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.     

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.     

Can Knowledge Improve Population Forecasts at Subcounty Levels?

Recent developments in urban and regional planning require more accurate population forecasts at subcounty levels, as well as a consideration of interactions among population growth, traffic flow, land use, and environmental impacts. However, the extrapolation methods, currently the most often used demographic forecasting techniques for subcounty areas, cannot meet the demand. This study tests a knowledge-based regression approach, which has been successfully used for forecasts at the national level, for subcounty population forecasting. In particular, this study applies four regression models that incorporate demographic characteristics, socioeconomic conditions, transportation accessibility, natural amenities, and land development to examine the population change since 1970 and to prepare the 1990-based forecast of year 2000 population at the minor civil division level in Wisconsin. The findings indicate that this approach does not outperform the extrapolation projections. Although the regression methods produce more precise projections, the least biased projections are often generated by one of the extrapolation techniques. The performance of the knowledge-based regression methods is discounted at subcounty levels by temporal instability and the scale effect. The regression coefficients exhibit a statistically significant level of temporal instability across the estimation and projection periods and tend to change more rapidly at finer geographic scales.     

Reconciling Forecasts of Infant Mortality Rates at National and Sub-National Levels: Grouped Time-Series Methods

Mortality rates are often disaggregated by different attributes, such as sex, state, education, religion, or ethnicity. Forecasting mortality rates at the national and sub-national levels plays an important role in making social policies associated with the national and sub-national levels. However, base forecasts at the sub-national levels may not add up to the forecasts at the national level. To address this issue, we consider the problem of reconciling mortality rate forecasts from the viewpoint of grouped time-series forecasting methods (Hyndman et al. in, Comput Stat Data Anal 55(9):2579–2589, 2011). A bottom-up method and an optimal combination method are applied to produce point forecasts of infant mortality rates that are aggregated appropriately across the different levels of a hierarchy. We extend these two methods by considering the reconciliation of interval forecasts through a bootstrap procedure. Using the regional infant mortality rates in Australia, we investigate the one-step-ahead to 20-step-ahead point and interval forecast accuracies among the independent and these two grouped time-series forecasting methods. The proposed methods are shown to be useful for reconciling point and interval forecasts of demographic rates at the national and sub-national levels, and would be beneficial for government policy decisions regarding the allocations of current and future resources at both the national and sub-national levels.     

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.     

Insights from longitudinal data on the earnings growth of U.S. foreign-born men

Does the growth in earnings of foreign-born men exceed that of U.S. natives? We use longitudinal data on earnings from a Social Security Administration (SSA) database matched to the 1994 March Current Population Survey to shed new light on this important issue. We also examine the trend over time in the foreign-born men's earnings growth and illuminate the various ways that SSA data can be used to explore the earnings patterns of immigrants.     

Statistically tested comparisons of the accuracy of forecasting methods for age-specific and sex-specific mortality and life expectancy

Although there are continuing developments in the methods for forecasting mortality, there are few comparisons of the accuracy of the forecasts. The subject of the statistical validity of these comparisons, which is essential to demographic forecasting, has all but been ignored. We introduce Friedman's test statistics to examine whether the differences in point and interval forecast accuracies are statistically significant between methods. We introduce the Nemenyi test statistic to identify which methods give results that are statistically significantly different from others. Using sex-specific and age-specific data from 20 countries, we apply these two test statistics to examine the forecast accuracy obtained from several principal component methods, which can be categorized into coherent and non-coherent forecasting methods.     

Evaluating the performance of the lee-carter method for forecasting mortality

Lee and Carter (LC) published a new statistical method for forecasting mortality in 1992. This paper examines its actual and hypothetical forecast errors, and compares them with Social Security forecast errors. Hypothetical historical projections suggest that LC tended to underproject gains, but by less than did Social Security. True e0 was within the ex ante 95% probability interval 97% of the time overall, but intervals were too broad up to 40 years and too narrow after 50 years. Projections to 1998 made after 1945 always contain errors of less than two years. Hypothetical projections for France, Sweden, Japan, and Canada would have done well. Changing age patterns of mortality decline over the century pose problems for the method.     

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.     

Increasing longevity and medicare expenditures

Official Medicare projections forecast that the elderly population will be less healthy and more costly over the next century. This prediction stems from the use of age as an indicator of health status: increases in longevity are assumed to increase demand for health care as individuals survive to older and higher-use ages. In this paper I suggest an alternative approach, in which time until death replaces age as the demographic indicator of health status. Increases in longevity are assumed to postpone the higher Medicare use and costs associated with the final decade of life. I contrast the two approaches, using mortality forecasts consistent with recent projections from the U.S. Census Bureau and the Social Security Administration. The time-until-death method yields significantly lower-cost forecasts. The hypothetical cost savings from improved health care small, however, relative to the size of the Medicare solvency problem caused by population aging.     

Using Subjective Expectations to Forecast Longevity: Do Survey Respondents Know Something We Don’t Know?

Old-age mortality is notoriously difficult to predict because it requires not only an understanding of the process of senescence-which is influenced by genetic, environmental, and behavioral factors-but also a prediction of how these factors will evolve. In this paper I argue that individuals are uniquely qualified to predict their own mortality based on their own genetic background, as well as environmental and behavioral risk factors that are often known only to the individual. Given this private information, individuals form expectations about survival probabilities that may provide additional information to demographers and policymakers in their challenge to predict mortality. From expectations data from the 1992 Health and Retirement Study (HRS), I construct subjective, cohort life tables that are shown to predict the unusual direction of revisions to U.S. life expectancy by gender between 1992 and 2004: that is, for these cohorts, the Social Security Actuary (SSA) raised male life expectancy in 2004 and at the same lowered female life expectancy, narrowing the gender gap in longevity by 25% over this period. Further, although the subjective life expectancies for men appear to be roughly in line with the 2004 life tables, the subjective expectations of women suggest that female life expectancies estimated by the SSA might still be on the high side.     

