International Journal of Forecasting 15 (1999) 259–271 Validation, probability-weighted priors, and information in stochastic forecasts * Shripad Tuljapurkar, Carl Boe Mountain View Research, 2251 Grant Road, Suite A, Los Altos, CA 94024, USA Accepted 17 November 1998 Abstract This paper addresses issues that arise in evaluating, making, and using stochastic forecasts of future fertility in the United States. We begin with Lee’s ARMA model which leads to prediction intervals that are more realistic and informative than point-wise forecasts or traditional scenario methods. The roles of historical information and expert judgment are analyzed in terms of their effects upon prediction uncertainty. Validation experiments suggest that the model performs well in characterizing the uncertainty of the US fertility experience. We argue that the long-run average of fertility, a key assumption of the model, operates on a time scale not probed by time-series methods and we use Bayesian priors to quantify the additional prediction uncertainty that comes from making the long-run fertility assumption.  1999 Elsevier Science B.V. All rights reserved. Keywords: United States; Fertility; Mortality; Population projection; Validation; ARMA Models; Stochastic forecasts 1. Introduction rates (Lee and Carter, 1992), a trend that appears to be continuing. In contrast, historical fertility is quite Stochastic population forecasts are a constructive volatile and there is little useful predictive theory for way of incorporating unavoidable uncertainty when fertility change, as discussed by Lee (1993). As a making projections of the future (for the reasons result, uncertainty about fertility dominates uncer- why, see Alho (In press) and Lee, 1996). Lee and tainty in forecasted population especially over long Tuljapurkar (1994) made stochastic forecasts of the forecast periods. United States population by combining time-series Our point of departure is a time-series approach to methods with cohort-component projections to pro- modeling US fertility due to Lee (1993) and used in vide probability intervals for the forecast. In this modified form by Lee and Tuljapurkar (1994). We paper we address issues that arise in evaluating, begin by reviewing historical features of the total making, and using stochastic forecasts of future fertility rate (TFR) in the US from 1917 to 1992 (see fertility in the United States. Mortality forecasts can Heuser (1976) for development of the rates used rely on the clear long-term historical trend in death here). We ignore here the important question of the age-distribution of fertility; the cited papers consider * ways of modeling age patterns. We indicate the roles Corresponding author. Tel.: 11-650-9371280; fax: 11-650- 9371284. of historical information and expert judgment in 0169-2070 / 99 / $ – see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S0169-2070(98)00082-X