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Statistical Modelling and Projection of Mortality for Australia

S.D. Constantinou

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Faculty of
Social, Human and Mathematical Studies                                                           University of Southampton, Highfield, Southampton,
SO17 1BJ

Abstract
The
purpose of this study is to establish which stochastic mortality models out of
the following three Lee-Carter, Renshaw-Haberman and the Cairns-Blake-Dowd has the best goodness of fit for Australia’s
male and female population. The data used was separate for males and female and
had details of Australians aged between 55 and 90 for the years 1921 to 2014. In
addition, the study will investigate which mortality model has the most
accurate forecasting ability. This will
mainly be done using StMoMo, 1 a package in R, the package is
capable of analysing Australia’s data and produce a range of graphs indicating
whether the morality model performs well in these areas. The finding of this
study highlight that the Renshaw-Haberman produces the best goodness of fit against
this data. However, for the models forecasting ability the results were not as
clear cut as no model outperformed the other two.

1. Introduction

 

1.1  Objective

In recent years, researchers have observed a decrease in
mortality rates with the passage of time. Trying to capture and represent this
observed decrease, various authors have proposed statistically based mortality
models, which they then use to model and forecast the improvements to mortality
in the future. Predicting the mortality rates of the future is essential, as
the changes have already had social and economic implications, and will
continue to do so. Therefore, having a full understanding of the best
ways to model and forecast mortality is of great importance.

 

The aim is to investigate the applicability of the Lee-Carter, Renshaw-Haberman and the Cairns-Blake-Dowd mortality
models to Australia’s mortality data for males and females aged between 55 and
90 from the years range of 1921 to 2014. The data for males and females was retrieved
separately from the Australia section of the Human Mortality Database 2.
A comparison on each model’s capability to capture the improvements in mortality will be completed, this will be done
by fitting each model to the Australian data, then evaluating its goodness of
fit using statistical methods. When comparing how multiple models perform in
relation to each other there are restrictions to the comparison as each model
has different conditions where they work best.  Furthermore, an analysis on each model’s
ability to predict the future distribution of mortality for Australia will be completed.
This will be assessed by back testing, meaning a forecast will be produced from
a time point in the past on data that is already available, then statistical
methods will be used to determine how accurate the forecast was.   confidence intervals on the future projected
distribution.

 

1.2 Background

Australia’s reductions in mortality have occurred due to a vast range of
factors, such as healthcare advances and increases in education. Similarly,
countries of similar affluence and development have experienced comparable
changes to reduce mortality rates. The key factors to these reductions are improvements
to medical care, mainly due to new technological advances, as well as improved
understanding of treatment, hence increasing the survival rates for illness and
disease. Specific examples that have a sizable impact include: the elderly
surviving cardiovascular
disease 3 and the prevention of death caused by premature birth
3 and pneumonia in infants. As previously mentioned, education also
plays an important role in mortality. The reasoning behind this is that an
increased level of education often leads to financial security, allowing the
person to have a more affluent lifestyle and to invest into preventive measures
such as vaccination.

The consequences of the
decreasing mortality rate have meant that there has been a surge in life
expectancy. This has led to the social impact of an ageing population; many other
countries of an equivalent wealth and development as Australia have also
experienced this with their own population demographics. The significance of
Australia’s ageing population is reinforced by the statistic that in 1968
Australia’s over 65’s made up 8% of its population, 4 compared to
the 15% recorded in 2016 4. The transformation in the demographics
has caused implications on Australia’s economy, as the amount of money that the government and companies spend on
pensions has increased dramatically. The Australian government has already
introduced the idea of a policy 4 that tackles this spending
increase, raising the retirement age from the current age of 65 4.
Thus, reinforcing that the issue is significant and demonstrates the need for
heavy research into modelling and forecasting mortality in Australia. Benefits
from finding a mortality model that produces accurate forecasts for mortality
would aid businesses offering staff pensions, as well as the Australian
government. The reason for this is it will allow the cost of providing pensions
in the future to be more accurately assessed as improvements to the
understanding of longevity will be built into calculations, and
thus would enable more precise estimation of the contributions required to fund
such future pensions.

