The relationship between longevity and lifespan variation

Abstract

This paper contributes to the current discussion on longevity by investigating the long-term dynamics of life expectancy, and lifespan variation. The analysis provides the expectiles of the life expectancy distribution, relating them to the values of lifespan variation measured by Keyfitz’s index. This allows us to study both the location and the spread of the life expectancy over time in light of the mortality compression process. Using cointegration analysis, we also explore the long-term dynamics between the two longevity measures in some selected developed countries. Overall, we found evidence of a negative relationship between longevity and lifespan variation, thus confirming previous studies. While, by detecting untracked behaviors taking into account the country-specific contribution over time, this study suggests that the strength of this relationship varies across life expectancy distribution’ sections in which the country is located (i.e. specifically expectile), and how fast the country converges towards best performances.

Appendix

According to Newey and Powell (1987), the expectiles of a distribution can be theoretically determined. We denote $e_p$ the theoretical expectile as a function of an asymmetry parameter p with $p\in (0,1)$. Therefore, we can calculate a set of expectiles by varying the asymmetry in the interval (0,1). Given a probability density function f(x), with distribution function F(x) and partial moment function G(x), respectively denoted by:

$$\begin{aligned} F(x)= & {} \int _{-\infty }^{x} f(u)du \end{aligned}$$

(6)

$$\begin{aligned} G(x)= & {} \int _{-\infty }^{x} uf(u)du, \end{aligned}$$

(7)

The theoretical expectile $e_p$ can be found minimizing the following expression (for continuous distributions).

$$\begin{aligned} \min _{e_p} \left( (1-p)\int _{-\infty }^{e_p} (u-e_p)^2f(u)du+p\int _{e_p}^{\infty }(u-e_p)^2f(u)du \right) \end{aligned}$$

(8)

As previously described, expectile regression works similarly to the quantile regression but uses the LAWS model. In fact, the theoretical quantile $q_p$ can be found minimizing the following expression (for continuous distributions):

$$\begin{aligned} \min _{q_p} \left( (1-p)\int _{-\infty }^{q_p} | u-q_p | f(u)du+p\int _{q_p}^{\infty } | u-q_p | f(u)du \right) \end{aligned}$$

The main difference between the above expression and Eq. 8 lies in the definition of the errors considered in the objective function: L2 norm for expectile versus L1 norm for quantile regression. After some algebra and using Eqs. 6 and 7 in the previous equation, we obtain the following solution:

$$\begin{aligned} e_p = \frac{(1-p)G(e_p)+p(m-G(e_p))}{(1-p)F(e_p)+p(1-F(e_p))} \end{aligned}$$

(9)

Where m is the mean of the distribution function F(x) and it is equal to $G(\infty )$. Solving the previous equation for p we obtain the following expression:

$$\begin{aligned} p = \frac{G(e_p)-e_pF(e_p)}{2(G(e_p)-e_pF(e_p))+(e_p-m))} \end{aligned}$$

(10)

After inversion, Eq. 10 allows to obtain the theoretical expectile $e_p$ for a given p.

Schnabel and Eilers (2009a) suggest to compare sample and theoretical distribution through the expectile-expectile plot (E-E plot), similarly to the well-known Q-Q plot, for a number of values of the asymmetry parameter p.