PAPP103 - s03: Demographic models: fertility

Parametric models of fertility (cont.)

Romaniuk’s three-parameter model

Another class of parametric models of fertility is that proposed by Romaniuk (1973). The model suggested is based on a Pearson Type-1 statistical distribution and takes the form

$$f\left(x\right)=T\cdot {\left(1+\frac{x}{M-\alpha }\right)}^{\left\{\frac{\left(M-\alpha \right)\left[\delta -2\left(A-\alpha \right)\right]}{\delta \left(A-M\right)}\right\}}\cdot {\left(1-\frac{x}{\delta -M+\alpha }\right)}^{\left\{\frac{\left(\delta -M+\alpha \right)\left[\delta -2\left(A-\alpha \right)\right]}{\delta \left(A-M\right)}\right\}}$$

In its most constrained form (that is, taking α and δ as constants) this is a 3-parameter model with parameters T, M and A.

The parameter T is a scaling parameter affecting the level of the fertility distribution.

The parameters A and M are the average age and the modal age of the schedule respectively. (Given the desired right-skewness of the distribution, one would anticipate that A would always be greater than M). A and M jointly determine the shape of the distribution, but in a rather complex way: their joint position along the age axis (relative to the starting age) determines the location (the youthfulness) of the fertility schedule, their separation (A-M, which is a measure of the skewness of the distribution) determines its spread.

α and δ represent the start and duration of the fertile age range (and hence are analogous to s and w in the generalised Brass polynomial), and are considered fixed (i.e. treated as constants), with fertility zero outside of this range.

If one uses the same parameters as the Brass polynomial (i.e. setting the starting age, α, to 15; the duration of reproduction, δ, to 33 years; the modal age, M, to 26 (i.e. 15+11) and the mean age, A, to 28.2 (=15+13.2)), Romaniuk’s model has exactly the same form as the original Brass polynomial.

A factor analysis of age-specific fertility schedules would show that three factors are needed to account for about 97% of observed variation in fertility schedules - therefore Romaniuk’s model, in its simplified form has the right number of parameters, but it is not an easy model to handle in practice.

Further, it is not particularly easy to fit, and to obtain good fits in practice the presumed constants: α and δ, will also need to be varied. The problem is that the location is determined jointly by α, M and A, while the spread is determined jointly by δ, M and A. Thus, the formal parameters and constants between them are not independent in their relationship to the first three principal components of variance in fertility distributions.