Lee and Tuljapurkar 1994 — Stochastic Population Forecasts for the United States Beyond High Medium and Low

Summary

Lee and Tuljapurkar develop the first fully stochastic, probabilistically consistent population projection for the United States, combining Lee-Carter stochastic mortality ($K_t$ as random walk with drift) with a constrained autoregressive moving average (ARMA) fertility model and Tuljapurkar's perturbation theory of random Leslie matrix products to generate joint probability intervals for all demographic quantities simultaneously. The key finding is that conventional high-medium-low (HML) scenario projections are mathematically incoherent: the same HML bounds imply a 95% confidence interval (CI) for total population size but must simultaneously correspond to a different — and inconsistent — probability level for dependency ratios, age-group sizes, and vital rates. The Census Bureau's 2065 intervals for the 65+ population are nearly three times too wide while their elderly dependency ratio interval is twelve times too narrow.

Key Claims

• HML scenarios cannot be probabilistically consistent across derived demographic quantities simultaneously; the same bounds that represent, say, a 90% CI for fertility must correspond to different probability levels for population totals vs. age-group sizes vs. dependency ratios.
• The quadratic perturbation approximation to random Leslie matrix products analytically computes means and variances of all outputs without Monte Carlo simulation; accurate to ~2% for population totals at 80 years and ~15% for variance (runs in minutes on a workstation vs. 10 days for the quartic approximation).
• Expected U.S. total population in 2065: ~398M, 95% CI 258M–642M; asymmetric because $P_t$ is lognormally distributed under multiplicative dynamics.
• Life expectancy in 2065: 86.1 years (95% CI: 81–90 years).
• Old-age dependency ratio in 2065: 0.47 (95% CI: 0.26–0.68); Census interval is 12× broader than Lee-Tuljapurkar; Social Security interval is consistent in sign but from a lower point forecast.
• Fertility uncertainty dominates after ~2030 for total population and youth dependency ratio; mortality uncertainty matters mainly for the elderly group through ~2025.
• Baby-boom age structure creates predictable demographic echoes — humps in each age-group forecast as cohorts pass through life stages — visible in all outputs.
• A logistic-bounded fertility model (constraining the total fertility rate (TFR) to [0, 4]) widens prediction intervals ~50% vs. the mean-constrained ARMA, because the nonlinear transform converts the normal $G_t$ distribution into a more rectangular $F_t$ distribution; the unconstrained mean-only model is preferred.

Concepts Introduced or Extended

• Probabilistic Population Forecasting
• Lee-Carter Model
• Stochastic Fertility Forecasting

Entities Mentioned

• Ronald Lee
• Shripad Tuljapurkar

Quotes

"Conventional population projections use 'high,' 'medium,' and 'low' scenarios to indicate uncertainty, but probability interpretations are rarely given, and in any event the resulting ranges for vital rates, births, deaths, age groups sizes, age ratios, and population size cannot possibly be probabilistically consistent with one another."

"Our interval for the elderly dependency ratio is fully 12 times as broad [as the Census Bureau's]! These items have major implications for policy, and these contrasting indications of uncertainty clearly show the limitations of the conventional scenario-based methods."

My Take

The paper's central insight — that HML scenarios are internally inconsistent across derived quantities — is devastating to standard practice and remarkably simple once stated. The quadratic approximation is elegant: rather than simulating sample paths, they analytically propagate second-order moments through the Leslie recursion, recovering most of the Monte Carlo accuracy at a fraction of the cost. The specific policy finding that official projections are simultaneously overconfident about age-group sizes and wildly underconfident about dependency ratios explains why Social Security actuarial solvency analyses underestimated risk for decades. The one important limitation is the assumption that fertility and mortality residuals are uncorrelated (reasonable for post-1940 U.S., not universal). Together with Lee-Carter (1992) and Lee-Tuljapurkar (1998), this forms the methodological spine of modern stochastic demographic forecasting.