Modeling and forecasting the time series of US fertility: age distribution, range, and ultimate level

Casino-based cash transfers and fertility among the Eastern Band of Cherokee Indians in North Carolina: A time-series analysis

Fertility decline remains a key concern among high-income countries. Prior research indicates that income supplementation through unconditional cash transfers (UCT) may correspond with increased fertility. We examine whether a casino-based UCT, in the form of per capita (percap) payments to members of the Eastern Band of Cherokee Indians (EBCI) corresponds with an acute increase in fertility. We use North Carolina vital statistics datasets from 1990 to 2006 and apply time-series analysis methods to examine the relation between specific months of percap payments (exposure) and monthly number of conceptions that result in live births (outcome) among the EBCI. We control for autocorrelation and monthly counts of births (arrayed by conception cohorts) among white women (ineligible for UCT receipt) in the study region. Results indicate an increase in conceptions that result in live births at 1 and 3 months after percap receipt among EBCI women aged ≥20 years (exposure month lag 1 coefficient = 1.74, p = 0.03; exposure month lag 3 coefficient = 1.60, p = 0.04). Exploratory analyses indicate that the observed fertility increase concentrates among primiparae EBCI women. We do not find any association between percap payment timing and births to EBCI women aged <20 years.

Projecting the Contribution of Assisted Reproductive Technology to Completed Cohort Fertility

Assisted reproductive technology (ART) is increasingly influencing the fertility trends of high-income countries characterized by a pattern of delayed childbearing. However, research on the impact of ART on completed fertility is limited and the extent to which delayed births are realized later in life through ART is not well understood. This study uses data from Australian fertility clinics and national birth registries to project the contribution of ART for cohorts of women that have not yet completed their reproductive life and estimate the role played by ART in the fertility 'recuperation' process. Assuming that the increasing trends in ART success rates and treatment rates continue, the projection shows that the contribution of ART-conceived births to completed fertility will increase from 2.1% among women born in 1968 to 5.7% among women born in 1986. ART is projected to substantially affect the extent to which childbearing delay will be compensated at older ages, suggesting that its availability may become an important factor in helping women to achieve their reproductive plans later in life.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11113-023-09765-3.

Projecting Spanish fertility at regional level: A hierarchical Bayesian approach

The transition from a demographic regime of high mortality and high fertility to one with low mortality and low fertility is universal and comes along with the process of socio-economic modernization. The Spanish total fertility rate has decreased to below replacement levels in the last decades. The decline has persisted since the 1960s and is diverse across the country. Based on that diversity, the use of population forecasts, not only at national but at regional levels, for planning purposes (governments and private sector) with large horizons has become a must to provide essential services. Using a Bayesian hierarchical model we constructed probabilistic fertility forecasts for Spain at the regional level. Although this approach is already issued by the United Nations little research has been done focusing on the Spanish subnational level. Our objective is to disaggregate the national projections of the total fertility rate for Spain into regional forecasts. The results of this research will show the model fitting, first to the national level and then using a multifaceted and continuous evolution of fertility over time, at the regional level, to check its convergence.

Robust Non-Parametric Mortality and Fertility Modelling and Forecasting: Gaussian Process Regression Approaches

A rapid decline in mortality and fertility has become major issues in many developed countries over the past few decades. An accurate model for forecasting demographic movements is important for decision making in social welfare policies and resource budgeting among the government and many industry sectors. This article introduces a novel non-parametric approach using Gaussian process regression with a natural cubic spline mean function and a spectral mixture covariance function for mortality and fertility modelling and forecasting. Unlike most of the existing approaches in demographic modelling literature, which rely on time parameters to determine the movements of the whole mortality or fertility curve shifting from one year to another over time, we consider the mortality and fertility curves from their components of all age-specific mortality and fertility rates and assume each of them following a Gaussian process over time to fit the whole curves in a discrete but intensive style. The proposed Gaussian process regression approach shows significant improvements in terms of forecast accuracy and robustness compared to other mainstream demographic modelling approaches in the short-, mid- and long-term forecasting using the mortality and fertility data of several developed countries in the numerical examples.

