Bayesian Forecasting of Cohort Fertility

Abstract

There are signs that fertility in rich countries may have stopped declining, but this depends critically on whether women currently in reproductive ages are postponing or reducing lifetime fertility. Analysis of average completed family sizes requires forecasts of remaining fertility for women born 1970–1995. We propose a Bayesian model for fertility that incorporates a priori information about patterns over age and time. We use a new dataset, the Human Fertility Database (HFD), to construct improper priors that give high weight to historically plausible rate surfaces. In the age dimension, cohort schedules should be well approximated by principal components of HFD schedules. In the time dimension, series should be smooth and approximately linear over short spans. We calibrate priors so that approximation residuals have theoretical distributions similar to historical HFD data. Our priors use quadratic penalties and imply a high-dimensional normal posterior distribution for each country's fertility surface. Forecasts for HFD cohorts currently aged 15–44 show consistent patterns. In the United States, Northern Europe, and Western Europe, slight rebounds in completed fertility are likely. In Central and Southern Europe, East Asia, and Brazil, there is little evidence for a rebound. Our methods could be applied to other forecasting and missing-data problems with only minor modifications.

KEY WORDS:

• Bayesian model
• Demographic forecasting
• Improper priors
• Penalized least squares

Notes

Omitting the age subscript for clarity, our freeze-slope regression model is θ_c−θ_c-h = hΔ for h = 0, …,4. The OLS slope estimator for any regression model without an intercept is Σxy/Σx²,which in this case is Σ_h[h·(θ_c−θ_c-h)]/Σ_h[h²], or

The freeze-slope residual for cohort c + 1 is then