TFR Predictions Based on Brownian Motion Theory

In stochastic population forecasts, the predictive distribution of the TFR is of central concern. Common time series models can be used to predict the TFR and its moments on the short run (up to 10 or 20 years), but on the long run (40-50 years) they result in excessively wide prediction intervals. The aim of this study is to analyse and apply a time series model for the TFR, which restricts the predicted values to a certain pre-specified interval. I will model the time series of log TFR-values as a Brownian motion with absorbing upper barrier. I will give and analyse expressions for the predictive distribution of the log of the TFR assuming it follows a Brownian motion with absorbing ceiling; expressions for the first and second moments of the predictive distribution of the log of the TFR. When the log of the TFR follows a random walk with absorbing ceiling, I find that the second moment of the predictive distribution for the long-run TFR in Norway is insensitive for ceiling levels beyond a threshold of approximately 3.4 children per woman. This conclusion holds for a fairly broad range of innovation variances. If the log of the TFR follows a random walk, sample paths that exceed approximately 3.4 children per woman may be rejected when simulating future fertility in Western countries. This will not have any major effect on the width of the long-term predictive distribution.