Projecting Population Change by Age and Birth Parity: the Third Generation of Population Projections

Abstract

Early population projections described future changes in total population and could foresee unsustainable population growth. Age-specific population projections could identify trends in population ageing and demographic dividends, and they have been widely used in recent decades owing to the efforts of collecting and estimating demographic data by age. In recent years, data are becoming available to allow for population projections by age and birth parity, which could help understanding future changes in family structures. To take advantage of the opportunity created by these increasingly accessible data, this paper extends the cohort component method to project populations by age and birth parity and provides an application for Canada.

Résumé

Les premières projections démographiques décrivaient les changements futurs de la population totale et pouvaient prévoir une croissance démographique non soutenable. Les projections démographiques par âge pourraient indiquer les tendances du vieillissement de la population et des dividendes démographiques, et elles ont été largement utilisées au cours des dernières décennies grâce aux efforts de collecte et d'estimation des données démographiques par âge. Ces dernières années, les données sur la fécondité par rang de naissance sont devenues de plus en plus disponibles, ce qui permet d’effectuer des projections démographiques par âge et rang de naissance. Ce type de projections pourrait aider à décrire les changements futurs dans les structures familiales. Pour profiter de l’opportunité créée par ces données de plus en plus accessibles, cet article étend la méthode des composantes pour projeter les populations par âge et rang de naissance. Nous proposons ainsi une application de cette méthode pour le Canada.

Appendices

Appendix 1. Formulas for the age-parity-specific population model

For women in the age group (a, a+1) and parity i at time t, the progression to parity i+1 between time t and time t+1 can be described through a life table approach with two decrements: death and giving birth to the (i + 1) th child, which are measured by age-specific death rates and age-parity-specific fertility rates, namely m(a) and f_i + 1(a), respectively. Here, time interval (t,t + 1) is omitted for the reason of making the expressions simpler.

Let the forces of mortality and fertility at parity i to be constant within each age and time interval. Then, the forces of mortality and fertility are m(a) and f_i + 1(a); and the probabilities of surviving from age a to age (a + x) are e^{−m(a) ⋅ x} and ${e}^{-{f}_i(a)\cdot x}$, respectively (see Pollard 1973). Further, assuming that death and delivering birth are independent, the declining⁴ number of women in parity i at age (a + x), namely ${l}_i^d\left(a+x\right)$, is

$$ {l}_i^d\left(a+x\right)={l}_i(a){e}^{-\left[m(a)+{f}_{i+1}(a)\right]x} $$

(5)

where l_i(a) is the number of women in parity i at age a. Subsequently, the declining person-years are

$$ {\displaystyle \begin{array}{l}{L}_i^d(a)={l}_i(a)\underset{0}{\overset{1}{\int }}{e}^{-\left[m(a)+{f}_{i+1}(a)\right]x} dx,\\ {}{L}_i^d\left(a+1\right)={l}_i(a){e}^{-\left[m(a)+{f}_{i+1}(a)\right]}\underset{0}{\overset{1}{\int }}{e}^{-\left[m\left(a+1\right)+{f}_{i+1}\left(a+1\right)\right]x} dx.\end{array}} $$

(6)

Consequently, the two-decrement ratio for a woman in parity i to survive from ages (a,a + 1) to ages (a + 1,a + 2), namely S_i(a), is

$$ {S}_i(a)=\frac{L_i^d\left(a+1\right)}{L_i^d(a)}=\frac{e^{-\left[m(a)+{f}_{i+1}(a)\right]}\underset{0}{\overset{1}{\int }}{e}^{-\left[m\left(a+1\right)+{f}_{i+1}\left(a+1\right)\right]x} dx}{\underset{0}{\overset{1}{\int }}{e}^{-\left[m(a)+{f}_{i+1}(a)\right]x} dx} $$

(7)