Synergy in fertility forecasting: improving forecast accuracy through model averaging

Accuracy in fertility forecasting has proved challenging and warrants renewed attention. One way to improve accuracy is to combine the strengths of a set of existing models through model averaging. The model-averaged forecast is derived using empirical model weights that optimise forecast accuracy at each forecast horizon based on historical data. We apply model averaging to fertility forecasting for the first time, using data for 17 countries and six models. Four model-averaging methods are compared: frequentist, Bayesian, model confidence set, and equal weights. We compute individual-model and model-averaged point and interval forecasts at horizons of one to 20 years. We demonstrate gains in average accuracy of 4–23% for point forecasts and 3–24% for interval forecasts, with greater gains from the frequentist and equal weights approaches at longer horizons. Data for England and Wales are used to illustrate model averaging in forecasting age-specific fertility to 2036. The advantages and further potential of model averaging for fertility forecasting are discussed. As the accuracy of model-averaged forecasts depends on the accuracy of the individual models, there is ongoing need to develop better models of fertility for use in forecasting and model averaging. We conclude that model averaging holds considerable promise for the improvement of fertility forecasting in a systematic way using existing models and warrants further investigation.

ACCOUNTING FOR UNCERTAINTY ABOUT PAST VALUES IN PROBABILISTIC PROJECTIONS OF THE TOTAL FERTILITY RATE FOR MOST COUNTRIES

Since the 1940s, population projections have in most cases been produced using the deterministic cohort component method. However, in 2015, for the first time, in a major advance, the United Nations issued official probabilistic population projections for all countries based on Bayesian hierarchical models for total fertility and life expectancy. The estimates of these models and the resulting projections are conditional on the UN's official estimates of past values. However, these past values are themselves uncertain, particularly for the majority of the world's countries that do not have longstanding high-quality vital registration systems, when they rely on surveys and censuses with their own biases and measurement errors. This paper extends the UN model for projecting future total fertility rates to take account of uncertainty about past values. This is done by adding an additional level to the hierarchical model to represent the multiple data sources, in each case estimating their bias and measurement error variance. We assess the method by out-of-sample predictive validation. While the prediction intervals produced by the extant method (which does not account for this source of uncertainty) have somewhat less than nominal coverage, we find that our proposed method achieves closer to nominal coverage. The prediction intervals become wider for countries for which the estimates of past total fertility rates rely heavily on surveys rather than on vital registration data, especially in high fertility countries.

A General Age-Specific Mortality Model With an Example Indexed by Child Mortality or Both Child and Adult Mortality

The majority of countries in Africa and nearly one-third of all countries require mortality models to infer the complete age schedules of mortality that are required to conduct population estimates, projections/forecasts, and other tasks in demography and epidemiology. Models that relate child mortality to mortality at other ages are important because almost all countries have measures of child mortality. A general, parameterizable component model (SVD-Comp) of mortality is defined using the singular value decomposition and calibrated to the relationship between child or child/adult mortality and mortality at other ages in the observed mortality schedules of the Human Mortality Database. Cross-validation is used to validate the model, and the predictive performance of the model is compared with that of the log-quadratic (Log-Quad) model, which is designed to do the same thing. Prediction and cross-validation tests indicate that the child mortality-calibrated SVD-Comp is able to accurately represent the observed mortality schedules in the Human Mortality Database, is robust to the selection of mortality schedules used for calibration, and performs better than the Log-Quad model. The child mortality-calibrated SVD-Comp can be used where and when child mortality is available but mortality at other ages is unknown.

Forecast accuracy hardly improves with method complexity when completing cohort fertility

Information on cohort fertility is critical for the understanding of population dynamics, but only in historical settings can it be calculated without forecasting. Several forecasting methods exist, but their strengths and weaknesses have not been evaluated. Relying on the Human Fertility Database, the largest high-quality fertility dataset to date, and the globally representative United Nations World Population Prospects, we present an assessment of all major methods that complete cohort fertility. This analysis is crucial to advance the understanding of benefits and drawbacks of state-of-the-art methods. We analyze forecast accuracy and uncertainty quantification, identify methodological breakthroughs, and uncover unresolved issues. This study constitutes an evaluation benchmark for cohort fertility forecasting and may inspire establishment of similar evaluation benchmarks in related fields.

