Essays on forecasting life expectancy and fiscal sustainability 2

1 | P a g e

ESSAY 1
FORECASTING LIFE EXPECTANCY
1.1. INTRODUCTION
Spurred by the well-established Lee-Carter (LC) approach (1992), there
have been renewed interests in mortality forecasting, see Booth (2006).
Sophisticated models and methods for forecasting life expectancy, such as the LC
model, can have significant impacts and contributions towards macroeconomics
research. Accurate demographic forecasts are important as they yield important
insights to issues relating to economic growth, life-cycle consumption, retirement
annuity, defined contribution plans and fiscal sustainability. See Heijdra and
Romp (2009), Lau (2009) and Alho et al. (2008), among others.

The LC method is based on a linear fit to an additive model of age-specific
death rates, with a fixed-age component and a time-varying component assuming
homoskedastic Gaussian error structure. It is computationally simple, yet
reasonably accurate in capturing trends and variations in mortality rates. Though
significantly deviated from previous methods, the LC model and its extensions
have been applied successfully to forecast life expectancy in developed countries,
including the USA and the G7 countries. See Lee and Miller (2002), Booth et al.
(2002) and Booth (2006).

The LC method and many other mortality models have commonly assumed
the variance of error terms to be constant over time i.e. a homoskedastic error
2 | P a g e

structure is adopted. This assumption may not be satisfactory; since for time series
data, it is often observed that volatility clustering are present, where large changes
are followed by large changes and small changes followed by small changes, an
evidence of heteroskedasticity. In the presence of time-varying volatility in

mortality data, the classical LC model will be inadequate.

Although the importance of time-varying volatility has been accounted for
in various macroeconomic time series data, to the best of our knowledge, there is
no study of volatility clustering in the LC model. This essay aims to fill this
current gap, and seek evidence of such time-variation in volatility using the well-
documented autoregressive conditional heteroskedasticity (ARCH)/generalized
ARCH (GARCH) models introduced respectively by Engle (1982) and Bollerslev
(1986). The countries included in this study are Japan, Australia, Hong Kong,
Taiwan, U.S. and U.K.

In addition to time-varying volatility, the presence of time-varying
correlations between the male and female mortality series of each country are also
of practical value and importance. While extending the LC method with univariate
GARCH model captures volatility clustering for individual series, volatility co-
movements between related series will require the use of Multivariate GARCH
(MGARCH) models.

To the best of our knowledge, there has also been no study of time-varying
correlation dynamics of mortality series using MGARCH-type models. We adopt
3 | P a g e

the use of such models with the LC approach to simultaneously examine the
presence of time-varying volatility and correlations in mortality forecasting. We
will adopt the use of both the Constant Conditional Correlation (CCC-) GARCH
model of Bollerslev (1990) and time-varying Conditional Correlation (VC-)
GARCH model of Tse and Tsui (2002). We then seek to compare the within-
sample forecast performance of the classical LC model with the proposed model
extensions in this paper in forecasting life expectancy at birth and at age 65.

This essay is structured as follows: Section 1.2 offers a literature review.
Section 1.3 describes the LC-GARCH model while section 1.4 summarizes the
data used and presents the estimation results. In section 1.5, we offer an overview
of the MGARCH models, and Section 1.6 describes the LC method extended with
CCC-GARCH and VC-GARCH models. Estimation results are then reported in
section 1.7. In section 1.8, we show that using the LC model with VC-GARCH
structure offers the largest improvement in within samples forecasts of life
expectancy at birth and at age 65. Concluding remarks are offered in the final
section.
4 | P a g e

1.2. LITERATURE REVIEW
While there are many approaches to forecast mortality rates, the LC method
is still one of the leading statistical models in the demographic literature for
mortality trend fitting and life expectancy projections since its development in
1992.

Proposed by Lee and Carter, it was initially used for projecting age-specific
mortality rates in the United States but has since been adopted by many other
countries as the basic mortality model for population projections. See applications
in, for example, Canada (Lee and Nault, 1993), Japan (Wilmoth, 1996) and the
seven most economically developed nations (G7) (Tuljapurkar et al., 2000).

