Coherent Mortality Forecasting: The Product-Ratio Method With Functional Time Series Models

When independence is assumed, forecasts of mortality for subpopulations are almost always divergent in the long term. We propose a method for coherent forecasting of mortality rates for two or more subpopulations, based on functional principal components models of simple and interpretable functions of rates. The product-ratio functional forecasting method models and forecasts the geometric mean of subpopulation rates and the ratio of subpopulation rates to product rates. Coherence is imposed by constraining the forecast ratio function through stationary time series models. The method is applied to sex-specific data for Sweden and state-specific data for Australia. Based on out-of-sample forecasts, the coherent forecasts are at least as accurate in overall terms as comparable independent forecasts, and forecast accuracy is homogenized across subpopulations.     

Modeling and forecasting populations by time series: The Swedish case

Time series analysis techniques are used to model and to forecast populations. An autoregressive (AR) and a moving average (MA) model, which seem to fit the population of Sweden very well, are found. Forecasts are calculated using both models and are compared with the forecasts obtained by other methods. This comparison is very favorable for the time series models. Although our study is confined to the mid-year population of Sweden, there are good reasons to expect that the technique can be successfully applied to other population parameters.     

Precision,bias, and uncertainty for state population forecasts: an exploratory analysis of time series models

Many researchers have used time series models to construct population forecasts and prediction intervals at the national level, but few have evaluated the accuracy of their forecasts or the out-of-sample validity of their prediction intervals. Fewer still have developed models for subnational areas. In this study, we develop and evaluate six ARIMA time series models for states in the United States. Using annual population estimates from 1900 to 2000 and a variety of launch years, base periods, and forecast horizons, we construct population forecasts for four states chosen to reflect a range of population size and growth rate characteristics. We compare these forecasts with population counts for the corresponding years and find precision, bias, and the width of prediction intervals to vary by state, launch year, model specification, base period, and forecast horizon. Furthermore, we find that prediction intervals based on some ARIMA models provide relatively accurate forecasts of the distribution of future population counts but prediction intervals based on other models do not. We conclude that there is some basis for optimism regarding the possibility that ARIMA models might be able to produce realistic prediction intervals to accompany population forecasts, but a great deal of work remains to be done before we can draw any firm conclusions.     

Evaluation of Alternative Cohort-Component Models for Local Area Population Forecasts

It is generally accepted by demographers that cohort-component projection models which incorporate directional migration are conceptually preferable to those using net migration. Yet net migration cohort-component models, and other simplified variations, remain in common use by both academics and practitioners because of their simplicity and low data requirements. While many arguments have been presented in favour of using one or other type of model, surprisingly little analysis has been undertaken to assess which tend to give the most accurate forecasts. This paper evaluates five cohort-component models which differ in the way they handle migration, four of which are well known, with one—a composite net migration model—being proposed here for the first time. The paper evaluates the performance of these five models in their unconstrained form, and then in a constrained form in which age–sex-specific forecasts are constrained to independent total populations from an extrapolative model shown to produce accurate forecasts in earlier research. Retrospective forecasts for 67 local government areas of New South Wales were produced for the period 1991–2011 and then compared to population estimates. Assessments of both total and age-specific population forecasts were made. The results demonstrate the superior performance of the forecasts constrained to total populations from the extrapolative model, with the constrained bi-regional model giving the lowest errors. The findings should be of use to practitioners in selecting appropriate models for local area population forecasts.     

Forecasting Mortality: A Parameterized Time Series Approach

This article links parameterized model mortality schedules with time series methods to develop forecasts of U.S. mortality to the year 2000. The use of model mortality schedules permits a relatively concise representation of the history of mortality by age and sex from 1900 to 1985, and the use of modern time series methods to extend this history forward to the end of this century allows for a flexible modeling of trend and the accommodation of changes in long-run mortality patterns. This pilot study demonstrates that the proposed procedure produces medium-range forecasts of mortality that meet the standard tests of accuracy in forecast evaluation and that are sensible when evaluated against the comparable forecasts produced by the Social Security Administration.     

Assessing Uncertainty in Small Area Forecasts: State of the Practice and Implementation Strategy

Forecasts are needed for everyday decisions and must be in the form of numbers. Yet forecasts invariably turn out to be different than the numbers that actually occur. Yet, most producers of forecasts only present a deterministic view of the future in the form of point predictions. However, the presence of uncertainty is inherent in management or policy decisions and there is often concern that benefits are overstated and risks are understated. Such concerns are difficult to address by providing only point forecasts with no assessment of their uncertainty. Having a better understanding of uncertainty can enhance the usefulness of forecasts and make the work of forecasting agencies an even more valuable product for planners, policy makers, and the public. The purpose of this paper is twofold. First, it presents an overview of the current state-of-the-practice is assessing forecast uncertainty. Second, it offers a guidelines and options for implementing and building uncertainty into small area forecasting processes. There are options for assessing forecasting uncertainty that can and should be implemented by most, if not all, producers of forecasts.     