Forecasts of completed fertility predict how many children will be born on average by women over their entire reproductive lifetime. These forecasts are important in informing public policy and influencing additional research in the social sciences. However, nothing is known about how to choose a forecasting method from a large basket of variants. We identified 20 major methods, with 162 variants altogether. The approaches range from naive freezing of current age-specific fertility rates to methods that use statistically sophisticated techniques or are grounded in demographic theory. We assess each method by evaluating the overall accuracy and if provided, uncertainty estimates using fertility data of all available birth cohorts and countries of the Human Fertility Database, which covers 1,096 birth cohorts from 29 countries. Across multiple measures of forecast accuracy, we find only four methods that consistently outperform the naive freeze rates method, and only two methods produce uncertainty estimates that are not severely downward biased. Among the top four, there are two simple extrapolation methods and two Bayesian methods. The latter are demanding in terms of input data, statistical techniques, and computational power but do not consistently complete cohort fertility more accurately at all truncation ages than simple extrapolation. This broad picture is unchanged if we base the validation on 201 United Nations countries and six world regions, including Africa, Asia, Europe, Latin America and the Caribbean, northern America, and Oceania.

A parsimonious characterization of change in global age-specific and total fertility rates

This study aims to understand trends in global fertility from 1950-2010 though the analysis of age-specific fertility rates. This approach incorporates both the overall level, as when the total fertility rate is modeled, and different patterns of age-specific fertility to examine the relationship between changes in age-specific fertility and fertility decline. Singular value decomposition is used to capture the variation in age-specific fertility curves while reducing the number of dimensions, allowing curves to be described nearly fully with three parameters. Regional patterns and trends over time are evident in parameter values, suggesting this method provides a useful tool for considering fertility decline globally. The second and third parameters were analyzed using model-based clustering to examine patterns of age-specific fertility over time and place; four clusters were obtained. A country’s demographic transition can be traced through time by membership in the different clusters, and regional patterns in the trajectories through time and with fertility decline are identified.

Bayesian Population Forecasting: Extending the Lee-Carter Method

In this article, we develop a fully integrated and dynamic Bayesian approach to forecast populations by age and sex. The approach embeds the Lee-Carter type models for forecasting the age patterns, with associated measures of uncertainty, of fertility, mortality, immigration, and emigration within a cohort projection model. The methodology may be adapted to handle different data types and sources of information. To illustrate, we analyze time series data for the United Kingdom and forecast the components of population change to the year 2024. We also compare the results obtained from different forecast models for age-specific fertility, mortality, and migration. In doing so, we demonstrate the flexibility and advantages of adopting the Bayesian approach for population forecasting and highlight areas where this work could be extended.

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.

Probabilistic forecasting using stochastic diffusion models, with applications to cohort processes of marriage and fertility

In this article, we show how stochastic diffusion models can be used to forecast demographic cohort processes using the Hernes, Gompertz, and logistic models. Such models have been used deterministically in the past, but both behavioral theory and forecast utility are improved by introducing randomness and uncertainty into the standard differential equations governing population processes. Our approach is to add time-series stochasticity to linearized versions of each process. We derive both Monte Carlo and analytic methods for estimating forecast uncertainty. We apply our methods to several examples of marriage and fertility, extending them to simultaneous forecasting of multiple cohorts and to processes restricted by factors such as declining fecundity.

Probabilistic projections of the total fertility rate for all countries

We describe a Bayesian projection model to produce country-specific projections of the total fertility rate (TFR) for all countries. The model decomposes the evolution of TFR into three phases: pre-transition high fertility, the fertility transition, and post-transition low fertility. The model for the fertility decline builds on the United Nations Population Division's current deterministic projection methodology, which assumes that fertility will eventually fall below replacement level. It models the decline in TFR as the sum of two logistic functions that depend on the current TFR level, and a random term. A Bayesian hierarchical model is used to project future TFR based on both the country's TFR history and the pattern of all countries. It is estimated from United Nations estimates of past TFR in all countries using a Markov chain Monte Carlo algorithm. The post-transition low fertility phase is modeled using an autoregressive model, in which long-term TFR projections converge toward and oscillate around replacement level. The method is evaluated using out-of-sample projections for the period since 1980 and the period since 1995, and is found to be well calibrated.

