# Mortality Forecasting: How Far Back Should We Look in Time?

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## Abstract

**:**

## 1. Introduction

## 2. Models for Comparison

#### 2.1. Notation

- ${d}_{x,t}$ as the observed number of deaths in calender year t aged x.
- ${e}_{x,t}$ as the exposure data that measure the average population in calendar year t aged x.
- ${m}_{x,t}$ as the central mortality rate, which reflects the death probability at age x in the middle of the year. It is calculated by:$${m}_{x,t}=\frac{{d}_{x,t}}{{e}_{x,t}}.$$
- ${q}_{x,t}$ as the initial mortality rate, which is the one-year death probability for a person who is aged exactly x at time t.

#### 2.2. CBD Model and a Local Linear Approach

- ${Y}_{it}=\mathrm{logit}\left({q}_{x,t}\right)$ and ${\u03f5}_{it}={\u03f5}_{x,t}$, where $i=x-a$ denotes age groups.
- ${X}_{i}=$$\left(\begin{array}{c}1\\ x-\overline{x}\end{array}\right).$
- ${\beta}_{t}=$$\left(\begin{array}{c}{\kappa}_{t}^{1}\\ {\kappa}_{t}^{2}\end{array}\right)$ where ${\kappa}_{t}^{1}$ and ${\kappa}_{t}^{2}$ are smooth functions of t.

#### 2.3. 2D LOP Model and 2D KS Model

#### 2.4. A Discussion on the Two Groups of Mortality Models

## 3. Case Study: GB Male Mortality Data from 1950–2016, Ages 50–89

#### 3.1. Data

#### 3.2. Fit Quality and Residual Plots

- The average error ($E1$), which is a measure of overall bias, is calculated as:$$E1=\frac{1}{NT}\sum _{x}\sum _{t}\frac{{\widehat{m}}_{x,t}-{m}_{x,t}}{{m}_{x,t}}.$$
- The absolute average error ($E2$), which measures the absolute size of the deviance, is calculated as:$$E2=\frac{1}{NT}\sum _{x}\sum _{t}\frac{|{\widehat{m}}_{x,t}-{m}_{x,t}|}{{m}_{x,t}}.$$
- The standard deviation of error ($E3$), which is an indicator of large deviance, is calculated as:$$E3=\sqrt{\frac{1}{NT}{\sum}_{x}{\sum}_{t}{\left(\frac{{\widehat{m}}_{x,t}-{m}_{x,t}}{{m}_{x,t}}\right)}^{2}}.$$

#### 3.3. Comparison of Forecasting Performance

#### 3.4. Robustness of Projections

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | For a summary of existing forecasting models, please see Booth and Tickle (2008). Mortality modelling and forecasting: A review of methods. Annals of Actuarial Science 3, 3–43. |

2 | For readers who want to read about the detailed derivation of the formula, please refer to: Dickson et al. (2009). Actuarial Mathematics for Life Contingent Risks. Cambridge University Press, London. |

3 | |

4 | |

5 | We have also considered the mortality experience of the U.S. and Luxembourg for the periods 1950–2016 and 1960–2014, respectively. The results are in line with the findings and conclusions in this paper. These additional results are available upon request. |

**Figure 1.**Residual plots for the (

**a**) CBD MLE model, (

**b**) CBD LLE model, (

**c**) 2D LOP model and (

**d**) 2D KS model based on GB male mortality data from 1950–2016, ages 50–89.

**Figure 2.**Three-year-ahead forecast from 2014–2016 for GB males aged (

**a**) 50, (

**b**) 60, (

**c**) 70 and (

**d**) 80 based on mortality data since 1950.

**Figure 3.**Five-year-ahead forecast from 2012–2016 for GB males aged (

**a**) 50, (

**b**) 60, (

**c**) 70 and (

**d**) 80 based on mortality data since 1950.

**Figure 4.**Ten-year-ahead forecast from 2007–2016 for GB males aged (

**a**) 50, (

**b**) 60, (

**c**) 70 and (

**d**) 80 based on mortality data since 1950.

**Figure 5.**Three-year-ahead forecast from 2014–2016 for GB males aged (

**a**) 50, (

**b**) 60, (

**c**) 70 and (

**d**) 80 based on mortality data since 1970.

**Figure 6.**Five-year-ahead forecast from 2012–2016 for GB males aged (

**a**) 50, (

**b**) 60, (

**c**) 70 and (

**d**) 80 based on mortality data since 1970.

**Figure 7.**Ten-year-ahead forecast from 2007–2016 for GB males aged (

**a**) 50, (

**b**) 60, (

**c**) 70 and (

**d**) 80 based on mortality data since 1970.

**Table 1.**Fitting results for GB male mortality data from 1950–2016, ages 50–89. CBD, Cairns–Blake–Dowd; LOP, Legendre orthogonal polynomial; KS, kernel smoothing.

