# Mortality Forecasting with an Age-Coherent Sparse VAR Model

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## Abstract

**:**

## 1. Introduction

## 2. The Factor-Based Model

## 3. The Vector-Autogression-Based Models

#### 3.1. The Sparse VAR Model

#### 3.2. The Coherent Sparse VAR Model

#### The Multi-Population Extension

## 4. Empirical Analysis

#### 4.1. Long-Term Analysis

#### 4.2. Out-of-Sample Forecast

#### 4.2.1. Robustness Analysis

#### 4.2.2. The Two-Population Extension

#### 4.2.3. Forecasting of Life Expectancy of Age 0

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LC | Lee–Carter model |

VAR | Vector Autoregression model |

CBD | Cairns–Blake–Dowd model |

LL | Li–Lee model |

STAR | Spatial-temporal Autoregression model |

SVAR | Sparse VAR model |

CSVAR | Coherent Sparse VAR model |

LASSO | Least Absolute Shrinkage and Selection Operator |

ENET | Elastic-net |

RMSFE | Root of Mean Squared Forecasting Error |

Variables of single population models: | |

${y}_{x,t}$ | Log central mortality rate at age x in year t |

${a}_{x}$ | The average mortality level at each age x |

${k}_{t}$ | The mortality index at time t |

${b}_{x}$ | The age-specific sensitivity of ${y}_{x,t}$ to changes in ${k}_{t}$ |

${\epsilon}_{x,t}$ | The normal error term |

$\Delta {\mathbf{Y}}_{\mathbf{t}}$ | The vector of differenced log central mortality rate |

$\mathbf{M}$ | Intercept vector of the VAR-type models |

$\mathbf{B}$ | Coefficients of $\Delta {\mathbf{Y}}_{\mathbf{t}-\mathbf{1}}$ in the VAR-type models |

${\hat{m}}_{x,h}$ | Forecast intercept term in the CSVAR model for age x at step h |

${\delta}_{h}\left({d}_{x}\right)$ | The hyperbolic parameter associated with ${\hat{m}}_{x,h}$ |

$\lambda $ | The ENET penalty |

Additional variables of joint population models: | |

${B}_{x}$ | Age effect of the common factor |

${K}_{t}$ | Period effect of the common factor |

## Appendix A. Additional Tables

Single Models | SVAR | CSVAR | ||||||

$\mathit{\lambda}$ | $\mathit{\lambda}$ | ${\mathit{d}}_{1}$ | b | |||||

UK | −11.66 | −9.26 | 0.2974 | 0.2184 | ||||

FR | −11.30 | −6.89 | 0.7426 | 0.0621 | ||||

Join Models | SVAR | CSVAR${}_{\mathbf{1}}$ | CSVAR${}_{\mathbf{2}}$ | |||||

$\mathit{\lambda}$ | $\mathit{\lambda}$ | ${\mathit{d}}_{\mathbf{1},\mathit{j}}$ | ${\mathit{b}}_{\mathit{j}}$ | ${\mathit{\lambda}}_{\mathit{j}}$ | ${\mathit{d}}_{\mathbf{1},\mathit{j}}$ | ${\mathit{b}}_{\mathit{j}}$ | ||

UK | −11.98 | −9.26 | 0.4458 | 0.1663 | −10.75 | 0.2638 | 0.2599 | |

FR | −11.98 | −9.26 | 0.7426 | 0.4789 | −11.16 | 0.3468 | 0.5311 |

_{j}) are reported in logarithms.

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1. | A maximum likelihood method may also be employed to calibrate the parameters (Renshaw and Haberman 2003). |

2. | Note that the choice of the length of test sample (one fifth) is common among existing studies. Adopting other popular alternatives such as the last third, fourth and tenth sample will lead to robust results. |

**Figure 4.**Mean forecast and the 95% prediction intervals vs. actual life expectancy at bitrh: 2001–2100.

${\mathit{R}\mathit{M}\mathit{S}\mathit{E}}_{\mathit{a}\mathit{l}\mathit{l},16}$ | Mean | Std. Dev. | ${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{3}$ | |
---|---|---|---|---|---|

Panel A: UK | |||||

LC | 0.1623 | 0.1454 | 0.0726 | 0.0912 | 0.1935 |

SVAR | 0.1209 | 0.1056 | 0.0592 | 0.0536 | 0.1571 |

CSVAR | 0.1106 | 0.1006 | 0.0463 | 0.0647 | 0.1321 |

Panel B: France | |||||

LC | 0.2159 | 0.1661 | 0.1387 | 0.0548 | 0.2601 |

SVAR | 0.1422 | 0.1210 | 0.0750 | 0.0670 | 0.1594 |

CSVAR | 0.1358 | 0.1143 | 0.0736 | 0.0584 | 0.1530 |

UK | France | |||||
---|---|---|---|---|---|---|

LC | SVAR | CSVAR | LC | SVAR | CSVAR | |

Five-year groups | 0.1674 | 0.1229 | 0.1150 | 0.2240 | 0.1429 | 0.1199 |

1970–2016 | 0.1507 | 0.1360 | 0.1163 | 0.1916 | 0.1428 | 0.1324 |

Smoothed rates | 0.1562 | 0.1069 | 0.0987 | 0.2102 | 0.1285 | 0.1140 |

${\mathit{R}\mathit{M}\mathit{S}\mathit{E}}_{\mathit{a}\mathit{l}\mathit{l},16}$ | Mean | Std. Dev. | ${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{3}$ | |
---|---|---|---|---|---|

Panel A: UK | |||||

LL | 0.1241 | 0.1053 | 0.0660 | 0.0530 | 0.1435 |

SVAR | 0.1207 | 0.1052 | 0.0594 | 0.0512 | 0.1569 |

CSVAR${}_{1}$ | 0.1159 | 0.1019 | 0.0555 | 0.0536 | 0.1354 |

CSVAR${}_{2}$ | 0.1110 | 0.1012 | 0.0459 | 0.0666 | 0.1348 |

Panel B: France | |||||

LL | 0.1559 | 0.1303 | 0.0860 | 0.0630 | 0.1770 |

SVAR | 0.1442 | 0.1220 | 0.0773 | 0.0664 | 0.1600 |

CSVAR${}_{1}$ | 0.1396 | 0.1178 | 0.0754 | 0.0631 | 0.1469 |

CSVAR${}_{2}$ | 0.1046 | 0.0906 | 0.0526 | 0.0521 | 0.1172 |

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**MDPI and ACS Style**

Li, H.; Shi, Y.
Mortality Forecasting with an Age-Coherent Sparse VAR Model. *Risks* **2021**, *9*, 35.
https://doi.org/10.3390/risks9020035

**AMA Style**

Li H, Shi Y.
Mortality Forecasting with an Age-Coherent Sparse VAR Model. *Risks*. 2021; 9(2):35.
https://doi.org/10.3390/risks9020035

**Chicago/Turabian Style**

Li, Hong, and Yanlin Shi.
2021. "Mortality Forecasting with an Age-Coherent Sparse VAR Model" *Risks* 9, no. 2: 35.
https://doi.org/10.3390/risks9020035