Empirical Prediction Intervals for County Population Forecasts

Population forecasts entail a significant amount of uncertainty, especially for long-range horizons and for places with small or rapidly changing populations. This uncertainty can be dealt with by presenting a range of projections or by developing statistical prediction intervals. The latter can be based on models that incorporate the stochastic nature of the forecasting process, on empirical analyses of past forecast errors, or on a combination of the two. In this article, we develop and test prediction intervals based on empirical analyses of past forecast errors for counties in the United States. Using decennial census data from 1900 to 2000, we apply trend extrapolation techniques to develop a set of county population forecasts; calculate forecast errors by comparing forecasts to subsequent census counts; and use the distribution of errors to construct empirical prediction intervals. We find that empirically-based prediction intervals provide reasonably accurate predictions of the precision of population forecasts, but provide little guidance regarding their tendency to be too high or too low. We believe the construction of empirically-based prediction intervals will help users of small-area population forecasts measure and evaluate the uncertainty inherent in population forecasts and plan more effectively for the future.     

The role of population size in the determination and prediction of population forecast errors: An evaluation using conifidence intervals for subcounty areas

Producers of population forecasts acknowledge the uncertainty inherent in trying to predict the future and should warn about the likely error of their forecasts. Confidence intervals represent one way of quantifying population forecast error. Most of the work in this area relates to national forecasts; although, confidence intervals have been developed for state and county forecasts. A few studies have examined subcounty forecast error, however, they only measured point estimates of error. This paper describes a technique for making subcounty population forecasts and for generating confidence intervals around their forecast error. It also develops statistical equations for calculating point estimates and confidence intervals for areas with different population sizes. A non-linear, inverse relationship between population size and forecast accuracy was found and we demonstrate the ability to accurately predict average forecast error and confidence intervals based on this relationship.     

Stability over time in the distribution of population forecast errors

A number of studies in recent years have investigated empirical approaches to the production of confidence intervals for population projections. The critical assumption underlying these approaches is that the distribution of forecast errors remains stable over time. In this article, we evaluate this assumption by making population projections for states for a number of time periods during the 20th century, comparing these projections with census enumerations to determine forecast errors, and analyzing the stability of the resulting error distributions over time. These data are then used to construct and test empirical confidence limits. We find that in this sample the distribution of absolute percentage errors remained relatively stable over time and data on past forecast errors provided very useful predictions of future forecast errors.     

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.     

Insights from the Evaluation of Past Local Area Population Forecasts

Local area population forecasts have a wide variety of uses in the public and private sectors. But not enough is known about the errors of such forecasts, particularly over the longer term (20 years or more). Understanding past errors is valuable for both forecast producers and users. This paper (i) evaluates the forecast accuracy of past local area population forecasts published by Australian State and Territory Governments over the last 30 years and (ii) illustrates the ways in which past error distributions can be employed to quantify the uncertainty of current forecasts. Population forecasts from the past 30 years were sourced from State and Territory Governments. Estimated resident populations to which the projections were compared were created for the geographical regions of the past projections. The key features of past forecast error patterns are described. Forecast errors mostly confirm earlier findings with regard to the relationship between error and length of projection horizon and population size. The paper then introduces the concept of a forecast ‘shelf life’, which indicates how far into the future a forecast is likely to remain reliable. It also illustrates how past error distributions can be used to create empirical prediction intervals for current forecasts. These two complementary measures provide a simple way of communicating the likely magnitude of error that can be expected with current local area population forecasts.     

On the validity of MAPE as a measure of population forecast accuracy

The mean absolute percent error (MAPE) is the summary measure most often used for evaluating the accuracy of population forecasts. While MAPE has many desirable criteria, we argue from both normative and relative standpoints that the widespread practice of exclusively using it for evaluating population forecasts should be changed. Normatively, we argue that MAPE does not meet the criterion of validity because as a summary measure it overstates the error found in a population forecast. We base this argument on logical grounds and support it empirically, using a sample of population forecasts for counties. From a relative standpoint, we examine two alternatives to MAPE, both sharing with it, the important conceptual feature of using most of the information about error. These alternatives are symmetrical MAPE (SMAPE) and a class of measures known as M-estimators. The empirical evaluation suggests M-estimators do not overstate forecast error as much as either MAPE or SMAPE and are, therefore, more valid measures of accuracy. We consequently recommend incorporating M-estimators into the evaluation toolkit. Because M-estimators do not meet the desired criterion of interpretative ease as well as MAPE, we also suggest another approach that focuses on nonlinear transformations of the error distribution.     

