Stochastic Population Forecasts Research Articles

Forecasting Population in an Uncertain World: Approaches, New Uses, and Troubling Limitations

AbstractThe long human lifespan enables long run forecasts of population size and age distribution. New methods include biodemographic research on upper limits to life expectancy and incorporation of early experiences affecting later life mortality such as smoking, obesity, and childhood health shocks. Some fertility forecasts incorporate education and quantum‐tempo insights. Statistical time series and Bayesian methods generate probabilistic forecasts. Yet recent decades have brought surprising changes in the economy, natural environment, and vital rates. In these changing circumstances we need new methods and the increasing use of probabilistic models and Bayesian methods incorporating outside information. The increasing use of microsimulation combined with aggregate forecasting methods is a very promising development enabling more detailed and heterogeneous forecasts. Some new uses of stochastic forecasts are interesting in themselves. Probabilistic mortality forecasts are used in finance and insurance, and a new Longevity Swap industry has been built on them. Random sample paths used to generate stochastic population forecasts can stress‐test public pension designs for fiscal stability and intergenerational equity. Population forecasting a few decades ago was a dull backwater of demographic research, but now it is increasingly important and is full of intellectual and technical challenges.

Stochastic population forecasting based on combinations of expert evaluations within the Bayesian paradigm.

This article suggests a procedure to derive stochastic population forecasts adopting an expert-based approach. As in previous work by Billari et al. (2012), experts are required to provide evaluations, in the form of conditional and unconditional scenarios, on summary indicators of the demographic components determining the population evolution: that is, fertility, mortality, and migration. Here, two main purposes are pursued. First, the demographic components are allowed to have some kind of dependence. Second, as a result of the existence of a body of shared information, possible correlations among experts are taken into account. In both cases, the dependence structure is not imposed by the researcher but rather is indirectly derived through the scenarios elicited from the experts. To address these issues, the method is based on a mixture model, within the so-called Supra-Bayesian approach, according to which expert evaluations are treated as data. The derived posterior distribution for the demographic indicators of interest is used as forecasting distribution, and a Markov chain Monte Carlo algorithm is designed to approximate this posterior. This article provides the questionnaire designed by the authors to collect expert opinions. Finally, an application to the forecast of the Italian population from 2010 to 2065 is proposed.

Stochastic population forecasts based on conditional expert opinions.

The paper develops and applies an expert-based stochastic population forecasting method, which can also be used to obtain a probabilistic version of scenario-based official forecasts. The full probability distribution of population forecasts is specified by starting from expert opinions on the future development of demographic components. Expert opinions are elicited as conditional on the realization of scenarios, in a two-step (or multiple-step) fashion. The method is applied to develop a stochastic forecast for the Italian population, starting from official scenarios from the Italian National Statistical Office.

Aggregation across countries in stochastic population forecasts

Stochastic methods of multi-state population modeling are less developed than methods for single states for two reasons. First, the structure of a multi-state population is inherently more complex than that of a single state because of state-to-state transitions. Second, estimates of cross-state correlations of the vital processes are a largely uncharted territory. Unlike multi-state lifetable theory, in forecasting applications the role of directed flows from state to state is often less important than the overall coherence of the assumptions concerning the vital processes. This is the case in the context of the European Union. Thus, a simplified approach is feasible, in which migration is represented by state-specific net numbers of migrants. This allows the use of existing single-state software, when simulations are suitably organized, in a multi-state setting. To address the second problem, we provide empirical estimates of cross-country covariances in the forecast uncertainty of fertility, mortality, and net migration. Together with point forecasts of these parameters that are coherent across countries, this produces coherent forecasts for aggregates of countries. The finding is that models for intermediate correlations are necessary for a proper accounting of forecast uncertainty at the aggregate level, in this case the European Union.

Stochastic population forecasts using functional data models for mortality, fertility and migration

Age–sex-specific population forecasts are derived through stochastic population renewal using forecasts of mortality, fertility and net migration. Functional data models with time series coefficients are used to model age-specific mortality and fertility rates. As detailed migration data are lacking, net migration by age and sex is estimated as the difference between historic annual population data and successive populations one year ahead derived from a projection using fertility and mortality data. This estimate, which includes error, is also modeled using a functional data model. The three models involve different strengths of the general Box–Cox transformation chosen to minimise out-of-sample forecast error. Uncertainty is estimated from the model, with an adjustment to ensure that the one-step-forecast variances are equal to those obtained with historical data. The three models are then used in a Monte Carlo simulation of future fertility, mortality and net migration, which are combined using the cohort-component method to obtain age-specific forecasts of the population by sex. The distribution of the forecasts provides probabilistic prediction intervals. The method is demonstrated by making 20-year forecasts using Australian data for the period 1921–2004. The advantages of our method are: (1) it is a coherent stochastic model of the three demographic components; (2) it is estimated entirely from historical data with no subjective inputs required; and (3) it provides probabilistic prediction intervals for any demographic variable that is derived from population numbers and vital events, including life expectancies, total fertility rates and dependency ratios.

