Lee and Carter 1992 — Modeling and Forecasting U.S. Mortality

Summary

Lee and Carter (1992) introduce a parsimonious bilinear model for forecasting age-specific mortality: $\ln m(x,t) = a_x + b_x k_t + \varepsilon_{x,t}$, where $a_x$ is the time-averaged log death rate at age $x$, $b_x$ captures each age's sensitivity to the overall trend, and $k_t$ is a latent mortality index estimated by singular value decomposition (SVD) on demeaned U.S. mortality data 1933–1987. The index $k_t$ is then modeled as an autoregressive integrated moving average, ARIMA(0,1,0), random walk with drift, yielding a central forecast of U.S. life expectancy at birth of 86.05 years in 2065 — 5.6 years above the Social Security Administration's (SSA) 1989 intermediate projection. The paper includes discussant comments by McNown (algebraic equivalence to independent exponential rate extrapolation) and Alho (confidence intervals understated because the flu dummy is excluded from error variance).

Key Claims

• Fitting to National Center for Health Statistics (NCHS) U.S. vital statistics 1933–1987 (22 age groups, annual data), the first singular vector pair accounts for 92.5% of the total variance in log age-specific death rates.
• $a_x$ (the age profile of log mortality) is U-shaped with its minimum around ages 10–14, following Gompertz regularities.
• $b_x$ (age-specific sensitivity to the mortality index) is highest at infant and child ages (0.09–0.11) and lowest at oldest ages (0.029–0.033 at ages 85+); this reflects faster historical proportional decline at young ages.
• After SVD, $k_t$ is re-estimated by the value that makes the model's implied life expectancy match the actual observed $e_0$ each year — correcting for Jensen's inequality bias from modeling log rates.
• $k_t$ declined approximately 364.5 units over 1933–1987; modeled as ARIMA(0,1,0) with drift: $k_t = k_{t-1} - 0.365 + 5.24 \cdot \text{flu} + e_t$, with $R^2 = 0.995$ and $\text{se} = 0.651$ (flu is a 1918 pandemic dummy).
• Central forecast: $k(2065) = -38.80$, implying $e_0 = 86.05$ years in 2065. The 95% confidence interval has half-width 5.6 years: [80.45, 91.65]. SSA's 1989 intermediate projection of $e_0 = 80.45$ in 2065 falls exactly at the lower bound of the Lee-Carter (LC) interval.
• By 2065, 46% of a birth cohort would survive to age 90 (vs. ~20% in 1987); 74% of all future person-years would accrue at ages 65+.
• McNown's comment: The LC model is algebraically equivalent to projecting each age-specific death rate at its own historical exponential trend rate independently — the bilinear structure is a compact notation for that procedure, not an independent constraint.
• Alho's comment: The 95% confidence intervals (CIs) are too narrow. The flu pandemic dummy absorbs a large one-time shock from $k_t$ but is excluded from $k_t$'s error variance; correcting for this would widen the CI by approximately 57%. Alho argues the intervals should be roughly twice as wide.

Concepts Introduced or Extended

• Lee-Carter Model
• Period Mortality
• SSA Mortality Forecasting

Entities Mentioned

• Ronald Lee
• Lawrence Carter
• Office of the Chief Actuary

Quotes

"It is useful to think of the $[k_t]$ series as a mortality index which, when combined with the age-specific schedule $b_x$, provides a one-dimensional summary of the level of mortality." (p. 660)

"While the projection of $k$ is based on standard stochastic time series methods...the resulting uncertainty in $e_0$ is quite small...the 95% confidence interval for the year 2065 has a half width of only 5.6 years." (p. 665)

"[The SSA's] medium projection of 80.45 for life expectancy at birth in 2065 is more pessimistic than our stochastic lower bound." (p. 665)

My Take

The LC model's elegance is also its limitation. Reducing all age-specific mortality dynamics to a single scalar $k_t$ discards the cross-age correlation structure that Bayesian vector autoregression (BVAR) and other multivariate methods exploit, and the reliance on a single factor means the model cannot distinguish between mortality shocks concentrated at particular age groups. McNown's algebraic equivalence critique does not invalidate the model — exponential extrapolation of each age-specific rate is a defensible baseline — but it deflates the apparent sophistication of the bilinear structure: the SVD is a compression device, not a separate identifying assumption. Alho's critique about the flu dummy is operationally important: excluding the pandemic dummy from $k_t$'s error variance artificially narrows the confidence intervals by roughly 57%. Meseguer (2008) empirically confirmed this directional claim at multi-decade horizons, finding that LC's 80% intervals achieve only 50–70% coverage for life expectancy (LE) and near-zero coverage for dependency ratios across 16 countries. Lee and Miller (2001) further showed that LC's trend-rate estimate is conservative — realized LE gains in several countries exceeded the model's forecasts. The model's 1965–2065 central forecast ($e_0 = 86.05$) exceeded SSA's projection by 5.6 years, a discrepancy later attributed by Lee (2003) to SSA's systematic underestimation of the pace of mortality improvement.