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Coherent Mortality Forecasting for Less Developed Countries^{ †}

^{1}

^{2}

^{3}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Mortality Models

## 3. The Rotation Algorithm

#### 3.1. Rotating the Age and Period Effects for Mortality Projections

#### 3.2. Determining the Weight Parameter

## 4. Empirical Analysis

#### 4.1. Mortality Data

#### 4.2. Empirical Results

#### 4.2.1. China

#### 4.2.2. Brazil

#### 4.2.3. Nigeria

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Variables of the Lee–Carter model: | |

$log{m}_{x,t}$ | 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 $log{m}_{x,t}$ to changes in ${k}_{t}$ |

${\epsilon}_{x,t}$ | The normal error term in the $log{m}_{x,t}$ process |

${\u03f5}_{t}$ | The normal error term in the ${k}_{t}$ process |

Additional variables of the Li–Lee models | |

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

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

${\nu}_{t}$ | The normal error term in the common factor |

${\alpha}_{1,i}$ | The age-specific sensitivity of log central mortality rate to the population-specific index |

## Appendix A. Optimal Logistic Function

**Table A1.**The AIC and the BIC ratios of the single and the double logistic function of the life expectancy gap for China, Brazil, and Nigeria. The lowest values of AIC and BIC for each country are marked in bold.

AIC | ||

Country | Single logistic | Double logistic |

China | −21.32 | −16.23 |

Brazil | 40.68 | −13.75 |

Nigeria | 154.14 | 140.36 |

BIC | ||

Country | Single logistic | Double logistic |

China | −13.22 | $-2.05$ |

Brazil | 48.79 | 0.43 |

Nigeria | 162.24 | 154.54 |

## Appendix B. The Weight Parameter of the Rotation Algorithm

## Appendix C. The Projected Life Expectancy at Birth

**Figure A4.**The average projected life expectancy of the benchmark countries and China using the rotation algorithm and the independent Lee–Carter model.

**Figure A5.**The average projected life expectancy of the benchmark countries and Brazil using the rotation algorithm and the independent Lee–Carter model.

**Figure A6.**The average projected life expectancy of the benchmark countries and Nigeria using the rotation algorithm and the independent Lee–Carter model.

## Notes

1 | The United Nations defines the less developed countries/regions as all regions of Africa, Asia (except Japan), Latin America and the Caribbean plus Melanesia, Micronesia and Polynesia, and the more developed countries/regions as all regions in Europe, Northern America, Australia, New Zealand, and Japan. For ease of exposition, we will use the word “country” to refer to any country or region. |

2 | The age-specific death rates are calculated using data from the 2017 revision of the World Population Prospects by the United Nations. The ranking is based on population statistics as of 1 June 2018. Source: https://www.census.gov/popclock/print.php?component=counter (accessed on 14 August 2021). |

3 | Besides these two models, there are many other linear extrapolation models consistent with our rotation algorithm, such as Cairns et al. (2006) and Hyndman and Ullah (2007), as well as Li et al. (2021) for a single population and Dowd et al. (2011), Hyndman et al. (2013), Li et al. (2019), Li and Lu (2018, 2019) for multiple populations. For summaries of linear extrapolation models, we refer to Booth et al. (2002), Cairns et al. (2011), and Li and Hardy (2011). |

4 | When convergence is achieved, the improvement rates of the logarithm of the age-specific mortality rates are the same between the modeled country and the benchmark countries. However, this does not necessarily lead to the same improvement rate of the life expectancy, due to Jensen’s inequality. |

5 | The data were collected from the 2017 revision of the World Population Prospects. We excluded 9 less developed countries/regions with life expectancy higher than 80 in 2010–2015, such as Hong Kong, Macao, and Singapore. The benchmark life expectancy was calculated using 10 more developed countries: Germany, Denmark, Finland, France, The Netherlands, Switzerland, Sweden, the UK, the US, and Japan. |

6 | The United Nations uses a simplified version of Equation (14) where ${k}_{1}$ is set to 0. |

7 | Source: http://www.mortality.org/ (accessed on 14 August 2021). |

8 | UN Source: http://www.un.org/en/development/desa/population/ (accessed on 14 August 2021). WHO Source: http://apps.who.int/gho/data/view.main.60340?lang=en (accessed on 14 August 2021). |

9 | Source: http://www.mortality.org/ (accessed on 14 August 2021). |

10 | UN Source: http://www.un.org/en/development/desa/population/ (accessed on 14 August 2021). WHO Source: http://apps.who.int/gho/data/view.main.60340?lang=en (accessed on 14 August 2021). |

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**Figure 1.**The logarithm of the age-specific central death rates for the five most populous countries, China, India, the US, Indonesia, and Brazil, in 1960 (

**left panel**) and 2015 (

**right panel**).

**Figure 2.**Life expectancy gaps against the life expectancy level for the less developed countries (in grey dots), the 50% quantile (dashed line), and the smoothed fit by the double logistic function (solid line).

**Figure 3.**The observed, fitted, and predicted life expectancy gaps for China, Brazil, and Nigeria using their respective optimal logistic function.

**Figure 4.**The historical period effects (

**left panel**) and the age effects (

**right panel**) of China and the benchmark populations.

**Figure 5.**The projected logarithm age-specific death rates (

**left panel**) and the remaining life expectancy at 65 (

**right panel**) of the average of the benchmark countries and China using different models. For the rotation algorithm, the 2.5%, 50%, and 97.5% quantile of the projected ${e}_{65}$ are plotted.

**Figure 6.**The historical period effects (

**left panel**) and the age effects (

**right panel**) of Brazil and the benchmark populations.

**Figure 7.**The projected logarithm age-specific death rates (

**left panel**) and the remaining life expectancy at 65 (

**right panel**) of the average of the benchmark countries and Brazil using different models. For the rotation algorithm, the 2.5%, 50%, and 97.5% quantile of the projected ${e}_{65}$ are plotted.

**Figure 8.**The historical period effects (

**left panel**) and the age effects (

**right panel**) of Nigeria and the benchmark populations.

**Figure 9.**The projected logarithm age-specific death rates (

**left panel**) and the remaining life expectancy at 65 (

**right panel**) of the average of the benchmark countries and Nigeria using different models. For the rotation algorithm, the 2.5%, 50%, and 97.5% quantile of the projected ${e}_{65}$ are plotted.

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**MDPI and ACS Style**

Li, H.; Lu, Y.; Lyu, P.
Coherent Mortality Forecasting for Less Developed Countries. *Risks* **2021**, *9*, 151.
https://doi.org/10.3390/risks9090151

**AMA Style**

Li H, Lu Y, Lyu P.
Coherent Mortality Forecasting for Less Developed Countries. *Risks*. 2021; 9(9):151.
https://doi.org/10.3390/risks9090151

**Chicago/Turabian Style**

Li, Hong, Yang Lu, and Pintao Lyu.
2021. "Coherent Mortality Forecasting for Less Developed Countries" *Risks* 9, no. 9: 151.
https://doi.org/10.3390/risks9090151