Demographic forecasting

Demographic forecasting

using functional data

analysis

Rob J Hyndman

Joint work with: Heather Booth, Han Lin Shang,Shahid Ullah, Farah Yasmeen.

Demographic forecasting using functional data analysis 1

Mortality rates


Fertility rates


Outline

1 A functional linear model

2 Bagplots, boxplots and outliers

3 Functional forecasting

4 Forecasting groups

5 Population forecasting

6 References


Outline






6 References

Demographic forecasting using functional data analysis A functional linear model 5

Some notation

Let yt ,x be the observed (smoothed) data in period tat age x , t = 1, . . . ,n .

yt ,x = ft(x)+ σt(x)εt ,x

ft(x) = µ(x)+K∑

k=1

βt ,k φk(x)+ et(x)


Estimate ft(x) using penalized regression splines.Estimate µ(x) as me(di)an ft(x) across years.Estimate βt ,k and φk(x) using (robust) functionalprincipal components.

εt ,xiid∼ N(0,1) and et(x)

iid∼ N(0,v(x)).

Some notation



ft(x) = µ(x)+K∑

k=1





iid∼ N(0,v(x)).

Some notation



ft(x) = µ(x)+K∑

k=1





iid∼ N(0,v(x)).

Some notation



ft(x) = µ(x)+K∑

k=1





iid∼ N(0,v(x)).

Some notation



ft(x) = µ(x)+K∑

k=1





iid∼ N(0,v(x)).

French mortality components


0 20 40 60 80 100

−6

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−1

Age

µ(x)

0 20 40 60 80 100

0.00

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Ageφ 1

(x)

t

β t1

1850 1900 1950 2000

−15

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05

10

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−0.

2−

0.1

0.0

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Age

φ 2(x

)t

β t2

1850 1900 1950 2000

−2

02

46

8

French mortality components


1850 1900 1950 2000

020

4060

8010

0 Residuals

Year

Age

Outline






6 References

Demographic forecasting using functional data analysis Bagplots, boxplots and outliers 8

French male mortality rates


0 20 40 60 80 100

−8

−6

−4

−2

0France: male death rates (1900−2009)

Age

Log

deat

h ra

te



0 20 40 60 80 100

−8

−6

−4

−2


Age

Log

deat

h ra

te

War years



0 20 40 60 80 100

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−6

−4

−2


Age

Log

deat

h ra

te

War years

Aims

1 “Boxplots” for functional data2 Tools for detecting outliers in

functional data

Robust principal components

Let {ft(x)}, t = 1, . . . ,n , be a set of curves.

å Each point in scatterplot represents one curve.å Outliers show up in bivariate score space.


1 Apply a robust principal component algorithm

ft(x) = µ(x)+n−1∑k=1

βt ,kφk(x)

µ(x) is median curve{φk (x)} are principal components{βt ,k } are PC scores

2 Plot βi ,2 vs βi ,1






ft(x) = µ(x)+n−1∑k=1

βt ,kφk(x)








ft(x) = µ(x)+n−1∑k=1

βt ,kφk(x)








ft(x) = µ(x)+n−1∑k=1

βt ,kφk(x)








ft(x) = µ(x)+n−1∑k=1

βt ,kφk(x)








ft(x) = µ(x)+n−1∑k=1

βt ,kφk(x)








ft(x) = µ(x)+n−1∑k=1

βt ,kφk(x)








ft(x) = µ(x)+n−1∑k=1

βt ,kφk(x)








ft(x) = µ(x)+n−1∑k=1

βt ,kφk(x)





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Scatterplot of first two PC scores

PC score 1

PC

sco

re 2



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Scatterplot of first two PC scores

PC score 1

PC

sco

re 2

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Functional bagplot

Bivariate bagplot due to Rousseeuw et al. (1999).Rank points by halfspace location depth.Display median, 50% convex hull and outerconvex hull (with 99% coverage if bivariatenormal).


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PC score 1

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Functional bagplot

Bivariate bagplot due to Rousseeuw et al. (1999).Rank points by halfspace location depth.Display median, 50% convex hull and outerconvex hull (with 99% coverage if bivariatenormal).Boundaries contain all curves inside bags.95% CI for median curve also shown.


