Coherent mortality forecasting using functional time series models
1. Coherent mortality
forecasting using functional
time series models
Coherent mortality forecasting 1
Rob J Hyndman
Australia: cohort life expectancy at age 50
10. Some notation
Let yt,x be the observed (smoothed) data in period t
at 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 principal
components.
εt,x
iid
∼ N(0,1) and et(x)
iid
∼ N(0,v(x)).
11. Some notation
Let yt,x be the observed (smoothed) data in period t
at 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 principal
components.
εt,x
iid
∼ N(0,1) and et(x)
iid
∼ N(0,v(x)).
12. Some notation
Let yt,x be the observed (smoothed) data in period t
at 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 principal
components.
εt,x
iid
∼ N(0,1) and et(x)
iid
∼ N(0,v(x)).
13. Some notation
Let yt,x be the observed (smoothed) data in period t
at 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 principal
components.
εt,x
iid
∼ N(0,1) and et(x)
iid
∼ N(0,v(x)).
14. Some notation
Let yt,x be the observed (smoothed) data in period t
at 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 principal
components.
εt,x
iid
∼ N(0,1) and et(x)
iid
∼ N(0,v(x)).
15. 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
0.050.100.150.20
Age (x)
φ1(x)
Year (t)
βt1
1920 1960 2000
−505
0 20 40 60 80
−0.15−0.050.050.15
Age (x)
φ2(x)
Year (t)
βt2
1920 1960 2000
−2.0−1.00.01.0
0 20 40 60 80
−0.10.00.10.2
Age (x)
φ3(x)
Year (t)
βt3
1920 1960 2000
−2−101
16. Australian male mortality model
Coherent mortality forecasting Functional forecasting 7
1940 1960 1980 2000
020406080100
Residuals
Year (t)
Age(x)
17. 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 main
regions of variation.
The scores {βt,k } are uncorrelated by
construction. So we can forecast each βt,k
using a univariate time series model.
Univariate ARIMA models can be used for
forecasting.
Coherent mortality forecasting Functional forecasting 8
18. 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 main
regions of variation.
The scores {βt,k } are uncorrelated by
construction. So we can forecast each βt,k
using a univariate time series model.
Univariate ARIMA models can be used for
forecasting.
Coherent mortality forecasting Functional forecasting 8
19. 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 main
regions of variation.
The scores {βt,k } are uncorrelated by
construction. So we can forecast each βt,k
using a univariate time series model.
Univariate ARIMA models can be used for
forecasting.
Coherent mortality forecasting Functional forecasting 8
20. 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 main
regions of variation.
The scores {βt,k } are uncorrelated by
construction. So we can forecast each βt,k
using a univariate time series model.
Univariate ARIMA models can be used for
forecasting.
Coherent mortality forecasting Functional forecasting 8
22. 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] = ˆµ(x) +
K
k=1
ˆβn+h,k
ˆφk (x)
Var[yn+h,x | y] = ˆσ2
µ (x) +
K
k=1
vn+h,k
ˆφ2
k (x) + σ2
t (x) + v(x)
23. 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
0.050.100.150.20
Age (x)
φ1(x)
Year (t)
βt1
1920 1980 2040
−20−15−10−505
0 20 40 60 80
−0.15−0.050.050.15
Age (x)
φ2(x)
Year (t)
βt2
1920 1980 2040
−10−8−6−4−202
0 20 40 60 80
−0.10.00.10.2
Age (x)
φ3(x)
Year (t)
βt3
1920 1980 2040
−2−101
24. Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10−8−6−4−20
Australia: male death rates (1921−2009)
Age
Logdeathrate
25. Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10−8−6−4−20
Australia: male death rates (1921−2009)
Age
Logdeathrate
26. Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10−8−6−4−20
Australia: male death rates forecasts (2010−2059)
Age
Logdeathrate
27. Forecasts of ft(x)
Coherent mortality forecasting Functional forecasting 11
0 20 40 60 80 100
−10−8−6−4−20
Australia: male death rates forecasts (2010 and 2059)
Age
Logdeathrate
80% prediction intervals
28. Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01234567
Australia: mortality sex ratio data
Age
Year
29. Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01234567
Australia: mortality sex ratio data
Age
Year
30. Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01234567
Australia: mortality sex ratio forecasts
Age
Year
31. Forecasts of mortality sex ratio
Coherent mortality forecasting Functional forecasting 12
0 20 40 60 80 100
01234567
Australia: mortality sex ratio forecasts
Age
Year
Male and female mortality
rate forecasts are
diverging.
32. Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting Forecasting groups 13
33. The problem
Let ft,j(x) be the smoothed mortality rate
for 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 behave
similarly.
Coherent forecasts do not diverge over
time.
Existing functional models do not
impose coherence.
Coherent mortality forecasting Forecasting groups 14
34. The problem
Let ft,j(x) be the smoothed mortality rate
for 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 behave
similarly.
Coherent forecasts do not diverge over
time.
Existing functional models do not
impose coherence.
Coherent mortality forecasting Forecasting groups 14
35. The problem
Let ft,j(x) be the smoothed mortality rate
for 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 behave
similarly.
Coherent forecasts do not diverge over
time.
Existing functional models do not
impose coherence.
Coherent mortality forecasting Forecasting groups 14
36. The problem
Let ft,j(x) be the smoothed mortality rate
for 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 behave
similarly.
Coherent forecasts do not diverge over
time.
Existing functional models do not
impose coherence.
Coherent mortality forecasting Forecasting groups 14
37. The problem
Let ft,j(x) be the smoothed mortality rate
for 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 behave
similarly.
Coherent forecasts do not diverge over
time.
Existing functional models do not
impose coherence.
Coherent mortality forecasting Forecasting groups 14
38. The problem
Let ft,j(x) be the smoothed mortality rate
for 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 behave
similarly.
Coherent forecasts do not diverge over
time.
Existing functional models do not
impose coherence.
Coherent mortality forecasting Forecasting groups 14
39. 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 the
first few coefficients (k = 1,2)
Non-stationary ARIMA forecasts will diverge.
Hence the mortality forecasts are not coherent.
Coherent mortality forecasting Forecasting groups 15
40. 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 the
first few coefficients (k = 1,2)
Non-stationary ARIMA forecasts will diverge.
Hence the mortality forecasts are not coherent.
Coherent mortality forecasting Forecasting groups 15
41. 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 the
first few coefficients (k = 1,2)
Non-stationary ARIMA forecasts will diverge.
Hence the mortality forecasts are not coherent.
Coherent mortality forecasting Forecasting groups 15
42. 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
0.050.100.150.20
Age
φ1(x)
Time
βt1
1950 2050
−25−15−505
0 20 40 60 80
−0.15−0.050.050.15
Age
φ2(x)
Time
βt2
1950 2050
−15−10−50
0 20 40 60 80
−0.10.00.10.2
Age
φ3(x)
Time
βt3
1950 2050
−2−101
43. 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
0.050.100.15
Age
φ1(x)
Time
βt1
1950 2050
−30−20−10010
0 20 40 60 80
−0.15−0.050.05
Age
φ2(x)
Time
βt2
1950 2050
−10−505
0 20 40 60 80
−0.4−0.20.0
Age
φ3(x)
Time
βt3
1950 2050
−1.0−0.50.00.51.0
44. Australian mortality forecasts
Coherent mortality forecasting Forecasting groups 18
0 20 40 60 80 100
−10−8−6−4−20
(a) Males
Age
Logdeathrate
0 20 40 60 80 100
−10−8−6−4−20
(b) Females
Age
Logdeathrate
45. Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratio
instead of the individual rates for each sex
separately.
pt(x) = ft,M (x)ft,F (x) and rt(x) = ft,M (x) ft,F (x).
Product and ratio are approximately
independent
Ratio should be stationary (for coherence) but
product can be non-stationary.
Coherent mortality forecasting Forecasting groups 19
46. Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratio
instead of the individual rates for each sex
separately.
pt(x) = ft,M (x)ft,F (x) and rt(x) = ft,M (x) ft,F (x).
Product and ratio are approximately
independent
Ratio should be stationary (for coherence) but
product can be non-stationary.
Coherent mortality forecasting Forecasting groups 19
47. Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratio
instead of the individual rates for each sex
separately.
pt(x) = ft,M (x)ft,F (x) and rt(x) = ft,M (x) ft,F (x).
Product and ratio are approximately
independent
Ratio should be stationary (for coherence) but
product can be non-stationary.
Coherent mortality forecasting Forecasting groups 19
48. Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratio
instead of the individual rates for each sex
separately.
pt(x) = ft,M (x)ft,F (x) and rt(x) = ft,M (x) ft,F (x).
Product and ratio are approximately
independent
Ratio should be stationary (for coherence) but
product can be non-stationary.
