Error models for official mortality forecasts

"The Office of the Actuary, U.S. Social Security Administration, produces alternative forecasts of mortality to reflect uncertainty about the future.... In this article we identify the components and assumptions of the official forecasts and approximate them by stochastic parametric models. We estimate parameters of the models from past data, derive statistical intervals for the forecasts, and compare them with the official high-low intervals. We use the models to evaluate the forecasts rather than to develop different predictions of the future. Analysis of data from 1972 to 1985 shows that the official intervals for mortality forecasts for males or females aged 45-70 have approximately a 95% chance of including the true mortality rate in any year. For other ages the chances are much less than 95%."     

Birth forecasting based on birth order probabilities,with application to U.S. data

"A model for birth forecasting based on prediction of the so-called 'birth order probabilities' is constructed. The relation between this model and recent models of fertility prediction is derived. Birth forecasts with approximate probability limits for the U.S. for the period 1983-1997 are generated. The performance of the proposed model in predicting future fertility is tested by fitting time series models to part of the available series (1917-1982) and ultimately generating birth forecasts for the remainder of the period, then comparing these forecasts with the actual data." The accuracy of the fertility forecasts made are compared with those made by other methods.     

The relationship between the length of the base period and population forecast errors

"The base period of a population forecast is the time period from which historical data are collected for the purpose of forecasting future population values. The length of the base period is one of the fundamental decisions made in preparing population forecasts, yet very few studies have investigated the effects of this decision on population forecast errors. In this article the relationship between the length of the base period and population forecast errors is analyzed, using three simple forecasting techniques and data from 1900 to 1980 for states in the United States. It is found that increasing the length of the base period up to 10 years improves forecast accuracy, but that further increases generally have little additional effect. The only exception to this finding is long-range forecasts of rapidly growing states, in which a longer base period substantially improves forecast accuracy for two of the forecasting techniques."     

The accuracy of population projections

A review of past population projection errors is presented as a means for constructing confidence intervals for future projections. The author first defines a statistic to measure projection errors independent of the size of population and the length of the projection period. A sample of U.S. and U.N. projections is used to show that the distributions of components of the error statistic are relatively stable. This information is then used to construct confidence intervals for the U.S. population up to the year 2000.     

Survival models for fertility evaluation

"Two proportional hazards models for cohort fertility evaluation are constructed. Time-dependent covariates describe sources of heterogeneity between and within women regarding fertility characteristics. In the first model, U.S. birth rates specific to maternal age, race, parity, and birth cohort are used as underlying hazard rates. Covariate effects are estimated by maximizing the full likelihood. In the second model, covariate effects are estimated via Cox regression with stratified underlying hazard rates regarded as unknown nuisance parameters." The authors illustrate the models "with an evaluation of the fertility histories of the wives of workers at a manufacturing plant with potential for hazardous exposure. Adjustments to the U.S. birth rates for maternal age and parity zero experience are required with the first approach. Then, despite differences in the model-specific estimation procedures, the point estimates of the exposure effect and the estimated standard errors from the two models are practically equivalent."     

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."     

University student enrollment forecasts by analysis structural ratios using ARIMA-methods

Summary Forecasts for the number of students in Germany are conducted by the Kultusministerkonferenz. They use a transition model which does not allow for prediction intervals and therefore lack a measure of uncertainty of the forecast. Since the uncertainty is high for such forecasts, this lack is of importance. In this paper, structural ratios, relating the number of university students to the population of the same age, are analyzed and forescasted using ARIMA-models with outliers. Multiplying these ratios with official population forecasts for Germany provides the future number of students, additionally giving prediction intervals. This number will increase from 1.94 million in 2002 to 2.35 million in 2015. The uncertainty of the forecast is high; the forecast interval in 2015 will range between 1.72 and 2.98 million at a 95% confidence level. Supported by the German Research Foundation (DFG). We are grateful to an anonymous referee for some helpful comments.     

Nonparametric Prediction Intervals Based on Ranked Set Samples

In this article, we develop nonparametric prediction intervals based on generalized ranked set samples using conditional as well as unconditional approaches. The predictions are developed for order statistics from a future sample as well as for order statistics from a future balanced ranked set sample. The effects of ranking errors on the coverage probabilities of these prediction intervals are also examined.     