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PC score 1

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Log

deat

h ra

te

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Functional bagplot


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Log

deat

h ra

te

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Functional HDR boxplot

Bivariate HDR boxplot due to Hyndman (1996).Rank points by value of kernel density estimate.Display mode, 50% and (usually) 99% highestdensity regions (HDRs) and mode.


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99% outer region


Bivariate HDR boxplot due to Hyndman (1996).Rank points by value of kernel density estimate.Display mode, 50% and (usually) 99% highestdensity regions (HDRs) and mode.


−10 −5 0 5 10 15

−2

02

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PC

sco

re 2

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90% outer region


Bivariate HDR boxplot due to Hyndman (1996).Rank points by value of kernel density estimate.Display mode, 50% and (usually) 99% highestdensity regions (HDRs) and mode.Boundaries contain all curves inside HDRs.


−10 −5 0 5 10 15

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02

46

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sco

re 2

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90% outer region

0 20 40 60 80 100

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−6

−4

−2

0

Age

Log

deat

h ra

te

191419151916191719181919

19401943194419451948



0 20 40 60 80 100

−8

−6

−4

−2

0

Age

Log

deat

h ra

te

191419151916191719181919

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Outline






6 References

Demographic forecasting using functional data analysis Functional forecasting 16

Functional time series model


ft(x) = µ(x)+K∑

k=1


The eigenfunctions φk(x) show the mainregions of variation.The scores {βt ,k } are uncorrelated byconstruction. So we can forecast each βt ,kusing a univariate time series model.Outliers are treated as missing values.Univariate ARIMA models are used forforecasting.




ft(x) = µ(x)+K∑

k=1






ft(x) = µ(x)+K∑

k=1






ft(x) = µ(x)+K∑

k=1






ft(x) = µ(x)+K∑

k=1




Forecasts


ft(x) = µ(x)+K∑

k=1



Forecasts


ft(x) = µ(x)+K∑

k=1


where vn+h ,k = Var(βn+h ,k | β1,k , . . . ,βn ,k)and y = [y1,x , . . . ,yn ,x ].Demographic forecasting using functional data analysis Functional forecasting 18

E[yn+h ,x | y] = µ̂(x)+K∑

k=1

β̂n+h ,k φ̂k(x)

Var[yn+h ,x | y] = σ̂2µ (x)+

K∑k=1

vn+h ,k φ̂2k(x)+ σ

2t (x)+ v(x)

Forecasting the PC scores


0 20 40 60 80 100

−6

−4

−2

0

Main effects

Age

Mea

n

0 20 40 60 80 100

0.05

0.10

0.15

0.20

AgeB

asis

func

tion

1

Interaction

Year

Coe

ffici

ent 1

1900 1940 1980 2020

−20

−10

05

1015

●

●

●

●

●

●

●

●

0 20 40 60 80 100

−0.

2−

0.1

0.0

0.1

0.2

Age

Bas

is fu

nctio

n 2

YearC

oeffi

cien

t 2

1900 1940 1980 2020

−6

−4

−2

02

●

●

●

●

●

●

●

●

Forecasts of ft(x)


0 20 40 60 80 100

−10

−8

−6

−4

−2


Age

Log

deat

h ra

te

Forecasts of ft(x)


0 20 40 60 80 100

−10

−8

−6

−4

−2


Age

Log

deat

h ra

te

Forecasts of ft(x)


0 20 40 60 80 100

−10

−8

−6

−4

−2

0France: male death forecasts (2010−2029)

Age

Log

deat

h ra

te

Forecasts of ft(x)


0 20 40 60 80 100

−10

−8

−6

−4

−2

0France: male death forecasts (2010 & 2029)

Age

Log

deat

h ra

te

80% prediction intervals

Fertility application


15 20 25 30 35 40 45 50

050

100

150

200

250

Australia fertility rates (1921−2009)

Age

Fer

tility

rat

e

Fertility model


15 20 25 30 35 40 45 50

05

1015

Age

Μ

15 20 25 30 35 40 45 50

−0.

3−

0.1

0.0

0.1

0.2

AgeΦ

1(x)

t

Βt1

1970 1980 1990 2000 2010

−10

−5

05

10

15 20 25 30 35 40 45 50

0.05

0.15

0.25

Age

Φ2(

x)t

Βt2

1970 1980 1990 2000 2010

−4

−2

02

46

8

Fertility model


1970 1980 1990 2000

1520

2530

3540

45Residuals

Year

Age

Fertility model


15 20 25 30 35 40 45 50

05

1015

Main effects

Age

Mea

n

15 20 25 30 35 40 45 50

−0.