Spatial weighting improves accuracy in small-area demographic forecasts of urban census tract populations

Existing research in small-area demographic forecasting suffers from two important limitations: (1) a paucity of studies that quantify patterns of error in either total or age/sex-specific estimates and (2) limited methodological innovation aimed specifically at improving the accuracy of such forecasts. This paper attempts to fill, in part, these gaps in existing research by presenting a comparative evaluation of the accuracy of standard and spatially-weighted Hamilton–Perry forecasts for urbanized census tracts within incorporated New Mexico municipalities. These comparative forecasts are constructed for a 10-year horizon (base 1 April 2000 and target 1 April 2010), then compared to the results of the 2010 Census in an ex post facto evaluation. Results are presented for the standard Hamilton–Perry forecasts as well as two sets that incorporate two common variants of spatial weights to improve forecast accuracy. Findings are discussed in the context of what is currently known about error in small-area demographic forecasts and with an eye toward continued innovations.     

Effect of aggregation on the estimation of trend in mortality

"There are three approaches to analyzing and forecasting age-specific mortality: (1) analyze age-specific data directly, (2) analyze each cause-specific mortality series separately and add the results, (3) analyze cause-specific mortality series jointly and add the results. We show that if linear models are used for cause-specific mortality, then the three approaches often give close results even when cause-specific series are correlated. This result holds for cross-correlations arising from random misclassification of deaths by cause, and also for certain patterns of systematic misclassification....The results are illustrated with U.S. age-specific mortality: (1) analyse age-specific mortality data from 1968-1985. In some cases the aggregate forecasts appear to be the more credible ones." This is a revised version of a paper originally presented at the 1990 Annual Meeting of the Population Association of America (see Population Index, Vol. 56, No. 3, Fall 1990, p. 407).     

Effects of targets and aggregation on the propagation of error in mortality forecasts

"Official forecasts of mortality depend on assumptions about target values for the future rates of decline in mortality rates. Smooth functions connect the jump-off (base-year) mortality to the level implied by the targets. Three alternative sets of targets are assumed, leading to high, middle, and low forecasts. We show that this process can be closely modeled using simple linear statistical models. These explicit models allow us to analyze the error structure of the forecasts. We show that the current assumption of perfect correlation between errors in different ages, at different forecast years, and for different causes of death, is erroneous. An alternative correlation structure is suggested, and we show how its parameters can be estimated from the past data. The effect of the level of aggregation on the accuracy of mortality forecasts is considered." The geographical focus is on the United States. (SUMMARY IN FRE)     

Are mortality projections always more pessimistic when disaggregated by cause of death?

"It is often observed that mortality projections are more pessimistic when disaggregated by cause of death. This article explores the generality and strength of this relationship under a variety of forecasting models. First, a simple measure of the pessimism inherent in cause-based mortality forecasts is derived. Second, it is shown that the pessimism of cause-based forecasts can be approximated using only data on the distribution of deaths by cause in two pervious time periods. Third, using Japanese mortality data during 1951-1990, the analysis demonstrates that the pessimism of cause-based forecasts can be attributed mainly to observed trends in mortality due to cancer and heart disease, with smaller contribution due to trends in stroke (women only), pneumonia/bronchitis, accidents, and suicide. The last point requires the important qualification, however, that observed trends in cancer and heart disease may be severely biased due to changes in diagnostic practice." (SUMMARY IN FRE)     

Social indicators and crime rate forecasting

This paper presents an attempt to use macrodynamic social indicators in a time series analysis of three crime categories-homicide; property and; robbery offenses in Israel. Earlier findings such as the relationship between homicide and unemployment, and density of population and property offenses are confirmed by the analysis. The models that are constructed are used later for forecasting and yield a satisfactory performance of at least two of the crime models.     

Effect of aggregation on the estimation of trend in mortality

There are three approaches to analyzing and forecasting age‐specific mortality: (1) analyze age‐specific data directly, (2) analyze each cause‐specific mortality series separately and add the results, (3) analyze cause‐specific mortality series jointly and add the results. We show that if linear models are used for cause‐specific mortality, then the three approaches often give close results even when cause‐specific series are correlated. This result holds for cross‐correlations arising from random misclassification of deaths by cause, and also for certain patterns of systematic misclassification. It need not hold, if one or more causes serve as “leading indicators”; for the remaining causes, or if outside information is incorporated into forecasting either through expert judgment or formal statistical modeling. Under highly nonlinear models or in the presence of modeling error the result may also fail. The results are illustrated with U.S. age‐specific mortality data from 1968–1985. In some cases the aggregate forecasts appear to be the more credible ones.     

