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Coherent mortality
forecasting using functional
time series models
Coherent mortality forecasting 1
Rob J Hyndman
Australia: cohort life expectancy at age 50
Year
Rem
aini
ng li
fe e
xpec
tanc
y
1920 1940 1960 1980 2000 2020 2040 2060
2025
3035
4045
50
Mortality rates
Coherent mortality forecasting 2
Mortality rates
Coherent mortality forecasting 3
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0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1970)
Age
Log
deat
h ra
te
Mortality rates
Coherent mortality forecasting 3
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0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1990)
Age
Log
deat
h ra
te
Mortality rates
Coherent mortality forecasting 3
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1921−2009)
Age
Log
deat
h ra
te
Mortality rates
Coherent mortality forecasting 3
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1921−2009)
Age
Log
deat
h ra
te
Mortality rates
Coherent mortality forecasting 3
0 20 40 60 80 100
12
34
Australia: mortality sex ratio (1921−2009)
Age
Sex
rat
io o
f rat
es: M
/F
Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting 4
Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting Functional forecasting 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)
Coherent mortality forecasting Functional forecasting 6
Estimate ft(x) using penalized regression splines.Estimate µ(x) as mean ft(x) across years.Estimate βt ,k and φk(x) using functional principalcomponents.
εt ,xiid∼ N(0,1) and et(x)
iid∼ N(0,v(x)).
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)
Coherent mortality forecasting Functional forecasting 6
Estimate ft(x) using penalized regression splines.Estimate µ(x) as mean ft(x) across years.Estimate βt ,k and φk(x) using functional principalcomponents.
εt ,xiid∼ N(0,1) and et(x)
iid∼ N(0,v(x)).
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)
Coherent mortality forecasting Functional forecasting 6
Estimate ft(x) using penalized regression splines.Estimate µ(x) as mean ft(x) across years.Estimate βt ,k and φk(x) using functional principalcomponents.
εt ,xiid∼ N(0,1) and et(x)
iid∼ N(0,v(x)).
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)
Coherent mortality forecasting Functional forecasting 6
Estimate ft(x) using penalized regression splines.Estimate µ(x) as mean ft(x) across years.Estimate βt ,k and φk(x) using functional principalcomponents.
εt ,xiid∼ N(0,1) and et(x)
iid∼ N(0,v(x)).
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)
Coherent mortality forecasting Functional forecasting 6
Estimate ft(x) using penalized regression splines.Estimate µ(x) as mean ft(x) across years.Estimate βt ,k and φk(x) using functional principalcomponents.
εt ,xiid∼ N(0,1) and et(x)
iid∼ N(0,v(x)).
Australian male mortality model
Coherent mortality forecasting Functional forecasting 7
0 20 40 60 80
−8
−7
−6
−5
−4
−3
−2
−1
Age (x)
µ(x)
0 20 40 60 80
0.05
0.10
0.15
0.20
Age (x)
φ 1(x
)
Year (t)
β t1
1920 1960 2000
−5
05
0 20 40 60 80
−0.
15−
0.05
0.05
0.15
Age (x)
φ 2(x
)
Year (t)
β t2
1920 1960 2000−2.
0−
1.0
0.0
1.0
0 20 40 60 80
−0.
10.
00.
10.
2
Age (x)
φ 3(x
)
Year (t)
β t3
1920 1960 2000
−2
−1
01
Australian male mortality model
Coherent mortality forecasting Functional forecasting 7
1940 1960 1980 2000
020
4060
8010
0 Residuals
Year (t)
Age
(x)
Functional time series model
yt ,x = ft(x)+ σt(x)εt ,x
ft(x) = µ(x)+K∑
k=1
βt ,k φk(x)+ et(x)
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.Univariate ARIMA models can be used forforecasting.
Coherent mortality forecasting Functional forecasting 8
Functional time series model
yt ,x = ft(x)+ σt(x)εt ,x
ft(x) = µ(x)+K∑
k=1
βt ,k φk(x)+ et(x)
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.Univariate ARIMA models can be used forforecasting.