3−

0.1

0.0

0.1

0.2

AgeB

asis

func

tion

1

Interaction

Year

Coe

ffici

ent 1

1970 1990 2010 2030

−10

05

1015

2025

15 20 25 30 35 40 45 50

0.05

0.15

0.25

Age

Bas

is fu

nctio

n 2

YearC

oeffi

cien

t 2

1970 1990 2010 2030

−4

−2

02

46

8

Forecasts of ft(x)


15 20 25 30 35 40 45 50

050

100

150

200

250


Age

Fer

tility

rat

e

Forecasts of ft(x)


15 20 25 30 35 40 45 50

050

100

150

200

250


Age

Fer

tility

rat

e

Forecasts of ft(x)


15 20 25 30 35 40 45 50

050

100

150

200

250

Australia fertility rates: 2010−2029

Age

Fer

tility

rat

e

Forecasts of ft(x)


15 20 25 30 35 40 45 50

050

100

150

200

250

Australia fertility rates: 2010 and 2029

Age

Fer

tility

rat

e

80% prediction intervals

Outline






6 References

Demographic forecasting using functional data analysis Forecasting groups 26

The problem

Let ft ,j(x) be the smoothed mortality rate for age x ingroup j in year t .

Groups may be males and females.Groups may be states within a country.Expected that groups will behave similarly.Coherent forecasts do not diverge over time.Existing functional models do not imposecoherence.


The problem




The problem




The problem




The problem




The problem




Forecasting the coefficients


ft(x) = µ(x)+K∑

k=1


We use ARIMA models for each coefficient{β1,j ,k , . . . ,βn ,j ,k }.The ARIMA models are non-stationary for thefirst few coefficients (k = 1,2)Non-stationary ARIMA forecasts will diverge.Hence the mortality forecasts are not coherent.




ft(x) = µ(x)+K∑

k=1






ft(x) = µ(x)+K∑

k=1




Male fts model


0 20 40 60 80

−8

−6

−4

−2

Age

µ M(x

)

0 20 40 60 80

0.05

0.10

0.15

Age

φ 1(x

)

Year

β t1

1960 2000

−10

−5

05

0 20 40 60 80

−0.

3−

0.1

0.0

0.1

0.2

Age

φ 2(x

)

Year

β t2

1960 2000

−2.

5−

1.5

−0.

50.

5

0 20 40 60 80

−0.

10.

00.

10.

2

Age

φ 3(x

)

Year

β t3

1960 2000

−1.

0−

0.5

0.0

0.5

Australian male death rates

Female fts model


0 20 40 60 80

−8

−6

−4

−2

Age

µ F(x

)

0 20 40 60 80

0.05

0.10

0.15

Age

φ 1(x

)

Year

β t1

1960 2000

−10

−5

05

0 20 40 60 80

−0.

10.

00.

10.

2

Age

φ 2(x

)

Year

β t2

1960 2000

−1.

0−

0.5

0.0

0.5

0 20 40 60 80

−0.

2−

0.1

0.0

0.1

Age

φ 3(x

)

Year

β t3

1960 2000

−0.

8−

0.4

0.0

0.4

Australian female death rates

Australian mortality forecasts


0 20 40 60 80 100

−10

−8

−6

−4

−2

(a) Males

Age

Log

deat

h ra

te

0 20 40 60 80 100

−10

−8

−6

−4

−2

(b) Females

Age

Log

deat

h ra

te

Mortality product and ratios

Key idea

Model the geometric mean and the mortality ratioinstead of the individual rates for each sexseparately.

pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =

√ft ,M(x)

/ft ,F(x).

Product and ratio are approximatelyindependent

Ratio should be stationary (for coherence) butproduct can be non-stationary.



Key idea



√ft ,M(x)

/ft ,F(x).





Key idea



√ft ,M(x)

/ft ,F(x).





Key idea



√ft ,M(x)

/ft ,F(x).




Mortality rates


Mortality rates


Model product and ratios


√ft ,M(x)

/ft ,F(x).

log[pt(x)] = µp(x)+K∑

k=1

βt ,kφk(x)+ et(x)

log[rt(x)] = µr(x)+L∑`=1

γt ,`ψ`(x)+wt(x).