Coherent mortality forecasting Forecasting groups 19
49. Product data
Coherent mortality forecasting Forecasting groups 20
0 20 40 60 80 100
−8−6−4−20
Australia: product data
Age
Logofgeometricmeandeathrate
50. Ratio data
Coherent mortality forecasting Forecasting groups 21
0 20 40 60 80 100
1234
Australia: mortality sex ratio (1921−2009)
Age
Sexratioofrates:M/F
51. 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,k φk (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
52. 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,k φk (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
53. 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,k φk (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
54. 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,k φk (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
55. Product model
Coherent mortality forecasting Forecasting groups 23
0 20 40 60 80
−8−6−4−2
Age
µP(x)
0 20 40 60 80
0.050.100.15
Age
φ1(x)
Year
βt1
1920 1980 2040
−20−10−505
0 20 40 60 80
−0.2−0.10.00.10.2
Age
φ2(x)
Year
βt2
1920 1980 2040
−1.5−0.50.51.0
0 20 40 60 80
−0.20−0.100.000.10
Age
φ3(x)
Year
βt3
1920 1980 2040
−4−2024
56. Ratio model
Coherent mortality forecasting Forecasting groups 24
0 20 40 60 80
0.10.20.30.4
Age
µR(x)
0 20 40 60 80
−0.10.00.10.2
Age
φ1(x)
Year
βt1
1920 1980 2040
−0.6−0.20.00.20.4
0 20 40 60 80
0.000.100.20
Age
φ2(x)
Year
βt2
1920 1980 2040
−2.0−1.00.01.0
0 20 40 60 80
−0.3−0.10.10.20.3
Age
φ3(x)
Year
βt3
1920 1980 2040
−0.4−0.20.00.20.4
60. Ratio forecasts
Coherent mortality forecasting Forecasting groups 28
0 20 40 60 80 100
01234567
Independent forecasts
Age
Sexratioofrates:M/F
0 20 40 60 80 100
01234567
Coherent forecasts
Age
Sexratioofrates:M/F
61. Life expectancy forecasts
Coherent mortality forecasting Forecasting groups 29
Life expectancy forecasts
Year
Age
1920 1960 2000 2040
707580859095
1920 1960 2000 2040
707580859095
Life expectancy difference: F−M
Year
Numberofyears
1960 1980 2000 2020
4567
62. 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,k φk (x) + et(x)
log[rt,j (x)] = µr,j (x) +
L
l=1
γt,l,j ψl,j (x) + wt,j (x).
Coherent mortality forecasting Forecasting groups 30
pt(x) and all rt,j (x)
are approximately
independent.
Ratios satisfy constraint
rt,1(x)rt,2(x)···rt,J (x) = 1.
log[ft,j (x)] = log[pt(x)rt,j (x)]
63. 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,k φk (x) + et(x)
log[rt,j (x)] = µr,j (x) +
L
l=1
γt,l,j ψl,j (x) + wt,j (x).
Coherent mortality forecasting Forecasting groups 30
pt(x) and all rt,j (x)
are approximately
independent.
Ratios satisfy constraint
rt,1(x)rt,2(x)···rt,J (x) = 1.
log[ft,j (x)] = log[pt(x)rt,j (x)]
64. 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,k φk (x) + et(x)
log[rt,j (x)] = µr,j (x) +
L
l=1
γt,l,j ψl,j (x) + wt,j (x).
Coherent mortality forecasting Forecasting groups 30
pt(x) and all rt,j (x)
are approximately
independent.
Ratios satisfy constraint
rt,1(x)rt,2(x)···rt,J (x) = 1.
log[ft,j (x)] = log[pt(x)rt,j (x)]
65. 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,k φk (x) + et(x)
log[rt,j (x)] = µr,j (x) +
L
l=1
γt,l,j ψl,j (x) + wt,j (x).
Coherent mortality forecasting Forecasting groups 30
pt(x) and all rt,j (x)
are approximately
independent.
Ratios satisfy constraint
rt,1(x)rt,2(x)···rt,J (x) = 1.
log[ft,j (x)] = log[pt(x)rt,j (x)]
66. 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,k φk (x) + et(x)
log[rt,j (x)] = µr,j (x) +
L
l=1
γt,l,j ψl,j (x) + wt,j (x).
Coherent mortality forecasting Forecasting groups 30
pt(x) and all rt,j (x)
are approximately
independent.