Assumptions for long-term stochastic population forecasts in 18 European countries

The aim of the ‘Uncertain Population of Europe’(UPE) project was to compute long-term stochastic (probabilistic) population forecasts for 18 European countries. We developed a general methodology for constructing predictive distributions for fertility, mortality and migration. The assumptions underlying stochastic population forecasts can be assessed by analysing errors in past forecasts or model-based estimates of forecast errors, or by expert judgement. All three approaches have been used in the project. This article summarizes and discusses the results of the three approaches. It demonstrates how the—sometimes conflicting—results can be synthesized into a consistent set of assumptions about the expected levels and the uncertainty of total fertility rate, life expectancy at birth of men and women, and net migration for 18 European countries.

Population Forecasts, Fiscal Policy, and Risk

This paper describes how stochastic population forecasts are used to inform and analyze policies related to government spending on the elderly, mainly in the context of the industrialized nations. The paper first presents methods for making probabilistic forecasts of demographic rates, mortality, fertility, and immigration, and shows how these are combined to make stochastic forecasts of population number and composition, using forecasts of the U.S. population by way of illustration. Next, the paper discusses how demographic models and economic models can be combined into an integrated projection model of transfer systems such as social security. Finally, the paper shows how these integrated models describe various dimensions of policy-relevant risk, and discusses the nature and implications of risk in evaluating policy alternatives.

Joint forecasts of Southern European fertility rates with non-stationary dynamic factor models

Fertility forecasting is the weak point of stochastic population forecasts. Changing trends account for large forecasting errors even a few years ahead. On the other hand, fertility trends have been shown to be common to different European countries. This paper explores the possibility of improving forecasts by jointly modelling total fertility rate (TFR) trends within relatively homogeneous clusters of countries. We propose different varieties of non-stationary dynamic factor models applied to Southern European countries. The forecasting performance of the common factor models is compared to alternative univariate and multivariate forecasting methods using data for the period 1950–2000. Joint forecasts show forecasting gains in terms of root mean square error of prediction (RMSE), particularly for longer forecast horizons. This corroborates the convenience of modelling fertility jointly for population forecasting.

Developing Official Stochastic Population Forecasts at the US Census Bureau

Summary The U.S. Census Bureau is approaching a critical decision regarding a major facet of its methodology for forecasting the United States population. In the past, it has relied on alternative scenarios, low, medium, and high, to reflect varying interpretations of current trends in fertility, mortality, and international migration to forecast population. This approach has limitations that have been noted in the recent literature on population forecasting. Census Bureau researchers are embarking on an attempt to incorporate probabilistic reasoning to forecast prediction intervals around point forecasts to future dates. The current literature offers a choice of approaches to this problem. We are opting to employ formal time series modeling of parameters of fertility, mortality, and international migration, with stochastic renewal processes. The endeavor is complicated by the administrative necessity to produce a large amount of racial and Hispanic origin detail in the population, as well as the ubiquitous cross‐categories of age and sex. As official population forecasts must meet user demand, we are faced with the added challenge of presenting and supporting the resulting product in a way that is comprehensible to users, many of whom have little or no comprehension of the technical forecasting literature, and are accustomed to simple, deterministic answers. We may well find a need to modify our strategy, depending on the realities that may emerge from the limitations of data, the administrative requirements of the product, and the diverse needs of our user community.

Assumptions on Fertility in Stochastic Population Forecasts*

Summary In recent years Statistics Netherlands has published several stochastic population forecasts. The degree of uncertainty of the future population is assessed on the basis of assumptions about the probability distribution of future fertility, mortality and migration. The assumptions on fertility are based on an analysis of historic forecasts of the total fertility rate (TFR), on time‐series models of observations of the TFR, and on expert knowledge. This latter argument‐based approach refers to the TFR distinguished by birth order. In the most recent Dutch forecast the 95% forecast interval of the total fertility rate in 2050 is assumed to range from 1.2 to 2.3 children per woman.

Time Series Based Errors and Empirical Errors in Fertility Forecasts in the Nordic Countries*

Summary We use ARCH time series models to derive model based prediction intervals for the Total Fertility Rate (TFR) in Norway, Sweden, Finland, and Denmark up to 2050. For the short term (5–10 yrs), expected TFR‐errors are compared with empirical forecast errors observed in historical population forecasts prepared by the statistical agencies in these countries since 1969. Medium‐term and long‐term (up to 50 years) errors are compared with error patterns based on so‐called naïve forecasts, i.e. forecasts that assume that recently observed TFR‐levels also apply for the future.

