Wilmoth 2005 — On the Relationship between Period and Cohort Mortality

Summary

Formal examination of the mathematical relationship between period and cohort mortality measures, including a taxonomy of six distinct mean-lifespan measures and a new "linear shift model" for mortality trends. Rejects the Bongaarts-Feeney (BF) claim that period life expectancy $e_0$ is distorted by mortality timing shifts, showing instead that their "tempo-adjusted $e_0$" is mathematically identical to CAL (cross-sectional average length of life) — a valid population-dynamics measure but not a correction. Derives the exact period-to-cohort speed conversion, validates the linear shift model empirically on Sweden 1751–2003, and clarifies that the period/cohort distinction is fundamentally about observation design, not bias.

Key Claims

• $e_0$ is not biased by tempo effects. The quantum/tempo distortion that affects the Total Fertility Rate (TFR) arises because TFR is a count (intensity) measure. Period life expectancy $e_0$ is itself already a timing (tempo) measure — the expected mean age at death of the synthetic cohort. No analogous distortion applies. The Bongaarts-Feeney diagnosis imports an argument from fertility that does not translate to mortality.
• Generalized tempo-adjusted $e_0$ = CAL. Bongaarts-Feeney's (2002) generalized adjustment, applied to mortality, yields a quantity mathematically identical to CAL (cross-sectional average length of life), independently derived by Brouard (1986) and Guillot (2003). CAL is the mean length of life in the population with constant births — a measure of population dynamics, not a bias correction to period life expectancy.
• Six-measure taxonomy (Table 1). The paper distinguishes six measures of mean lifespan: (1) $e_0$ — pure period, depends only on death rates at t; (2) $e_0^* = \text{CAL}$ — relative size of constant-birth population, depends on cohort survival; (3) $e_0'$ (MAD) — mean age at death in constant-birth population; (4) $e_0^{BF}$ — Bongaarts-Feeney approximation, no clear standalone interpretation; (5) expected cohort life expectancy (LE) — linear-shift projection for cohort born at t; (6) achieved cohort LE — linear-shift projection for cohort whose mean age at death is attained at t. Among these, $e_0^{BF}$ is characterized as "least interesting."
• Why $e_0$ > CAL under declining mortality. Period survival probabilities $\ell_p(x,t)$ are conditioned on current (favorable) mortality rates; cohort survival probabilities $\ell_c(x,t-x)$ are conditioned on past (higher) mortality rates for x years ago. Under improvement, the cohort-based measure CAL is necessarily lower than the period measure $e_0$. This is not a bias — the two measures answer genuinely different questions.
• Period-to-cohort speed conversion. If period $e_0$ rises at rate $r$ per calendar year, cohort $e_0$ rises at $s = r/(1-r)$ per birth year. Example: if period $e_0$ improves at $r = 0.20$ years/year, cohort $e_0$ improves at $s = 0.25$ years/birth-year — 25% steeper from the cohort perspective. This is an exact consequence of the coordinate transformation τ = t − x.
• Linear shift model. A new class of mortality models — "percentile models" — specifying that each percentile of the age-at-death distribution shifts linearly over time. Key results: period $e_0(t) = e_0^\circ + \bar{r}t$ (linear in calendar year); cohort $e_0(\tau) = e_0^{\circ*} + \bar{s}\tau + \text{correction term}$ (linear in birth year, slightly steeper); $\text{CAL} \approx \tfrac{1}{2}(e_0' + e_0'')$. The model also implies: cohort $e_0$ lies approximately halfway between period $e_0$ and period $e_0$ estimated at the mean age of death.
• Smoothing requirement for CAL. Computing CAL empirically requires a complete set of cohort survival probabilities — including probabilities for cohorts not yet observed at time t. Any practical calculation must smooth data across adjacent cohort years. This smoothing requirement is distinct from $e_0$'s calculation, which uses only current period data.
• Swedish empirical validation. The linear shift model fits Sweden 1751–2003 (Human Mortality Database (HMD) data) closely: period $e_0$ grows linearly at $\bar{r} \approx 0.17$ years/year (full period) with short-run fluctuations. CAL runs approximately 2–4 years below $e_0$ over most of the record, tracking period improvement with a lag.
• Cohort percentile slopes. The paper defines $s_c(x,\tau)$ as the pace of cohort-to-cohort shift in the percentile of the age-at-death distribution at age x: $s_c = \frac{d}{d\tau}\ell_c(x,\tau) / \varphi_c(x,\tau) = -\frac{d}{d\tau}H_c(x,\tau) / \mu_c(x,\tau)$, where $H$ is the cumulative hazard. These slopes quantify how mortality improvement at specific ages propagates into cohort life expectancy.
• Bongaarts-Feeney approximation fails. The BF formula $e_0^{BF}(t) \approx e_0(t) / (1 - \dot{r}(t))$ requires an estimate of the rate of change of $e_0$, which requires smoothing and produces further ambiguity. Even setting aside the conceptual objection, the formula lacks a clear demographic interpretation except under special cases. Guillot (2003, 2005) showed that the exact version equals CAL; the BF approximation therefore approximates CAL but does so less precisely and without the clear interpretation CAL carries.

Concepts Introduced or Extended

• Period vs. Cohort Life Expectancy — Core conceptual framework; formal taxonomy of six distinct lifespan measures
• Cross-Sectional Average Length of Life (CAL) — Shown to be equivalent to generalized tempo-adjusted $e_0$; independently derived by Brouard (1986) and Guillot (2003)
• Tempo Effects in Mortality — Negative result: tempo effects that distort TFR do not apply to $e_0$; the distinction between count and tempo measures is formalized
• Linear Shift Model — New class of percentile-based mortality trend models; yields exact period-cohort speed relationship and connects $e_0$ to cohort $e_0$ analytically
• Period Mortality — Updated: period-cohort relationship formalized; $e_0$ > CAL under declining mortality explained; Swedish empirical validation documented

Entities Mentioned

• John R. Wilmoth
• Human Mortality Database

Quotes

"The generalized tempo-adjusted life expectancy as defined by Bongaarts and Feeney equals CAL, as defined by Brouard and Guillot. … [I]t is a measure of population dynamics rather than a corrected period life expectancy."

"Life expectancy as a period measure is not subject to tempo distortion in the same sense that the TFR is distorted, because it is a measure of timing rather than quantum."

"The rate of improvement in cohort life expectancy is always greater than the corresponding rate of improvement in period life expectancy."

My Take

The paper's key contribution is disambiguating a confused debate: Bongaarts and Feeney imported a valid fertility argument (TFR counts births; tempo shifts distort counts) into a mortality context where it doesn't hold ($e_0$ is already a timing measure). Wilmoth's proof that the "correction" equals CAL reframes the issue constructively — CAL is worth computing, but as a population-dynamics measure in its own right, not as a corrected $e_0$. The six-measure taxonomy is the paper's most useful practical tool: it forces clarity about which question is being asked before any calculation begins. The Swedish empirical section confirms that CAL tracks $e_0$ with a multi-year lag under sustained improvement — directly useful for understanding why period and cohort estimates diverge in projections.

One limitation: the linear shift model assumes linearity in the percentile of the age-at-death distribution, not in log death rates (as Lee-Carter assumes). This is a different and in some ways more natural parametrization, but its empirical performance relative to Lee-Carter on out-of-sample forecasting is not tested here.