The discussion concerning fertility in Singapore during the post-Second World War period has been confined to consideration of the reduction in total period fertility. Although a focus on period fertility makes good sense in planning for school and university admissions, for maternal and child health care services, for future national service intakes and so on, it is easy, yet potentially misleading, to go one stage further and deduce that since total period fertility has been below the replacement level for a few years, the population is not replacing itself.

Ni Bhrolchain (1992) has provided a trenchant critique of the cohort perspective. It certainly appears that the period approach to fertility analysis, by way of period parity progression rather than through the increasingly discredited total period fertility, is reclaiming some of the ground lost to the cohort perspective advocated by Ryder (1968). However, it seems clear that the question of whether a population is actually replacing itself can be much more effectively answered from a cohort perspective of fertility. It is one of the key aims of this paper to provide such a viewpoint from which Singapore's projected fertility experience can be judged. Obviously, if total period fertility remains below the replacement level for a long period of time, then this fact will eventually be reflected in complete cohort fertility also falling below the replacement level. The question to be addressed in this paper is whether total period fertility has yet been below the replacement level in Singapore long enough for this to be likely to happen in the short- to medium-term future.

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* The author of this paper may be reached at Skandia International Insurance Corp., #13-10 Ocean Building, 10 Collyer Quay, Singapore 049315.

In a recent paper, Street (1995) calculated complete cohort fertility rates for 18 generations of Singapore women whose childbearing years from age 15 to 44 spanned the post-war period from 1946 to 1992. These cohorts were born between the years 1931 and 1948. In this earlier paper, the concept of partial cohort fertility was suggested as a firm base from which reasonably reliable estimates of future complete cohort fertility could be determined. It is the purpose of the paper to provide these estimates.

At any point in time, for each cohort of women below age 44 (the assumed upper age limit of a woman's childbearing years), partial cohort fertility is defined to be the accumulated achieved cohort fertility from age 15 (the assumed lower age limit of a woman's childbearing years) to the current age. Thus, for example, for women aged n (n < 44) in 1992, partial cohort fertility would be:

n

PCFn,92 = E ASFRi,92-n+i

i=15

where, ASFRi,92-n+i is the age-specific fertility rate (ASFR) at age i in the calendar year 1992-n+i.

In Singapore, it is interesting to differentiate these rates by race; although the majority (about 78 per cent) of the population is Chinese, there are significant minorities, namely Malay (about 14 per cent) and Indian (about 7 per cent). The projections in this paper will therefore be made by race as well as for the aggregate population.

Partial cohort fertility rates in Singapore for the year 1992 are shown in the accompanying table and graphs. Figure 1 shows partial cohort fertility for the aggregate Singapore population and figure 2 by race.

It has been shown by Street (1995) that, up to and including 1992, no cohorts had completed their childbearing with fewer than 2.1 children on average and that this is true not only for Singapore's population as a whole but also for the three racial groups considered separately. But the question remains regarding those cohorts who were still within their childbearing years at the end of 1992. How can we best project complete cohort fertility for these women? In fact, the question to be addressed in this paper is somewhat less ambitious than this. Our attention will be focused only on the 10 further cohorts who were aged from 34 to 43 years in 1992.

For any such projection, it would seem sensible to base the estimates on as much firm data as it is possible to obtain. It follows that, since partial cohort figures are already available for the cohorts aged between 34 and 43 at the end of 1992, in order to predict the complete cohort fertility for these women, all that is needed are the following estimates, using as before the notation ASFRi,j to denote the age-specific rate at age i in calendar year j

Cohort year of birth	Further estimates necessary for complete cohort fertility to be projected
1949	ASFR44,93
1950	ASFR43,93	+ ASFR44,94
1951	ASFR42,93	+ ASFR43,94	+ ASFR44,95
...	...	...	...
...	...	...	...
1958	ASFR35,93	+ ASFR36,94	+ ASFR37,95	+ ...	+ ASFR44,02

It is clear, therefore, that in order to project complete cohort fertility for the next 10 generations of females, it is necessary to estimate certain period fertility rates over the next 10 years. However, the only estimates necessary will be of period fertility (at single ages) ranging from estimates for ages 35 to 44 for the year 1993, to an estimate for age 44 for the year 2002. The appropriate sums shown in the above array would then be added to the corresponding 1992 partial cohort fertility rates to obtain projections of complete cohort fertility for the 10 further cohorts.

