To develop high-low bounds for population projections, traditional demographic forecasts assume fertility is always high or always low. Such bounds are related to bounds on average annual fertility up to year t rather than to individual year bounds in time series forecasts, and so the autocorrelation of time series forecast errors is important in practical applications. This paper develops methods for using time series methods to make constrained long term forecasts of fertility. Specifically, age-time variations in fertility are modeled with a single time-varying parameter, or fertility index; upper and lower bounds on the total fertility rate are imposed by forecasting an inverse logistic transform of the fertility index; the long run level of the fertility forecast is also constrained to equal a prespecified level. The principal interest is in the variance and the autocorrelation structure of the forecast errors. Based on these for the U.S.A. The paper concludes: (1) the probability interval for average fertility up to time t begins to contract after about 50 years, but only very slightly; (2) the probability interval for average fertility up to year 2065 is about three-fifths as wide as that for single year fertility in 2065, but is still far wider than the band for official forecasts; (3) realizations of the simple ARMA (1,0,1) forecast model exhibit long fluctuations something like actual fertility in industrial nations; (4) the model of fertility age patterns fits poorly at older ages, but may be adequate for present purposes.