Functional data modelling approach for analysing and predicting trends in incidence rates--an application to falls injury

SUMMARY

Policy decisions about the allocation of current and future resources should be based on the most accurate predictions possible. A functional data analysis (FDA) approach improves the understanding of current trends and future incidence of injuries. FDA provides more valid and reliable long-term predictions than commonly used methods.

INTRODUCTION

Accurate information about predicted future injury rates is needed to inform public health investment decisions. It is critical that such predictions derived from the best available statistical models to minimise possible error in future injury incidence rates.

METHODS

FDA approach was developed to improve long-term predictions but is yet to be widely applied to injury epidemiology or other epidemiological research. Using the specific example of modelling age-specific annual incidence of fall-related severe head injuries of older people during 1970-2004 and predicting rates up to 2024 in Finland, this paper explains the principles behind FDA and demonstrates their superiority in terms of prediction accuracy over the more commonly reported ordinary least squares (OLS) approach.

RESULTS

Application of the FDA approach shows that the incidence of fall-related severe head injuries would increase by 2.3-2.6-fold by 2024 compared to 2004. The FDA predictions had 55% less prediction error than traditional OLS predictions when compared to actual data.

CONCLUSIONS

In summary, FDA provides more accurate predictions of long-term incidence trends than commonly used methods. The production of FDA prediction intervals for future injury incidence rates gives likely guidance as to the likely accuracy of these predictions.

Population-based prediction of trauma volumes at a Level 1 trauma centre

OBJECTIVE

With an ageing US population, the demographics of traumatic injuries are being significantly altered. Census projections predict that the number of Americans over age 65 will double in the next 20 years. We used stochastic methods to forecast trauma admissions in order to predict the effects of such demographic changes at our trauma centre.

METHODS

Age- and sex-related rates of traumatic admission were determined using population statistics and trauma registry data from 1994 to 1999. These rates were then projected from 2000 to 2025 based on both the Lee-Carter and random walk with drift methods. Stochastic population projections were made and paired with the projected trauma rates, allowing estimation of total trauma volume.

RESULTS

Trauma rates were predicted to increase for most age groups. Trauma admissions are predicted to increase 57% by 2024. By 2019, 50% of trauma admissions will be 60 or older.

CONCLUSIONS

Our trauma volume is expected to increase 57% by 2024, an increase of 2% per year. More of this volume will consist of elderly patients, potentially requiring increased health-care resources.

An approach to forecasting health expenditures, with application to the U.S. Medicare system

OBJECTIVE

To quantify uncertainty in forecasts of health expenditures.

STUDY DESIGN

Stochastic time series models are estimated for historical variations in fertility, mortality, and health spending per capita in the United States, and used to generate stochastic simulations of the growth of Medicare expenditures. Individual health spending is modeled to depend on the number of years until death.

DATA SOURCES/STUDY SETTING

A simple accounting model is developed for forecasting health expenditures, using the U.S. Medicare system as an example.

PRINCIPAL FINDINGS

Medicare expenditures are projected to rise from 2.2 percent of GDP (gross domestic product) to about 8 percent of GDP by 2075. This increase is due in equal measure to increasing health spending per beneficiary and to population aging. The traditional projection method constructs high, medium, and low scenarios to assess uncertainty, an approach that has many problems. Using stochastic forecasting, we find a 95 percent probability that Medicare spending in 2075 will fall between 4 percent and 18 percent of GDP, indicating a wide band of uncertainty. Although there is substantial uncertainty about future mortality decline, it contributed little to uncertainty about future Medicare spending, since lower mortality both raises the number of elderly, tending to raise spending, and is associated with improved health of the elderly, tending to reduce spending. Uncertainty about fertility, by contrast, leads to great uncertainty about the future size of the labor force, and therefore adds importantly to uncertainty about the health-share of GDP. In the shorter term, the major source of uncertainty is health spending per capita.

CONCLUSIONS

History is a valuable guide for quantifying our uncertainty about future health expenditures. The probabilistic model we present has several advantages over the high-low scenario approach to forecasting. It indicates great uncertainty about future Medicare expenditures relative to GDP.

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)

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)