Coherent mortality forecasting Functional forecasting 8
Functional time series model
yt ,x = ft(x)+ σt(x)εt ,x
ft(x) = µ(x)+K∑
k=1
βt ,k φk(x)+ et(x)
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.Univariate ARIMA models can be used forforecasting.
Coherent mortality forecasting Functional forecasting 8
Functional time series model
yt ,x = ft(x)+ σt(x)εt ,x
ft(x) = µ(x)+K∑
k=1
βt ,k φk(x)+ et(x)
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.Univariate ARIMA models can be used forforecasting.
Coherent mortality forecasting Functional forecasting 8
Forecasts
yt ,x = ft(x)+ σt(x)εt ,x
ft(x) = µ(x)+K∑
k=1
βt ,k φk(x)+ et(x)
Coherent mortality forecasting Functional forecasting 9
Forecasts
yt ,x = ft(x)+ σt(x)εt ,x
ft(x) = µ(x)+K∑
k=1
βt ,k φk(x)+ et(x)
where vn+h ,k = Var(βn+h ,k | β1,k , . . . ,βn ,k)and y = [y1,x , . . . ,yn ,x ].
Coherent mortality forecasting Functional forecasting 9
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
Coherent mortality forecasting Functional forecasting 10
0 20 40 60 80
−8
−7
−6
−5
−4
−3
−2
−1
Age (x)
µ(x)
0 20 40 60 80
0.05
0.10
0.15
0.20
Age (x)
φ 1(x
)
Year (t)
β t1
1920 1980 2040
−20
−15
−10
−5
05
0 20 40 60 80
−0.
15−
0.05
0.05
0.15
Age (x)
φ 2(x
)
Year (t)
β t2
1920 1980 2040
−10
−8
−6
−4
−2
02
0 20 40 60 80
−0.
10.
00.
10.
2
Age (x)
φ 3(x
)
Year (t)
β t3
1920 1980 2040
−2
−1
01
Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1921−2009)
Age
Log
deat
h ra
te
Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates (1921−2009)
Age
Log
deat
h ra
te
Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates forecasts (2010−2059)
Age
Log
deat
h ra
te
Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10
−8
−6
−4
−2
0Australia: male death rates forecasts (2010 and 2059)
Age
Log
deat
h ra
te
80% prediction intervals
Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01
23
45
67
Australia: mortality sex ratio data
Age
Year
Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01
23
45
67
Australia: mortality sex ratio data
Age
Year
Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01
23
45
67
Australia: mortality sex ratio forecasts
Age
Year
Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01
23
45
67
Australia: mortality sex ratio forecasts
Age
Year
Male and female mortalityrate forecasts arediverging.
Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting Forecasting groups 13
The problem
Let ft ,j(x) be the smoothed mortality ratefor age x in group j in year t .
Groups may be males and females.Groups may be states within a country.Expected that groups will behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
The problem
Let ft ,j(x) be the smoothed mortality ratefor age x in group j in year t .
Groups may be males and females.Groups may be states within a country.Expected that groups will behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
The problem
Let ft ,j(x) be the smoothed mortality ratefor age x in group j in year t .
Groups may be males and females.Groups may be states within a country.Expected that groups will behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
The problem
Let ft ,j(x) be the smoothed mortality ratefor age x in group j in year t .
Groups may be males and females.Groups may be states within a country.Expected that groups will behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
The problem
Let ft ,j(x) be the smoothed mortality ratefor age x in group j in year t .
Groups may be males and females.Groups may be states within a country.Expected that groups will behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
The problem
Let ft ,j(x) be the smoothed mortality ratefor age x in group j in year t .