Alternative models for the heterogeneity of mortality risks among the aged

The authors examine how sensitive the estimates of heterogeneity in the mortality risks in a population are to the choices of two types of function, "one describing the age-specific rate of increase of mortality risks for individuals and the other describing the distribution of mortality risks across individuals." U.S. data from published Medicare mortality rates for the period 1968-1978 are used to analyze total mortality among the aged. "In addition, national vital statistics data for the period 1950-1977 were used to analyze adult lung cancer mortality. For these data, the estimates of structural parameters were less sensitive to reasonable choices of the heterogeneity distribution (gamma vs. inverse Gaussian) than to reasonable choices of the hazard rate function (Gompertz vs. Weibull)."     

Assessing Macro Uncertainty in Real-Time When Data Are Subject To Revision

Model-based estimates of future uncertainty are generally based on the in-sample fit of the model, as when Box–Jenkins prediction intervals are calculated. However, this approach will generate biased uncertainty estimates in real time when there are data revisions. A simple remedy is suggested, and used to generate more accurate prediction intervals for 25 macroeconomic variables, in line with the theory. A simulation study based on an empirically estimated model of data revisions for U.S. output growth is used to investigate small-sample properties.     

Classification by race and Spanish origin in the 1980 census and its impact on white and nonwhite rates

A basic change concerning the racial classification of persons of Spanish origin used in the 1980 U.S. census is examined for its impact on white and nonwhite population counts, particularly in urban areas. "Arrest rates by race for central city Phoenix together with 1980 census data by race and ethnicity for Phoenix and 11 other central cities are used to illustrate the substantive effect of changes in the white and 'other race' counts produced by this change in procedure." The authors consider "remedies for the problems faced by those using published census data..., and one possibility for creating comparable rates is presented. Closely related complications created by the failure of the Office of Management and Budget to arrive at a single, logical statistical standard for the classification of U.S. residents by race and ethnicity are also identified."     

Scenarios,uncertainty and conditional forecasts of the world population

This study is concerned with the methods available for the forecasting of future trends in the world's population. Particular attention is given to the problem of the uncertainties that these forecasts include. "The purpose of this paper is to show how subjective and data-based probabilistic assessments of error can be combined, to give a user a realistic assessment of the uncertainty of demographic forecasts, and to apply these concepts to forecasts of the world population. Moreover, we shall show how conditional forecasts can provide a simple conceptual framework in which to view scenarios. They can be particularly useful in the evaluation of proposed policies. Indeed, the so-called environmental impact assessments...that are now mandatory in many countries for major construction projects typically contain elements of conditional forecasting." The concepts discussed are illustrated by comparing a scenario of future global population growth prepared at the Institute of Applied Systems Analysis with a UN population projection.     

An evaluation of population projection errors for census tracts

"In this article we evaluate the accuracy and bias of projections of total population and population by age group for census tracts in three counties in Florida. We use [U.S. census] data from 1970 and 1980 and several simple extrapolation techniques to produce projections for 1990; we then compare these projections with 1990 census counts and evaluate the differences. For the total sample, we find mean absolute errors of 17%-20% for the three most accurate techniques for projecting total population and find no indication of overall bias. For individual age groups, mean absolute errors range from 20%-29%." This is a revised version of a paper presented at the 1993 Annual Meeting of the Population Association of America.     

State space modeling of multiple time series

Time series methods offer the possibility of making accurate forecasts even when the underlying structural model is unknown, by replacing the structural restrictions needed to reduce sampling error and improve forecasts with restrictions determined from the data. While there has been considerable success with relatively simple univariate time series modeling procedures, the complex interrela- tionships possible with multiple series requite more powerful techniques.Based on the insights of linear systems theory, a multivariate state space methos for both stationary and nonstationary problems is described and related to ARMA models. The states or dynamic factors of the procedure are chosen to be robust in the presence of model misspecification, in constrast to ARMA models which lack this property. In addition, by treating th emidel choice as a formal approximation problem certain new optimal properties of the procedure with respect to specification are established; in particular, it is shown that no other model of equal or smaller order fits the observed autocovariance sequence any better in the sense of a Hankel norm. Finally, in the treatment of nonstationary series, a natural decomposition into long run and short run dynamics results in easily implemented two step procedures that use characteristics of the data to identify and model trend and cycle components that correspond to cointegration and error correction models. Applications include annualo U.S. GNP and money stock growth rates, monthly California beef prices and inventories, and monthly stock prices for large retailers.     

