San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 The Cohort Projection Model Including Marital Status

The Cohort Survival Projection Model (CSPM) is a useful method for projecting population. The populations in the age groups in the base period are projected ahead one projection period, say five years, on the basis of survival rates. This gives all population age groups except the lowest, say 0 to 4 years of age. The future level for this age group is based upon fertility rates which are age-specific. The fertility rates give the number of births. The births are divided between male and female taking into account the sex ratio of births, which is 105 males per 100 females.

The fertility rates should be based upon the marital status of the female population in each age group.

The extension of the CSPM to take into account the marital status of the female population is given below. It is most convenient to express the model in matrix form. Let Xi,t be a two dimensional column vector of the female population in age group i in time period t with 1 denoting the unmarried state and 2 the married state. Thus

#### F1,i,t Xi,t    = F2,i,t

Now let Pi be a 2×2 matrix that gives the proportions of the female population in each marital status group that would be in the two statues for the next time period. These proportions are due to both survival factors and marriage and divorce rates. Thus

#### Xi,t+1 = PiXi,t

Written out the above relation is

#### |F1,t+1| |p1,1p1,2||F1,t| |         |  =|          ||      | |F2,t+1| |p2,1p2,2||F2,t|

A matrix Pi may be considered to be the product of a matrix Si representing survival proportions and a matrix Mi that represents marital status transition proportions.

## Births

The population of infants, 0 to 4 years of age for a five-year time period, is the product of fertility and survival rates, which are mother's age and marital status specific. The total number of babies born and survive for mothers in age group i are then

#### Bi,t+1 = f1,iF1,i,t + f2,iF2,i,t

The number of female babies is then computed on the basis of the ratio of male biths to female births, the sex ratio. For a sex ratio of 1.05 then the proportion of births that are female is 1/2.05.

The computation of the number of females in the first age group is equivalent to multiplying a two dimensional row vector Yi to the two dimensional column vector of female population Xi.

## The Matrix Form of the Model

The model can be expressed in matrix form, with Xt representing the column vector of the female population for time period t stacked by age group and marital status, as

#### Xt+1 = QXt

where the matrix Q is of the form

#### Y1Y2Y3Y4…Yn 0P100…0 00P20…0 ……………0 000……Pn-1 000……Pn

where n is the number of age groups. The last age group is different from the others in as much as survivors of this age group remain in that age group.

After the female population has been projected the male population can be projected from the base period populations and survival rates. The population of male babies can be derived from the projeced female populations and births. The male population can be projected by marital status as well as age but it is not necessary to do so. The marital status of females was necessary to allow for the different fertility rates of the two statuses.

## Generalization

Although the method as was presented was limited to two marital statuses there was no necessity for the subclassification to be limited to two or that they be limited to marital statuses.