The LC model was designed to examine the long-term patterns in the
natural logarithm of central death rates using a single index of mortality. It
essentially describes the logarithmically transformed age-specific central rate of
death as a sum of an age-specific component that is independent of time, and the
product of a time-varying parameter (also known as the mortality index that
summarizes the general level of mortality) and an additional age-specific
component that represents how rapidly or slowly mortality at each age varies when

the mortality index changes.

The advantages of the parsimonious LC model lie in its simplicity. This is
because once data is fitted to the model and the model parameter values are
estimated, they are held fixed as constants and only the mortality index needs to be
5 | P a g e

forecasted. The LC model also adopts an extrapolative method where forecasts are
carried out using time series methods and historical information. With the use of
logarithms, it also allows mortality rates to decrease exponentially without the need
for restrictions.

Like many other models, the LC model has its set of limitations as well. As
it is based on extrapolation methods, its forecast accuracy fares unfavorably when
historical data fails to hold in the future and/or structural changes occur. It also
does not account for any changes in social economic factors, such as medical
advancement, lifestyle changes etc. Furthermore, the LC model assumes a constant
variance for the residual term; which can potentially be another limitation since it
constrains the model's ability to capture the volatility of series if changes across
periods are far from being constant.

While many extensions to the LC model have likewise assumed a constant
variance for the residual term, the presence of time-variation in volatility has been
detected in mortality series and some studies have questioned the assumption of
using constant variances. Among them, Renshaw & Haberman (2003a, b) used
heteroskedastic Poisson error structures in their mortality forecast, where ordinary
least-squares regression is replaced with Poisson regression for the death counts.
Cossette et al (2007) suggested the use of a Binomial regression model where the
annual number of deaths is assumed to follow a Binomial distribution and the death
probability is expressed as a function of the force of mortality. Delwarde, Denuit

6 | P a g e

and Partrat (2007) also proposed the use of Negative Binomial Distribution to
account for the presence of heterogeneity in mortality.

The use of ARCH and GARCH models (Engle, 1982; Bollerslev, 1986) has
been proven to be capable in capturing the existence of non-constant variances in
many applications. Nonetheless, among these extensions, the use of
ARCH/GARCH models in demography forecasting is limited. While Keilman and
Pham (2004) incorporated ARCH-type structure with the use of ARIMA models in
their forecasts of fertility, mortality and net migration for 18 countries in the
European Economic Area, their study did not attempt to incorporate the use of
ARCH/GARCH models with the LC model. This is our first objective in this
essay.

Univariate ARCH/GARCH models face two restrictions; firstly it does not
accommodate the asymmetric effects of positive and negative shocks and secondly,
it assumes independence between conditional volatilities across different groups.
In financial series, it has been established that volatility in the returns of financial
variables exhibit an asymmetric character where negative shocks contribute more
to volatility than positive shocks of the same magnitude (see for instance, Nelson,
1991 and Glosten, Jagannathan and Runkle, 1993). It is not known if such
asymmetric effects are also present in mortality data and this is the second
objective of this essay which seeks to address the limitation of using univariate
ARCH/GARCH models. Hence, further to extending the LC model with
7 | P a g e

ARCH/GARCH type models, we will also examine the presence of asymmetric
effects in mortality series using the LC model extended with EGARCH.

Other variations of the LC method included the recent works of Girosi and
King (2008), who adopted the Baynesian approach and Markov Chain Monte Carlo
estimation to improve on mortality forecasting. They developed a general
Baynesian hierarchical framework for forecasting different demographic variables
and incorporated exogenously measured covariates as proxies for systematic causes
of death. Their methodology have not only generalized the LC model to an
analysis involving several principal components; it also uses additional information
about regularities along the dimensions of age, sex, country and death causes to
improve on forecasting results. The framework, however, only models a single
population in isolation, and also requires a lot of extensive information which may
not be easily available in many countries.

The use of multivariate GARCH models allows one to account for the
presence of co-movements across related series, which makes it possible to model
separate series jointly and incorporate their interactions. In the financial sector, the
dependence in co-movements across asset returns is important since the co-
movements or covariance of assets in a portfolio provides critical information for
asset pricing. Using MGARCH models allow us to extract such covariance
information and improve the analysis for asset pricing models, hedging, portfolio
selection and Value-at-Risk forecasts. Studies have found the variances of
financial time series to be interacting, see for instance, Cifarelli and Paladino
8 | P a g e

(2005) who studied the linkages in equity markets. MGARCH models have been
used to examine the volatility and correlation transmissions and spillover effects in
contagion studies as well, see Tse and Tsui (2002).