Because the values of m(a) and f_i + 1(a) are small, (7) can be written approximately as

$$ {\displaystyle \begin{array}{c}{S}_i(a)\approx \frac{\left[1-m(a)-{f}_{i+1}(a)\right]\underset{0}{\overset{1}{\int }}\left[1-m\left(a+1\right)\cdotp x-{f}_{i+1}\left(a+1\right)\cdotp x\right] dx}{\underset{0}{\overset{1}{\int }}\left(1-m(a)\cdotp x-{f}_{i+1}(a)\cdotp x\right] dx}\\ {}\approx \frac{\left[1-m(a)-{f}_{i+1}(a)\right]\cdotp \left[1-0.5\cdotp m\left(a+1\right)-0.5\cdotp {f}_{i+1}\left(a+1\right)\right]}{1-0.5\cdotp m(a)-0.5\cdotp {f}_{i+1}(a)}\\ {}\approx \frac{\left[1-m(a)\right]\cdotp \left[1-0.5\cdotp m\left(a+1\right)\right]}{1-0.5\cdotp m(a)}+{f}_{i+1}(a)\cdotp \frac{\partial {S}_i(a)}{\partial {f}_{i+1}(a)}\left|{}_{f_{i+1}(a)=0,{f}_{i+1}\left(a+1\right)=0}\right.+\\ {}+f\left(a+1\right)\cdotp \frac{\partial {S}_i(a)}{\partial {f}_{i+1}\left(a+1\right)}\left|{}_{f_{i+1}(a)=0,{f}_{i+1}\left(a+1\right)=0}\right.\\ {}\approx \frac{\left[1-m(a)\right]\cdotp \left[1-0.5\cdotp m\left(a+1\right)\right]}{1-0.5\cdotp m(a)}-0.5\cdotp \frac{1-0.5\cdotp m\left(a+1\right)\Big]}{{\left[1-0.5\cdotp m(a)\right]}^2}\cdotp {f}_{i+1}(a)-\\ {}-0.5\cdotp \frac{\left[1-m(a)\right]}{1-0.5\cdotp m(a)}\cdotp {f}_{i+1}\left(a+1\right).\end{array}} $$

(8)

Neglecting the product terms between fertility and mortality rates, (8) is further simplified as:

$$ {S}_i(a)\approx s(a)-\frac{f_{i+1}(a)+{f}_{i+1}\left(a+1\right)}{2}. $$

(9)

In (9), $s(a)=\frac{\left[1-m(a)\right]\cdotp \left[1-0.5\times m\left(a+1\right)\right]}{1-0.5\times m(a)}$ is the survival ratio under mortality decrement. The difference between s(a) and the survival ratios based other life-table assumptions are negligible.

When the requirement on accuracy is high, (7) can be implemented by using more accurate but complex formulas. Since (9) is intuitively understandable, it is adopted in this paper.

Specifying now the survival ratio and fertility rate for time interval (t,t + 1) as s(a,t) and f_i(a, t), and using (9) for i = 0, the equation for parity 0 is obtained as

$$ {p}_0\left(a+1,t+1\right)={p}_0\left(a,t\right)\cdotp \left\{s\left(a,t\right)-\left[{f}_1\left(a,t\right)+{f}_1\left(a+1,t\right)\right]/2\right\} $$

(10)

where p_i(a, t) represents the number of women aged (a,a + 1) in parity i at time t. The number of women in parity 0 declined due to mortality is p₀(a, t) ⋅ [1 − s(a, t)]; and these women exited the total population. The number of women declined due to delivering the first birth is p₀(a, t) ⋅ [f₁(a, t) + f₁(a + 1, t)]/2; these women, however, did not exit the total population but entered parity 1 at time (t + 1). Thus, the equation of parity 1 is obtained by applying (9) for i = 1 and keeping the balance of total population as

$$ {\displaystyle \begin{array}{l}{p}_1\left(a+1,t+1\right)={p}_1\left(a,t\right)\cdotp \left\{s\left(a,t\right)-\left[{f}_2\left(a,t\right)+{f}_2\Big(a+1,t\Big)\right]/2\right\}+\\ {}+{p}_0\left(a,t\right)\cdotp \left[{f}_1\left(a,t\right)+{f}_1\Big(a+1,t\Big)\right]/2.\end{array}} $$

(11)

