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Bayesian Population Projections for the United Nations

Adrian E. Raftery, Leontine Alkema, Patrick Gerland

arXiv:1405.4708v1stat.ME

TL;DR

UN population projections are widely used, but their deterministic standard method does not quantify uncertainty. The paper develops Bayesian hierarchical projections of fertility and life expectancy, then combines them with cohort-component projections; the approach performed reasonably well out of sample and was used experimentally by the UN.

  • Problem

    The UN’s Medium population projection provides a single value without uncertainty, while its Low and High variants are scenarios without probabilistic interpretation.

    Source: §1 · p. 2

  • Method

    Bayesian hierarchical models generate posterior predictive trajectories for fertility and life expectancy, which feed the cohort-component method to produce distributions for future population quantities.

    Source: §5 · pp. 8–9

  • Results

    The method performed reasonably well in out-of-sample predictive performance.

    Source: §5 · p. 9

  • Takeaways & Limitations

    The UN used the method for experimental probabilistic projections and considered it for official projections in WPP 2014.

    Source: §5 · p. 10

    The method omits uncertainty in base population, sex ratio at birth, age-specific fertility and mortality, and net migration, and is not adapted to countries with generalized HIV/AIDS epidemics.

    Source: §5 · p. 9

Abstract

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The United Nations regularly publishes projections of the populations of all the world's countries broken down by age and sex. These projections are the de facto standard and are widely used by international organizations, governments and researchers. Like almost all other population projections, they are produced using the standard deterministic cohort-component projection method and do not yield statements of uncertainty. We describe a Bayesian method for producing probabilistic population projections for most countries which are projections that the United Nations could use. It has at its core Bayesian hierarchical models for the total fertility rate and life expectancy at birth. We illustrate the method and show how it can be extended to address concerns about the UN's current assumptions about the long-term distribution of fertility. The method is implemented in the R packages bayesTFR, bayesLife, bayesPop and bayesDem.

1. INTRODUCTION

The UN’s country population projections are a widely used standard, but its Medium projection gives a single value without uncertainty and its Low and High variants are not probabilistic. This section motivates and describes an ongoing effort to develop Bayesian probabilistic projections for UN use.

1. INTRODUCTION

  • UN projections are a de facto standard used for planning, social and health research, development-goal monitoring, and as inputs to other forecasts.The cited applications include climate-change impact forecasts (Intergovernmental Panel on Climate Change (2007); Seto, G¨uneral and Hutyra (2012)).Source: §1 · p. 1
  • The UN uses a deterministic cohort-component method based on an age-structured accounting identity for population, births, deaths, and migration.The identity relates the population at time t+1 to the population at time t, births, deaths, immigration, and emigration (Whelpton, 1936; Leslie, 1945; Preston, Heuveline and Guillot (2001)).Source: §1 · p. 1
  • The UN’s Medium projection gives one future population value without uncertainty, while its Low and High variants are alternative scenarios without probabilistic interpretation.The variants use total fertility rates half a child below or above the Medium projection.Source: §1 · p. 2
  • The proposed ongoing effort develops probabilistic projections for UN use, building on prior time-series methods and the work of Lee and collaborators.The cited work includes Lee and Carter (1992), Lee and Tuljapurkar (1994), and Lee (2011).Source: §1 · p. 2

2. CURRENT UN POPULATION PROJECTION METHODOLOGY

The current UN method projects population by age using fertility, survival, and migration, with future rates specified for each projection period. It projects overall fertility and mortality before converting them to age-specific rates.

2.1 Cohort Component Projection Method

  • The cohort-component method advances age-group populations using births, survival, and net migration, and recursively applies the projection matrix to obtain population projections.The formulation also extends to two-sex populations; future age-specific fertility, mortality, and migration must be specified, and uncertainty in these quantities accounts for most uncertainty about future population.Source: §2.1 · pp. 2–3

2.2 Projecting Mortality and Fertility Rates

  • For most countries, the UN projects overall fertility and mortality levels and converts them to age-specific rates.Life expectancy gains are projected with a deterministic double logistic function, whose country parameters are selected by a UN analyst from five predetermined patterns.Source: §2.2 · p. 3, Figure 1 · p. 3
  • The UN projects total fertility rate decrements with a double logistic function and holds fertility at 1.85 children; below that level at projection start, it increases by 0.05 children per five-year period until reaching 1.85.Age-specific rates are derived from past patterns or model life tables, and High and Low variants shift future total fertility rates by half a child.Source: §2.2 · p. 3

3. BAYESIAN POPULATION PROJECTIONS

The paper replaces the UN’s deterministic assumptions with Bayesian hierarchical models that generate probabilistic trajectories for fertility and life expectancy, then feed those trajectories into cohort-component projections. Across 159 countries, the projections cover just under 90% of the world’s 2010 population; Egypt’s example highlights a projected decline in the potential support ratio amid substantial uncertainty about future population size.

