TFR Predictions Based on Brownian Motion Theory

Authors

• Keilman Nico

DOI: https://doi.org/10.23979/fypr.44977

Keywords:

time series model, TFR, population forecasts

Abstract

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.