2. Statistical Mortality Models

2.1 Lee-Carter

The first model that I will be evaluating is the
1992 Lee-Carter model 5, this model is the most well-known
mortality model. Lee-Carter is a two-factor model 6; the factors
it considers when modelling and forecasting are age and time/period 6.
The fact that the model added this second factor of time/period meant that the
Lee-Carter could be used to spot age patterns 6, which previously
was a disadvantage of one-factor models. The model structure is as follows:

 

where is the central mortality rate at age  in year 6. The  term is average
log-mortality at age . The  and the  terms are estimators, originally
formed from Singular Value Decomposition 6 which is applied to find the least square solution. Nowadays the  and the  terms are calculated using a GLM approach 7.  The term measures the response at age  to change in the overall level of
mortality over time 6. Whilst,  represents the overall level of
mortality in year,  is the error term at each age  time  6. For the Lee-Carter model it
is also assumed that is a random walk with drift 8
where,

The term monitors trends and is known
as the drift parameter 8, whereas is new error term 8. In addition, there is two constraints on
the  and  terms that help ensure the model uniqueness; the
constraints are such that the  and 6. The
Lee-Carter model has been used by many others since its early release to
develop many other mortality models.

 

2.2 Renshaw-Haberman

The 2006 Renshaw-Haberman
model 9 is an extension of the Lee-Carter model. The model differs
from Lee-Carter as it includes the cohort effect 6 as an additional factor, and thus making it a three-factor model. The
cohort effect determines whether there is a relationship between the date of
birth and the mortality rate. This is useful as it can the help to recognise
the associated health risk that people born within a given date range may
encounter. This advantage comes with a price as the third factor makes the
model more complex. The Renshaw-Haberman model structure is as follows:

 

where the term has the same meaning as the  in the Lee-Carter model 8, and is the average mortality at each age . The term is
the age interaction with time at each age  and is the
age interaction with the cohort at each age . It has
been decided that due to the complexity of having  varying in R that the  term will be treated as a constant with a
value of 1. The  term accounts for the
period effect whilst  accounts for the cohort effect 8. Finally, is also
the same as Lee-Carter for this model and is the error term value at each age  and year  10. The model
takes the  and  parameters to model two random walks with drifts 8 where:

 

      and       

 

The and  terms monitor trends and is known
as the drift parameter 7, whereas and  are error terms 8. The Renshaw-Haberman model, like the Lee-Carter, has the same two
conditions on and  8. Moreover, there are a further
two conditions on the and  parameters,
such that   and 10. These
conditions were applied to
avoid identification problems 10 whilst not having a negative impact on how
well the model functions.

 

2.3 Cairns-Blake-Dowd         
The 2006 Cairns-Blake-Dowd model, 11 also known as the CBD model was the first of many
Cairns models. The CBD model, is a three-factor model which include the
age-time effect. The addition of the age-time effect 12 means the
model will produce cleaner residual plots 12 and it is also
another layer of protection that prevents important factors being missed or
unexplained. A strength with the model is that the three-factor model creates a
non-trivial correlation structure 12. The non-trivial correlation
structure means the CBD model captures the fact that at the older ages (55
plus) the  function is essentially linear in age. This therefore why the CBD
model is not applicable at the younger ages as this observed feature of the function
being effectively linear in age is not present. The Cairns-Blake-Dowd model specification is as
follows: 

 

 

where  and
the term is the mean age of the considered range
of ages 13.  denotes the effect of the general time and  denotes the age-specific time effect 12. Finally, like the other two models, the
 is the error term. The  and parameters are the time indexes, 13 where:

and.

where and are
drift parameters like with the other models.  and  are normal variables 8. Unlike
the Lee-carter and Renshaw-Haberman, the CBD model doesn’t have any constraints
added to its parameters; the reason for this is that there is no identifiability problem for
this model 10. After the CBD model was released, an extension to
the model was released a few years later that accounted for the cohort effect;
however, in this study I will only be investigating the original model.

3. Data and Software

The data collected from the Human Mortality database was
separate for males and females and had information for Australians aged 55-90
from years 1921-2014. To
retrieve any data from the Human Mortality Database, it is required to sign up
and create an account. To import this data into R, installation of two packages
is necessary: StMoMo and Demography 14. Once installed, the
following code displayed below allows R to access an individual’s Human
Mortality Database account to collect data:

StMomodata

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