Simple versus complex models: evaluation,accuracy, and combining

"This paper argues that it is premature to decide whether simple forecasting models in demography are more (or less) accurate than complex models and whether causal models are more (or less) accurate than noncausal models. It is also too early to say under what conditions one type of model can outperform another. The paper also questions the wisdom of searching for a single best model or approach. It suggests that combining forecasts may improve accuracy." (SUMMARY IN FRE)     

Effects of targets and aggregation on the propagation of error in mortality forecasts

Official forecasts of mortality depend on assumptions about target values for the future rates of decline in mortality rates. Smooth functions connect the jump‐off (base‐year) mortality to the level implied by the targets. Three alternative sets of targets are assumed, leading to high, middle, and low forecasts. We show that this process can be closely modeled using simple linear statistical models. These explicit models allow us to analyze the error structure of the forecasts. We show that the current assumption of perfect correlation between errors in different ages, at different forecast years, and for different causes of death, is erroneous. An alternative correlation structure is suggested, and we show how its parameters can be estimated from the past data.

The effect of the level of aggregation on the accuracy of mortality forecasts is considered. It is not clear whether or not age‐ and cause‐specific analyses have been more accurate in the past than analyses based on age‐specific mortality alone would have been. The major contribution of forecasting mortality by cause appears to have been in allowing for easier incorporation of expert opinion rather than in making the. data analysis more accurate or the statistical models less biased.     

Time series analysis and stochastic forecasting: An econometric study of mortality and life expectancy

Methods for time series modeling of mortality and stochastic forecasting of life expectancies are explored, using Canadian data. Consideration is given first to alternative indexes of aggregate mortality. Age-sex group system models are then estimated. Issues in the forecasting of life expectancies are discussed and their quantitative implications investigated. Experimental stochastic forecasts are presented and discussed, based on nonparametric, partially parametric, and fully parametric methods, representing alternatives to the well known Lee-Carter method. Some thoughts are offered on the interpretation of historical data in generating future probability distributions, and on the treatment of demographic uncertainty in long-run policy planning. All correspondence to Frank T. Denton. This paper is a revised version of one presented at the Annual Congress of the European Society for Population Economics, Athens, Greece, June 2001. The underlying work was carried out as part of the SEDAP (Social and Economic Dimensions of an Aging Population) Research Program supported by the Social Sciences and Humanities Research Council of Canada, Statistics Canada and the Canadian Institute for Health Information. Ronald Lee provided comments that were very helpful in revising an earlier version of the paper. We thank him and participants at the ESPE session at which that version was presented. We thank also the Journal's anonymous referees. Responsible editor: Junsen Zhang.     

香港人口死亡率演变及其未来发展

与其它发展地区类似 ,二战以后香港人口死亡率已经经历了显著的下降 ,达到了一个非常低的水平。在这种极低水平的现状下 ,香港人口死亡率进一步下降的空间还有多大呢 ?本文基于香港人口死亡率历史数据 ,探讨了其演变趋势 ,同时利用Lee -Carter模型对香港未来 5 0年分性别的人口死亡率进行了预测。如果我们将预测结果与最近的官方预测数据进行比较 ,可以发现 ,本文Lee -Carter模型预测的未来香港人口死亡率下降趋势比官方预测结果要乐观     

Variability of Travel Time,Congestion, and the Cost of Travel

The variability of travel time modifies the rush hour traffic and the cost of commuting. The bottleneck model of road congestion with fixed peak-load demand is augmented of an additive random delay. When individuals have (α-β-γ) preferences, there exists a unique Nash equilibrium. The variability of travel time leads to departure rates that are spread more evenly over the rush hour than when travel times are deterministic. This equilibrium mechanism mitigates both congestion and the cost of unreliability. This implies that “single-traveler models,” which treat congestion as an exogenous phenomenon, overestimate the value of reliability for the rush hour. The application with the uniform or with the exponential distribution helps appraise the overestimation.     

Modeling and Forecasting Health Expectancy: Theoretical Framework and Application

Life expectancy continues to grow in most Western countries; however, a major remaining question is whether longer life expectancy will be associated with more or fewer life years spent with poor health. Therefore, complementing forecasts of life expectancy with forecasts of health expectancies is useful. To forecast health expectancy, an extension of the stochastic extrapolative models developed for forecasting total life expectancy could be applied, but instead of projecting total mortality and using regular life tables, one could project transition probabilities between health states simultaneously and use multistate life table methods. In this article, we present a theoretical framework for a multistate life table model in which the transition probabilities depend on age and calendar time. The goal of our study is to describe a model that projects transition probabilities by the Lee-Carter method, and to illustrate how it can be used to forecast future health expectancy with prediction intervals around the estimates. We applied the method to data on the Dutch population aged 55 and older, and projected transition probabilities until 2030 to obtain forecasts of life expectancy, disability-free life expectancy, and probability of compression of disability.