Much research has been done on attempting to develop a general mathematical formulation for the fertility curve. Probably the best known models are Brass's Modified Gompertz Method (1980) and Coale and Trussel's Model Fertility Curves (1974). However, the aim in the present paper is much more mundane than to try to develop a general mathematical model for "fertility"; it is rather to establish a mathematical representation that appears to be suitable for the limited projection of the specific data that are available on Singapore.

The projection function

The following graphs indicate that period fertility would appear to be much more amenable than cohort fertility to such modelling. The first series of six graphs show typical aggregate population cohort fertility curves, from which it will be seen that it would be difficult indeed to try to develop a general mathematical expression for Singapore's cohort fertility. The curves show a definite skewness to the right but little smoothness.

However, if the curves of total period fertility are considered, it will be seen from the following six plots that as the fertility-weighted average age has increased over the past 15 years, so the fertility curve has become more symmetrical although there is still a slight tendency for skewness to the right. It will also be observed that the graphs show a much more well-behaved appearance and could possibly be approximated by some form of mathematical expression. But which mathematical function should be chosen?

The first attempt was to fit a log-normal distribution to the data. Figure 3 (page 66) shows a log-normal curve fitted to the 1992 aggregate population fertility data with the actual fertility rates plotted as "+" signs.

For the central ages from about 20 to 40, the fit is quite good, but at the extreme ages the fitted log-normal distribution understates fertility at the youngest ages and overstates it at the highest ages.

It is clear that the fit at the extreme ages needs to be improved and the next step would be to observe what sort of fit a normal distribution would provide. Figure 4 (page 66) shows a normal curve fitted to the same 1992 aggregate population data.

Here, for the extreme ages at 20 and below and at 40 and above, the normal fit is quite good, but at the central ages the normal curve understates fertility for ages up to the mode of the actual fertility distribution and overstates it for ages above the mode.

These same general characteristics are also apparent in the fitted normal and log-normal curves to the 1992 data for the three racial groups as the graphs on page 67 show.

It will be recalled from what was mentioned previously that, in order to project complete cohort fertility for the next 10 cohorts of women, only estimates are required of period fertility at ages 35+t to 44 in each of the years 1993+t to 2002, respectively, where 0 < t < 9. Thus, focus is on the fertility curve in its final one-third range, where for all races, as has been seen, it seems to tend towards normality. However, the approach to normality differs between the aggregate population and between the three races. The fertility curves for Malays and Indians appear to become more or less normal at ages 35 and above, whereas those for the Chinese and for the aggregate population tend to follow the log-normal curve until the late thirties and the normal curve thereafter.

For the purposes of the projections for the period 1993 to 2002, it was assumed that, at all ages from 35 to 44, the normal curve provided a suitable model for Malay and Indian fertility. For the Chinese, it was assumed that the log-normal curve was appropriate to age 37 and the normal curve thereafter. For the aggregate population, it was assumed that the normal curve was appropriate for ages from 40 to 44, but that from 35 to 39 a blend of the normal and log-normal curves would be more suitable. After some trial and error, coefficients of (t/5)2 and 1 - (t/5)2 at age 35+t (0 < t < 4) were adopted for the normal and log-normal components, respectively.

These fitted curves, together with actual fertility rates at ages 35 and above, are shown in the following graphs for the aggregate population and the three racial groups for the years 1983 and 1992. It should perhaps be mentioned at this point that all the curves here were fitted by the method of moments.

It is noticeable that in both years illustrated, there is a broad improvement in the quality of the fit as the population size increases from Indian to Malay to Chinese to the aggregate population.

The projections parameters

Having decided upon the general shapes to be adopted for the projection of the various racial and aggregate population fertility curves, the next step is to project the various parameters that are required. The first requirement here is estimates of total period fertility for the three racial groups and the aggregate population for the 10 years from 1993 to 2002. The second requirement is to give consideration to the necessary parameters for the projected normal and log-normal probability distributions.

