Incorporating model uncertainty into fertility schedule estimates for population forecasting

Jonathan Forster, University of Southampton
Jakub Bijak, University of Southampton
Peter W. F. Smith, University of Southampton

Population estimation and forecasting via the cohort-component approach requires, as a key input, an estimate or projection of the age-specific fertility rates over the corresponding estimation or forecast period. A wide range of models have been proposed for estimating fertility schedules; however, different models not only yield different best estimates but also generate different prediction intervals. In this paper, we develop a Bayesian statistical approach to the quantification of fertility schedule uncertainty that incorporates model uncertainty. The Bayesian approach under model uncertainty updates a prior probability distribution over the models (in the form of probabilities or weights assigned to models) to a posterior distribution, in light of observed data. The posterior distribution accounts for how well the various models fit the observed historical data, and is used explicitly in weighting the models in projections. This approach is sometimes referred to as ‘Bayesian model averaging’. Although the principles of Bayesian inference (including under model uncertainty) are straightforward, practical methodology for incorporating probabilistic model uncertainty into estimates and forecasts of fertility schedules is currently underdeveloped. In this paper, we provide such a methodology. Our approach is illustrated on data from England and Wales. Using a selection of different plausible models, we present the estimated fertility schedules provided by each model, and illustrate how the posterior model probabilities are computed, together with the resulting forecast arising from integrating over the models to account for model uncertainty. The integrated projection uncertainty provides a coherent and more realistic assessment of uncertainty than any corresponding analysis based upon a single model. We also discuss how `model-averaged' fertility schedules can be combined with similarly integrated mortality forecasts in an overall probabilistic population projection.