CBD Model: MLE | CBD Model: LLE | 2D LOP Model | 2D KS Model | |
---|---|---|---|---|

$E1(\%)$ | 0.64 | $-0.07$ | 0.07 | 0.03 |

$E2(\%)$ | 4.23 | 4.42 | 2.94 | 1.75 |

$E3(\%)$ | 5.73 | 5.52 | 3.63 | 2.34 |

**Table 2.**Forecasting results of the CBD model for GB males aged 50–89 based on mortality data since 1950. RW, random walk; LL, local linear.

CBD Model: RW Forecast | CBD Model: LL Forecast | |||||
---|---|---|---|---|---|---|

Forecast horizon | $E1(\%)$ | $E2(\%)$ | $E3(\%)$ | $E1(\%)$ | $E2(\%)$ | $E3(\%)$ |

3 | $-4.22$ | 8.13 | 10.16 | $-4.17$ | 7.08 | 8.89 |

5 | $-4.40$ | 7.44 | 9.23 | $-4.24$ | 7.37 | 9.32 |

10 | 3.58 | 8.15 | 9.73 | $-2.88$ | 6.36 | 8.11 |

**Table 3.**Forecasting results of the 2D LOP model and the 2D KS model for GB males aged 50–89 based on mortality data since 1950.

2D LOP Model | 2D KS Model | |||||
---|---|---|---|---|---|---|

Forecast horizon | $E1(\%)$ | $E2(\%)$ | $E3(\%)$ | $E1(\%)$ | $E2(\%)$ | $E3(\%)$ |

3 | $-3.83$ | 4.71 | 5.62 | $-0.61$ | 2.39 | 3.14 |

5 | $-2.03$ | 4.54 | 5.69 | $-3.59$ | 4.53 | 5.53 |

10 | 8.87 | 9.02 | 10.25 | 1.97 | 3.80 | 5.06 |

**Table 4.**Forecasting results of the CBD model for GB males aged 50–89 based on mortality data since 1970.

CBD Model: RW Forecast | CBD Model: LLE Forecast | |||||
---|---|---|---|---|---|---|

Forecast horizon | $E1(\%)$ | $E2(\%)$ | $E3(\%)$ | $E1(\%)$ | $E2(\%)$ | $E3(\%)$ |

3 | $-5.43$ | 8.34 | 10.80 | $-4.83$ | 7.11 | 8.73 |

5 | $-6.33$ | 8.06 | 10.42 | $-6.46$ | 7.04 | 9.10 |

10 | $-3.02$ | 7.42 | 8.97 | $-4.45$ | 6.94 | 9.52 |

**Table 5.**Forecasting results of the 2D LOP model and the 2D KS model for GB males aged 50–89 based on mortality data since 1970.

2D LOP Model | 2D KS Model | |||||
---|---|---|---|---|---|---|

Forecast horizon | $E1(\%)$ | $E2(\%)$ | $E3(\%)$ | $E1(\%)$ | $E2(\%)$ | $E3(\%)$ |

3 | $-2.36$ | 3.70 | 4.50 | $-0.61$ | 2.39 | 3.14 |

5 | $-0.47$ | 4.64 | 5.76 | $-3.59$ | 4.53 | 5.53 |

10 | 5.25 | 7.06 | 9.98 | 1.97 | 3.80 | 5.06 |

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**MDPI and ACS Style**

Li, H.; O’Hare, C.
Mortality Forecasting: How Far Back Should We Look in Time? *Risks* **2019**, *7*, 22.
https://doi.org/10.3390/risks7010022

**AMA Style**

Li H, O’Hare C.
Mortality Forecasting: How Far Back Should We Look in Time? *Risks*. 2019; 7(1):22.
https://doi.org/10.3390/risks7010022

**Chicago/Turabian Style**

Li, Han, and Colin O’Hare.
2019. "Mortality Forecasting: How Far Back Should We Look in Time?" *Risks* 7, no. 1: 22.
https://doi.org/10.3390/risks7010022