PAPP103 - s03: Demographic models: fertility

Parametric models of fertility (cont.)

Brass’ fertility polynomial

Around 1962, Brass devised a parametric model - his fertility polynomial. He used this widely in developing many of his indirect estimation methods. The polynomial is described in greater detail in Brass (1975).

In this model, the age-specific fertility rate, f(x) at exact age x is given by:

$$f\left(x\right)=c\left(x-s\right){\left(s+33-x\right)}^2$$ $$\text{for }s<x<s+33$$

Since the integral of the f(x) values between s and s+33 is the TFR, we can express c in terms of the TFR:

$${\displaystyle \mathrm{TFR}=\underset{s}{\overset{s+33}{\int }}f(x)}dx={\displaystyle \underset{s}{\overset{s+33}{\int }}c\left(x-s\right){\left(s+33-x\right)}^{2}dx}$$ $$c=\frac{TFR}{{\displaystyle \underset{s}{\overset{s+33}{\int }}\left(x-s\right){\left(s+33-x\right)}^{2}dx}}$$

Once the integration is complete, this expression is independent of the value of s. In fact, it can be simplified to the relationship

$$c=\frac{12}{{33}^4}TFR=\frac{TFR}{98826.75}\approx .00001012TFR$$

This is a two-parameter model. The constant c is a level parameter, proportional to the TFR, while s is a location parameter, representing the age at which fertility begins. Fertility is assumed to cease 33 years later, at age (s+33). At ages under age s and above age s+33, f(x) is fixed at zero