Lawrence R. Carter

Modeling and Forecasting U.S. Mortality

Abstract Time series methods are used to make long-run forecasts, with confidence intervals, of age-specific mortality in the United States from 1990 to 2065. First, the logs of the age-specific death rates are modeled as a linear function of an unobserved period-specific intensity index, with parameters depending on age. This model is fit to the matrix of U.S. death rates, 1933 to 1987, using the singular value decomposition (SVD) method; it accounts for almost all the variance over time in age-specific death rates as a group. Whereas e 0 has risen at a decreasing rate over the century and has decreasing variability, k(t) declines at a roughly constant rate and has roughly constant variability, facilitating forecasting. k(t), which indexes the intensity of mortality, is next modeled as a time series (specifically, a random walk with drift) and forecast. The method performs very well on within-sample forecasts, and the forecasts are insensitive to reductions in the length of the base period from 90 to 30 ...

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.

Joint Forecasts of U.S. Marital Fertility, Nuptiality, Births, and Marriages Using Time Series Models

Abstract This article presents a new approach to forecasting U.S. marital fertility, nuptiality, births, and marriages. The analysis represents a wedding of demographic and statistical time series in models amenable to Box-Jenkins techniques of model identification, estimation, diagnosis, and forecasting. The models demonstrate the advantages in this approach in forecasting both rates and events as opposed to the common practice of simply forecasting events. Using the best models of indexes of fertility and nuptiality, forecasts of births and first marriages are made for the U.S. for the years 1983–2000. Analyses of these forecasts are made with discussions of their demographic realism in terms of their forecast confidence intervals.

FORECASTING U.S. MORTALITY:. A Comparison of Box-Jenkins ARIMA and Structural Time Series Models

This article compares two methodologies for modeling and forecasting statistical time series models of demographic processes: Box-Jenkins ARIMA and structural time series analysis. The Lee-Carter method is used to construct nonlinear demographic models of U.S. mortality rates for the total population, gender, and race and gender combined. Single time varying parameters of k, the index of mortality, are derived from these model and fitted and forecasted using the two methodologies. Forecasts of life expectancy at birth, eo, are generated from these indexes of k. Results show marginal differences in fit and forecasts between the two statistical approaches with a slight advantage to structural models. Stability across models for both methodologies offers support for the robustness of this approach to demographic forecasting.

Long-Run Relationships in Differential U.S. Mortality Forecasts by Race and Sex: Tests for Co-integration

There is considerable interest in demography in comparing mortality by race and by sex to determine the magnitude of the differentials, to understand why they exist and to detect if they are changing over time. The question is if they will converge, and if so, how soon. For race, nowhere is that interest so evident as in the controversy over the well-researched black/white mortality crossover. A persistent decline in the age of crossover signals convergence, while advancing age of crossover indicates mortality divergence. Even so, white and nonwhite life expectancies have been converging generally over time. A similar concern is shown with sex differences since they have displayed a marked increase in the later twentieth century. The confounding of race and sex in mortality analyses invites the desegregation of the U.S. population into four race-sex specific groups, and an investigation of their pairings to understand their possible relationships well into the future. This paper does so by extending the mortality analysis by Lee and Carter (1992) for the total U.S. population and Carter and Lee (1992) for U.S. sex differentials. The basic approach is to examine some life table functions derived from forecasts of mortality for white males, white females, nonwhite males and nonwhite females using the Lee-Carter method (Lee and Carter 1992). We focus on puzzling patterns of life expectancy forecasts that show white and nonwhite life expectancies at birth initially continuing their historic decline, but then reversing themselves and increasing in the latter part of the forecast period.

Imparting structural instability to mortality forecasts: Testing for sensitive dependence on initial conditions with innovations

This article explores a nontraditional approach to examining the problem of forecast uncertainty in extrapolative demographic models. It builds on prior research on stochastic time series forecast models, but diverges to examine their deterministic counterparts. The focus here is an examination of the structural integrity of the Lee‐Carter (1992) method applied to mortality forecasts. I investigate the nonlinear dynamics of the Lee‐Carter method, particularly its sensitive dependence of the forecasts on the initial conditions of the model. I examine the Lee‐Carter nonlinear demographic model, mx,t — exp (ax+ bxkt + ex,t), which is decomposed using SVD to derive a single time‐varying linear index of mortality, kt. From a 90 year time series of kt, forty nine 40 year realizations are sampled. These realizations are modeled and estimated using Box‐Jenkins techniques. The estimated parameters of these realizations and the first case of each of the samples are the initial conditions for the iterations of nonlinearized transformation of k, to exp (kt). The terminal year for each of the 49 iterated series is 2065. The deterministic nonlinear dynamics of this system of 49 iterated series is investigated by testing its Lyapunov exponents as a nonparametric diagnostic of a one dimensional dynamical system. The exponents are all negative, indicating that chaos is not prevalent in this system. The nonexistence of chaos suggests stability in the model and reaffirms the predictability of this one dimensional map. Augmenting the iterations of the initial conditions with additive stochastic innovations, {et, t ≥ 1}, creates a stochastic dynamical system of the form, k t = k t,−1 — c + ϕ flu +et. Here, et is treated as a surrogate for some unanticipated time series event (e.g. an epidemic) that impacts the deterministic map. Gaussian white noise innovations do not move the iterations far from equilibrium and only for short time intervals. So, stepping the mean of the innovations by .01 produces stable Lyapunov exponents until the mean equals .35 where some of the exponents are positive. At this point, deterministic chaos is evident, implying instability in the forecasts. The substantive implications of this instability are discussed.