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International Workshop on Population Projections using Census Data 14 – 16 January 2013 Beijing, China

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Session II: Main population projection methods The basic accounting identity of Demography Projections by mathematical formulae Projections by the cohort-component method

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The basic accounting identity of Demography Also known as Balance Equation Populations can only change through births, deaths and immigration or emigration It follows that two populations at two different points in time are related to each other by the components of change (births, deaths, migration).

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The basic accounting identity of Demography Closed population P(t+n) = P(t) + Births(t, t+n) - Deaths(t, t+n) If a population is closed, it can only change through births and deaths The difference between births and deaths is known as natural change. The world as a whole is a closed population. Closed populations are easier to model and project.

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The basic accounting identity of Demography Closed population P(t+n) = P(t) + Births(t, t+n) - Deaths(t, t+n) Births(t, t+n) = P(t+n)- P(t) + Deaths(t, t+n) Deaths(t, t+n) = P(t)- P(t+n) + Births(t, t+n)

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The basic accounting identity of Demography Open population P(t+n)=P(t) + Births(t, t+n) - Deaths(t, t+n) + Immigration(t, t+n) – Emigration(t,t+n) The difference between immigration and emigration is known as net migration If a population is open it can only change through births and deaths and immigration and emigration Most countries have open populations.

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Projections by mathematical formulae Often used for total populations Often applied for ad-hoc or preliminary calculations (back of the envelope) There are a number of mathematical functions useful: – Linear function – Exponential function – Logistic function

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Projections by mathematical formulae Linear function True if populations grow by the same absolute amount every year P(t = 0)=100 P(t = 1) = P(t=0) + 10=110 P(t = 2) = P(t=1) + 10=120 P(t = 3) = P(t=2) + 10=110 P(t) = *t y = a + b*x

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Projections by mathematical formulae Exponential function Assumes constant growth rate of population Similar to compounded interest

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Projections by mathematical formulae Logistic function An exponential growth approaching a limit.

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Projections by mathematical formulae

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Linear function well suited for short time intervals, ad-hoc calculations Exponential function reflects the compounding nature of natural growth, but must be used with caution => exponential trap Logistic can be superior to simple exponential, but needs knowledge about an absolute limit

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Projections by the cohort-component method The most popular method for projecting populations Advisable for larger populations; often used for national populations and subnational populations at the first administrative level. Cohort component method is based on age- structured populations and age-structured components of change.

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Projections by the cohort-component method Most population projections are carried out with five year age groups and five year time periods. Three steps: 1.Project survivors of initial/base population at the end of the period 2.Project births to the population during the period 3.Project survivors of births to the end of the period

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Projections by the cohort-component method Is based on the balance equation, but by age (and sex): P(x+n, t+n) =P(x, t) - Deaths(x, t..t+n) + NetMigration(x, t..t+n) x = 0, 5, … ; n=5 x = 0, 1, 2, 3, … ; n=1 Special treatment for surviving births and last, open-ended age group (shown later)

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Projections by the cohort-component method Instead of absolute number of births and deaths by age, demographer prefer to formulate and implement the cohort- component method by age-specific rates. Births => Age-specific fertility rates Deaths => Survivor ratios [Migration in absolute numbers preferred]

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P(x+n, t+n) = P(x, t) – Deaths(x, t..t+n) P(x+n, t+n) = P(x, t) * P LT (x+n, t+n) /P LT (x, t) P(x+n, t+n) = P(x, t) * L(x+n, t+n) /L(x, t) P(x+n, t+n) = P(x, t) * SR(x, t..t+n) Projections by the cohort-component method: Closed population

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Cohort-Component Method Notation I Time, location, etc. Age Duration/ Length Indicator symbol Sex

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Cohort-Component Method Notation II

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Cohort-Component Method Notation III ½ => n/2

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Cohort-Component Method I Cohorts already born Projecting the people still alive n years later, the population aged x to x+n in year t is multiplied by the corresponding survivor ratio to yield the population aged x+n to x+2n in year t+n. This forward survives a cohort.

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Cohort-Component Method II New Cohorts: Births

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Cohort-Component Method III Open-ended Age groups All population projections will have to handle a final, open-ended age group. The open ended-age group used to be set to start at age 80 or 85 in the past. With people surviving to higher ages, the open- ended age groups should also set to higher ages Rule of thumb: Last, open-ended group should be smaller that the preceding, closed, age group!

