Linear Shift Model

Definition

The linear shift model is a class of mortality trend models in which each percentile of the age-at-death distribution shifts linearly over calendar time (or birth year). This contrasts with "rate models" (like Lee-Carter) that specify trends in age-specific death rates. Under the linear shift model, period life expectancy grows linearly in calendar time and cohort life expectancy grows linearly in birth year, with cohort improvement always steeper than period improvement by the factor $1/(1-r)$ .

Key Ideas

Percentile models vs. rate models. Traditional mortality models (Lee-Carter, Gompertz extrapolations) specify trends in age-specific mortality rates $\mu(x,t)$ . "Percentile models" instead specify trends in the distribution of the age at death. The linear shift model is the simplest percentile model: all percentiles of the age-at-death distribution shift at the same constant rate $r$ per calendar year.
Linear period $e_0$ . Under the linear shift model, $e_0(t) = e_0^\circ + \bar{r}t$ — life expectancy grows exactly linearly in calendar time. This is consistent with the empirical observation of near-linear best-practice life expectancy trends documented by Oeppen and Vaupel (2002) and Lee (2003).
Linear cohort $e_0$ , with steeper slope. Cohort $e_0(\tau) = e_0^{*\circ} + \bar{s}\tau$ + correction term, where the correction is small. The cohort improvement rate $\bar{s}$ is related to the period improvement rate $\bar{r}$ by $\bar{s} = \bar{r}/(1-\bar{r})$ . Under $\bar{r} = 0.20$ years/year, $\bar{s} = 0.25$ years/birth-year — $25\%$ steeper. Under $\bar{r} = 0.25$ years/year, $\bar{s} = 0.33$ years/birth-year — $33\%$ steeper.
Period-to-cohort speed conversion: $s = r/(1-r)$ . This is an exact result under the linear shift model, not an approximation. It follows from the coordinate change $\tau = t - x$ : a shift in the age-at-death distribution that moves at speed $r$ in calendar time appears to move at speed $s = r/(1-r)$ when indexed by birth year $\tau$ . For observed $r$ in the range $0.15$ – $0.25$ years/year, the cohort improvement is approximately $20$ – $33\%$ faster.
Cross-Sectional Average Length of Life (CAL) under the linear shift model. $\text{CAL} \approx \tfrac{1}{2}(e_0'(t) + e_0''(t))$ , where $e_0'$ is the mean age at death in the constant-birth population under period rates and $e_0''$ is the achieved cohort life expectancy (LE) for the cohort whose mean age at death falls at time $t$ . This places CAL approximately midway between the two cohort-linked period measures, and below period $e_0$ .
Swedish empirical validation. Wilmoth (2005) fits the linear shift model to Sweden 1751–2003 using Human Mortality Database (HMD) data. Period $e_0$ grows approximately linearly at $\bar{r} \approx 0.17$ years/year over the full period, with short-run deviations. CAL runs $2$ – $4$ years below period $e_0$ across the record.
Cohort percentile slopes $s_c(x,\tau)$ . Under the linear shift model, the cohort percentile slope $s_c(x,\tau) = \frac{d}{d\tau}\ell_c(x,\tau) / \phi_c(x,\tau)$ is constant across ages and equals $\bar{s}$ . This simplification — uniform shift across all ages — is the model's key assumption. Empirically, the shift is not perfectly uniform across ages, which motivates a more general version.

How It Works

The core assumption: there exists a function $d(t)$ such that the age-at-death distribution at time $t$ is a shifted version of some baseline distribution: $F(x,t) = F_0(x - d(t))$ , where $d(t) = \bar{r}t$ . This means if the $p$ -th percentile of the age-at-death distribution was $x_p$ in year $0$ , it is $x_p + \bar{r}t$ in year $t$ .

Under this assumption:

Period survival: $\ell_p(x,t) = 1 - F_0(x - \bar{r}t) \;\Rightarrow\; e_0(t) = e_0^\circ + \bar{r}t$
Cohort survival: $\ell_c(x,\tau) = 1 - F_0(x - \bar{r}(\tau+x)) = 1 - F_0(x(1-\bar{r}) - \bar{r}\tau) \;\Rightarrow\; e_0(\tau) \approx e_0^\circ/(1-\bar{r}) + \bar{r}/(1-\bar{r})\cdot\tau = e_0^{*\circ} + \bar{s}\tau$

The coordinate change $\tau = t-x$ is what introduces the $(1-\bar{r})$ denominator and makes cohort improvement faster than period improvement.

Why It Matters

Analytical tractability. The linear shift model is one of the few mortality trend models that yields closed-form expressions for both period and cohort $e_0$ and their relationship to CAL. This makes it useful for deriving intuitions that must otherwise be obtained by simulation.
Empirical plausibility. The near-linear trend in best-practice $e_0$ is one of the most robust stylized facts in demography (Oeppen and Vaupel 2002; Lee 2003). The linear shift model is the simplest model consistent with this fact, unlike Lee-Carter (which implies eventual deceleration in $e_0$ gains).
Period vs. cohort discrepancy. The $s = r/(1-r)$ conversion quantifies how much cohort improvement exceeds period improvement. For Old-Age, Survivors, and Disability Insurance (OASDI) solvency, using period $e_0$ trend extrapolation implicitly understates future cohort longevity by the factor $1/(1-\bar{r})$ — relevant to the known optimism bias in Social Security Administration (SSA) forecasts. See SSA Mortality Forecasting.

Open Questions

The linear shift model assumes a uniform shift across all age-percentiles. In practice, improvement rates differ by age (especially at very old ages). A generalized version with age-specific shift rates is more realistic but loses closed-form tractability.
Whether the linear shift model outperforms Lee-Carter or its extensions in out-of-sample mortality forecasting has not been formally tested. See SSA Mortality Forecasting for the out-of-sample forecasting literature.