Projected total period fertility

There has been a small but definite increase in total period fertility since the Government's new population policy was introduced (Yap, 1995). In 1992, total period fertility stood at 2.60, 1.56, 1.95 and 1.76 for the Malays, Chinese, Indian and aggregate population, respectively.

Projections were made on three bases for total period fertility. Firstly, it was assumed that for the next 10 years it would remain at these 1992 levels. Secondly, and rather more optimistically (at least for the Indians and Chinese), it was assumed that for each racial group total period fertility would approach 2.1 linearly over a 20-year period. Thirdly, and perhaps unrealistically optimistic, it was assumed that each race would achieve the 2.1 level in 10 years. It should be pointed out that, for the Malays, the second and third of these assumptions imply declining total period fertility.

Projected model parameters

The first assumption regarding fertility-weighted average age was that, for each race, it would remain at the same level as in 1992, namely at 27.95, 29.59, 27.97 and 29.08 years for the Malays, Chinese, Indians and aggregate population, respectively. Since 1986, however, all races have shown an increase in fertility-weighted average age; respective mean annual increments of about 0.05, 0.16, 0.03 and 0.14 years have been experienced. The second assumption was that these annual increases would continue until 2002.

For the Malays, the standard deviation of the fertility schedule has increased by an average annual amount of about 0.014 since 1986. The Chinese, Indian and aggregate population standard deviations have, however, shown reductions over this period by 0.033, 0.010 and 0.022 per annum on average, respectively. The first projection basis for standard deviations was to assume that they remained at their 1992 levels until 2002, and the second was that each year they changed by these same 1986-1992 average amounts.

The log-normal parameters for the Chinese and aggregate population projections were selected to be consistent with these average ages and standard deviations.

Therefore, there are 12 separate projections as follows: (three fertility assumptions) x (two mean age assumptions) x (two standard deviation

assumptions).

Total period fertility (TFF)	Fertility-weighted mean age	Standard deviation of fertility-weighted age
(1)	(2)	(3)
1992 TPF	1992 mean ages	1992 standard deviations
TPF to 2.10 by year 2012	Mean ages increasing by 1986- 1992 average	Standard deviations chang- ing by 1986-1992
TPF to 2.10 by year 2002

The results of the projections on these 12 bases are shown below with the notation A/B/C denoting a projection using bases A, B and C from columns (1), (2) and (3) of the tableau immediately above, respectively.

