Booth Maindonald and Smith 2002 — Age-Time Interactions in Mortality Projections Applying Lee-Carter to Australia

Published version: Booth, H., J. Maindonald, and L. Smith. (2002). "Applying Lee-Carter Under Conditions of Variable Mortality Decline." Population Studies 56(3): 325–336. This source page covers the Australian National University (ANU) Working Paper version (revised 2 August 2002); the published title differs from the working paper title.

Summary

Booth, Maindonald, and Smith adapt the Lee-Carter model to Australian mortality data (1907–1999), arguing that the widely acknowledged problems of $k(t)$ non-linearity and $b(x)$ instability are jointly caused by inappropriate fitting-period selection. They introduce an objective selection criterion — the ratio $R(S)$ of log-linear to log-additive chi-squared fit — and identify 1968 as the structural break and optimal starting year for Australia. Within the 1968–1999 optimal period, both problems are substantially resolved; the model produces a forecast of $e_0 = 92.0$ years by 2050, well above the official Australian Bureau of Statistics (ABS) projection of $84.9$ years.

Key Claims

Australian $k(t)$ departs from linearity over 1907–1999; $b(x)$ is correspondingly unstable across sub-periods. These are not independent problems: both stem from fitting over an inappropriate (too long) period.
$R(S) = \chi^2_{\text{loglin}}(S) / \chi^2_{\text{logadd}}(S)$ — the ratio of chi-squared from a log-linear fit to chi-squared from a log-additive (LC) fit over a starting period $S$ — is an objective criterion for selecting the fitting period. The marginal version $RD(S)$ identifies the structural break year directly. $R(S)$ near $1$ indicates a good LC fit.
For Australia, $RD(S)$ identifies 1968 as the structural break; the optimal fitting period is 1968–1999.
The year 1950 — Lee and Miller's "simple and satisfactory solution" for most countries — is among the worst starting years for Australia.
Within the 1968–1999 period, $b(x)$ stability is substantially improved, jointly resolving the two key Lee-Carter shortcomings.
$k(t)$ is re-estimated by Poisson quasi-maximum likelihood, adjusting to match observed death counts $D(x,t)$ each year (Maindonald 1984 deviance criterion), rather than the LC default of life expectancy matching.
An expanded singular value decomposition (SVD) (up to three singular value pairs) captures cohort-period effects, but higher-order terms are not readily extrapolatable; the practical forecast rests on the first term alone.
Australian $B(x)$ by era: 1907–1949 = young-age improvement dominant; 1950–1967 = $\approx$ age 30 dominant; 1968–1999 = old ages and childhood dominant. The post-1968 shift to old-age improvement explains the higher forecasts from the optimal period.
Forecast (both sexes, 1968–1999 base): $e_0 = 92.0$ years by 2050 (from $79.6$ in 1999); $95\%$ prediction interval (PI): $85.1$ – $99.8$ . The ABS official projection of $84.9$ years falls just inside the lower bound.
Carter and Prskawetz (2001) is critiqued for examining $b(x)$ instability heuristically without first verifying $k(t)$ linearity; they "refrain from drawing definite conclusions" because their approach lacks a theoretical model for $b(x)$ .

Concepts Introduced or Extended

Lee-Carter Model

Entities Mentioned

Quotes

"The choice of fitting period is not independent of the assumption of linearity in k(t)."

"Providing that the fitting period is chosen appropriately, both problems — linearity of k(t) and invariance of b(x) — are substantially resolved."

"The year 1950 selected by Lee and Miller as a 'simple and satisfactory solution' for most countries is, for Australia, among the worst starting years."

My Take

The $R(S)/RD(S)$ criterion is the paper's most durable contribution: it converts an ad hoc judgment about when "the linear regime starts" into a reproducible, data-driven procedure. The finding that 1950 is among the worst starting years for Australia is a sharp empirical rebuke to the Lee-Miller universal recommendation. The 92-year forecast is provocative — the $95\%$ PI barely overlaps the ABS projection — but the methodology's rationale is principled: forecast only from the regime that demonstrably fits. The main limitation acknowledged by the authors is that neither the second nor third SVD term is extrapolatable, so the practical forecast still rests on the first term alone.