Evaluating Population Forecast Accuracy: A Regression Approach Using County Data

Many studies have evaluated the impact of differences in population size and growth rate on population forecast accuracy. Virtually all these studies have been based on aggregate data; that is, they focused on average errors for places with particular size or growth rate characteristics. In this study, we take a different approach by investigating forecast accuracy using regression models based on data for individual places. Using decennial census data from 1900 to 2000 for 2,482 counties in the US, we construct a large number of county population forecasts and calculate forecast errors for 10- and 20-year horizons. Then, we develop and evaluate several alternative functional forms of regression models relating population size and growth rate to forecast accuracy; investigate the impact of adding several other explanatory variables; and estimate the relative contributions of each variable to the discriminatory power of the models. Our results confirm several findings reported in previous studies but uncover several new findings as well. We believe regression models based on data for individual places provide powerful but under-utilized tools for investigating the determinants of population forecast accuracy.     

An Empirical Analysis of the Effect of Length of Forecast Horizon on Population Forecast Errors

Many studies have found that population forecast errors generally increase with the length of the forecast horizon, but none have examined this relationship in detail. Do errors grow linearly, exponentially, or in some other manner as the forecast horizon becomes longer? Does the error-horizon relationship differ by forecasting technique, launch year, size of place, or rate of growth? Do alternative measures of error make a difference? In this article we address these questions using two simple forecasting techniques and population data from 1900 to 1980 for states in the United States. We find that in most instances there is a linear or nearly linear relationship between forecast accuracy and the length of the forecast horizon, but no consistent relationship between bias and the length of the horizon. We believe that these results provide useful information regarding the nature of population forecast errors.     

MAPE-R: a rescaled measure of accuracy for cross-sectional subnational population forecasts

Accurately measuring a population and its attributes at past, present, and future points in time has been of great interest to demographers. Within discussions of forecast accuracy, demographers have often been criticized for their inaccurate prognostications of the future. Discussions of methods and data are usually at the centre of these criticisms, along with suggestions for providing an idea of forecast uncertainty. The measures used to evaluate the accuracy of forecasts also have received attention and while accuracy is not the only criterion advocated for evaluating demographic forecasts, it is generally acknowledged to be the most important. In this paper, we continue the discussion of measures of forecast accuracy by concentrating on a rescaled version of a measure that is arguably the one used most often in evaluating cross-sectional, subnational forecasts, Mean Absolute Percent Error (MAPE). The rescaled version, MAPE-R, has not had the benefit of a major empirical test, which is the central focus of this paper. We do this by comparing 10-year population forecasts for U.S. counties to 2000 census counts. We find that the MAPE-R offers a significantly more meaningful representation of average error than MAPE in the presence of substantial outlying errors, and we provide guidelines for its implementation.     

Accounting for migration in cohort-component projections of state and local populations

Migration is the most difficult component of state and local population growth to forecast accurately because it is more volatile than either births or deaths, and subject to much larger fluctuations within a short period of time. In addition, migration rates can be based on several different measures of migration and the base population. The choice of the appropriate base population has received little attention from demographic researchers, but can have a tremendous impact on population projections. In this article, I develop three different models for projecting migration, each using a different denominator for migration rates. Population projections for ten states are made, using identical data and cohort component techniques, except for the different formulations of migration rates. Differences among the three sets of projections are noted, and conclusions are drawn regarding their usefulness as forecasts of population growth.     

International Migration, Sex Ratios, and the Socioeconomic Outcomes of Nonmigrant Mexican Women

This article assesses whether international migration from Mexico affects the marital, fertility, schooling, and employment outcomes of the Mexican women who remain behind by exploiting variation over time as well as across Mexican states in the demographic imbalance between men and women. I construct a gauge of the relative supply of men for women of different age groups based on state-level male and female population counts and the empirically observed propensity of men of specific ages to marry women of specific ages. Using Mexican census data from 1960 through 2000, I estimate a series of models in which the dependent variable is the intercensus change in an average outcome for Mexican women measured by state and for specific age groups, and the key explanatory variable is the change in the relative supply of men to women in that state/age group. I find that the declining relative supply of males positively and significantly affects the proportion of women who have never been married as well as the proportion of women who have never had a child. In addition, states experiencing the largest declines in the relative supply of men also experience relatively large increases in female educational attainment and female employment rates. However, I find little evidence that women who do marry match to men who are younger or less educated than themselves.     