1992 age	Malay	Chinese	Indian	Total
34	0.356	0.238	0.232	0.252
35	0.263	0.172	0.164	0.186
36	0.188	0.120	0.113	0.133
37	0.131	0.080	0.075	0.090
38	0.088	0.049	0.048	0.058
39	0.056	0.029	0.029	0.035
40	0.035	0.016	0.017	0.020
41	0.020	0.008	0.009	0.011
42	0.010	0.004	0.004	0.005
43	0.004	0.001	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.345	0.250	0.235	0.260
35	0.255	0.180	0.166	0.191
36	0.183	0.125	0.114	0.136
37	0.127	0.083	0.075	0.093
38	0.086	0.051	0.048	0.059
39	0.055	0.030	0.029	0.036
40	0.034	0.017	0.017	0.021
41	0.019	0.009	0.009	0.011
42	0.010	0.004	0.004	0.005
43	0.004	0.001	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.334	0.263	0.237	0.268
35	0.247	0.189	0.168	0.197
36	0.178	0.131	0.115	0.140
37	0.124	0.086	0.076	0.095
38	0.084	0.053	0.048	0.061
39	0.054	0.031	0.029	0.036
40	0.033	0.017	0.017	0.021
41	0.019	0.009	0.009	0.011
42	0.010	0.004	0.004	0.005
43	0.004	0.001	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.376	0.291	0.240	0.299
35	0.277	0.210	0.170	0.220
36	0.198	0.146	0.117	0.156
37	0.138	0.097	0.077	0.106
38	0.092	0.060	0.049	0.068
39	0.059	0.035	0.030	0.041
40	0.036	0.019	0.017	0.023
41	0.021	0.010	0.009	0.012
42	0.010	0.004	0.004	0.006
43	0.004	0.001	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.364	0.308	0.243	0.309
35	0.269	0.221	0.172	0.227
36	0.193	0.153	0.118	0.161
37	0.134	0.101	0.078	0.109
38	0.090	0.062	0.050	0.070
39	0.058	0.036	0.030	0.042
40	0.036	0.020	0.017	0.024
41	0.020	0.010	0.009	0.013
42	0.010	0.004	0.004	0.006
43	0.004	0.001	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.352	0.325	0.245	0.320
35	0.260	0.233	0.174	0.234
36	0.187	0.160	0.119	0.165
37	0.130	0.105	0.079	0.112
38	0.088	0.065	0.050	0.071
39	0.057	0.038	0.030	0.043
40	0.035	0.021	0.018	0.024
41	0.020	0.010	0.009	0.013
42	0.010	0.005	0.004	0.006
43	0.004	0.002	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.363	0.225	0.228	0.242
35	0.269	0.162	0.161	0.178
36	0.193	0.112	0.110	0.127
37	0.134	0.074	0.073	0.086
38	0.090	0.046	0.046	0.055
39	0.058	0.027	0.028	0.033
40	0.036	0.015	0.016	0.019
41	0.020	0.008	0.009	0.010
42	0.010	0.004	0.004	0.005
43	0.004	0.001	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.352	0.236	0.231	0.250
35	0.261	0.169	0.163	0.183
36	0.187	0.117	0.112	0.130
37	0.131	0.077	0.074	0.088
38	0.088	0.047	0.047	0.057
39	0.057	0.028	0.029	0.034
40	0.035	0.015	0.017	0.020
41	0.020	0.008	0.009	0.010
42	0.010	0.004	0.004	0.005
43	0.004	0.001	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.340	0.247	0.233	0.257
35	0.253	0.177	0.165	0.188
36	0.182	0.122	0.113	0.134
37	0.127	0.080	0.074	0.090
38	0.086	0.049	0.047	0.058
39	0.056	0.029	0.029	0.035
40	0.034	0.016	0.017	0.020
41	0.020	0.008	0.009	0.011
42	0.010	0.004	0.004	0.005
43	0.004	0.001	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.384	0.275	0.236	0.287
35	0.283	0.197	0.167	0.211
36	0.203	0.136	0.114	0.149
37	0.141	0.090	0.075	0.101
38	0.095	0.055	0.048	0.065
39	0.061	0.032	0.029	0.039
40	0.037	0.018	0.017	0.022
41	0.021	0.009	0.009	0.012
42	0.011	0.004	0.004	0.006
43	0.004	0.001	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.371	0.290	0.239	0.297
35	0.275	0.207	0.169	0.217
36	0.197	0.143	0.115	0.154
37	0.137	0.094	0.076	0.104
38	0.092	0.057	0.048	0.066
39	0.060	0.033	0.029	0.040
40	0.036	0.018	0.017	0.023
41	0.021	0.009	0.009	0.012
42	0.010	0.004	0.004	0.006
43	0.004	0.001	0.002	0.002

1992 age	Malay	Chinese	Indian	Total
34	0.358	0.305	0.241	0.307
35	0.266	0.217	0.171	0.224
36	0.192	0.149	0.117	0.158
37	0.134	0.097	0.077	0.106
38	0.090	0.059	0.049	0.068
39	0.058	0.035	0.030	0.040
40	0.036	0.019	0.017	0.023
41	0.020	0.010	0.009	0.012
42	0.010	0.004	0.004	0.006
43	0.004	0.001	0.002	0.002

For the Malays, certain of the parameter assumptions are significantly different from those for the other races. Firstly, unlike for the Chinese and Indians, progress of total period fertility towards the replacement level implies a reduction in fertility rates since the Malay rate in 1992 was 2.60; thus, all projections other than P1.1, P2.1, P3.1 and P4.1 assume declining fertility for the Malays. Secondly, projections P3.1 to P4.3 assume a "changing" standard deviation of the fertility curve. For the Malays, this implies an increasing standard deviation whereas for the other races the standard deviation is assumed to decline.