MGARCH models estimate a conditional covariance matrix comprising
time-varying conditional volatilities and correlations. It has been proven in many
financial applications that modeling the dynamics of the covariance matrix using a

multivariate approach yields better results than working with separate univariate
models for each individual series. The importance of accounting for co-
movements using MGARCH models is similarly valid for other markets. In the
area of mortality, the presence of co-movements between the male and female
population is potentially large.

The MGARCH literature include several types of models including VEC
model by Bollerslev, Engle and Wooldridge (1988), Baba-Engle-Kraft-Kroner
(BEKK) model by Engle and Kroner (1995), CCC-GARCH model by Bollerslev
(1990) and time-varying conditional correlation GARCH models (VC-GARCH and
DCC-GARCH models by Tse and Tsui (2002) and Engle (2002) respectively. In
the area of demography, the extension of the LC model with MGARCH models has
not been attempted in the literature and this is the third gap which we seek to
address in this essay. Such an extension will allow us to examine the presence of
co-movements between the mortality series of male and female populations.

9 | P a g e

1.3. EMPIRICAL MODEL
1.3.1. LC Model
The LC Model is widely used in mortality forecasting and life expectancy
forecasting. It was introduced in 1992 and has been used by the United States
Social Security Administration, the US Census Bureau, and the United Nations.

The structure proposed by Lee and Carter (1992) is as follows:












1.1


 




1.2

where 

is the central death rate for an individual aged at time t; 

is the
additive age-specific constant displaying the average shape across age of the
mortality schedule; 

describes the relative sensitivity of the mortality at age to
changes in the general level; 

is the error term which reflects any age-specific
historical influences not captured in the model and 


is a time-specific index of the
general level of mortality.



is modeled to follow the random walk model with drift
1
as described in
       

is assumed to be independent, identically
distributed Normal




 For different values of 

, the fitted model defines a set
of central death rates, which can then be used to derive a life table conveniently.

1
Lee and Carter (1992) claimed that other ARIMA models might be preferable for different set of
data, but the random walk model with drift is mostly used in practice.
10 | P a g e

When 

changes linearly with time, mortality at each age changes at a constant
exponential rate accompanied by the age-specific constants varying by age.

As equation (1.1) is over-parameterized, 

is taken to be the arithmetic
mean of 

over time while the sums of 

and 

are normalized to one and
zero respectively, so as to ensure a unique solution. Furthermore, given that all the
parameters on the right-hand side of equation (1.1) are unobservable, the LC model
requires a two-stage estimation procedure. In stage one, the Singular Value
Decomposition (SVD) estimation procedure is applied to the matrix of 




 to obtain estimates of 

and

. Following stage one, the time series of

in
equation (1.2) is re-estimated in stage two by solving the following















1.3

where 

is the total number of deaths in time t, and 

is the exposure to risk of
an individual aged x in time t. The re-estimation of 

in stage two ensures that the
mortality schedules fitted will reconcile with the actual total number of deaths and
the population age distributions.


11 | P a g e

1.3.2. Introduction to ARCH/GARCH models
The importance of accounting for ARCH/GARCH effects has been
established in many empirical studies. Although volatility is not directly

observable, it has some commonly observed characteristics and facts. The various
stylized facts confirmed by numerous studies include volatility clustering and
persistence, fat-tailed behavior, mean reversion and leverage effect. The basic
ARCH/GARCH model (Engle, 1982; Bollerslev, 1986) and its subsequent
extensions have proven to be capable of capturing the existence of time-varying
volatility and account for these characteristics.

In the basic ARCH/GARCH model, the conditional variance is expressed as
a weighted average of its long run mean value, as well as shocks and the level of
conditional variance in previous periods. A GARCH process is an infinite-order
ARCH process with a lag structure imposed on the coefficients.

