Lectures 2, 3Variance in Death and Mortality Decline

Shripad Tuljapurkar

Ryan D. EdwardsQueens College & Grad Center CUNY

MORTALITY LEVELS, DECLINES ARE ASSESSED IN TERMS OF e0

e0 = LIFE EXPECTANCY AT BIRTH

= AVERAGE AGE AT DEATH

= E(T) where T = Random age at death

Density of T is (.)

1750 1800 1850 1900 1950 2000 205010

20

30

40

50

60

70

80

90

YEAR

SWEDEN LIFE EXPECTANCY AT BIRTH

MORTALITY CHANGE

THE DETAILS ARE MESSY

•Year to year decline irregular

•Persistent, puzzling differentials

•Cause of death structure difficult to understand & to predict

•Poor understanding of causal relationship to driving forces

•Startling reversibility -- the Former Soviet Union

BUT…

IN THE AGGREGATE (i.e., age/sex)

OVER THE LONG-TERM ( >40 years)

IN HIGHLY INDUSTRIALIZED NATIONS

THERE APPEARS TO BE A

Simple, general (?) pattern of decline

log m(x,t) = s a(x) k(t) + r b(x) g(t) + …

Singular Values s > r > … > 0

IF s >> r > …

THEN

DOMINANT TEMPORAL PATTERN IS

k(t)

% VARIANCE EXPLAINED IS

s2/(s2 + r2 + …)

Lee Carter (US)

Tulja, Li , Boe (G7)

In every G-7 country

ONE TEMPORAL COMPONENT

EXPLAINS OVER 92 % OF CHANGE IN log m(x,t)

m(x,t) = central death rate

G-7 = Canada, France, Germany, Italy, Japan, UK, US

Period = 1950 TO 1994

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995-20

-15

-10

-5

0

5

10

15

20

CanadaFranceGermanyItalyJapanUKUS

LEE CARTER MORTALITY

log ( , ) log ( , 1) ( ) ( ) ( )m x t m x t z b x e t b x

OEPPEN-VAUPEL

Best-in-world life expectancy has risen in a straight line for 160 years, as shown

by

0 20 40 60 80 100 1200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Age

Pro

b D

yin

g q

x*lx

SWEDEN Prob Death by Age 1875

0 20 40 60 80 100 1200

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Age

Pro

b D

yin

g q

x*lx

SWEDEN Prob Death by Age 2000

0

before befor

Age at death

Prob{die age A} {age at death|die age A}

Prob{die age A} {age at death|die age A}

( )

e

after after

T

p e E

p e E

E Te p e p e

Death – young death before age A, – adult death after age A

p

p

0

Age at death

dominated by

T

e e

Most death – adult death after age A

2 20 0

Var( )

( ) ( )

T p V p V

p e e p e e

Variance in age at death –

young death, adult death

From young deathFrom adult death

+

Age at death

=V Var( | die after age )

T

T A

Most variance in death – variance in adult death after age A

0 20 40 60 80 100 1200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Infant Mortality – leave out Mode

S10 – Variance of Age at Death if Die after Age 10

“ADULT” DEATHS AGES > 10 YEARS

CAPTURES MOST VARIANCE IN AGE OF DEATH

V(10) = VAR (AGE AT DEATH | DIE AT AGE > 10)

S(10) = √ V(10) = STANDARD DEVIATION IN AGE AT ADULT

DEATH.

US

JapanSweden

Age a

( ) ( ) ( )a a l a Conditional distribution --- die after age 10

0

2

0 2

2 0''

0

Mode of ( ) is at

( )( ) ( ) exp

2

( )

| ( ) |

( )o

a a

a aa a

a

a

0

Gompertz Slope and Variance in Death

1

( )= aa e

Did β change through history? Is it still changing?

15 20 25 30 35 40 45 50 5518.5

19

19.5

20

20.5

21

21.5

22

Sweden Female S(20) VS e0. 1751-1891

50 55 60 65 70 7512

14

16

18

20

22

24

Sweden Female S(20) VS e0. 1891-1953

73 74 75 76 77 78 79 80 81 82 8311.8

12

12.2

12.4

12.6

12.8

13

Sweden Female S(20) VS e0. 1954-2003

σ DECREASED and

β INCREASED

through the first half of the 20th century everywhere*

σ is still DECREASING and

β INCREASING

in Sweden

Forecasting Models

Bongaarts

0 0( ) ( 1)

( ) ( 1)

a t a t D

t t

Forecasting Models

Lee-Carter

0

log ( , ) log ( , 1) ( )

( ) 0 ( ) 1

( ) peaks at age Mode of ( , )a

a t a t z b x

b a b a

b a a a t

00 0 2

Mode and variance ( , ) projected by Lee-Carter

2 ( 1)( ) ( 1)

( 1)

( ) ( 1) if ( / ) 0 at mode

a t

z b a ta t a t

t

t t db dx

Shape of b(x) at ages past mode could reverse this – case of Japan

Role of T and V(T) (adult death)

Annuities, Life insurance

Longevity bonds

Risk – life cycle savings and consumption

Risk – societal pension risk

Optimization without constant environments – economic models

Variance in age of adult death

= Var. betwen groups +

Var. within groups

Var. within groups

AV

0 10 20 30 40 50 600

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 10 20 30 40 50 600

0.005

0.01

0.015

0.02

0.025

Can racial differentials explain high U.S. S10?

14

15

16

17

18

19

1970 1975 1980 1985 1990

African Am. S10 White S10 USA S10Canada S10 France S10

S10 in the U.S. by race; compare Canada, France

African Americans & Whites

-10

-8

-6

-4

-2

0

2

0 10 20 30 40 50 60 70 80 90 100 110

African Americans

Whites

Source: Berkeley Mortality Database (NCHS data for 1981)

Lo

g m

ort

alit

yA

ges

at

dea

th

0

0.01

0.02

0.03

0.04

0 10 20 30 40 50 60 70 80 90 100 110

African Americans = 70.7

S = 17.4

Whites = 75.8

S = 15.2

WHAT ELSE MAKES US SPECIAL?

“EXTERNAL CAUSES OF DEATH” (Homicide, suicide, violence, other)

SEPARATE OUT EXTERNAL DEATHS, FIND S10 FOR WHAT’S LEFT

FACT: Education & Income affect Mortality Risk

BUT: Variance within educational/income groups??

USUAL Q: how much Mortality when Educ

HH income and age at death using the NLMS

0.00

0.01

0.02

0.03

15 25 35 45 55 65 75 85 95

Lif

e t

ab

le d

eath

s (

den

sit

ies)

Lowest 20% of HH income = 71.92 = 16.8

Upper 80% of HH income

= 77.42 = 14.4

Source: first year of NLMS, roughly 1981

Education and age at death using the NLMS

0.00

0.01

0.02

0.03

15 25 35 45 55 65 75 85 95

Lif

e t

ab

le d

eath

s (

den

sit

ies)

Less than high school

= 72.92 = 16.7

High school graduate = 78.02 = 14.6

Source: first year of NLMS, roughly 1981

WHAT ABOUT AGGREGATE INEQUALITY?

DOES INCOME INEQUALITY IMPLY

INEQUALITY IN AGE AT DEATH?

Risk factors?

Epidemics of risk factors (obesity, smoking, alcohol)?

Comparative analysis

( )a

Age a

Age a

( )l a

Age a

( ) ( ) ( )a a l a