Interpreting the projections

For the Chinese and the aggregate population, the order of the projections going from highest to lowest is:

P2.3>P2.2>P4.3>P2.1>P4.2>P4.1>P1.3>P1.2>P3.3>P1.1>P3.2>P3.1 ______(1)

For the Indians, the order is identical except that P4.1 and P1.3 are interchanged.

For the Malays, the order is different, reflecting the points mentioned above. The order of the projection is as follows:

P4.1>P2.1>P4.2>P2.2>P3.1>P4.3>P1.1>P3.2>P2.3>P1.2>P3.3>P1.3_____ (2)

However, there are similarities between (1) and (2). It will be observed that P4.1, P4.2 and P4.3 appear in the upper half of both (1) and (2), as also does P2.2. Similarly, P1.1, P1.2, P1.3, P3.2 and P3.3 all appear in the lower half of (1) and (2).

The Chinese, Indians and aggregate population

If groups of projections are denoted as G1 = (P1.1,P1.2,P1.3), G2 = (P2.1,P2.2,P2.3), G3 = (P3.1,P3.2,P3.3) and G4 = (P4.1,P4.2,P4.3), it is obvious that within any group, Pn.3 is greater than Pn.2> and Pn.2 is greater than Pn.1 since the only parameter which is changing is a reducing total period fertility assumption from Pn.3 to Pn.2 to Pn.1.

Most relationships between Pn.k and Pm.k can easily be explained by considering the nature of the parameters which change from one to the other. For example, P1.1 is greater than P3.1 since the only difference here is a declining standard deviation which will reduce the spread of the fertility distribution in the second projection and thus reduce the projected rates at the older ages. The only one of these relationships which calls for particular mention is that between G1 and G4. Going from G1 to G4, fertility values are simultaneously being increased with a higher average age and reduced with a smaller standard deviation. As will be evident, the reduction due to the declining standard deviations is more than offset by the increasing average age assumption since P4.1 is greater than P1.1, P4.2 is greater than P1.2 and P4.3 is greater than P1.3.

More interesting are certain comparisons between Pn.k and Pm.j where k =/= j. For example, consider P1.3 and P2.1. Here, the first assumes an increase of total period fertility to the replacement level within 10 years, the average age and standard deviation remaining at 1992 levels. The second, however, assumes an increase in average age with total period fertility and standard deviation constant at their 1992 values. It is perhaps surprising that P2.1 is greater than P1.3, indicating that an increase in average age can outweigh a fairly rapid rise in total period fertility.

Even more notable is the fact that P4.1 is greater than P1.3, since even with the reduction in P4.1 that will result from the inclusion of a declining standard deviation assumption, it is still higher than the G1 projection with the highest fertility rates.

The Malays

Because of the different nature of the assumptions made for the Malays, the relationship between various projections requires a modified interpretation. Here, within any group we shall have Pn.1 being greater than Pn.2 and Pn.2 being greater than Pn.3 because of the declining fertility assumption. On the other hand, the assumed increases in both average age and in standard deviation will have the effect of inflating projected fertility rates.

Particularly noteworthy for the Malays is the fact that P4.3 is greater than P1.1, showing again that, despite a rapidly declining total period fertility rate reducing to 2.1 in 10 years, the increasing average age and standard deviation assumed in P4.3 still places it ahead of P1.1 which assumes continuation of the 1992 status quo.

If these projected values are added to the partial cohort fertility rates given in the section of this paper under the subheading "partial cohort fertility in 1992" on pages 60-62, we shall obtain projected values of complete cohort fertility for 10 further cohorts of women. In order to provide an idea of the range within which these projections lie, the following table and graphs show projected values of complete cohort fertility for each race and for the aggregate population on two bases: firstly "high", on the highest of the projections given above in the section under the subheading "projecting fertility" starting on pages 62 and secondly "low", on the lowest. Provided that Singapore's total period fertility does not actually decline within the 35 to 44 age group over the next 10 years, which is probably unlikely, the actual outcomes for the Chinese, the Indians and the aggregate population should be within the range of the two values shown.