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Cohort-Component Method III Open-ended Age groups

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Cohort-component method Cohorts already born [t, t+5] tt+5 Age nLxnLxnSxnSxnPxnPxn P x L05L0n S 0 = 5 L 5 / 5 L 05P05P L55L5n S 5 = 5 L 10 / 5 L 55P55P55 P 5 = 5 P 0 * 5 L 5 / 5 L L 10n S 10 = 5 L 15 / 5 L 105 P 105 P 10 = 5 P 5 * 5 L 10 / 5 L 5 …………… x - x+5 5Lx5Lxn S 0 = 5 L x+5 / 5 L x5Px5Px5 P x = 5 P x-5 * 5 L x / 5 L x-5 … L 705 P 705 P 70 = 5 P 65 * 5 L 70 / 5 L L 755 P 75 = 5 P 70 * 5 L 75 / 5 L

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Cohort-component method Cohorts already born [t, t+5] tt+5 Age nLxnLxnSxnSxnPxnPxn P x L05L0n S 0 = 5 L 5 / 5 L 05P05P L55L5n S 5 = 5 L 10 / 5 L 55P55P55 P 5 = 5 P 0 * 5 L 5 / 5 L L 10n S 10 = 5 L 15 / 5 L 105 P 105 P 10 = 5 P 5 * 5 L 10 / 5 L 5 …………… x - x+5 5Lx5Lxn S 0 = 5 L x+5 / 5 L x5Px5Px5 P x = 5 P x-5 * 5 L x / 5 L x-5 … L 705 P 705 P 70 = 5 P 65 * 5 L 70 / 5 L L 755 P 75 = 5 P 70 * 5 L 75 / 5 L

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Cohort-component method Cohorts already born [t, t+5] tt+5 Age nLxnLxnSxnSxnPxnPxn P x L05L0n S 0 = 5 L 5 / 5 L 05P05P L55L5n S 5 = 5 L 10 / 5 L 55P55P55 P 5 = 5 P 0 * 5 L 5 / 5 L L 10n S 10 = 5 L 15 / 5 L 105 P 105 P 10 = 5 P 5 * 5 L 10 / 5 L 5 …………… x - x+5 5Lx5Lxn S 0 = 5 L x+5 / 5 L x5Px5Px5 P x = 5 P x-5 * 5 L x / 5 L x-5 … L 70n S 70 = 5 L 75 / 5 L 705 P 705 P 70 = 5 P 65 * 5 L 70 / 5 L L 755 P 75 = 5 P 70 * 5 L 75 / 5 L

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Cohort-component method Last, open ended age group [t, t+5] tt+5 Age nLxnLxnSxnSxnPxnPxn P x L05L0n S 0 = 5 L 5 / 5 L 05P05P L55L5n S 5 = 5 L 10 / 5 L 55P55P55 P 5 = 5 P 0 * 5 L 5 / 5 L L 10n S 10 = 5 L 15 / 5 L 105 P 105 P 10 = 5 P 5 * 5 L 10 / 5 L 5 5 P 15 = 5 P 10 * 5 L 15 / 5 L 10 … x - x+5 5Lx5Lxn S x = 5 L x+5 / 5 L x5Px5Px5 P x = 5 P x-5 * 5 L x / 5 L x-5 … L 755 P ∞ L 80∞ S 80 = T 80 /T 75∞ P 80∞ P 80 =[ 5 P 75 + ∞ P 80 ]* ∞ S 80

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Cohort-component method First age group (surviving births) [t, t+5] tt+5 Age nLxnLxnSxnSxnPxnPxn P x L05L0n SB 0 = 5 L 0 /l 05 P 0 = B * 5 L 0 /l 0

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Projections by the cohort-component method Open population Consider (net) migration! P(x+n, t+n) = P(x, t) – Deaths(x, t..t+n) + NetMigration(x, t…t+n) ? What about the deaths (births) happening to migrants during the projection period while in the country?

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Cohort-Component Method IV Cohorts already born + migration Open populations have two additional elements of change: emigration and immigration. Emigration is generated by the population under study and can be subjected to standard demographic analysis (outmigration rates. …). Immigration is an external event, generated by populations outside the country. Calculating demographic rates is meaningless.

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Cohort-Component Method IV Cohorts already born + migration Because immigration is better formulated as absolute numbers, net international migration is also mostly formulated in terms of absolute numbers. Challenge: Migration is a continuous process that also affects the births and deaths occurring in the country during the projection period.

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Cohort-Component Method IV Cohorts already born + migration One convenient approach: Split number migrants during a period into halves. Half of the migrants are assumed to enter the population at the beginning of the projection interval, thereby fully exposed to risks of dying and giving births. Half of the migrants are assumed to enter the population at the end of the projection interval, thereby not exposed to risks of dying and giving births

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Cohort-Component Method IV Cohorts already born + migration Half of net- migration

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Cohort-Component Method IV Births + migration

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Births are calculated as for closed population, PLUS – The contribution of migrants to births, usually assumed same fertility as the non-migrant population. Immigration adds additional births: Births Emigration (negative net-migration) subtracts births (those births that would have been born if the migrants would not have moved): Births

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