In the right-hand side of (11), the first term is the number of women declined due to mortality and fertility of parity 1, which is a result of two-decrement process and is obtained from applying (9) for i = 1; and the second term is the number of women entered parity 1 from parity 0 due to delivered the first birth, which is a result of keeping the balance of total population. In general, the equation of parity i is obtained for the same reason as

$$ {\displaystyle \begin{array}{l}{p}_i\left(a+1,t+1\right)={p}_i\left(a,t\right)\cdotp \left\{s\left(a,t\right)-\left[{f}_{i+1}\left(a,t\right)+{f}_{i+1}\Big(a+1,t\Big)\right]/2\right\}+\\ {}+{p}_{i-1}\left(a,t\right)\cdotp \left[{f}_i\left(a,t\right)+{f}_i\left(a+1,t\right)\right]/2.\end{array}} $$

(12)

The equation of the last open parity, however, is different, because women do not exit this parity when delivering an additional child. Thus, the fertility decrement in (9) is excluded, and the result is

$$ {p}_{m+}\left(a+1,t+1\right)={p}_{m+}\left(a,t\right)\cdotp s\left(a,t\right)+{p}_{m-1}\left(a,t\right)\cdotp \left[{f}_m\left(a,t\right)+{f}_m\left(a+1,t\right)\right]/2. $$

(13)

Appendix 2. Stability of the age-parity-specific population model

The stable population theory is often attributed to Lotka (1939). Nonetheless, the basic equation and the stable property, which guarantees the age structure of a population converge to a constant one if vital rates remain constant and there is no migration, were published earlier in two papers written by Lotka and his colleagues (Sharpe and Lotka 1911; Dublin and Lotka 1925). Citing these two papers, Linder (1938, p.144) generalized Lotka’s equation and stability property to include dimensions other than age. Linder’s work was written in the form of the Leslie model and is applied to the age-parity-specific population model presented in this paper as indicated below.

Consider the situation of over-time constant vital rates and zero migration, in which equation (2) above (the birth equation of the age-parity-specific population model) can be written as:

$$ {b}_i(t)=\sum \limits_{a={a}_{\mathrm{min}}}^{a_{\mathrm{max}}}{f}_i(a)\cdotp \left[{p}_{i-1}\left(a,t\right)+{p}_{i-1}\left(a,t+1\right)\right]/2 $$

(14)

In the form of the Leslie model, equation (14) becomes ${b}_i(t)=\sum \limits_{a={a}_{\mathrm{min}}}^{a_{\mathrm{max}}}{f}_i(a)\cdotp {p}_{i-1}\left(a,t\right)$, to which Linder’s procedure directly applies. However, [p_i − 1(a, t) + p_i − 1(a, t + 1)]/2 measures the person-years in (a,a + 1) and (t,t + 1) more accurately than p_i − 1(a, t) does, and it thus requires an indication that Linder’s procedure is still valid.

By applying equation (1), the survival equation of the age-parity-specific population model, to variable p_i − 1(a, t + 1), equation (14) can be written in a form that contains populations only at time t:

$$ {\displaystyle \begin{array}{c}{b}_i(t)=\sum \limits_{a={a}_{\mathrm{min}}}^{a_{\mathrm{max}}}{f}_i(a)\cdotp {p}_{i-1}\left(a,t\right)/2+\\ {}+\sum \limits_{a={a}_{\mathrm{min}}}^{a_{\mathrm{max}}}{f}_i(a)\cdotp {p}_{i-1}\left(a-1,t\right)\left\{s(a)-\left[{f}_i(a)+{f}_i\left(a+1\right)\right]/2\right\}/2\\ {}+\sum \limits_{a={a}_{\mathrm{min}}}^{a_{\mathrm{max}}}{f}_i(a)\cdotp {p}_{i-2}\left(a-1,t\right)\left[{f}_{i-1}(a)+{f}_{i-1}\left(a+1\right)\right]/2\Big\}/2.\end{array}} $$

(15)