Ratios satisfy constraint
rt,1(x)rt,2(x)···rt,J (x) = 1.
log[ft,j (x)] = log[pt(x)rt,j (x)]
67. 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,k φk (x) +
L
=1
γt, ,j ψ ,j (x) + zt,j (x)
68. 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,k φk (x) +
L
=1
γt, ,j ψ ,j (x) + zt,j (x)
69. 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,k φk (x) +
L
=1
γt, ,j ψ ,j (x) + zt,j (x)
70. 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,k φk (x) +
L
=1
γt, ,j ψ ,j (x) + zt,j (x)
71. 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,k φk (x) +
L
=1
γt, ,j ψ ,j (x) + zt,j (x)
72. Li-Lee method
Li & Lee (Demography, 2005) method is a special
case of our approach.
ft,j (x) = µj (x) + βtφ(x) + γt,j ψj (x) + et,j (x)
where f is unsmoothed log mortality rate, βt is a
random 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 correlated
with each other, and so independent models are
not appropriate.
Coherent mortality forecasting Forecasting groups 32
73. Li-Lee method
Li & Lee (Demography, 2005) method is a special
case of our approach.
ft,j (x) = µj (x) + βtφ(x) + γt,j ψj (x) + et,j (x)
where f is unsmoothed log mortality rate, βt is a
random 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 correlated
with each other, and so independent models are
not appropriate.
Coherent mortality forecasting Forecasting groups 32
74. Li-Lee method
Li & Lee (Demography, 2005) method is a special
case of our approach.
ft,j (x) = µj (x) + βtφ(x) + γt,j ψj (x) + et,j (x)
where f is unsmoothed log mortality rate, βt is a
random 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 correlated
with each other, and so independent models are
not appropriate.
Coherent mortality forecasting Forecasting groups 32
75. Li-Lee method
Li & Lee (Demography, 2005) method is a special
case of our approach.
ft,j (x) = µj (x) + βtφ(x) + γt,j ψj (x) + et,j (x)
where f is unsmoothed log mortality rate, βt is a
random 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 correlated
with each other, and so independent models are
not appropriate.
Coherent mortality forecasting Forecasting groups 32
76. Li-Lee method
Li & Lee (Demography, 2005) method is a special
case of our approach.
ft,j (x) = µj (x) + βtφ(x) + γt,j ψj (x) + et,j (x)
where f is unsmoothed log mortality rate, βt is a
random 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 correlated
with each other, and so independent models are
not appropriate.
Coherent mortality forecasting Forecasting groups 32
77. Outline
1 Functional forecasting
2 Forecasting groups
3 Coherent cohort life expectancy forecasts
4 Conclusions
Coherent mortality forecasting Coherent cohort life expectancy forecasts 33
78. 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 some
year t.
Cohort life expectancy: let mx = mx,t+x for birth
cohort in year t.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
79. 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 some
year t.
Cohort life expectancy: let mx = mx,t+x for birth
cohort in year t.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
80. 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 some
year t.
Cohort life expectancy: let mx = mx,t+x for birth
cohort in year t.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
81. 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 some
year t.
Cohort life expectancy: let mx = mx,t+x for birth
cohort in year t.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 34
82. Cohort life expectancy
Because we can forecast mx,t we can
estimate the mortality rates for each
birth cohort (using actual values when
they are available).
We can simulate future mx,t in order to
estimate the uncertainty associated
with ex.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 35
83. Cohort life expectancy
Because we can forecast mx,t we can
estimate the mortality rates for each
birth cohort (using actual values when
they are available).
We can simulate future mx,t in order to
estimate the uncertainty associated
with ex.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 35
84. Cohort life expectancy
Coherent mortality forecasting Coherent cohort life expectancy forecasts 36
Age
0
20
40
60
80
Year
1950
2000
2050
2100
2150
Logdeathrate
−10
−5
85. 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,k φk (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
86. 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,k φk (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
87. 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,k φk (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
88. 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,k φk (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
89. Cohort life expectancy
Coherent mortality forecasting Coherent cohort life expectancy forecasts 38
Australia: cohort life expectancy at age 50
Year
Remaininglifeexpectancy
1920 1940 1960 1980 2000 2020 2040 2060
20253035404550
90. 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")
91. Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 40
Australian female cohort e50
Year
Remainingyears
1920 1940 1960 1980 2000
2628303234363840
92. Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 41
Australian female cohort e50: Data to 1955
Year
Remainingyears
1920 1940 1960 1980 2000
2628303234363840
93. Forecast accuracy evaluation
Compute age 50 remaining cohort life
expectancy with a rolling forecast origin
beginning in 1921.
