Lévy CARMA models for shocks in mortality


Recent literature on mortality modeling suggests to include in the dynamics of mortality rates the effects of time, age, the interaction of these two and a term for possible shocks. In this paper we investigate models that use Legendre polynomials for the inclusion of age and cohort effects. In order to capture the dynamics of the shock term it is suggested to consider continuous autoregressive moving average (CARMA) models due to their flexibility in reproducing different autoregressive profiles of the shock term. In order to validate the proposed model, different life tables are considered. In particular the male life tables for New Zealand, Taiwan and Japan are used for the presentation of in-sample fitting. Empirical analysis suggests that the inclusion of more flexible models such as higher-order CARMA(p,q) models leads to better in-sample fitting.

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  1. 1.

    As observed in Loregian et al. (2012), in literature, several parametrizations for the Variance Gamma are available. In this paper, we use the parametrization considered by Madan and Seneta (1990), but we recall that it is always possible to switch from a parametrization to another through a proper parameter transformation.

  2. 2.

    The data on Human Mortality Database (2014) are collected yearly.

  3. 3.

    This is the same dataset used in Ahmadi and Gaillardetz (2015).

  4. 4.

    This is the null hypothesis.

  5. 5.

    This is the alternative hypothesis.

  6. 6.

    CARMA-BM(2,1) and CARMA-BM(2,0) are two nested models and therefore the LR-test is used.


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Proof of Proposition 1

Proof of Proposition 1


Starting from the explicit solution of the state-space process \(X_t\) in (7), we write the CARMA(p,q) model \(Y_t\) as follows:

$$\begin{aligned} Y_{t}=\mathbf {b}^{\intercal }e^{A\left( t-s\right) }X_{s}+\int _{s}^{t}\mathbf {b}^{\intercal }e^{A\left( t-u\right) }\mathbf {e}\text{ d }Z_{u},\ \ \forall t>s. \end{aligned}$$

As a first step, we observe that the quantity \(g\left( t-u\right) :=\mathbf {b}^{\intercal }e^{A\left( t-u\right) }\mathbf {e}\mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \) can be written as follows:

$$\begin{aligned} g\left( t-u\right) =\mathbf {b}^{\intercal }Re^{\Lambda (t-u)}R^{-1}\mathbf {e}\mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \end{aligned}$$

where \(\Lambda \) is a diagonal matrix whose elements are the eigenvalues of matrix A and R is the matrix of right eigenvectors of A, i.e.,

$$\begin{aligned} R= & {} \left[ \lambda _{j}^{i-1}\right] _{i,j=1}^{p}\\ \Lambda= & {} {\left\{ \begin{array}{ll} \lambda _{j} &{} j=i\\ 0 &{} j\ne i \end{array}\right. }\ i,j=1,\ldots ,p. \end{aligned}$$

As in Brockwell et al. (2011), we have:

$$\begin{aligned} R^{-1}\mathbf {e}= & {} \left[ \frac{1}{a^{\prime }\left( \lambda _{1}\right) },\frac{1}{a^{\prime }\left( \lambda _{2}\right) },\ldots ,\frac{1}{a^{\prime }\left( \lambda _{p}\right) }\right] ^{T} \\ Re^{\Lambda (t-u)}= & {} \left[ \begin{array}{cccc} e^{\lambda _{1}} &{}\quad e^{\lambda _{2}} &{}\quad \ldots &{}\quad e^{\lambda _{p}}\\ \lambda _{1}e^{\lambda _{1}} &{}\quad \lambda _{2}e^{\lambda _{2}} &{}\quad \ldots &{}\quad \lambda _{p}e^{\lambda _{p}}\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \lambda _{1}^{p-1}e^{\lambda _{1}} &{}\quad \lambda _{2}^{p-1}e^{\lambda _{2}} &{}\quad \ldots &{}\quad \lambda _{p}^{p-1}e^{\lambda _{p}} \end{array}\right] . \end{aligned}$$

The kernel \(g\left( t-u\right) \) in (20) can be written as:

$$\begin{aligned} g\left( t-u\right)= & {} \mathbf {b}^{\intercal }\left[ \begin{array}{cccc} e^{\lambda _{1}\left( t-u\right) } &{} e^{\lambda _{2}\left( t-u\right) } &{} \ldots &{} e^{\lambda _{p}\left( t-u\right) }\\ \lambda _{1}e^{\lambda _{1}\left( t-u\right) } &{}\quad \lambda _{2}e^{\lambda _{2}\left( t-u\right) } &{}\quad \ldots &{}\quad \lambda _{p}e^{\lambda _{p}\left( t-u\right) }\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \lambda _{1}^{p-1}e^{\lambda _{1}\left( t-u\right) } &{}\quad \lambda _{2}^{p-1}e^{\lambda _{2}\left( t-u\right) } &{}\quad \ldots &{}\quad \lambda _{p}^{p-1}e^{\lambda _{p}\left( t-u\right) } \end{array}\right] \left[ \begin{array}{c} \frac{1}{a^{\prime }\left( \lambda _{1}\right) }\\ \frac{1}{a^{\prime }\left( \lambda _{2}\right) }\\ \vdots \\ \frac{1}{a^{\prime }\left( \lambda _{p}\right) } \end{array}\right] \mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \\= & {} \mathbf {b}^{\intercal }\left[ \begin{array}{c} \underset{i=1}{\overset{p}{\sum }}\frac{1}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }\\ \underset{i=1}{\overset{p}{\sum }}\frac{\lambda _{i}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }\\ \vdots \\ \underset{i=1}{\overset{p}{\sum }}\frac{\lambda _{i}^{p-1}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) } \end{array}\right] \mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \\= & {} \left( \underset{i=1}{\overset{p}{\sum }}\frac{b_{0}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }+\underset{i=1}{\overset{p}{\sum }}\frac{b_{1}\lambda _{i}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }+\cdots +\underset{i=1}{\overset{p}{\sum }}\frac{b_{p-1}\lambda _{i}^{p-1}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }\right) \\&\times \mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \\= & {} \left( \underset{i=1}{\overset{p}{\sum }}\frac{b_{0}+b_{1}\lambda _{i}+\cdots +b_{p-1}\lambda _{i}^{p-1}}{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }\right) \mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) \\= & {} \underset{i=1}{\overset{p}{\sum }}\left[ \frac{b\left( \lambda _{i}\right) }{a^{\prime }\left( \lambda _{i}\right) }e^{\lambda _{i}\left( t-u\right) }\right] \mathrm {1}_{\left\{ s\le u\le t\right\} }\left( u\right) . \end{aligned}$$

The process \(Y_{t}\) in (5) becomes the result in (8). \(\square \)

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Hitaj, A., Mercuri, L. & Rroji, E. Lévy CARMA models for shocks in mortality. Decisions Econ Finan 42, 205–227 (2019). https://doi.org/10.1007/s10203-019-00248-9

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  • Force of mortality
  • CARMA(p
  • q) model
  • Lévy process

JEL Classification

  • C02
  • C53
  • G22