Groups may be males and females.Groups may be states within a country.Expected that groups will behavesimilarly.Coherent forecasts do not diverge overtime.Existing functional models do notimpose coherence.Coherent mortality forecasting Forecasting groups 14
Forecasting the coefficients
yt ,x = ft(x)+ σt(x)εt ,x
ft(x) = µ(x)+K∑
k=1
βt ,k φk(x)+ et(x)
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.
Coherent mortality forecasting Forecasting groups 15
Forecasting the coefficients
yt ,x = ft(x)+ σt(x)εt ,x
ft(x) = µ(x)+K∑
k=1
βt ,k φk(x)+ et(x)
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.
Coherent mortality forecasting Forecasting groups 15
Forecasting the coefficients
yt ,x = ft(x)+ σt(x)εt ,x
ft(x) = µ(x)+K∑
k=1
βt ,k φk(x)+ et(x)
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.
Coherent mortality forecasting Forecasting groups 15
Male fts model
Coherent mortality forecasting Forecasting groups 16
0 20 40 60 80
−8
−7
−6
−5
−4
−3
−2
−1
Age
µ(x)
0 20 40 60 80
0.05
0.10
0.15
0.20
Age
φ 1(x
)
Time
β t1
1950 2050
−25
−15
−5
05
0 20 40 60 80
−0.
15−
0.05
0.05
0.15
Age
φ 2(x
)
Time
β t2
1950 2050
−15
−10
−5
0
0 20 40 60 80
−0.
10.
00.
10.
2
Age
φ 3(x
)
Time
β t3
1950 2050
−2
−1
01
Female fts model
Coherent mortality forecasting Forecasting groups 17
0 20 40 60 80
−8
−6
−4
−2
Age
µ(x)
0 20 40 60 80
0.05
0.10
0.15
Age
φ 1(x
)
Time
β t1
1950 2050
−30
−20
−10
010
0 20 40 60 80
−0.
15−
0.05
0.05
Age
φ 2(x
)
Time
β t2
1950 2050
−10
−5
05
0 20 40 60 80
−0.
4−
0.2
0.0
Age
φ 3(x
)
Time
β t3
1950 2050
−1.
0−
0.5
0.0
0.5
1.0
Australian mortality forecasts
Coherent mortality forecasting Forecasting groups 18
0 20 40 60 80 100
−10
−8
−6
−4
−2
0(a) Males
Age
Log
deat
h ra
te
0 20 40 60 80 100
−10
−8
−6
−4
−2
0
(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.Coherent mortality forecasting Forecasting groups 19
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.Coherent mortality forecasting Forecasting groups 19
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.Coherent mortality forecasting Forecasting groups 19
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.Coherent mortality forecasting Forecasting groups 19
Product data
Coherent mortality forecasting Forecasting groups 20
0 20 40 60 80 100
−8
−6
−4
−2
0Australia: product data
Age
Log
of g
eom
etric
mea
n de
ath
rate
Ratio data
Coherent mortality forecasting Forecasting groups 21
0 20 40 60 80 100
12
34
Australia: mortality sex ratio (1921−2009)
Age
Sex
rat
io o
f rat
es: M
/F
Model product and ratios
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√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).
Coherent mortality forecasting Forecasting groups 22
Model product and ratios
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√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).
Coherent mortality forecasting Forecasting groups 22
Model product and ratios
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√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).
Coherent mortality forecasting Forecasting groups 22
Model product and ratios
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√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).
Coherent mortality forecasting Forecasting groups 22
Product model
Coherent mortality forecasting Forecasting groups 23
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
1920 1980 2040
−20
−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
1920 1980 2040
−1.
5−
0.5
0.5
1.0
0 20 40 60 80
−0.
20−
0.10
0.00
0.10
Age
φ 3(x
)
Year
β t3
1920 1980 2040
−4
−2
02
4
Ratio model
Coherent mortality forecasting Forecasting groups 24
0 20 40 60 80
0.1
0.2
0.3
0.4
Age
µ R(x
)
0 20 40 60 80
−0.
10.
00.
10.
