CAFÉ JUAN VALDÉZ
2.7 ANÁLISIS DE ACONDICIONAMIENTOS PREVIOS APLICADOS AL PROYECTO:
The evolution of mortality studies has started a few centuries ago. A number of pioneering works were carried out as early in the 15th century. In 1693, a famous
astronomer, Edmond Halley developed the first life table based on applications to life contingencies. Subsequently, in 1725, Abraham De Moivre proposed the first mathematical formula of mortality modelling.
lx =k 1− x 86 , f or 12≤x≤86 (2.1)
where lx is the number of individuals still alive at age x last birthday from an
original pool,l0of individuals, andkis a constant. He assumed that all individuals
must die before the age of 86. For more details, please see Bowers et al. (1997).
A century later, Gompertz (1825) proposed the new mortality law in 1825. Theo- retically, the Gompertz’s idea on the term of f orceof mortality can be accurately expressed in the model. Denoting the force of mortality by µx, Gompertz’s law is
as follows:
µx =αexp (βx) (2.2)
where α and β are positive parameters and x denotes the age3. More discussions
about mortality laws can be found in Bowers et al. (1997). What is more, Heligman and Pollard (1980) proposed a class of formulae which aimed to represent the age pattern of mortality over the whole span of life4.
Over 160 years later and with advances in computing techniques, Lee and Carter (1992) proposed the new mortality model that has been widely accepted and re- ferred to in mortality studies. They remarkably proposed a simple stochastic model reflecting the reduction of the annual log age-specific death rates through a time-dependent index, ln(mx,t) =b1x+b 2 xk 1 t +x,t (2.3)
wheremx,t is the central death rate, the ratio between the number of people aged
x who died in year t, and the exposure to risk of the average population aged x
in year t. Factor b1
x describes the average age-specific mortality, that ensures the
basic shape of the mortality curve over ages is in line with historical observations,
3By focusing on the old ages, Gompertz (1825) has weakened the model by not representing
the mortality over the whole lifetime span.
4Heligman and Pollard (1980) proposed a sum of three terms representing different compo-
nents of mortality:
m(x) =A(x+B)C +DeE(lnx−lnF)2+ GH
x
(1 +GHx),
factor k1
t represents the changes in the mortality level, whilst factor b2x describes
the decline in mortality at age x. It explains how rapidly/slowly mortality rates decline in response to kt1. The error term at age x and time t, is x,t. The LC
model describes the smooth and gradual decline of mortality rates over time. This has become debatable as the variance of mortality rates exponentially grows in time. Following the LC model, a series of extensive studies were conducted by many researchers including Lee and Miller (2001), Booth et al. (2002), Renshaw and Haberman (2006), Hyndman and Ullah (2007), Cairns et al. (2009, 2011), Plat (2009), O’Hare and Li (2012) building on the extrapolation approach and improving the mortality modelling.
To be more precise, Renshaw and Haberman (2006) modified the LC model by adding a cohort effect parameter into the formula:
ln(mx,t) = b1x+b 2 xk 1 t +b 3 xγt−x+x,t (2.4)
where γt−x models the cohort effect. Their model provides a better fit to the
historical data, where the cohort effect was observed in the past in a particular country with the best results for b3x = 1. However, this model suffers from lack of robustness and has a trivial correlation structure (Cairns et al., 2009, 2011). Notwithstanding this criticism, Haberman and Renshaw (2011) then conquer this argument by justifying that issues observed in these studies were a result of the fitting procedure used to obtain parameter estimates. This statement has also been confirmed in a more recent paper by Hunt and Villegas (2015).
Particularly, Cairns et al. (2006b) proposed a two-factor model of mortality:5 Sub-
sequently, in their extended research using data from England and Wales, and United States, they observed that the fitted cohort effect appears to have a trend in the year of birth. This suggested that the cohort effect compensates for a lack of a second age-period effect as it attempts to capture the cohort effect from the data. Hence, in 2009, Cairns et al. improved the the two-factor model by adding
5Cairns et al. (2006b):
log( qx,t 1−qx,t
) =kt1+ (x−x¯)k2t+x,t,
where qx is the probability that a person agedxdies within the next year (qx,t ≈1−e−mx,t),
¯
xis the mean age in the sample range and (k1
t, kt2) are assumed to be a bivariate random walk
the second age cohort effect. This improvement captures the cohort effect as an additional effect on top of the two-factor (age and period) effects.
Following this, Plat (2009) incorporated the cohort and age-period effects6 to the
LC model. His model provides significant and better results since it implies the importance of younger ages in modelling the mortality experience of a population.
Subsequently, O’Hare and Li (2012) extended the Plat (2009) model by adapting the non-linear profile of mortality at lower ages:
ln(mx,t) =b1x+k 1 t + (¯x−x)k 2 t + (¯x−x) + + ([¯x−x]+)2kt3+γt−x+x,t (2.5) where k2
t factor allows changes in mortality to vary between ages reflecting the
historical observation that improvement rates can differ for different age classes,
k3
t models the effects specific to the lower ages, γt−x models the cohort effect,
(¯x−x)+ = max(¯x−x,0), and ¯x is the average of age considered. Their model provides a better fit for the range of countries considered and shows flexibility to fit the mortality rates of a wider range of ages. What is more important, this model does not lose any of the benefits of the previous stochastic models.
The number of wide-ranging mortality studies has increased significantly due to the rapid increase of longevity and the need to better understand it. That increase in all age-intervals is primarily due to several factors such as socio-economic, med- ical improvements, lifestyle, living environment, climate change and many oth- ers (Granados, 2008; Granados and Ionides, 2011; Hanewald, 2011; French and O’Hare, 2013, 2014; Niu and Melenberg, 2014; Cairns et al., 2016; Seklecka et al., 2017b). Socio-economic factor has been discussed and agreed widely as one of the important factors that affect longevity (Bhargava et al., 2001; Batchvarov and Gakwaya, 2006; Renton et al., 2012; Preston, 2007; Granados, 2008; Granados and Ionides, 2011; Hanewald, 2011). According to the Institute and Faculty of Actuaries (IFoA), as socio-economic status increases, the life expectancy is also increases. Among other things, socio-economic status can affect a person’s ability to access adequate medical care and participate in healthier lifestyle habits like exercising more, smoking less and maintaining a healthy weight.
6Plat (2009) model specification is given by
ln(mx,t) =b1x+k 1 t+ (¯x−x)k 2 t + (¯x−x) +k3 t +γt−x+x,t,
In accessing the growth of economics and identifying long-term trends in a partic- ular country, factors like GDP, unemployment rate, income and wages, consumer price index, currency strength and interest rates are among those indicators that being studied in observing how the economy changes over time. These indicators are also known as lagging indicators, reflect the economy’s historical performance and changes. These indicators are only identifiable after an economic trend or pattern has already been established. Among these indicators, changes in GDP has been widely used by many economists in measuring the economy’s current health in a particular country. When GDP increases, it is a sign the economy is strong. Moreover, GDP is a key determinant to most businesses as to decide and adjust their expenditures on inventory, payroll, and other investments based on GDP output.
This study analyses GDP as one of those influencing factors that affect mortality. This study employs a correlational studies between GDP and mortality. Corre- lational studies is more important and useful than other methods as it can be widely used for many variables especially for variables that cannot be simply ma- nipulated for ethical reasons like human malnutrition. Additionally, as this study involves with variables that definitely cannot be manipulated like birth, sex and age, thus the application of correlational studies is deemed appropriate as the scientific knowledge concerning them must be based on correlation evidence.