A low order GARCH model is preferred over a higher order ARCH model
since the former is more parsimonious and allow for a slower, exponential decay
rate. A lower order  process is generally preferred to higher order
models since the latter may have many local maxima and minima. The most
popular and widely used model within the GARCH family thus far is still
the  model, where three parameters in the conditional variance
equation are often adequate and good enough for model fit and short term
forecasts. Many empirical studies have supported the superiority of the
12 | P a g e

 model. According to Hansen and Lunde (2001), it is difficult to find
a volatility model that outperforms the simple 





The potential of ARCH/GARCH models has been proven in financial

applications, but it can be applied in any markets which require the forecasting of
volatility. Since the introduction of ARCH/GARCH models, the GARCH family
of models have been extensively used in many different areas to model volatility
and forecasting, ranging from the earlier applications in asset pricing, risk
management, GDP growth rates, interest rates, stock index returns, option pricing,
foreign exchange rates, to the more recent applications in agriculture, internet
traffic, etc. See such applications in Christoffersen and Diehold (2000), Yang,
Haigh and Leatham (2001) and Zhou, He and Sun (2006).



13 | P a g e

1.3.3. LC-GARCH Model Extension
We propose two model modifications and enhancements in this section.
First of all, the LC model assumes a random walk with drift specification in
equation (1.2), which may overly restrictive. Hence, instead of modeling 

using a
random walk with drift process, we allow 

to follow a more general formulation.
We adopt a stationary autoregressive process of order p, AR(p), where we allow
the data to determine the order of p and values of 

in the conditional mean
equation of (1.4).

Next, rather than a homoskedastic error structure, we assume that the
residual term 


follows the well-documented  structure pioneered by
Bollerslev (1986). Conditional on

, i.e. the information set available at time t-
1, the conditional variance equation of 

is specified below.

Conditional mean equation











1.4

Conditional variance equation





















1.5

14 | P a g e

where 





and 

is a sequence of identically and independently distributed
normal random variables with zero mean and unit variance.

The GARCH specification in (1.5) implies that the best predictor of

conditional variance in the next period is a weighted average of its long-run mean
value, the variance predicted for this period, as well as all shocks arriving in this
period that is captured by the most recent squared residual. Since last period's
shocks enter the conditional-variance equation (1.5) in squared terms, shocks of the
same magnitude will produce the same level of volatility irrespective of their sign,
which means that negative and positive return shocks from the previous period will
contribute equally to current period's volatility.

Constraints 





 and sum of all 

 

are
imposed to ensure a positive conditional variance and a weakly stationary
unconditional variance. In particular, 

measures the extent to which a shock in
this period feeds into next period's volatility, and the sum of parameters 




measures the persistence of any volatility shocks. The greater it is, the greater is
the persistence of shocks to volatility and the longer it takes to converge to the

unconditional variance.

Over the long run, forecasts of conditional variance will exhibit mean
reversion towards its unconditional variance. Forecast made m periods ahead will
converge to the unconditional variance 







so long as 

 


15 | P a g e

The value 

 

 determines the speed of convergence to w; the larger the
value, the longer the convergence period.

As a check against the adequacy of the LC-GARCH model, we apply the
LjungBox test on the standardized residuals and squared standardized residuals to
check if the autocorrelations of the series are statistically different from zero. The
test statistic is

 







1.6

where n is the sample size, 


is the sample autocorrelation at lag k, and h is the
number of lags being tested. For a    he critical region for
rejecting the hypothesis of randomness is


, where 


  -
quantile of the chi-square distribution with h degrees of freedom. It is necessary to
perform the Ljung-Box tests on the standardized residuals and squared
standardized residuals because the presence of any autocorrelation will violate the
ordinary least squares (OLS) assumption that the residual terms are uncorrelated.
While this does not bias the OLS coefficient estimates, this implies that the
standardized residuals tend to be underestimated (and the t-statistics overestimated)
when the autocorrelations of the residual terms at low lags are positive.

16 | P a g e

1.4. DATA AND ESTIMATION RESULTS OF LC-GARCH MODELS
1.4.1. Data
The data set used for this study consists of annual mortality tables by sex
for Australia (1945-2007), Japan (1955-2007), Hong Kong (1971-2008), Taiwan
(1970-2008), U.K. (1945-2006) and U.S. (1941-2006). There are a total of 111 (0,
1, 2 110+) ages for each of the countries, with the exception of Hong Kong where
only 101 ages (0, 1, 2 100+) are available. And except for Hong Kong which is
obtained from the Census and Statistics Department of Hong Kong
2
, the rest are
culled from the Human Mortality Database
3
available at the University of
California, Berkeley.