{γt ,`} restricted to be stationary processes:either ARMA(p ,q) or ARFIMA(p ,d ,q).No restrictions for βt ,1, . . . ,βt ,K .Forecasts: fn+h |n ,M(x) = pn+h |n(x)rn+h |n(x)

fn+h |n ,F(x) = pn+h |n(x)/rn+h |n(x).




√ft ,M(x)

/ft ,F(x).


k=1

βt ,kφk(x)+ et(x)


γt ,`ψ`(x)+wt(x).






√ft ,M(x)

/ft ,F(x).


k=1

βt ,kφk(x)+ et(x)


γt ,`ψ`(x)+wt(x).






√ft ,M(x)

/ft ,F(x).


k=1

βt ,kφk(x)+ et(x)


γt ,`ψ`(x)+wt(x).




Product model


0 20 40 60 80

−8

−6

−4

−2

Age

µ P(x

)

0 20 40 60 80

0.05

0.10

0.15

Age

φ 1(x

)

Year

β t1

1960 2000

−10

−5

05

0 20 40 60 80

−0.

2−

0.1

0.0

0.1

0.2

Age

φ 2(x

)

Year

β t2

1960 2000−2.

5−

1.5

−0.

50.

5

0 20 40 60 80

−0.

2−

0.1

0.0

0.1

Age

φ 3(x

)

Year

β t3

1960 2000

−0.

6−

0.2

0.2

0.6

Ratio model


0 20 40 60 80

0.1

0.2

0.3

0.4

0.5

Age

µ R(x

)

0 20 40 60 80

−0.

100.

000.

100.

20

Age

φ 1(x

)

Year

β t1

1960 2000

−0.

50.

00.

5

0 20 40 60 80

−0.

050.

050.

150.

25

Age

φ 2(x

)

Year

β t2

1960 2000

−0.

6−

0.2

0.0

0.2

0.4

0 20 40 60 80

−0.

20.

00.

10.

20.

30.

4

Age

φ 3(x

)

Year

β t3

1960 2000

−0.

3−

0.1

0.1

0.3

Product forecasts


0 20 40 60 80 100

−10

−8

−6

−4

−2

Age

Log

of g

eom

etric

mea

n de

ath

rate

Ratio forecasts


0 20 40 60 80 100

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Age

Sex

rat

io: M

/F

Coherent forecasts


0 20 40 60 80 100

−10

−8

−6

−4

−2

(a) Males

Age

Log

deat

h ra

te

0 20 40 60 80 100

−10

−8

−6

−4

−2

(b) Females

Age

Log

deat

h ra

te

Ratio forecasts


0 20 40 60 80 100

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Independent forecasts

Age

Sex

rat

io o

f rat

es: M

/F

0 20 40 60 80 100

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Coherent forecasts

Age

Sex

rat

io o

f rat

es: M

/F

Life expectancy forecasts


Life expectancy forecasts

Year

Age

1960 1980 2000 2020

7075

8085

1960 1980 2000 2020

7075

8085Life expectancy difference: F−M

Year

Num

ber

of y

ears

1960 1980 2000 2020

24

68

Coherent forecasts for J groups

pt(x) = [ft ,1(x)ft ,2(x) · · · ft ,J(x)]1/J

and rt ,j(x) = ft ,j(x)/pt(x),


k=1

βt ,kφk(x)+ et(x)

log[rt ,j(x)] = µr ,j(x)+L∑

l=1

γt ,l ,jψl ,j(x)+wt ,j(x).


pt(x) and all rt ,j(x)are approximatelyindependent.

Ratios satisfy constraintrt ,1(x)rt ,2(x) · · ·rt ,J(x) = 1.

log[ft ,j(x)] = log[pt(x)rt ,j(x)]


pt(x) = [ft ,1(x)ft ,2(x) · · · ft ,J(x)]1/J



k=1

βt ,kφk(x)+ et(x)


l=1







pt(x) = [ft ,1(x)ft ,2(x) · · · ft ,J(x)]1/J



k=1

βt ,kφk(x)+ et(x)


l=1







pt(x) = [ft ,1(x)ft ,2(x) · · · ft ,J(x)]1/J



k=1

βt ,kφk(x)+ et(x)


l=1







pt(x) = [ft ,1(x)ft ,2(x) · · · ft ,J(x)]1/J



k=1

βt ,kφk(x)+ et(x)


l=1







µj(x) = µp(x)+µr ,j(x) is group mean

zt ,j(x) = et(x)+wt ,j(x) is error term.