TFR Predictions Based on Brownian Motion Theory

In stochastic population forecasts, the predictive distribution of the TFR is of centralconcern. Common time series models can be used to predict the TFR and itsmoments on the short run (up to 10 or 20 years), but on the long run (40-50 years)they result in excessively wide prediction intervals. The aim of this study is toanalyse and apply a time series model for the TFR, which restricts the predictedvalues to a certain pre-specified interval.I will model the time series of log TFR-values as a Brownian motion with absorbingupper barrier. I will give and analyse expressions for the predictive distribution of the log of the TFR assuming itfollows a Brownian motion with absorbing ceiling; expressions for the first and second moments of the predictive distribution ofthe log of the TFR.When the log of the TFR follows a random walk with absorbing ceiling, I find thatthe second moment of the predictive distribution for the long-run TFR in Norwayis insensitive for ceiling levels beyond a threshold of approximately 3.4 childrenper woman. This conclusion holds for a fairly broad range of innovation variances.If the log of the TFR follows a random walk, sample paths that exceed approximately3.4 children per woman may be rejected when simulating future fertility in Westerncountries. This will not have any major effect on the width of the long-termpredictive distribution.

Death and Taxes: Longer life, consumption, and social security

We analyze in three steps the influence of the projected mortality decline on the long run finances of the Social Security System. First, on a theoretical level, mortality decline adds person years of life which are distributed across the life cycle. The interaction of this distribution with the age distribution of labor earnings minus consumption, or of taxes minus benefits, partially determines the corresponding steady state financial consequences of mortality decline. The effect of mortality decline on population growth rates also matters, but is negligible in low mortality populations. Second, examination of past mortality trends in the United States and of international trends in low mortality populations, suggests that mortality will decline faster than foreseen by the Social Security Administration's forecasts. Third, we combine the work of the first two parts in dynamic simulations to examine the implications of mortality decline and of alternative forecasts of mortality for the finances of the social security system. Also, we use stochastic population forecasts to assess the influence of uncertainty about mortality decline on uncertainty about finances; we find that uncertainty about fertility still has more important implications than uncertainty about mortality, contrary to sensitivity tests in the official forecasts.

Stochastic Population Forecasts for the United States: Beyond High, Medium, and Low

"This article presents and implements a new method for making stochastic population forecasts that provide consistent probability intervals. We blend mathematical demography and statistical time series methods to estimate stochastic models of fertility and mortality based on U.S. data back to 1900 and then use the theory of random-matrix products to forecast various demographic measures and their associated probability intervals to the year 2065. Our expected total population sizes agree quite closely with the Census medium projections, and our 95 percent probability intervals are close to the Census high and low scenarios. But Census intervals in 2065 for ages 65+ are nearly three times as broad as ours, and for 85+ are nearly twice as broad. In contrast, our intervals for the total dependency and youth dependency ratios are more than twice as broad as theirs, and our ratio for the elderly dependency ratio is 12 times as great as theirs. These items have major implications for policy, and these contrasting indications of uncertainty clearly show the limitations of the conventional scenario-based methods."

Stochastic Population Forecasts for the United States: Beyond High, Medium, and Low

Abstract 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. This article presents and implements a new method for making stochastic population forecasts that provide consistent probability intervals. We blend mathematical demography and statistical time series methods to estimate stochastic models of fertility and mortality based on U.S. data back to 1900 and then use the theory of random-matrix products to forecast various demographic measures and their associated probability intervals to the year 2065. Our expected total population sizes agree quite closely with the Census medium projections, and our 95 percent probability intervals are close to the Census high and low scenarios. But Census intervals in 2065 for ages 65+ are nearly three times as broad as ours, and for 85+ are nearly twice as broad. In contrast, our intervals for the total dependency and youth dependency ratios are more than twice as broad as theirs, and our ratio for the elderly dependency ratio is 12 times as great as theirs. These items have major implications for policy, and these contrasting indications of uncertainty clearly show the limitations of the conventional scenario-based methods.

Stochastic population forecasts and their uses

The properties and uses of stochastic forecasts are discussed here. For linear stochastic projections, we show how the computation of forecast moments and the statistical distribution of forecasts depend on the multiplicative and autoregressive structure of the dynamics. Both scalar and vector projection methods are discussed, and their similarities are explored. Next we discuss the uses of stochastic forecasts, arguing that it is important to relate forecasts to the specific decision-making criteria of particular forecast users. The example of Social Security is used to show how a dynamic programming approach may be used to explore alternative decisions in a probabilistic context.

Stochastic demographic forecasting

This paper describes a particular approach to stochastic population forecasting, which is implemented for the USA through 2065. Statistical time series methods are combined with demographic models to produce plausible long run forecasts of vital rates, with probability distributions. The resulting mortality forecasts imply gains in future life expectancy that are roughly twice as large as those forecast by the Office of the Social Security Actuary. The method for forecasting fertility is useful mainly for providing estimates of the variance and autocorrelation in fertility forecast errors, and not for making stand-alone point forecasts. The forecasts of the probability distributions of the vital rates can then be used in a stochastic Leslie matrix to produce stochastic population forecasts. These provide probability distributions for the quantities forecast, reflecting lower bound estimates of uncertainty. Resulting stochastic forecasts of the elderly population, elderly dependency ratios, and payroll tax rates for health, education and pensions are presented.