2
Age
φ 1(x
)
Year
β t1
1920 1980 2040
−0.
6−
0.2
0.0
0.2
0.4
0 20 40 60 80
0.00
0.10
0.20
Age
φ 2(x
)
Year
β t2
1920 1980 2040−2.
0−
1.0
0.0
1.0
0 20 40 60 80
−0.
3−
0.1
0.1
0.2
0.3
Age
φ 3(x
)
Year
β t3
1920 1980 2040
−0.
4−
0.2
0.0
0.2
0.4
Product forecasts
Coherent mortality forecasting Forecasting groups 25
0 20 40 60 80 100
−10
−8
−6
−4
−2
Age
Log
of g
eom
etric
mea
n de
ath
rate
Ratio forecasts
Coherent mortality forecasting Forecasting groups 26
0 20 40 60 80 100
12
34
Age
Sex
rat
io: M
/F
Coherent forecasts
Coherent mortality forecasting Forecasting groups 27
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
Coherent mortality forecasting Forecasting groups 28
0 20 40 60 80 100
01
23
45
67
Independent forecasts
Age
Sex
rat
io o
f rat
es: M
/F
0 20 40 60 80 100
01
23
45
67
Coherent forecasts
Age
Sex
rat
io o
f rat
es: M
/F
Life expectancy forecasts
Coherent mortality forecasting Forecasting groups 29
Life expectancy forecasts
Year
Age
1920 1960 2000 2040
7075
8085
9095
1920 1960 2000 2040
7075
8085
9095Life expectancy difference: F−M
Year
Num
ber
of y
ears
1960 1980 2000 2020
45
67
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),
log[pt(x)] = µp(x)+K∑
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).
Coherent mortality forecasting Forecasting groups 30
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)]
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),
log[pt(x)] = µp(x)+K∑
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).
Coherent mortality forecasting Forecasting groups 30
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)]
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),
log[pt(x)] = µp(x)+K∑
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).
Coherent mortality forecasting Forecasting groups 30
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)]
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),
log[pt(x)] = µp(x)+K∑
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).
Coherent mortality forecasting Forecasting groups 30
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)]
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),
log[pt(x)] = µp(x)+K∑
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).
Coherent mortality forecasting Forecasting groups 30
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)]
Coherent forecasts for J groups
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 .
Coherent mortality forecasting Forecasting groups 31
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)
Coherent forecasts for J groups
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 .
Coherent mortality forecasting Forecasting groups 31
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)
Coherent forecasts for J groups
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 .
Coherent mortality forecasting Forecasting groups 31
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)
Coherent forecasts for J groups
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 .
Coherent mortality forecasting Forecasting groups 31
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)
Coherent forecasts for J groups
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 .
Coherent mortality forecasting Forecasting groups 31
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)
Li-Lee method
Li & Lee (Demography, 2005) method is a specialcase of our approach.
ft ,j(x) = µj(x)+ βtφ(x)+γt ,jψj(x)+ et ,j(x)
where f is unsmoothed log mortality rate, βt is arandom walk with drift and γt ,j is AR(1) process.
No smoothing.Only one basis function for each part,Random walk with drift very limiting.AR(1) very limiting.The γt ,j coefficients will be highly correlatedwith each other, and so independent models arenot appropriate.Coherent mortality forecasting Forecasting groups 32
Li-Lee method
Li & Lee (Demography, 2005) method is a specialcase of our approach.
ft ,j(x) = µj(x)+ βtφ(x)+γt ,jψj(x)+ et ,j(x)
where f is unsmoothed log mortality rate, βt is arandom walk with drift and γt ,j is AR(1) process.
No smoothing.Only one basis function for each part,Random walk with drift very limiting.AR(1) very limiting.The γt ,j coefficients will be highly correlatedwith each other, and so independent models arenot appropriate.Coherent mortality forecasting Forecasting groups 32
Li-Lee method
Li & Lee (Demography, 2005) method is a specialcase of our approach.
ft ,j(x) = µj(x)+ βtφ(x)+γt ,jψj(x)+ et ,j(x)
where f is unsmoothed log mortality rate, βt is arandom walk with drift and γt ,j is AR(1) process.