On the other hand, if Malay total period fertility increases rather than remaining the same or reducing, the actual outcomes could be higher than those shown.

Year born	Malay		Chinese		Indian		Total
	High	Low	High	Low	High	Low	High	Low
1948	2.745	2.745	2.259	2.259	2.800	2.800	2.354	2.354
1949	2.567	2.567	2.147	2.147	2.779	2.779	2.238	2.238
1950	2.591	2.590	2.084	2.083	2.669	2.669	2.186	2.185
1951	2.541	2.539	2.033	2.031	2.436	2.435	2.125	2.123
1952	2.523	2.519	2.036	2.030	2.283	2.282	2.119	2.114
1953	2.478	2.471	1.970	1.959	2.188	2.186	2.051	2.042
1954	2.497	2.486	1.934	1.915	2.312	2.308	2.039	2.023
1955	2.384	2.366	1.912	1.881	2.260	2.254	2.000	1.974
1956	2.466	2.441	1.892	1.844	2.182	2.174	1.987	1.949
1957	2.468	2.432	1.887	1.816	2.074	2.062	1.982	1.926
1958	2.518	2.468	1.885	1.785	2.025	2.008	1.983	1.906

The first point that should be mentioned here is that this table and the graphs shown hereafter include complete cohort fertility for the 1948 cohort. According to our definition of cohort fertility, this cohort had already completed its childbearing at the end of 1992. However, it has been included in order to provide a known starting point for the projected values.

As far as the aggregate population is concerned, the next four cohorts, i.e. those born from 1949 to 1952, should achieve the replacement level of complete cohort fertility.

For the aggregate population, the 1953 cohort shows a significant reduction compared with the 1952 cohort and thereafter projected complete cohort fertility continues to decline: to a low for the 1957 cohort of 1.982 on the "high" projection or reducing throughout on the "low" basis to 1.906 for the 1958 cohort.

For the Malays, even on the "low" basis, all cohorts are projected to complete childbearing with complete cohort fertility well above the replacement level. The rates decline to a low point of around 2.37 for the cohort born in 1955 and then begin to increase again. A noticeable feature of the rates is the small difference between most of the "low" and "high" projections.

In stark contrast to that of the Malays, Chinese complete cohort fertility appears set to remain above the replacement level for only one further cohort: the 1949 generation of women. Therefore for the Chinese, even with the most optimistic of scenarios, namely a total period fertility rate increasing to 2.1 by the year 2002 and an increasing fertility-weighted average age, it would appear inevitable that not only will complete cohort fertility fall below the replacement level for the cohorts born after 1949, but also it is likely to remain at that level for a good number of years. There is a hint that, on the "high" basis, it could perhaps reach a minimum for the 1958 cohort but on the "low" basis, which assumes a continuation of 1992 total period fertility, no increase in fertility-weighted average age and a continuing decline in standard deviations, the situation would appear to be bleak indeed for a return to a replacement level of complete cohort fertility even in the medium-term future.

This "low" scenario is, however, probably too pessimistic. Not only does it assume a continuation of total period fertility at its 1992 level (not in itself an unreasonable proposition viewed in the context of the relatively static Chinese experience over the past 15 or so years) but also a reducing standard deviation of the fertility curve without an accompanying increase in fertility-weighted average age. Under present circumstances, the combination of these last two features is unlikely. Thus, the "low" curve should be viewed here, and indeed for the Indian and aggregate populations as well, as the minimum values for complete cohort fertility below which, barring any reductions in total period fertility, it is most unlikely to fall.

It will be noted that, by the time the 1958 cohort reaches age 44, the difference between the Malay "high" and "low" curves is only about 2 per cent. For the Chinese, however, the difference is somewhat larger at 5.5 per cent. In Redington's (1952) graphic terms, the "funnel of doubt" is expanding far more quickly for the Chinese.