A multi-population evaluation of the Poisson common factor model for projecting mortality jointly for both sexes

Mortality forecasts are critically important inputs to the consideration of a range of demographically-related policy challenges facing governments in more developed countries. While methods for jointly forecasting mortality for sub-populations offer the advantage of avoiding undesirable divergence in the forecasts of related populations, little is known about whether they improve forecast accuracy. Using mortality data from ten populations, we evaluate the data fitting and forecast performance of the Poisson common factor model (PCFM) for projecting both sexes’ mortality jointly against the Poisson Lee–Carter model applied separately to each sex. We find that overall the PCFM generates the more desirable results. Firstly, the PCFM ensures that the projected male-to-female ratio of death rates at each age converges to a constant in the long run. Secondly, using out-of-sample analysis, we find that the PCFM provides more accurate projection of the sex ratios of death rates, with the advantage being greater for longer-term forecasts. Thus the PCFM offers a viable and sensible means for coherently forecasting the mortality of both sexes. There are also significant financial implications in allowing for the co-movement of mortality of females and males properly.     

Disaggregation in population forecasting: do we need it? And how to do it simply

"We have described a method for reducing the dimensionality of the forecasting problem by parsimoniously modeling the evolution over time of the age schedules of vital rates. This method steers a middle course between forecasting aggregates and forecasting individual age specific rates: we reduce the problem to forecasting a single parameter for fertility and another one for mortality. We have described a number of refinements and extensions of those basic methods, which preserve their underlying structure and simplicity. In particular, we show how one can fit the model more simply, incorporate lower bounds to the forecasts of rates, disaggregate by sex or race, and prepare integrated forecasts of rates for a collection of regions. We also discuss alternate approaches to forecasting the estimated indices of fertility and mortality, including state-space methods. These many versions of the basic method have yielded remarkably similar results." (SUMMARY IN FRE)     

Lifespan Disparity as an Additional Indicator for Evaluating Mortality Forecasts

Evaluating the predictive ability of mortality forecasts is important yet difficult. Death rates and mean lifespan are basic life table functions typically used to analyze to what extent the forecasts deviate from their realized values. Although these parameters are useful for specifying precisely how mortality has been forecasted, they cannot be used to assess whether the underlying mortality developments are plausible. We therefore propose that in addition to looking at average lifespan, we should examine whether the forecasted variability of the age at death is a plausible continuation of past trends. The validation of mortality forecasts for Italy, Japan, and Denmark demonstrates that their predictive performance can be evaluated more comprehensively by analyzing both the average lifespan and lifespan disparity—that is, by jointly analyzing the mean and the dispersion of mortality. Approaches that account for dynamic age shifts in survival improvements appear to perform better than others that enforce relatively invariant patterns. However, because forecasting approaches are designed to capture trends in average mortality, we argue that studying lifespan disparity may also help to improve the methodology and thus the predictive ability of mortality forecasts.     

Long‐Term Population Projections and the US Social Security System

According to a report recently issued by the Technical Panel for the US Social Security Administration, the long‐term financial outlook for the system is worse than previously thought. The worsening projected by the panel in the long‐run funding imbalance of the Social Security System is mostly due to the recommendation by the panel to add an extra four years to the currently projected increase in life expectancy by 2075: from 81.8 years to 85.9 years. The panel recommended no change in the current intermediate projected long‐run TFR of 1.9 and net immigration of 900,000 persons per year. The recommendation to increase the projected gains in life expectancy was based on international trends as well as on historical trends in the United States and the absence of biological evidence ruling out such gains. Industrial countries have a history of under‐predicting the growth of their elderly population, and it is expected that large mortality adjustments may be needed in the projections for public pension programs also in industrial countries other than the United States.     

The development and use of demographic models

In this review, we first examine two classical demographic models - conventional life tables and stable populations - and a modern generalization of stable population theory; we then discuss mathematical models of conception and birth. These models involve purely mathematical relations in formal demography as opposed to empirical regularities. Next we consider model age schedules of mortality, nuptialitiy, marital fertility, fertility, and migration that are explicitly based on such empirical patterns. We close this empirical section with a discussion of model stable populations, which are based on model life tables. We next examine the use of demographic models in forecasting future mortality, nuptiality, and fertility and in population projection. Following a discussion of microsimulation models, which gives us the opportunity to mention model age schedules of post partum amenorrhoea and of sterility, we close with observations about the purposes and uses of demographic models.     

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.     

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.     

Trends in Mortality Decrease and Economic Growth

The vast literature on extrapolative stochastic mortality models focuses mainly on the extrapolation of past mortality trends and summarizes the trends by one or more latent factors. However, the interpretation of these trends is typically not very clear. On the other hand, explanation methods are trying to link mortality dynamics with observable factors. This serves as an intermediate step between the two methods. We perform a comprehensive analysis on the relationship between the latent trend in mortality dynamics and the trend in economic growth represented by gross domestic product (GDP). Subsequently, the Lee-Carter framework is extended through the introduction of GDP as an additional factor next to the latent factor, which provides a better fit and better interpretable forecasts.