Based on the published life tables, we follow the first stage of the LC
method to estimate 



and 

by country and by sex, using the method of
Singular Value Decomposition (SVD). During the estimation procedure,
additional conditions have to be imposed to secure a unique solution since SVD
often yields multiple solutions. In particular, constraints




and 




are imposed, with vector 

computed as the average of 

over time.

In the second stage, we re-estimate the values of 

by matching the fitted
life expectancy at birth with actual life expectancy. Details of such estimation
procedures are available in Lee and Carter (1992) and Lee and Miller (2002).
Finally, extrapolation of 

using the fitted models enables us to obtain forecasts of

2
See
3
See
17 | P a g e

future mortality rates, which can then be used to compute forecasts of life

expectancy at various ages.

1.4.2. Descriptive Statistics
Table 1.1 first exhibits some descriptive statistics and unit root tests for the
time-varying mortality index, 

by country and by sex. Under standard normal
distributions, the skewness and kurtosis of any series should be 0 and 3
respectively. The descriptive statistics revealed that the sample skewness is
positive for Japan, Hong Kong males and females of U.S. and U.K. but negative
for the rest.

The plots of 

in Figure 1.1 suggest its non-stationary behavior. We
further apply the Augmented Dickey-Fuller (ADF) (Dickey & Fuller, 1981) and
Phillips-Perron (PP) (Phillips & Peron, 1988) unit root tests on 

and the results in
Table 1.1 confirm that 

are non-stationary at the 5% level of significance for all
the countries. As 

are I(1) series, we obtained its first difference, denoted as 

,
to fulfill stationary conditions.

18 | P a g e

Table 1.1. Descriptive Statistics and Unit Root Tests for Time-Varying Mortality
Index
Country
Japan
Australia
Hong Kong
Gender
F
M
F
M
F
M
Descriptive
Statistics
Mean
Median
Maximum
Minimum
Std deviation
Skewness
Kurtosis

JB test

20.5
19.1
108.9
-64.2
52.7
0.04
1.8

3.4



-22.7
-30.7
69.2
-88.1
44.6
0.5
2.2

3.4


25.9
39.7
79.6
-53.5

38.0
-0.6
2.2

5.6


40.1
55.1
77.9
-31.1
32.4
-0.6
2.0

6.8


-16.1
-11.4
48.9
-79.3
39.0
-0.09
1.8

2.2


8.1

8.2
60.0
-37.3
29.9
0.08
1.8

2.4
Unit root test
Augmented DF
PP

0.3
-4.7

-2.9
-1.8

1.0
-0.9

2.2
-3.0

-0.5
-3.3

-0.4
-4.6

Country
Taiwan
USA
UK
Gender
F
M
F
M
F
M
Descriptive
Statistics
Mean
Median
Maximum
Minimum
Std deviation
Skewness
Kurtosis

JB test



2.9
6.6
40.4
-41.6
22.4

-0.2
2.1

1.8



1.9
1.1
32.7
-28.3
16.0
-0.03
2.4

0.7


-15.8
-14.3
55.7
-67.2
33.2
0.3
2.0

3.7


-6.7

-0.4
36.8
-52.6
25.6
-0.2
1.9

3.7


-6.0
-1.5
64.8
-74.5
36.0
0.02
2.3

1.4


16.1
25.8
60.4
-50.2
29.7
-0.6
2.3

5.1

Unit root test
Augmented
DF
PP


0.6
-1.5

-0.6
-2.1

-2.3
-3.1

-0.8
-2.0

-1.1
-2.1

4.0
-1.0
Notes:
The Augmented ADF test refers to the case where the regression equation only contains
the intercept.
For the PP test, the regression equation included both the intercept and time trend.