A striking feature of the Indian graph is the virtual absence of any funnel at all: the funnel of doubt seems to have been transformed into a narrow passageway of certainty. This feature is a refection of the traditionally small proportion of Indian complete cohort fertility which is contributed by the late phase and which is projected to continue over the next 10 years. It will be observed that, on any of the projection bases above, Indian complete cohort fertility will reduce to below the replacement level for the 1957 cohort.

The aim of this paper has been to describe one way in which complete cohort fertility rates may be projected into the future, with the underlying purpose, essentially, of addressing the issue of population replacement. It is certainly not claimed that this is the only way or indeed necessarily the best way to make such projections but, based as it is on the secure foundation of partial cohort fertility, it possesses the advantage of being grounded in objectively determined past fertility performance. The extent to which the projections are wrong can only be equal to the extent to which projections of single-age fertility rates for ages 35 to 44 next year, to age 44 in 10 years' time, can be wrong. They always will be wrong, of course, but the final decade of the childbearing years tends to be the least significant. Moreover, the method has the advantage that the further one projects into the future, the smaller are the number of fertility rates required. Therefore, the extent of any error should be small.

This concluding section of the paper seeks to justify the use of cohort fertility measures in the approach to answering questions concerning population replacement and ends with a very brief review, for those who feel that they might like to adopt the general approach outlined here, of some practical issues and of a method that could be used where the available data are not as comprehensive as they are in Singapore.

Why cohort fertility? - the replacement issue

Virtually all published information on fertility in Asia refers to period fertility. There are good reasons for this. Not only are cohort measures much more troublesome to calculate but in many, if not most ways, policy makers and planners will, as already mentioned, be far more concerned with the trend of total period fertility, or probably more accurately, with the number of births, from year to year.

What purpose does the measurement of cohort fertility serve? According to Ni Bhrolchain (1992), very little apparently. However, quite apart from its intrinsic interest, it could be argued that, at least when considering the replacement question, cohort fertility gives a greater insight than does total period fertility into whether a population's fertility experience has been adequate to achieve replacement. It is, therefore, with this replacement issue in mind that this paper has been written.

In Singapore before the mid-1970s and in many Asian countries today, it could be safely inferred from period fertility schedules that the population's experience was sufficient to achieve replacement - total period fertility of above 3 per annum coupled with moderate levels of mortality can safely be assumed to generate sufficient births to ensure replacement. The problem arises when total period fertility falls to below the replacement level (assumed in a low mortality environment to be 2.10) as it has been in Singapore since 1976.

It is in fact meaningless to ask: "At the present fertility level, is the population replacing itself?" And this is true whether or not the answer is expressed in terms of total period fertility or complete cohort fertility. In the period domain, the current year's total period fertility experience tells us little or nothing about replacement, influenced as it can be by timing distortions caused perhaps by incentives or disincentives introduced by the authorities to influence fertility levels, or, as in Chinese communities, by whether the current year happens to be the "Year of the Dragon" which is considered particularly auspicious for having children or the "Year of the Tiger" which is considered inauspicious for having girls, as discussed by Saw (1990). However, it is obviously equally as erroneous to declare on the basis of the complete cohort fertility of those women who have just attained the age of 45 that the population is or is not replacing itself. What we can say of relevance about these women however -- and this is an objective fact -- is whether or not they have, as a cohort, produced sufficient children to replace themselves.

It is over time that the replacement of a population will occur and it is therefore only over time that an answer to the replacement question may be suggested. This obviously requires an averaging of fertility rates over a period of years. It can be claimed that the average of the past 2N years total period fertility will provide a satisfactory answer. However, there are three problems with this approch: two theoretical and one practical. Firstly, it is necessary to think carefully about what is being averaged. The question of population replacement is a question of whether on average real cohorts are replacing themselves. Why then use the synthetic cohort approach of the period method if data on real cohorts are available? Secondly, any averaging over the past 2N years will be centred at a time N years ago. While a larger value of 2N will clearly tend to reduce the influence of the type of period distortions mentioned above, it will, however, at the same time reduce the immediacy and relevance of the resulting number by removing it further back in time from current experience. Thirdly, it is worth pointing out how misleading in practice such an average can be; over the 20 years from 1974 to 1993, the average of annual total period fertility for Singapore's aggregate population has been 1.79, i.e. well below the replacement level, yet, as we have seen, no cohorts have yet completed their childbearing with fewer than 2.1 children. Therefore, historical total period fertility averages appear to be of little value when addressing the replacement issue.

