(1)
The Inverse Projection method has been developed simultaneously by Ronald Lee (1974) and Etienne Van de Walle (1974). It estimates population age structure at time Tn from total births, deaths and population numbers over the period To - Tn , where 1 ... n are one-year or five-year periods.
The method sets a hypothetical age structure at time To , and projects this population forward, using a set of model life tables. The mortality level is chosen as to equal total deaths over each period. Total births are used immediately, and total population numbers are used for estimating net migration at each period. Net migration will then be redistributed over age groups using a migration age profile.
The five to five years projection from To to Tn is then:
(1)
where:
pa,t is population of five year age group a in time t ( a from 0 to 95 )
La,t-5 is number living at age group a of life table at work over time period going from t-5 to t
( a from -5 to 90 ; L-5 = total births)
ma-5 is the part of net migration from age group a-5 to age group a in total net migration ( Sm = 1 )
MIGt-5 is total net migration over time period going from t-5 to t .
Total net migration is given by:
(2)
where:
TPt is total population in time t
Bt-5 is total births over time period going from t-5 to t
Dt-5 is total deaths over time period going from t-5 to t .
Total population and total deaths satisfy:
( a from 0 to 95 ) (3)
( a from -5 to 90 ; p-5 = Bt-5 ) (4)
The age migration profile ( ma ) is supposed to be independent of time and generally concentrated, at least as far as international migration is concerned, over ages from 10 till 30.
The life tables are chosen amongst model life tables, like those of Coale and Demeny (1983).
In most cases, the Inverse Projection is used to reconstruct population history onto recent population data like censuses. The aim of the reconstruction is then to estimate the population age structure in Tn as close as possible to census data. The parameters to improve this fit are:
Thorough knowledge of the population, of its history, its climatic, social and economic environment as of its data registration, is the best guarantee for success. Unfortunately, this cannot always be the case.
Reconstruction from vital rates
Sometimes, we do not dispose of total births, deaths and population, but we may have estimations of birth, death and total increase rates. This is the case of the Vietnam reconstruction, where we dispose of total births, deaths and population of the Cochinchine region from 1913 till 1937. The other regions show deficient registration. So we may start the reconstruction of Vietnam's population by the hypothesis (A) that Cochinchina's vital rates are valid for all of Vietnam.
Equation 4 will then read:
( a from -5 to 90 ; p-5 = Bt-5 ) (5)
and total births are given by:
( a from -5 to 90 ; p-5 = Bt-5 ) (6)
Calculation will need several iterations, because total deaths and births can be calculated only after estimating population age structure, which needs, in turn, total deaths and births. However, convergence is reached quickly.
We dispose also of Cochinchina's net migration rates for 1913-1937. The annual mean rate calculated from registered data is next to 1/1000, which is insignificant in proportion to vital rates. If Cochinchina's net migration may be neglected, all-over Vietnam's net migration may be so all the more. This has been hypothesis (B): zero net migration for Vietnam 1913-1989. Of course, this will not have been the exact truth for Vietnam. Yet, reconstruction results (see below) match very well to this supposition.
Mortality patterns
We should expect Vietnam's mortality pattern to be close to the South Tables of Coale and Demeny's model life tables, for these tables are based on the "warmest" countries they disposed of. Yet, first reconstruction results suggest this is not very likely. Figure 1 to 3 show best reconstruction results with South and with West model tables.
Figure 1. Vietnam Total Population (Thousands) : census data, best South and West reconstruction.
Figure 2. Vietnam registered and reconstructed death and birth rates ; registered values (bars), best West reconstructed values (plain lines), best South reconstructed values (dotted lines) ; rates for 1000.
Figure 3. Vietnam 1989 census (thick line) ; reconstructed by West tables (plain normal) and by South tables (dotted line). (Millions)
West and South model life tables come up with very close results for Vietnam's 1989 census reconstruction. But the West reconstruction observe also registered vital rates whereas the South reconstruction diverge considerably from registered rates. Moreover, the West reconstruction deviates less from registered total population before 1950. It seems reasonable to give preference to the West model life tables.
East and North reconstruction has been attempted, but did not improve the West estimations.
Further parameters
Male-female relative mortality level : after some try and error, we fixed the relative life expectancy levels at 106 % for women and 94 % for men.
Total population fit in 1989 : given the under-registration of young girls at the 1989 census, the reconstruction did not look for the best fit to total population in 1989, but reconstructed total population at 15-59 year of age. These ages are considered to have been faithfully registered.