Focusing on the populations by age and parity, equation (15) can be reorganized as

$$ {b}_i(t)=\sum \limits_{a={a}_{\mathrm{min}}}^{a_{\mathrm{max}}}{f}_i(a)\cdotp {p}_{i-1}\left(a,t\right)/2+\sum \limits_{a=\left({a}_{\mathrm{min}}-1\right)}^{\left({a}_{\mathrm{max}}-1\right)}{g}_{1i}\left(a+1\right)\cdotp {p}_{i-1}\left(a,t\right)+\sum \limits_{a=\left({a}_{\mathrm{min}}-1\right)}^{\left({a}_{\mathrm{max}}-1\right)}{g}_{2i}\left(a+1\right)\cdotp {p}_{i-2}\left(a,t\right) $$

(16)

where g_1i(a) and g_2i(a) are functions of the constant vital rates and contain f_i(a) as a factor, and are positive at reproductive ages and zero otherwise. Let the total number of births in (t-a, t-a + 1) be B(t-a). Then, p_i(a, t)/B(t − a) is determined by the constant viral rates and can be defined as the probability of a female birth to be in parity i at ages (a,a + 1), namely Pr(a, i| birth):

$$ \Pr \left(a,i| birth\right)={p}_i\left(a,t\right)/B\left(t-a\right) $$

(17)

Applying equation (17) to (16), a special case of Linder’s generalized equation is obtained as follows:

$$ {\displaystyle \begin{array}{c}{b}_i(t)=\sum \limits_{a={a}_{\mathrm{min}}}^{a_{\mathrm{max}}}0.5\cdotp {f}_i(a)\cdotp \Pr \left(a,i-1| birth\right)\cdotp B\left(t-a\right)\\ {}+\sum \limits_{a=\left({a}_{\mathrm{min}}-1\right)}^{\left({a}_{\mathrm{max}}-1\right)}{g}_{1i}\left(a+1\right)\cdotp \Pr \left(a,i-1| birth\right)\cdotp B\left(t-a\right)\\ {}+\sum \limits_{a=\left({a}_{\mathrm{min}}-1\right)}^{\left({a}_{\mathrm{max}}-1\right)}{g}_{2i}\left(a+1\right)\cdotp \Pr \left(a,i-2| birth\right)\cdotp B\left(t-a\right)\\ {}=\left\{\sum \limits_{a={a}_{\mathrm{min}}}^{a_{\mathrm{max}}}0.5\cdotp {f}_i(a)\cdotp \Pr \left(a,i-1| birth\right)+\sum \limits_{a=\left({a}_{\mathrm{min}}-1\right)}^{\left({a}_{\mathrm{max}}-1\right)}{g}_{1i}\left(a+1\right)\cdotp \Pr \left(a,i-1| birth\right)\right.\\ {}\left.+\sum \limits_{a=\left({a}_{\mathrm{min}}-1\right)}^{\left({a}_{\mathrm{max}}-1\right)}{g}_{2i}\left(a+1\right)\cdotp \Pr \left(a,i-2| birth\right)\right\}\times B\left(t-a\right).\end{array}} $$

(18)

In equation (18), summing up the functions in the large bracket over all the parities gives a function of age a, namely ϕ(a):

$$ {\displaystyle \begin{array}{l}\phi (a)=\sum \limits_{i=1}^m0.5\cdotp {f}_i(a)\cdotp \Pr \left(a,i-1| birth\right)+\sum \limits_{i=1}^m{g}_{1i}\left(a+1\right)\cdotp \Pr \left(a,i-1| birth\right)\\ {}+\sum \limits_{i=2}^m{g}_{2i}\left(a+1\right)\times \Pr \left(a,i-2| birth\right).\end{array}} $$

(19)

It is clear that ϕ(a) is positive in (a_min − 1, a_max) and zero otherwise. Since the minimal reproductive age is itself flexible, ϕ(a) is equivalent to the net maternity function of Lotka’s equation. Subsequently, summing for all birth orders in equation (18), the total number of births becomes a discrete form of Lotka’s equation (see Pollard 1973; p.24):

$$ B(t)=\sum \limits_{a=\left({a}_{\mathrm{min}}-1\right)}^{a_{\mathrm{max}}}\phi (a)\times B\left(t-a\right). $$

(20)

By applying Lotka’s procedure for proving that the continuous version of equation (20) is stable, B(t) will converge to a stable state regardless of initial conditions. If vital rates are constant in equation (18), b_i(t) and population size at all ages and parities will also converge to stable states.