{γt ,`} restricted to be stationary processes:either ARMA(p ,q) or ARFIMA(p ,d ,q).

No restrictions for βt ,1, . . . ,βt ,K .


log[ft ,j(x)] = log[pt(x)rt ,j(x)] = log[pt(x)] + log[rt ,j ]

= µj(x)+K∑

k=1

βt ,kφk(x)+L∑`=1

γt ,`,jψ`,j(x)+ zt ,j(x)








= µj(x)+K∑

k=1










= µj(x)+K∑

k=1










= µj(x)+K∑

k=1










= µj(x)+K∑

k=1



Outline






6 References

Demographic forecasting using functional data analysis Population forecasting 45

Demographic growth-balance equation


Pt+1(x +1) = Pt(x)−Dt(x ,x +1)+Gt(x ,x +1)Pt+1(0) = Bt −Dt(B ,0) +Gt(B ,0)

x = 0,1,2, . . . .

Pt(x) = population of age x at 1 January, year tBt = births in calendar year t

Dt(x ,x +1) = deaths in calendar year t of persons aged x atthe beginning of year t

Dt(B ,0) = infant deaths in calendar year tGt(x ,x +1) = net migrants in calendar year t of persons aged

x at the beginning of year tGt(B ,0) = net migrants of infants born in calendar year t





x = 0,1,2, . . . .

Pt(x) = population of age x at 1 January, year tBt = births in calendar year t

Dt(x ,x +1) = deaths in calendar year t of persons aged x atthe beginning of year t

Dt(B ,0) = infant deaths in calendar year tGt(x ,x +1) = net migrants in calendar year t of persons aged

x at the beginning of year tGt(B ,0) = net migrants of infants born in calendar year t


Key ideas

Build a stochastic functional model for eachof mortality, fertility and net migration.Treat all observed data as functional (i.e.,smooth curves of age) rather than discretevalues.Use the models to simulate future sample

paths of all components giving the entire agedistribution at every year into the future.Compute future births, deaths, net migrants.and populations from simulated rates.Combine the results to get age-specificstochastic population forecasts.


Key ideas




Key ideas




Key ideas




Key ideas




The available data

In most countries, the following data are

available:

Pt(x) = population of age x at 1 January, year tEt(x) = population of age x at 30 June, year tBt(x) = births in calendar year t to females of age xDt(x) = deaths in calendar year t of persons of age x

From these, we can estimate:

mt(x) = Dt(x)/Et(x) = central death rate incalendar year t ;ft(x) = Bt(x)/E F

t (x) = fertility rate for females ofage x in calendar year t .


The available data


available:






The available data


available:






Net migration

We need to estimatemigration data based ondifference in population numbers afteradjusting for births and deaths.


Gt(x ,x +1) = Pt+1(x +1)− Pt(x)+Dt(x ,x +1)Gt(B ,0) = Pt+1(0) −Bt +Dt(B ,0)

x = 0,1,2, . . . .

Note: “net migration” numbers also include errors

associated with all estimates. i.e., a “residual”.


Net migration




x = 0,1,2, . . . .




Net migration




x = 0,1,2, . . . .




Net migration




x = 0,1,2, . . . .




Net migration


0 20 40 60 80 100

−20

000

2000

4000

6000

8000 Australia: male net migration (1973−2008)

Age

Net

mig

ratio

n

Net migration


0 20 40 60 80 100

−20

000

2000

4000

6000

8000 Australia: female net migration (1973−2008)

Age

Net

mig

ratio

n

Stochastic population forecasts

Component models

Data: age/sex-specific mortality rates, fertilityrates and net migration.Models: Functional time series models formortality (M/F), fertility and net migration (M/F)assuming independence between componentsand coherence between sexes.Generate random sample paths of eachcomponent conditional on observed data.Use simulated rates to generate Bt(x),D Ft (x ,x +1), DM

t (x ,x +1) for t = n +1, . . . ,n +h ,assuming deaths and births are Poisson.