3. BAYESIAN POPULATION PROJECTIONS

  • The method generates posterior predictive trajectories for fertility and life expectancy and uses them in the cohort-component method to produce distributions for future population quantities.For 159 countries, the projections used 2,000 fertility trajectories and 2,000 joint female and male life-expectancy trajectories for each five-year period from 2010 to 2100, alongside current UN migration assumptions.Source: §3 · p. 4, §3.3 · p. 6

3.1 Bayesian Fertility Projection Model

  • For Phase II fertility declines, the model adds heteroscedastic error, allows country parameters to vary continuously, and estimates them hierarchically from a world distribution.Hierarchical borrowing stabilizes estimation when country data are sparse: most countries have at most 12 five-year periods for estimating a curve with five parameters.Source: §3.1 · p. 4
  • For Phase III, the model uses a first-order autoregression around a long-term mean of 2.1 children, the approximate replacement fertility level.Phase III is the post-fertility-transition period; by the paper’s definition, 21 countries had entered it by 2010.Source: §3.1 · pp. 4–5

3.2 Bayesian Life Expectancy Projection Model

  • For female life expectancy, the model makes expected gains stochastic, allows country-specific double-logistic parameters to vary continuously, and draws them from common world distributions.The asymptotic increase parameter is restricted to less than 1.15 years per five-year period, based on the stated historical upper bound.Source: §3.2 · p. 5
  • Female and male life expectancy are projected jointly by first modeling female life expectancy and then modeling the gap between them with t-distributed errors.The gap is restricted to no more than 18 years, slightly above the highest value observed to date.Source: §3.2 · p. 5

3.3 Bayesian Population Projections

  • The projections covered 159 countries, just under 90% of the world’s 2010 population, excluding 38 countries with generalized HIV/AIDS epidemics and 30 areas with fewer than 100,000 people.Source: §3.3 · p. 6
  • A backtest using 1950–1990 data to project 1990–2010 found reasonably accurate projections and reasonably well-calibrated intervals for population, fertility, and female and male life expectancy.Source: §3.3 · p. 6
  • Egypt’s potential support ratio is projected to fall from 10.7 to 1.4 by 2100, with an 80% interval of 1.0–2.0; the decline is projected for developing countries as well as developed ones.The authors characterize the overall trend as essentially inevitable despite uncertainty about the extent of the eventual decline.Source: §3.3 · p. 6

4. ACCOMMODATING CONTROVERSY ABOUT ULTIMATE FERTILITY LEVEL VIA BAYES

The section relaxes the fixed replacement-level fertility assumption with a hierarchical Phase III model, whose country-specific asymptotes yield a notably lower Singapore projection while leaving most countries similar to the WPP 2010 model.

  • The method’s predictive median supplied the UN’s deterministic TFR projection in WPP 2010, while probabilistic projections were being considered for future WPP editions.Source: §4 · p. 6
  • The WPP 2010 projections anticipated slower fertility decline in Africa and gradual fertility increases in Europe, and projected 10 billion people by 2100 versus 9 billion in WPP 2008 by 2050.Source: §4 · p. 6
  • Basten, Coleman and Gu (2012) challenged the UN’s assumed fertility recovery toward 2.1 for five advanced East Asian economies.Source: §4 · p. 7
  • For the 21 Phase III countries, the posterior world-mean asymptote essentially excludes values below 1.6, while most projections resemble the fixed-2.1 WPP 2010 model.Source: Figure 4 · p. 8, §4 · p. 8
  • Singapore’s hierarchical-model projection asymptotes at 1.5 rather than 2.1, supporting the possibility of country-specific long-term fertility levels.Source: Figure 4 · p. 8, §4 · p. 8
  • The proposed model lets both μ and ρ vary across Phase III countries; its otherwise diffuse priors restrict the world mean asymptote to at most 2.1.Source: §4 · p. 8

5. DISCUSSION

The discussion presents Bayesian hierarchical fertility and mortality models feeding a cohort-component projection, with the fertility model already adopted by the UN. It emphasizes borrowing information across countries while noting omitted uncertainty sources and remaining limits for aggregation.

  • Bayesian hierarchical models project country fertility and female and male life expectancy; posterior predictive samples become age-specific rates and feed the cohort-component method.Source: §5 · pp. 8–9
  • The method omits uncertainty in base populations, sex ratio at birth, age patterns, and migration, and is not adapted to countries with generalized HIV/AIDS epidemics.Source: §5 · p. 9
  • For regional probabilistic projections, the model does not capture possible extra between-country fertility correlation, although it appears to account for most life-expectancy correlation.Source: §5 · p. 9
  • Hierarchical modeling borrows strength across countries to address limited country-level data and estimation instability, while priors incorporate external information on life-expectancy growth.Source: §5 · p. 9
  • Other Bayesian population-projection work has focused mainly on mortality; some approaches rely on covariates or age-specific mortality data that may be unavailable across countries.Source: §5 · p. 9
  • The UN adopted the TFR model as the basis for deterministic WPP 2010 projections and issued projections using these methods experimentally in 2012.Source: §5 · p. 10
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