Measurement of Hispanic ethnicity in the U.S. census: an evaluation based on latent-class analysis

"Two models, the U.S. census model and the latent-class model, are compared in their application to evaluating measurements of ethnicity. Although the census approach assumes that the response categories of a questionnaire item correspond to groups in the population, the latent-class approach seeks to assess whether any set of response categories can represent observed ethnic heterogeneity. Data collected using the 1990 census Hispanic-origin question and other instruments for measuring ethnicity suggest that the latent-class approach is superior whenever the response categories are not known to be valid. In particular, using the latent-class model, this article rejects the census model's assumption of a single dimension of meaning underlying responses to the Hispanic-origin question."     

Empirical Bayes procedures for stabilizing maps of U.S. cancer mortality rates

"The geographic mapping of age-standardized, cause-specific death rates is a powerful tool for identifying possible etiologic factors, because the spatial distribution of mortality risks can be examined for correlations with the spatial distribution of disease-specific risk factors. This article presents a two-stage empirical Bayes procedure for calculating age-standardized cancer death rates, for use in mapping, which are adjusted for the stochasticity of rates in small area populations. Using the adjusted rates helps isolate and identify spatial patterns in the rates. The model is applied to sex-specific data on U.S. county cancer mortality in the white population for 15 cancer sites for three decades: 1950-1959, 1960-1969, and 1970-1979. Selected results are presented as maps of county death rates for white males."     

基于均衡关系的中国人口死亡率预测模型

人口死亡率反映了人口的死亡程度,准确预测死亡率是人口科学及人口经济学研究的重点之一,同时也是长寿风险测量的重要数据基础。基于Lee-Carter模型,探索中国大陆与台湾地区死亡率的相关性,通过协整分析考虑两地死亡率的长期均衡关系,创新性地建立基于相关性的向量误差修正模型(VECM),克服传统自回归移动平均模型(ARIMA)使用有限数据进行预测的局限性;均方预测误差作为检验标准,结果表明:基于VECM模型的预测效果比传统的预测效果更佳;基于中国大陆地区和台湾地区的死亡率长期均衡关系,可以为两地联合长寿债券的定价提供重要参考。     

Some coverage error models for census data

"Alternative models are presented for representing coverage error in surveys and censuses of human populations. The models are related to the capture-recapture models used in wildlife applications and to the dual-system models employed in the vital events literature. Estimation methodologies are discussed for one of the coverage error models." After a discussion of the theory underlying the methodology, "distinctions are made between two kinds of error: (a) sampling error and (b) error associated with the model. An example involving data from the 1980 U.S. census is presented. The problem of adjusting census and survey data for coverage error is also discussed."     

Bootstrap Prediction Intervals for Factor Models

We propose bootstrap prediction intervals for an observation h periods into the future and its conditional mean. We assume that these forecasts are made using a set of factors extracted from a large panel of variables. Because we treat these factors as latent, our forecasts depend both on estimated factors and estimated regression coefficients. Under regularity conditions, asymptotic intervals have been shown to be valid under Gaussianity of the innovations. The bootstrap allows us to relax this assumption and to construct valid prediction intervals under more general conditions. Moreover, even under Gaussianity, the bootstrap leads to more accurate intervals in cases where the cross-sectional dimension is relatively small as it reduces the bias of the ordinary least-squares (OLS) estimator.     

Forecasting births in post-transition population: stochastic renewal with serially correlated fertility

Time series analysis of fertility can improve demographic forecasts. The optimal forecast and its variance for births to an age-structured population subject to serially correlated random fertility are developed. The general case in which the fertility process had arbitrary autoregressive structure is dealt with and then the 4 special cases of white noise, 1st-order autoregressive, 2nd-order autoregressive, and random walk are considered. Consequently, it is determined that the predictions and their variances are highly sensitive to the autoregressive structure of fertility and, therefore, if stochastic models are to be used for prediction, they must emphasize this aspect of the problem. Preliminary empirical efforts to model the time series of U.S. fertility from 1917 to 1972 proved unsuccessful, but it is obvious that at least a 2nd-order autoregressive scheme is require d. The analysis proveded should be helpful in: 1) any application of the procedures requires a successful parameterization of the fertility process; 2) fertility variations could be decomposed into the effects of nuptiality and marital fertility and then births and marriages could be jointly predicted; and 3) the simplifying approximations should be dropped and each age-specific rate could be analyzed and predicted.