No smoothing.Only one basis function for each part,Random walk with drift very limiting.AR(1) very limiting.The γt ,j coefficients will be highly correlatedwith each other, and so independent models arenot appropriate.Coherent mortality forecasting Forecasting groups 32
Li-Lee method
Li & Lee (Demography, 2005) method is a specialcase of our approach.
ft ,j(x) = µj(x)+ βtφ(x)+γt ,jψj(x)+ et ,j(x)
where f is unsmoothed log mortality rate, βt is arandom walk with drift and γt ,j is AR(1) process.
No smoothing.Only one basis function for each part,Random walk with drift very limiting.AR(1) very limiting.The γt ,j coefficients will be highly correlatedwith each other, and so independent models arenot appropriate.Coherent mortality forecasting Forecasting groups 32
Li-Lee method
Li & Lee (Demography, 2005) method is a specialcase of our approach.
ft ,j(x) = µj(x)+ βtφ(x)+γt ,jψj(x)+ et ,j(x)
where f is unsmoothed log mortality rate, βt is arandom walk with drift and γt ,j is AR(1) process.
No smoothing.Only one basis function for each part,Random walk with drift very limiting.AR(1) very limiting.The γt ,j coefficients will be highly correlatedwith each other, and so independent models arenot appropriate.Coherent mortality forecasting Forecasting groups 32
Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting Coherent cohort life expectancy forecasts 33
Life expectancy calculation
Using standard life table calculations:
For x = 0,1, . . . ,ω −1: qx =mx /(1+(1−ax)mx)
`x+1 = `x(1−qx)Lx = `x [1−qx(1−ax)]Tx = Lx + Lx+1 + · · ·+ Lω−1 + Lω+ex = Tx /Lx
where ax = 0.5 for x ≥ 1 and a0 taken from Coale et al (1983).qω+ = 1, Lω+ = lx /mx , and Tω+ = Lω+.
Period life expectancy: let mx =mx ,t for someyear t .Cohort life expectancy: let mx =mx ,t+x for birthcohort in year t .Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
Life expectancy calculation
Using standard life table calculations:
For x = 0,1, . . . ,ω −1: qx =mx /(1+(1−ax)mx)
`x+1 = `x(1−qx)Lx = `x [1−qx(1−ax)]Tx = Lx + Lx+1 + · · ·+ Lω−1 + Lω+ex = Tx /Lx
where ax = 0.5 for x ≥ 1 and a0 taken from Coale et al (1983).qω+ = 1, Lω+ = lx /mx , and Tω+ = Lω+.
Period life expectancy: let mx =mx ,t for someyear t .Cohort life expectancy: let mx =mx ,t+x for birthcohort in year t .Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
Life expectancy calculation
Using standard life table calculations:
For x = 0,1, . . . ,ω −1: qx =mx /(1+(1−ax)mx)
`x+1 = `x(1−qx)Lx = `x [1−qx(1−ax)]Tx = Lx + Lx+1 + · · ·+ Lω−1 + Lω+ex = Tx /Lx
where ax = 0.5 for x ≥ 1 and a0 taken from Coale et al (1983).qω+ = 1, Lω+ = lx /mx , and Tω+ = Lω+.
Period life expectancy: let mx =mx ,t for someyear t .Cohort life expectancy: let mx =mx ,t+x for birthcohort in year t .Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
Life expectancy calculation
Using standard life table calculations:
For x = 0,1, . . . ,ω −1: qx =mx /(1+(1−ax)mx)
`x+1 = `x(1−qx)Lx = `x [1−qx(1−ax)]Tx = Lx + Lx+1 + · · ·+ Lω−1 + Lω+ex = Tx /Lx
where ax = 0.5 for x ≥ 1 and a0 taken from Coale et al (1983).qω+ = 1, Lω+ = lx /mx , and Tω+ = Lω+.
