter and Prskawetz extend the Lee-Carter (LC) model by fitting 30 rolling 24-year submatrices (a sliding singular value decomposition (SVD) window) to Austrian single-age mortality data 1947–1999, yielding time-varying realizations of the $a_x$, $b_x$, and $k_t$ components. They find that $a_x$ and $k_t$ are relatively stable across submatrices while $b_x$ is not: pre-1970 sensitivity is concentrated at young and middle ages, post-1970 it shifts increasingly to old ages, revealing a structural break around 1968. The extended method improves life expectancy fit by $\approx 0.3$ years for females and substantially for $e_{60}$ in both sexes; forecasts to 2050 are sensitive to base-period choice, especially for males ($+3.5$ years by 2050 from the most-recent subsample vs. the full series).

Key Claims

• $b_x$ instability: The age-sensitivity profile is NOT constant over the full observation window. Pre-1970 submatrices show high $b_x$ at young/middle ages; post-1970 submatrices show high $b_x$ at old ages. This structural shift in age-specific improvement rates $\approx 1968$ is empirically confirmed for Austria.
• $a_x$ and $k_t$ stability: In contrast, the average log-mortality schedule ($a_x$) and the mortality index ($k_t$) are relatively stable across the 30 submatrices, making $b_x$ the locus of model instability.
• Extended LC fit improvement: Using the most-recent 1976–1999 subsample for the $b_x$ profile (with the full series' $k_t$ drift for forecasting) reduces the observed-vs-estimated $e_0$ gap by $\approx 0.3$ years for females. Improvement for $e_{60}$ is larger and more consistent across both sexes.
• Base-period sensitivity: Forecasting from the full series (1947–1999) vs. the latest subsample (1976–1999) yields a difference of $+1.3$ years for females and $+3.5$ years for males by 2050. The male subsample forecast exits the full-series $95\%$ confidence interval (CI) within the first decade of the 21st century.
• Austrian $e_0$ forecasts (full series, to 2050): Female $81 \to 88$, male $75 \to 82$. Official Statistics Austria forecast (Hanika 2000) lies within the LC $95\%$ CI across the entire horizon.
• Method: Wilmoth (1993) weighted SVD; normalizations $\sum b_x^2 = 1$, $\sum k_t = 0$; Box-Jenkins random walk with drift for $k_t$; forecasts bias-adjusted following Bell (1997).

Concepts Introduced or Extended

• Lee-Carter Model ($b_x$ instability — empirical evidence; shortcoming #2 tested on Austrian data)
• Period Mortality

Entities Mentioned

• Lawrence Carter
• Alexia Prskawetz
• Ronald Lee
• John R. Wilmoth
• William R. Bell

Quotes

"Since a first investigation of these components shows A_{x,s} and K_{t_s} to be relatively stable, the focus here was on the components of B_{x,s}."

"The fact that we observe very high sensitivity at very low ages also in the latter periods might be due to the small numbers of people in these age groups, which means that even a modest mortality decline will have a pronounced effect."

"Our simulations should be taken as evidence that the choice of the base period may not be negligible — particularly in the case of the data on Austrian males we use here."

My Take

The key contribution is providing direct, well-visualized empirical evidence (the contour plots of $B_{x,s}$) that $b_x$ is not stable over time — specifically the shift from young/middle-age improvement toward old-age improvement around 1968 in Austria. This is the clearest empirical demonstration of Lee-Carter shortcoming #2. The limitation is that the extended method is descriptive rather than inferential: no formal statistical test for structural breaks in $b_x$ is offered, and the rolling-window selection of base period is ad hoc. The paper is also Austria-specific; the applicability to other countries is suggested but not demonstrated. That said, the finding aligns with Rau et al. (2006)'s cross-national evidence of accelerating old-age improvement, suggesting the Austrian pattern is not idiosyncratic. The base-period sensitivity results are the most practically important for forecasters: the difference of $+3.5$ years for males by 2050 is substantial, and the fact that the male subsample forecast exits the full-series CI so quickly ($\approx 2010$) is a strong warning against uncritical use of the full historical series.