# Two-Population Mortality Forecasting: An Approach Based on Model Averaging

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## Abstract

**:**

## 1. Introduction

## 2. Two-Population Mortality Models

#### 2.1. Model Estimation

#### 2.2. Stochastic Factor Assumptions

- ${\kappa}_{t}^{(i,m)}={\mu}^{(i,m)}+{\kappa}_{t-1}^{(i,m)}+{Z}_{t}^{(i,m)},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}i=1,2,3$
- ${\kappa}_{t}^{(i,f)}={\kappa}_{t}^{(i,m)}+{\varphi}^{(i,f)}({\kappa}_{t-1}^{(i,f)}-{\kappa}_{t-1}^{(i,m)})+{Z}_{t}^{(i,f)},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}i=1,2,3$

- ${\kappa}_{t}=\mu +{\kappa}_{t-1}+{Z}_{t},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}$

- ${\gamma}_{u}^{(m)}=(1+{\varphi}^{(m)}){\gamma}_{u-1}^{(m)}-{\varphi}^{(m)}{\gamma}_{u-2}^{(m)}+{Y}_{u}^{(m)}$
- ${\gamma}_{u}^{(f)}={\gamma}_{u}^{(m)}+{\varphi}^{(f)}({\gamma}_{u-1}^{(f)}-{\gamma}_{u-1}^{(m)})+{Y}_{u}^{(f)}$

## 3. Model-Averaging Approaches

- Equal weights (EW):$${\lambda}_{l}^{EW}=\frac{1}{L},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}l=1,\dots ,L.$$
- Proportional weights (PW):$${\lambda}_{l}^{PR}=\frac{\frac{1}{{g}_{l}}}{{\sum}_{k=1}^{L}\frac{1}{{g}_{k}}},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}l=1,\dots ,L$$
- Weights based on the the softmax function (SM):$${\lambda}_{l}^{SM}=\frac{exp\{-{g}_{l}\}}{{\sum}_{k=1}^{L}exp\{-{g}_{k}\}},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}l=1,\dots ,L.$$The concept is similar to the proportional weights model-averaging approach, but here we penalise less the models with poor performance in the validation period and reward less the models with good performance. See also Benchimol et al. (2016) for a similar formulation.
- Weights based on trimming (TR):$${\lambda}_{l}^{TR}=\left\{\begin{array}{cc}\frac{1}{\widehat{L}},\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}l\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{among}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\widehat{L}\phantom{\rule{4.pt}{0ex}}\mathrm{best}\phantom{\rule{4.pt}{0ex}}\mathrm{models}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{validation}\phantom{\rule{4.pt}{0ex}}\mathrm{period},\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

## 4. Data

## 5. Implementation

#### Step-By-Step Procedure

- First stage
- 1.1
- We fit the two-population models (LC, RH, CBD, PLAT, M6, M7, M8, CF, and ACF) on the period $[{t}_{0},{t}_{s}-10]$ (training period 1) using the StMoMo package; see Villegas et al. (2018). Notice that this implies h, i.e., the length of the validation period, is set equal to 10.
- 1.2
- We simulate mortality rates for the period $[{t}_{s}-9,{t}_{s}]$ (validation period) using the models fitted in 1.1.
- 1.3
- 1.4
- We repeat steps 1.2 and 1.3 1000 times, and we obtain the forecasted truncated life expectancy and Gini index as the average of these for each model.
- 1.5
- We calculate the MAFE as the difference between forecasted truncated life expectancy and Gini index calculated in 1.4 and the historical ones (Formula (10)) for each model.
- 1.6

