# The Future of Australian Mortality by rraul

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```									The Future of Australian Mortality
Heather Booth Rajesh Chauhan John Maindonald Len Smith Australian National University APA conference 29/11 to 1/12/2000

Background
• Lee and Carter (1992): USA • Tuljapurkar et al (2000): G7 countries • ABS projections 1999-2101

Lee and Carter: method
• No subjective judgements involved • Based on past mortality rates

The model
ln m x ,t = a x + bx kt + ε x ,t .................(1)
ln m x ,t kt

( )

( ) is a matrix of log age specific death rates

is the index of improvement in mortality over time (‘dominant temporal signal’) ax and bx are the age specific constants; indicate the rate of decline in specific age group is the error term of the equation with mean 0 and variance ε x ,t σ∈2

• Adjust data prior to fitting the model • Extension to age 105+
– relational logistic model fitted to ages 50 to 85+ (Himes, Preston & Condran 1994)

• Normalisation
– adjust ln[m(x,t)] by subtracting the mean over time of ln[m(x,t)]

Singular Value Decomposition
• Solution found by SVD method • First left and right vectors and leading value of SVD provide a unique solution:
– dominant temporal signal, k(t) – fixed relative age effects, b(x)

Singular Value Decomposition

X = TDA′
Where: X is the matrix of normalised log death rates T is the matrix of time effect vectors A is the matrix of age effect vectors D is the diagonal matrix of singular values

ARIMA fit
• ARIMA (auto-regressive integrated moving average) used to project future signal • Random walk with drift, z ˆ ˆ • k(t + 1) = k(t) − z + εt • Fitted to k 1950-1994 • Projected k 1995-2050 • Confidence limits of projection estimated from error of fit

Lee and Carter: application
• US 1900-1989; projected to 2065 • Life expectancy > official projections • Adopted widely in North America

Tuljapurkar et al: application
• G7 countries: Canada, France, Germany, Italy, Japan,UK,US • 1950-1994 data; projected to 2050 • ‘Universal’ pattern: exponential decline at all ages • e(0) < official in short term, but > official in long term

ABS projections 1999-2101
• First period 1997-2002
– increase of 0.30 per annum in male e(0) – increase of 0.22 per annum in female e(0)

• Declining rate of increase thereafter • By 2051
– male e(0) = 83.3 years – female e(0) = 86.6 years

• past age pattern of change to 2027

ABS projected e(0) 1997-2052
86 85 84 83 82 81 80 79 78 1990

2010

2030

2050

2070

The Australian study
• Apply Lee-Carter method to Australian data following Tuljapurkar et al. • Compare Australia with G7 analysis • Compare new projections with ABS projections, 1999-2050

Results
• 95 per cent of variance explained by first vectors and value of SVD

Values of k: G7 + Australia
25 20 15 10 5 0 1950 -5 -10 -15 -20 -25

France Italy UK Australia

1955

1960

1965

1970

1975

1980

1985

1990

1995

Values of b: G7 + Australia
0.16

0.14

France Italy UK Australia

Germany Japan US

0.12

0.1

0.08

0.06

0.04

0.02

0 -20 0 20 40 60 80 100 120

Observed and projected e(0)
Both Sexes, Australia, 1950-2050
100.0 95.0 90.0 85.0

e (0,0)

80.0 75.0 70.0 65.0 60.0 1940

1960

1980

2000 Time

2020

2040

2060

Discussion
• Differences between Australia and other countries • Differences between Lee-Carter and official projections • Further research:
– by sex, by cause, by cohort – limits to life expectancy – improvements in methodology

Comparison of ABS and L-C e(0)
100

95

ABS Lee-Carter

90

85

80

75

70 1990 2000 2010 2020 2030 2040 2050 2060

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