Compare against actual cohort life
expectancy where available.
Compute 80% prediction interval actual
coverage.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 42
94. Forecast accuracy evaluation
Compute age 50 remaining cohort life
expectancy with a rolling forecast origin
beginning in 1921.
Compare against actual cohort life
expectancy where available.
Compute 80% prediction interval actual
coverage.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 42
95. Forecast accuracy evaluation
Compute age 50 remaining cohort life
expectancy with a rolling forecast origin
beginning in 1921.
Compare against actual cohort life
expectancy where available.
Compute 80% prediction interval actual
coverage.
Coherent mortality forecasting Coherent cohort life expectancy forecasts 42
96. Forecast accuracy evaluation
Coherent mortality forecasting Coherent cohort life expectancy forecasts 43
5 10 15 20 25
0.00.20.40.60.81.01.2
Mean Absolute Forecast Errors
Forecast horizon
Years
1 2 3 4
Male
Female
99. Some conclusions
New, automatic, flexible method for
coherent forecasting of groups of
functional time series.
Suitable for age-specific mortality.
Based on geometric means and ratios,
so interpretable results.
More general and flexible than existing
methods.
Easy to compute prediction intervals for
any computable statistics.
Coherent mortality forecasting Conclusions 45
100. Some conclusions
New, automatic, flexible method for
coherent forecasting of groups of
functional time series.
Suitable for age-specific mortality.
Based on geometric means and ratios,
so interpretable results.
More general and flexible than existing
methods.
Easy to compute prediction intervals for
any computable statistics.
Coherent mortality forecasting Conclusions 45
101. Some conclusions
New, automatic, flexible method for
coherent forecasting of groups of
functional time series.
Suitable for age-specific mortality.
Based on geometric means and ratios,
so interpretable results.
More general and flexible than existing
methods.
Easy to compute prediction intervals for
any computable statistics.
Coherent mortality forecasting Conclusions 45
102. Some conclusions
New, automatic, flexible method for
coherent forecasting of groups of
functional time series.
Suitable for age-specific mortality.
Based on geometric means and ratios,
so interpretable results.
More general and flexible than existing
methods.
Easy to compute prediction intervals for
any computable statistics.
Coherent mortality forecasting Conclusions 45
103. Some conclusions
New, automatic, flexible method for
coherent forecasting of groups of
functional time series.
Suitable for age-specific mortality.
Based on geometric means and ratios,
so interpretable results.
More general and flexible than existing
methods.
Easy to compute prediction intervals for
any computable statistics.
Coherent mortality forecasting Conclusions 45
104. Selected references
Hyndman, Ullah (2007). “Robust forecasting of mortality
and fertility rates: A functional data approach”.
Computational Statistics and Data Analysis 51(10),
4942–4956
Hyndman, Shang (2009). “Forecasting functional time
series (with discussion)”. Journal of the Korean
Statistical Society 38(3), 199–221
Hyndman, Booth, Yasmeen (2013). “Coherent mortality
forecasting: the product-ratio method with functional
time series models”. Demography 50(1), 261–283
Booth, Hyndman, Tickle (2013). “Prospective Life Tables”.
Computational Actuarial Science, with R. ed. by
Charpentier. Chapman & Hall/CRC, 323–348
Hyndman (2013). demography: Forecasting mortality,
fertility, migration and population data. v1.16.
cran.r-project.org/package=demography
Coherent mortality forecasting Conclusions 46
105. Selected references
Hyndman, Ullah (2007). “Robust forecasting of mortality
and fertility rates: A functional data approach”.
Computational Statistics and Data Analysis 51(10),
4942–4956
Hyndman, Shang (2009). “Forecasting functional time
series (with discussion)”. Journal of the Korean
Statistical Society 38(3), 199–221
Hyndman, Booth, Yasmeen (2013). “Coherent mortality
forecasting: the product-ratio method with functional
time series models”. Demography 50(1), 261–283
Booth, Hyndman, Tickle (2013). “Prospective Life Tables”.
Computational Actuarial Science, with R. ed. by
Charpentier. Chapman & Hall/CRC, 323–348
Hyndman (2013). demography: Forecasting mortality,
fertility, migration and population data. v1.16.
cran.r-project.org/package=demography
Coherent mortality forecasting Conclusions 46
¯ Papers and R code:
robjhyndman.com
¯ Email: Rob.Hyndman@monash.edu