It would seem that a far more promising approach is to take the average of successive complete cohort fertility rates over a period of 2N years starting N years ago. These values will be for the N cohorts who have most recently completed their childbearing together with the projected values for the next N cohorts to do so. Such an average will be centred at the present time and will, moreover, comprise the actual and reliably projected lifetime reproductive performance of real cohorts of women, which is after all the fundamental component of the process by which a population replaces itself.

A further advantage of the cohort approach, and the use in particular of partial cohort rates, is that, for those cohorts of women still within the childbearing years, it is a simple matter to show what future fertility performance from each cohort is necessary in order that they will achieve the replacement level. In this way it is possible to assess informally, in the light of past experience and without the need to perform projections, how likely this would be.

Practical problems

While in many Asian and Pacific societies total period fertility rates are certainly well above the replacement level, the success of the family planning movement in others, for example Australia, China, Hong Kong, Japan, New Zealand, the Republic of Korea, Singapore and Thailand, has taken the total period fertility rate to a level of 2.1 or lower. In these societies and in others where total period fertility is beginning to approach the replacement level, concern will quite naturally turn at some stage to the question of replacement and to the longer term prospect, ultimately, of population decline. Admittedly, an equally significant problem in those countries, which have seen a large reduction in fertility over recent years, will be the economic and social implications of the ageing of those who were born in the high fertility era, owing to the resulting rapidly increasing proportion of the population in the older age groups. None the less, at some point, concern is likely to be expressed, as by Singapore's Prime Minister Goh Chok Tong (1986), that the replacement question is of fundamental importance and is an issue which has to be addressed.

As will be clear from this paper, it is suggested that this subject is best approached from fertility calculated on a cohort basis and it is, of course, here that problems are likely to arise, since in many countries the data are just not available to enable cohort rates to be calculated easily. In Singapore, the annual number of births by the sex of the child and by the age (and for most years by the race) of the mother are available for the entire period since 1946. With this information, together with annual female population numbers, the calculation of single-age fertility rates for each calendar year is a technically simple (yet tedious) task. Appropriate series of these rates are then summed in order to provide complete cohort fertility rates for those generations of women who have completed their childbearing, and partial cohort fertility rates for those women below age 44. Where such information is not available, but, say, reasonably reliable period fertility rates in the normal five-year age groupings have been published for a good number of years, these may be interpolated to provide schedules of single-age fertility rates for each calendar year from which partial cohort fertility rates may be derived to form the foundation for the projections outlined here.

Next, it should be pointed out that there is nothing sacrosanct about the normal and log-normal distributions used here to model the total period fertility curve. Whatever distributions are appropriate in practice clearly depends upon their general suitability in a given context for modelling period fertility over the final 10-year age span from 35 to 44. The use here of the normal and log-normal curves was a result not only of their fairly good fit to the Singapore data but also because of their general familiarity and the fact that only three parameters are required for estimation purposes, namely the total period fertility rate and the two parameters of each probability distribution. However, in other contexts different distributions might well be more appropriate.

Also, nothing has been carved in stone regarding the assumptions that have been made regarding the changes in the three parameters over the next 10 years. In particular, there would appear to be little justification for assuming that total period fertility will tend towards the 2.1 replacement level in the longer term. In many ways -- although it has comforting implications -- this assumption seems to be used by some demographers as a convenient means to avoid the (admittedly disturbing) prospect of ultimate extinction either by overcrowding or disappearance.

Finally, although projections have been made here for only the next 10 cohorts, there is no theoretical objection against projections for a larger number of cohorts. However, the problems with this are firstly that increased care would be required in choosing the distribution(s) which are suitable for modelling period fertility over a greater range of ages and secondly that a consequence of longer term projections is that the width of the "funnel of doubt" will inevitably expand progressively the further we try to look into the future.

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For further information on this material please contact: loftus.unescap@un.org