Period life expectancy: let mx =mx ,t for someyear t .Cohort life expectancy: let mx =mx ,t+x for birthcohort in year t .Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
Cohort life expectancy
Because we can forecast mx ,t we canestimate the mortality rates for eachbirth cohort (using actual values whenthey are available).We can simulate future mx ,t in order toestimate the uncertainty associatedwith ex .
Coherent mortality forecasting Coherent cohort life expectancy forecasts 35
Cohort life expectancy
Because we can forecast mx ,t we canestimate the mortality rates for eachbirth cohort (using actual values whenthey are available).We can simulate future mx ,t in order toestimate the uncertainty associatedwith ex .
Coherent mortality forecasting Coherent cohort life expectancy forecasts 35
Cohort life expectancy
Coherent mortality forecasting Coherent cohort life expectancy forecasts 36
Age
0
20
40
60
80Year
1950
2000
2050
2100
2150
Log death rate
−10
−5
Simulate future mortality rates
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√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 ,`} and {βt ,k } simulated.{et(x)} and {wt(x)} bootstrapped.Generate many future sample paths for ft ,M(x)and ft ,F(x) to estimate uncertainty in ex .
Coherent mortality forecasting Coherent cohort life expectancy forecasts 37
Simulate future mortality rates
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√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 ,`} and {βt ,k } simulated.{et(x)} and {wt(x)} bootstrapped.Generate many future sample paths for ft ,M(x)and ft ,F(x) to estimate uncertainty in ex .
Coherent mortality forecasting Coherent cohort life expectancy forecasts 37
Simulate future mortality rates
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√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 ,`} and {βt ,k } simulated.{et(x)} and {wt(x)} bootstrapped.Generate many future sample paths for ft ,M(x)and ft ,F(x) to estimate uncertainty in ex .
Coherent mortality forecasting Coherent cohort life expectancy forecasts 37
Simulate future mortality rates
pt(x) =√ft ,M(x)ft ,F(x) and rt(x) =
√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 ,`} and {βt ,k } simulated.{et(x)} and {wt(x)} bootstrapped.Generate many future sample paths for ft ,M(x)and ft ,F(x) to estimate uncertainty in ex .
Coherent mortality forecasting Coherent cohort life expectancy forecasts 37
Cohort life expectancy
Coherent mortality forecasting Coherent cohort life expectancy forecasts 38
Australia: cohort life expectancy at age 50
Year
Rem
aini
ng li
fe e
xpec
tanc
y
1920 1940 1960 1980 2000 2020 2040 2060
2025
3035
4045
50
Complete code
Coherent mortality forecasting Coherent cohort life expectancy forecasts 39
library(demography)
# Read data
aus <- hmd.mx("AUS","username","password","Australia")
# Smooth data
aus.sm <- smooth.demogdata(aus)
#Fit model
aus.pr <- coherentfdm(aus.sm)
# Forecast
aus.pr.fc <- forecast(aus.pr, h=100)
# Compute life expectancies
e50.m.aus.fc <- flife.expectancy(aus.pr.fc, series="male",
age=50, PI=TRUE, nsim=1000, type="cohort")
e50.f.aus.fc <- flife.expectancy(aus.pr.fc, series="female",
age=50, PI=TRUE, nsim=1000, type="cohort")
Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 40
Australian female cohort e50
Year
Rem
aini
ng y
ears
1920 1940 1960 1980 2000
2628
3032
3436
3840
Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 41
Forecast accuracy evaluation
Compute age 50 remaining cohort lifeexpectancy with a rolling forecast originbeginning in 1921.Compare against actual cohort lifeexpectancy where available.Compute 80% prediction interval actualcoverage.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 42
Forecast accuracy evaluation
Compute age 50 remaining cohort lifeexpectancy with a rolling forecast originbeginning in 1921.Compare against actual cohort lifeexpectancy where available.Compute 80% prediction interval actualcoverage.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 42
Forecast accuracy evaluation