19 | P a g e

Figure 1.1. Plots of Time-Varying Mortality Index








-100
-50
0
50
100
150
1955
1959
1963
1967
1971
1975
1979
1983
1987
1991
1995
1999
2003
2007

Japan Females
-100
-50
0
50
100
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Japan Males
-60
-10
40
90
1945
1950
1955
1960
1965
1970
1975
1980

1985
1990
1995
2000
2005
Australia Females
-50
0
50
100
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Australia Males
-100
-50
0
50
100
1971

1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007
Hong Kong Females
-50
0
50
100
1971
1974
1977
1980
1983
1986
1989
1992
1995
1998
2001
2004
2007

Hong Kong Males
-50
0
50
1970
1973
1976
1979
1982
1985
1988
1991
1994
1997
2000
2003
2006
Taiwan Females
-40
-20
0
20
40
1970
1973
1976
1979
1982
1985
1988

1991
1994
1997
2000
2003
2006
Taiwan Males
20 | P a g e







The descriptive statistics for 

by country and sex is given in Table 1.2.
The sample skewness is negative for Japan males and females in Hong Kong, USA
and UK. Except for Japan males, the sample kurtosis for all the remaining series
exceed 3 and the Jarque-Bera (JB) statistic fails to support normality at 5%
significance level for Japan females, USA, Taiwan and UK. Other than the visual
plots of 

revealing their stationary behavior, the ADF and PP unit root tests
performed on 

also rejected the null hypothesis of non-stationary at 5% level of
significance for all countries. Given that 


are stationary, we will be replacing


in equation (1.4) for all the countries with the use of 

.

-80
-30
20
70
1941
1946
1951
1956
1961
1966
1971
1976
1981
1986
1991
1996
2001
2006
USA Females
-60
-10
40
90

1941
1946
1951
1956
1961
1966
1971
1976
1981
1986
1991
1996
2001
2006
USA Males
-100
-50
0
50
100
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990

1995
2000
2005
UK Females
-100
-50
0
50
100
1945
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
UK Males
21 | P a g e

Equation (1.4) is hence re-written as



 











1.7



Table 1.2. Descriptive Statistics & Unit Root Tests for Time-Varying Mortality
Index After First Differencing
Country
Japan
Australia
Hong Kong
Gender
F
M
F
M
F
M
Descriptive
Statistics
Mean

Median
Maximum
Minimum
Std deviation
Skewness
Kurtosis

JB test



-2.95
-3.44
11.9
-9.21
3.60
1.63
7.86

74.2



-2.98
-3.19
1.91
-10.4
2.91
-0.55
2.98

2.60



-2.13
-3.06
9.71
-9.57
4.14
0.64
3.06

4.22


-1.41
-1.36
8.60
-7.95
3.70
0.58
3.06

3.47



-3.44
-2.92

2.54
-14.8
4.21
-0.73
3.21

3.39



-2.39
-3.13
9.95
-11.6
4.22
0.67
4.02

4.38
Unit root test
Augmented
DF
PP


-5.95
-5.92

-6.31
-7.05

-9.47
-9.74

-7.39
-8.82

-6.24
-15.3

-7.61
-8.72

Country
Taiwan
USA
UK
Gender
F
M
F
M
F
M
Descriptive
Statistics
Mean
Median
Maximum
Minimum

Std deviation
Skewness
Kurtosis

JB test



-2.16
-2.36
5.66
-7.76
2.44
0.47
4.70

6.00


-1.61
-1.88
7.59
-7.61
2.74
1.04
5.36

15.6

-1.89
-1.88
5.63
-10.2
2.48
-0.24
5.20

13.7


-1.25
-1.55
3.77
-4.74
2.01
0.77
3.23

6.49



-2.24
-2.02
3.60
-14.1
3.19
-0.81
4.80

15.0


-1.64
-1.81
10.7
-8.66
3.04
0.90
6.54

40.0
22 | P a g e

Unit root test
Augmented
DF
PP


-6.20
-6.31


-5.42
-5.34


-10.4

-10.3

-6.87
-6.90

-10.8
-10.7

-8.90
-14.5


Figure 1.2 presents the plots of the mortality indexes after first differencing.
We observed across the countries that the amplitude of the series tend to change
over time and the phenomenon of volatility clustering (large changes followed by
large changes and small changes followed by small changes) is evident.

