1 Annex III: Miscellaneous

This Annex assembles documentation and formulas not elsewhere covered.

1.1 Abacus

The Abacus population projection software of the United Nations Population Division used the following variant of the cohort-component model¹. This implementation applies the split-migration option for all closed age groups, except the first one. For the first age group, half the migrants of this age group are added at the end of the projection period, that is without exposing them to the risk of dying.

${}_nP_x^{t + n} = {}_nP_{x - n}^{t + n}*{}_nS_{x - n}^{t.t + n} + {1 \over 2}\left( {{}_nM_{x - n}^{t,t + n}*{}_nS_{x - n}^{t.t + n} + {}_nM_x^{t,t + n}} \right);x = n,2n,...,z - n$

${}_\infty P_z^{t + n} = \left( {{}_nP_{z - n}^t + {}_nP_z^t} \right)*{}_\infty S_z^{t,t + n} + {1 \over 2}\left[ {\left( {{}_nM_{z - n}^{t,t + n} + {}_\infty M_z^{t,t + n}} \right)*{}_\infty S_z^{t,t + n} + {}_\infty M_z^{t,t + n}} \right]$

${}_nP_0^{t + n} = {B^{t,t + n}}*{{{}_nL_0^{t,t + n}} \over {n*{l_0}}} + {1 \over 2}*{}_nM_0^{t,t + n}$

1.2 Rules of thumb: Splitting factors

Preston, Keyfitz, and Schoen² (1972) suggested the following approximation for the separation factor for those less than 1 year of age: ${}_1{a_o} = 0.07 + 1.7*{}_1{m_o}$ .

Coale and Demeny introduced a system of regression coefficients to determine separation factors for age groups 0-1 and 1-5 (1963, 20; see also Coale et al. 1983, 20-21) that are anchored at values of ${}_1{q_0}$ , e.g the probability of dying between ages 0 and 1 (exact ages). Preston et al. adapted this approach to the age-specific mortality rates for age groups 0 to 1 ( ${}_1{m_0}$ ) because this indicator seems to be more common in life tables constructions (2001, 47-48).

I calculated a more realistic functional form of the ${}_5{a_0}$ for the age group 0+5 based on the corresponding age-specific mortality rates for that age group ( ${}_5{m_0}$ ). Employing the updated Coale/Demeny Model Life Table prepared by the Population Division of the United Nations (UNPD 2015) for the “West” family of life tables, the following parameters were obtained:

Males:

${}_5{a_0} = \left\{ {\matrix{ {{\rm{ - 0}}{\rm{.7427}}{}_5{{\rm{m}}_0}{\rm{ + 0}}{\rm{.9966}}} \hfill & {{}_5{m_0} \ge 0.06} \hfill \cr {{\rm{4768}}{\rm{.1*}}{}_5{{\rm{m}}_0}^3{\rm{ - 735}}{\rm{.36*}}{}_5{{\rm{m}}_0}^2{\rm{ + 38}}{\rm{.319*}}{}_5{{\rm{m}}_0}{\rm{ + 0}}{\rm{.2703}}} \hfill & {{}_5{m_0} < 0.06} \hfill \cr } } \right.$

Females

${}_5{a_0} = \left\{ {\matrix{ {{\rm{ - 0}}{\rm{.8145*}}{}_5{{\rm{m}}_0}{\rm{ + 1}}{\rm{.1002}}} \hfill & {{}_5{m_0} \ge 0.06} \hfill \cr {{\rm{4825}}{\rm{.5*}}{}_5{{\rm{m}}_0}^3{\rm{ - 768}}{\rm{.63*}}{}_5{{\rm{m}}_0}^2{\rm{ + 40}}{\rm{.828*}}{}_5{{\rm{m}}_0}{\rm{ + 0}}{\rm{.3264}}} \hfill & {{}_5{m_0} < 0.06} \hfill \cr } } \right.$

2 Bibliography

Coale, Ansley J, and Paul Demeny. 1966. “Regional Model Life Tables and Stable Populations.” In Regional Model Life Tables and Stable Populations, 871. Princeton, NJ: Princeton University Press.

Coale, Ansley J, Paul Demeny, and Barbara Vaughan. 1983. Regional Model Life Tables and Stable Populations (Second Edition). Studies in Population. New York, NY; London, England: Academic Press. https://www.sciencedirect.com/science/book/9780121770808.

Pollard, A. H., Farhat Yusuf, and G.N Pollard (1990). Demographic Techniques. Third edition. Pergamon Press

Preston, S., N. Keyfitz, and R. Schoen. 1972. Causes of Death: Life Tables for National Populations. New York: Seminar Press.

Preston, Samuel H, Patrick Heuveline, and Michel Guillot. 2001. Demography: Measuring and Modeling Population Processes. Malden MA, USA; Oxford, UK: Blackwell Publishers.

Swanson, D.A. and J.S. Siegel (2004) The Methods and Materials of Demography, Second Edition, Emerald Group Publishing.

UNPD (2015). Extended Model Life Tables (Version 1.3). United Nations Population Division, New York. https://esa.un.org/unpd/wpp/Download/Other/MLT/.

The Abacus projection software, programmed in FORTRAN, was in use up to the 2012 Revision of the United Nation’s global population projections.↩
Cited from Siegel and Swanson (2004, p. 308, equation 13.5)↩