Compute age 50 remaining cohort lifeexpectancy with a rolling forecast originbeginning in 1921.Compare against actual cohort lifeexpectancy where available.Compute 80% prediction interval actualcoverage.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 42
Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 43
5 10 15 20 25
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Mean Absolute Forecast Errors
Forecast horizon
Year
s
1 2 3 4
Male
Female
Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 43
5 10 15 20 25
020
4060
8010
080% prediction interval coverage
Forecast horizon
Per
cent
age
cove
rage
1 2 3 4
Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting Conclusions 44
Some conclusions
New, automatic, flexible method forcoherent forecasting of groups offunctional time series.Suitable for age-specific mortality.Based on geometric means and ratios,so interpretable results.More general and flexible than existingmethods.Easy to compute prediction intervals forany computable statistics.Coherent mortality forecasting Conclusions 45
Some conclusions
New, automatic, flexible method forcoherent forecasting of groups offunctional time series.Suitable for age-specific mortality.Based on geometric means and ratios,so interpretable results.More general and flexible than existingmethods.Easy to compute prediction intervals forany computable statistics.Coherent mortality forecasting Conclusions 45
Some conclusions
New, automatic, flexible method forcoherent forecasting of groups offunctional time series.Suitable for age-specific mortality.Based on geometric means and ratios,so interpretable results.More general and flexible than existingmethods.Easy to compute prediction intervals forany computable statistics.Coherent mortality forecasting Conclusions 45
Some conclusions
New, automatic, flexible method forcoherent forecasting of groups offunctional time series.Suitable for age-specific mortality.Based on geometric means and ratios,so interpretable results.More general and flexible than existingmethods.Easy to compute prediction intervals forany computable statistics.Coherent mortality forecasting Conclusions 45
Some conclusions
New, automatic, flexible method forcoherent forecasting of groups offunctional time series.Suitable for age-specific mortality.Based on geometric means and ratios,so interpretable results.More general and flexible than existingmethods.Easy to compute prediction intervals forany computable statistics.Coherent mortality forecasting Conclusions 45
Selected referencesHyndman, Ullah (2007). “Robust forecasting of mortalityand fertility rates: A functional data approach”.Computational Statistics and Data Analysis 51(10),4942–4956Hyndman, Shang (2009). “Forecasting functional timeseries (with discussion)”. Journal of the KoreanStatistical Society 38(3), 199–221Hyndman, Booth, Yasmeen (2013). “Coherent mortalityforecasting: the product-ratio method with functionaltime series models”. Demography 50(1), 261–283Booth, Hyndman, Tickle (2013). “Prospective Life Tables”.Computational Actuarial Science, with R. ed. by
Charpentier. Chapman & Hall/CRC, 323–348Hyndman (2013). demography: Forecasting mortality,fertility, migration and population data. v1.16.cran.r-project.org/package=demography
Coherent mortality forecasting Conclusions 46
Selected referencesHyndman, Ullah (2007). “Robust forecasting of mortalityand fertility rates: A functional data approach”.Computational Statistics and Data Analysis 51(10),4942–4956Hyndman, Shang (2009). “Forecasting functional timeseries (with discussion)”. Journal of the KoreanStatistical Society 38(3), 199–221Hyndman, Booth, Yasmeen (2013). “Coherent mortalityforecasting: the product-ratio method with functionaltime series models”. Demography 50(1), 261–283Booth, Hyndman, Tickle (2013). “Prospective Life Tables”.Computational Actuarial Science, with R. ed. by
Charpentier. Chapman & Hall/CRC, 323–348Hyndman (2013). demography: Forecasting mortality,fertility, migration and population data. v1.16.cran.r-project.org/package=demography
Coherent mortality forecasting Conclusions 46
å Papers and R code:robjhyndman.com
å Email: [email protected]