Figure 1.2. Plots of Time-Varying Mortality Index After First Differencing

Japan - Females
-10
-5
0
5
10
1956 1966 1976 1986 1996 2006
Japan-Males
-11
-9
-7
-5
-3
-1
1
3
1956 1966 1976 1986 1996 2006
Australia-Females
-10
-8
-6
-4
-2
0
2
4

6
8
10
1946 1956 1966 1976 1986 1996 2006
Australia-Males
-10
-8
-6
-4
-2
0
2
4
6
8
10
1946 1956 1966 1976 1986 1996 2006
23 | P a g e

Hong Kong-Females
-16
-14
-12
-10
-8
-6
-4
-2
0
2
1972 1982 1992 2002
Hong Kong-Males
-12
-10
-8

-6
-4
-2
0
2
4
6
8
10
12
1972 1982 1992 2002
Taiwan-Females
-8
-6
-4
-2
0
2
4
6
1971 1981 1991 2001
Taiwan-Males
-8
-6
-4
-2
0
2
4
6

8
1971 1981 1991 2001
USA-Females
-12
-10
-8
-6
-4
-2
0
2
4
6
1942 1952 1962 1972 1982 1992 2002
USA-Males
-5
-3
-1
1
3
1942 1952 1962 1972 1982 1992 2002
UK-Females
-16
-14
-12
-10
-8
-6
-4
-2

0
2
4
1946 1956 1966 1976 1986 1996 2006
UK-Males
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
1946 1956 1966 1976 1986 1996 2006
24 | P a g e

1.4.3. Estimation Results
The parameters of the conditional mean and conditional variance equations
in (1.7) and (1.5) respectively are estimated jointly using maximum likelihood
technique. Based on log-likelihood values, we find that either an AR(0) or AR(1)
model for the conditional mean equation is adequate among all the countries.

The estimation results of (1.7) for 

contradicted the use of the random
walk with drift model in the LC method. Diebold and Nerlove (1989) have argued

against the use of a random walk model to safeguard from any specification error.
They pointed that the use of a low order AR model can account for any potential
non-captured weakly serial correlation in the series.

Table 1.3 displays the estimation results
4
of LC model with GARCH effect
and the Ljung-Box Q statistics (Ljung & Box, 1978) as diagnostic checks against
the presence of serial correlation in the standardized residuals and squared
standardized residuals.

In the LC model, 

indexes the time variation of mortality rates. Hence, an
increase in 

will raise death rates for each age group, while death rates across
ages will fall if 

declines. Based on equation (1.7), negative values obtained for


is declining. Since 

is defined as the difference of 


and 

, this is equivalent to a fall in mortality index and death rates over time.

4
It is found that the conditional mean equation follows either an AR(0) or AR(1) model. In
addition, the conditional variance equation mostly follows the GARCH (0,1) structure. We skip
reporting estimates of the ARCH parameter as they are statistically insignificant at the 5% level.
25 | P a g e

Due to the use of logarithm in the LC model, the death rates for each age rate will
change at an exponential rate as mortality rate changes. As a result, even if
mortality index declines and become negative, there is no concern that negative
death rates will occur.

The intercept, 

, of the GARCH structure are positive for all the countries
examined, which is consistent with the non-negativity of the conditional variance.
The estimated parameters for 

are also positive and significant at the 5% level of
significance, which supports our case for modeling conditional volatility in
mortality series. Since 

is less than unity for all the samples, the variance is
mean reverting, although different countries require varying timeframes. The size
of 

is largest for Taiwan, indicating that the degree of volatility persistence for
Taiwan is the greatest.

Table 1.3. Estimated Parameters of LC-Univariate GARCH Model

Country
Japan
Australia
Hong Kong
Gender
F
M
F
M
F
M





0


1

-4.00
(0.40)
-0.26
(0.09)
3.76
(1.12)
0.54
(0.27)

-3.35
(0.45)
-0.36
(0.12)
3.74
(1.10)
0.47
(0.24)
-2.19
(0.45)
-0.28
(0.13)
12.74
(2.80)
0.35
(0.18)
-1.57
(0.33)
-0.07
(0.18)
9.51
(2.11)
0.35
(0.16)
-4.11
(0.86)
-0.17
(0.15)
9.53
(2.68)

0.57
(0.27)
-3.03
(0.68)
-0.21
(0.13)
7.81
(2.14)
0.49
(0.25)
Log-
Likelihood

-123.5

-120.8

-244.6

-229.6

-101.5

-98.14
Diagnostic
checks
Q(4)
Q
2
(4)

0.14
3.20


4.49
3.42


2.27
1.31


14.0
3.69


2.60
3.02


3.77
1.76