International Workshop on Population Projections using Census Data

International Workshop on

Population Projectionsusing Census Data

14 – 16 January 2013Beijing, China

Session II:Main population projection methods

• The basic accounting identity of Demography• Projections by mathematical formulae• Projections by the cohort-component method

http://unstats.un.org/unsd/demographic/meetings/wshops/China2013/list_of_docs.htm

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).


• 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.


• 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)


• 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.

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


• 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) = 100 + 10*t• y = a + b*x


• Exponential function• Assumes constant growth rate of population• Similar to compounded interest


• Logistic function• An exponential growth approaching a limit.


1800 1850 1900 1950 2000 2050 21000

2

4

6

8

10

12

14

16

18

20

World population fitted by different functions (1927-2012)

Population linear exponential Logistic


• 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

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.


• 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 period2. Project births to the population during the period3. Project survivors of births to the end of the period


• 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)


• 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]

• P(x+n, t+n) = P(x, t) – Deaths(x, t..t+n)• P(x+n, t+n) = P(x, t) * PLT(x+n, t+n) /PLT(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

Cohort-Component MethodNotation I

s tn xI

Time, location, etc.

AgeDuration/Length

Indicator symbol

Sex

Cohort-Component MethodNotation II

Survivors to exact age xPerson-years lived for persons

in age group x to x+n (life table population)T Person-years lived above age xS Survivor ratios for population in age group x to x+n

x

n x

x

n x

lL

to survive to age group x+n to x+2nF Age-specific fertility rate for age group x to x+nB Births to mothers in age group x to x+nP Population in age group x to x+n

SRB= Sex ratio at birth

n x

n x

n x

Cohort-Component MethodNotation II

S Survivor ratios

=

n x

n x n

n x

LL

0

0

SB Survivor ratio of births

=*

n

nLn l

SRB Sex ratio of births (masculinity ratio)

=

m

m

f

BB

Cohort-Component MethodNotation III

PYL Person-years lived (real population)

1=2

tn x

t t nn x n xP P

B Births; B Births (male); B Births (female)

1B * ; B *1 1

if SRB is based on 100 females

100B * ; B *100 100

m f

mm f

m m

mm f

m m

SRBB BSRB SRB

SRBB BSRB SRB

½ => n/2

Cohort-Component Method ICohorts 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.

*

*

t n t tx n x n x

tt n x nx t

n x

P P S

LP

L

Cohort-Component Method IINew Cohorts: Births

5

55

2

5 5

*

* *2

* * *2

* * *2

t t f tn x n x n x

t f t f tn x x x

tt f t f t n xx x x t

n x n

t f t f t tx x x x

B F PYL

nF P P

LnF P PL

nF P P S

Cohort-Component Method IINew Cohorts: Births

1.05* *1 1.05 1

1 1* *1 1.05 1

t tn x

x

m tm t t t

m t

f t t tm t

B B

SRBB B BSRB

B B BSRB

Cohort-Component Method IIIOpen-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!

Cohort-Component Method IIIOpen-ended Age groups

5 *t

t t tn n t

n

t t

t tn n

TP P P

T

T PT P

Cohort-component methodCohorts already born

[t, t+5] [t, t+5] t t+5

Age nLx nSx nPx nPx+5

0-5 5L0 nS0= 5L5/5L0 5P0

5-10 5L5 nS5= 5L10/5L5 5P5 5P5 = 5P0 * 5L5/5L0

10-15 5L10 nS10= 5L15/5L10 5P10 5P10 = 5P5 * 5L10/5L5

… … … … …x - x+5 5Lx nS0= 5Lx+5/5Lx 5Px 5Px = 5Px-5 * 5Lx/5Lx-5

…70-75 5L70 5P70 5P70 = 5P65 * 5L70/5L65

75-80 5L75 5P75 = 5P70 * 5L75/5L70

80+


[t, t+5] [t, t+5] t t+5


0-5 5L0 nS0= 5L5/5L0 5P0

5-10 5L5 nS5= 5L10/5L5 5P5 5P5 = 5P0 * 5L5/5L0

10-15 5L10 nS10= 5L15/5L10 5P10 5P10 = 5P5 * 5L10/5L5


…70-75 5L70 5P70 5P70 = 5P65 * 5L70/5L65

75-80 5L75 5P75 = 5P70 * 5L75/5L70

80+


[t, t+5] [t, t+5] t t+5


0-5 5L0 nS0= 5L5/5L0 5P0

5-10 5L5 nS5= 5L10/5L5 5P5 5P5 = 5P0 * 5L5/5L0

10-15 5L10 nS10= 5L15/5L10 5P10 5P10 = 5P5 * 5L10/5L5


…70-75 5L70 nS70= 5L75/5L70 5P70 5P70 = 5P65 * 5L70/5L65

75-80 5L75 5P75 = 5P70 * 5L75/5L70

80+

Cohort-component methodLast, open ended age group

[t, t+5] [t, t+5] t t+5


0-5 5L0 nS0= 5L5/5L0 5P0

5-10 5L5 nS5= 5L10/5L5 5P5 5P5 = 5P0 * 5L5/5L0

10-15 5L10 nS10= 5L15/5L10 5P10 5P10 = 5P5 * 5L10/5L5

5P15 = 5P10 * 5L15/5L10

…x - x+5 5Lx nSx= 5Lx+5/5Lx 5Px 5Px = 5Px-5 * 5Lx/5Lx-5

…75-80 5L75 5P75

80+ ∞L80 ∞S80= T80/T75 ∞P80 ∞P80=[5P75+∞P80]*∞S80

Cohort-component methodFirst age group (surviving births)

[t, t+5] [t, t+5] t t+5


0-5 5L0 nSB0= 5L0/l0 5P0 = B * 5L0/l0


• 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?

Cohort-Component Method IVCohorts 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.


• 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.


• 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


51 1* *2 2

t n t t t tn x n n x n x n x x nP P NM S NM

Half of net-migration

Half of net-migration

Cohort-Component Method IVBirths + migration

1* *21* *2

t t f t t f tn x n x n x n x n x

t f t f tn x n x n x

B F PYL F NM

F PYL NM

Cohort-Component Method IVBirths + migration

• 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

International Workshop on Population Projections using Census Data

Documents

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