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.     

不同预测模型下西藏自治区人口发展状况及对策研究

针对西藏自治区人口数据的有限性、数据序列的不平滑性等特点,分别采用一元回归、马尔萨斯模型、lo-gistic模型、GM(1,1)模型等4种方法,利用西藏自治区1980—2009年统计年鉴数据进行人口预测,综合考虑各种方案预测值,确定西藏自治区2010-2030年的人口数,在此基础上探讨基于人口增长的西藏经济发展模式与对策。结果表明,几种预测模型中平均相对误差率最小的是一元线性预测模型Ⅲ,仅0.2611%,马尔萨斯预测模型Ⅲ的平均相对误差率最大,也只有2.0767%;从预测结果看,Logistic模型的预测结果最为保守,仅为339.61万人,马尔萨斯模型预测最为乐观,高达380.10万人,未来相关研究中,可应用Logistic模型预测值作为下限,马尔萨斯模型预测值为上限;为综合应用四种模型的优势,研究中综合平均四个不同模型的预测结果,则西藏2010年人口将达到294.07万人,2015年达到311.34万人,2020年328.81万人,2025年346.54万人,2030年364.57万人,未来20年总人口增长趋势逐渐加快,区域发展中有一定人口压力。     

Forecast accuracy of australian subnational population projections

Despite the considerable resources devoted to making demographic projections in Australia over the past two decades, there have been few attempts to evaluate the performance of these projections in terms of forecast accuracy. This paper first considers the role of accuracy amongst other objectives of projection activity. Accepting accuracy as a legitimate goal, we then assess the performance of 48 sets of population projections and forecasts for states and territories of Australia prepared since 1970. Projection accuracy is assessed by reference to length of forecast horizon, population size and rate of growth. We also examine the main sources of forecast error in selected projections for each state and compare the performance of past projections with alternatives based on simple extrapolation of contemporary population trends.     

The uncertain timing of reaching 8 billion, peak world population, and other demographic milestones

We present new probabilistic forecasts of the timing of the world's population reaching 8 billion, the world's peak population, and the date at which one-third or more of the world's population would be 60+ years old. The timing of these milestones, as well as the timing of the Day of 7 Billion, is uncertain. We compute that the 60 percent prediction interval for the Day of 8 Billion is between 2024 and 2033. Our figures show that there is around a 60 percent chance that one-third of the world's population would be 60+ years old in 2100. In the UN 2010 medium variant, that proportion never reaches one-third. As in our past forecasts (Lutz et al. 2001, 2008), we find the chance that the world's population will peak in this century to be around 84 percent and the timing of that peak to be highly uncertain. Focal days, like the Day of 7 Billion, play a role in raising public awareness of population issues, but they give a false sense of the certainty of our knowledge. The uncertainty of the timing of demographic milestones is not a constant of nature. Understanding the true extent of our demographic uncertainty can help motivate governments and other agencies to make the investments necessary to reduce it.     

Predictability,complexity, and catastrophe in a collapsible model of population,development, and environmental interactions

"More and more population forecasts are being produced with associated 95 percent confidence intervals. How confident are we of those confidence intervals? In this paper, we produce a simulated dataset in which we know both past and future population sizes, and the true 95 percent confidence intervals at various future dates. We use the past data to produce population forecasts and estimated 95 percent confidence intervals using various functional forms. We, then, compare the true 95 percent confidence intervals with the estimated ones. This comparison shows that we are not at all confident of the estimated 95 percent confidence intervals." (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.     

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.     

Bayesian Probabilistic Projections of Life Expectancy for All Countries

We propose a Bayesian hierarchical model for producing probabilistic forecasts of male period life expectancy at birth for all the countries of the world to 2100. Such forecasts would be an input to the production of probabilistic population projections for all countries, which is currently being considered by the United Nations. To evaluate the method, we conducted an out-of-sample cross-validation experiment, fitting the model to the data from 1950–1995 and using the estimated model to forecast for the subsequent 10 years. The 10-year predictions had a mean absolute error of about 1 year, about 40 % less than the current UN methodology. The probabilistic forecasts were calibrated in the sense that, for example, the 80 % prediction intervals contained the truth about 80 % of the time. We illustrate our method with results from Madagascar (a typical country with steadily improving life expectancy), Latvia (a country that has had a mortality crisis), and Japan (a leading country). We also show aggregated results for South Asia, a region with eight countries. Free, publicly available R software packages called bayesLife and bayesDem are available to implement the method.