Component models





Component models





Component models




Simulation

Demographic growth-balance equation used to getpopulation sample paths.



x = 0,1,2, . . . .

10000 sample paths of population Pt(x), deathsDt(x) and births Bt(x) generated fort = 2004, . . . ,2023 and x = 0,1,2, . . . ,.This allows the computation of the empiricalforecast distribution of any demographicquantity that is based on births, deaths andpopulation numbers.


Simulation




x = 0,1,2, . . . .



Simulation




x = 0,1,2, . . . .



Forecasts of TFR


Forecast Total Fertility Rate

Year

TF

R

1920 1940 1960 1980 2000 2020

2000

2500

3000

3500

Population forecastsForecast population: 2028

Population ('000)

Age

Male Female

150 100 50 0 50 100 150

010

3050

7090

010

3050

7090

020

4060

8010

0

020

4060

8010

0

Age

20092009

Forecast population pyramid for 2028, along with80% prediction intervals. Dashed: actual populationpyramid for 2009.


Population forecasts


● ● ●●

●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●●

●●

●

1980 1990 2000 2010 2020 2030

810

1214

16Total females

Year

Pop

ulat

ion

(mill

ions

)

Population forecasts


● ● ●●

●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

●●

●●

●

1980 1990 2000 2010 2020 2030

810

1214

16Total males

Year

Pop

ulat

ion

(mill

ions

)

Old-age dependency ratio


Old−age dependency ratio forecasts

Year

ratio

1920 1940 1960 1980 2000 2020

0.10

0.15

0.20

0.25

0.30

Advantages of stochastic simulation approach

Functional data analysis provides a way offorecasting age-specific mortality, fertility andnet migration.Stochastic age-specific cohort-componentsimulation provides a way of forecasting manydemographic quantities with predictionintervals.No need to select combinations of assumedrates.True prediction intervals with specifiedcoverage for population and all derivedvariables (TFR, life expectancy, old-agedependencies, etc.)











Outline






6 References

Demographic forecasting using functional data analysis References 58

Selected referencesHyndman, Shang (2010). “Rainbow plots, bagplots and boxplots forfunctional data”. Journal of Computational and Graphical Statistics19(1), 29–45Hyndman, Ullah (2007). “Robust forecasting of mortality and fertilityrates: A functional data approach”. Computational Statistics andData Analysis 51(10), 4942–4956Hyndman, Shang (2009). “Forecasting functional time series (withdiscussion)”. Journal of the Korean Statistical Society 38(3),199–221Shang, Booth, Hyndman (2011). “Point and interval forecasts ofmortality rates and life expectancy : a comparison of ten principalcomponent methods”. Demographic Research 25(5), 173–214Hyndman, Booth (2008). “Stochastic population forecasts usingfunctional data models for mortality, fertility and migration”.International Journal of Forecasting 24(3), 323–342Hyndman, Booth, Yasmeen (2012). “Coherent mortality forecasting:the product-ratio method with functional time series models”.Demography, to appearHyndman (2012). demography: Forecasting mortality, fertility,migration and population data.cran.r-project.org/package=demography


cran.r-project.org/package=demography

Selected referencesHyndman, Shang (2010). “Rainbow plots, bagplots and boxplots forfunctional data”. Journal of Computational and Graphical Statistics19(1), 29–45Hyndman, Ullah (2007). “Robust forecasting of mortality and fertilityrates: A functional data approach”. Computational Statistics andData Analysis 51(10), 4942–4956Hyndman, Shang (2009). “Forecasting functional time series (withdiscussion)”. Journal of the Korean Statistical Society 38(3),199–221Shang, Booth, Hyndman (2011). “Point and interval forecasts ofmortality rates and life expectancy : a comparison of ten principalcomponent methods”. Demographic Research 25(5), 173–214Hyndman, Booth (2008). “Stochastic population forecasts usingfunctional data models for mortality, fertility and migration”.International Journal of Forecasting 24(3), 323–342Hyndman, Booth, Yasmeen (2012). “Coherent mortality forecasting:the product-ratio method with functional time series models”.Demography, to appearHyndman (2012). demography: Forecasting mortality, fertility,migration and population data.cran.r-project.org/package=demography


å Papers and R code:robjhyndman.com

å Email: [email protected]

cran.r-project.org/package=demography

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