- Second stage
- 2.1
- We repeat step 1.1 using the period $[{t}_{0},{t}_{s}]$ (training period 2) instead of $[{t}_{0},{t}_{s}-10]$.
- 2.2
- We repeat steps 1.2 and 1.3 using the period $[{t}_{s}+1,{t}_{n}]$ (test period) instead of $[{t}_{s}-9,{t}_{s}]$.
- 2.3
- We repeat step 2.2 10,000 times4 and we obtain the forecasted truncated average life expectancy and Gini index as the average of these for each model.
- 2.4
- For each model-averaging approach, we carry out 1 simulation from a multinomial distribution with parameters equal to 10,000, 9, and the vector of the weights obtained in 1.6. The result of this simulation will be a vector with 9 elements, which sum to 10,000, that represent the number of truncated life expectancy and Gini index trajectories that are considered in the model-averaging approach from the 9 two-population models.
- 2.5
- Using the results of the simulation at point 2.4 as parameters, we resample by bootstrapping from the truncated life expectancy and Gini index trajectories obtained in step 2.3, and we average them using Formula (5) obtaining the forecasted truncated life expectancy and Gini index for all the model-averaging approaches. Similarly, we take the 5th and 95th percentile to build the 90% confidence forecasting intervals for the two metrics. See Figure 2 for an example of forecasted life expectancy and Gini index using the model-averaging approach with equal weights.
- 2.6
- We calculate the MAFE as the difference between the forecasted truncated life expectancy and the Gini index calculated in 2.3 and 2.5 with the observed ones. Similarly, we compare the confidence forecasting intervals of the two metrics with the observed values in order to obtain the interval forecast accuracy that represents the proportion of times in which the observed truncated life expectancy and Gini index fall within the respective prediction intervals.

## 6. Results

#### 6.1. Rolling Test Period

#### 6.2. Fixed Test Period

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | Probabilities of death ${q}_{x}$ can be calculated from the corresponding mortality rates ${m}_{x}$ by using the relation ${q}_{x}={m}_{x}/(1+\frac{1}{2}{m}_{x})$, and vice versa, ${m}_{x}={q}_{x}/(1-\frac{1}{2}{q}_{x})$. |

2 | For consistency, we use the Poisson distribution assumption coupled with the log-link function and mortality rates for models such as M6, M7, and M8, which usually are presented under a binomial assumption coupled with the logit-link function and probabilities of death. |

3 | The formula used here is an approximation for the Gini index at age 55, truncated at age 90, which includes an additional term accounting for the survival past age 90. |

4 | We carry out more simulations than in the first stage since here we consider the interval forecast accuracy in addition to the MAFE. |

5 | These percentages have been calculated as the ratio of the number of cases in which each model is the best over the total number of cases considered (260). |

6 | These percentages have been calculated as the ratio of the number of cases in which each model is the best over the total number of cases considered (200). |

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**Figure 1.**An illustration of the training, validation, and test periods to train and evaluate the models.

**Figure 2.**Example of forecasted truncated life expectancy and Gini index, the respective 90% prediction intervals, and the observed values of the two metrics (points). Training period: 1950–1979. Test period: 1980–1994. Country: England and Wales. Model-averaging approach: equal weights.

**Figure 3.**Summary of the MAFEs by model. Results for individual models and corresponding weighted average. Rolling test period case.

**Figure 4.**Model with the lowest MAFE by period and country. Results for individual models and model-averaging approaches. Rolling test period case.

**Figure 5.**Summary of the MAFEs by model. Results for individual models and corresponding weighted average. Fixed test period case.

**Figure 6.**Model with the lowest MAFE by period and country. Results for individual models and model-averaging approaches. Fixed test period case.

**Table 1.**Summary of multi-population models used. Here ${\kappa}_{t}^{(i,p)}$, $i=1,2,3$, ${\kappa}_{t}$, and ${\kappa}_{t}^{(p)}$ are time-varying stochastic factors; ${\gamma}_{t-x}^{(p)}$ are cohort-related stochastic factors; ${\beta}_{x}^{(i,p)}$, $i=1,2,3$, and ${\beta}_{x}^{(2)}$ are age-specific parameters; $\overline{x}=\frac{1}{m+1}{\sum}_{i=0}^{m}{x}_{i}$ is the mean age over the population age range; ${\widehat{\sigma}}_{x}^{2}=\frac{1}{m+1}{\sum}_{i=0}^{m}{({x}_{i}-\overline{x})}^{2}$ is the age variance; and finally ${x}_{c}^{(p)}$ is an arbitrary fixed age.

Model | $ln({m}_{x,t}^{(p)})$ |

1. Lee–Carter model (LC) | ${\beta}_{x}^{(1,p)}+{\beta}_{x}^{(2,p)}{\kappa}_{t}^{(2,p)}$ |

2. Renshaw–Haberman model (RH) | ${\beta}_{x}^{(1,p)}+{\beta}_{x}^{(2,p)}{\kappa}_{t}^{(2,p)}+{\gamma}_{t-x}^{(p)}$ |

3. Cairns–Blake–Dowd model (CBD) | ${\kappa}_{t}^{(1,p)}+{\kappa}_{t}^{(2,p)}(x-\overline{x})$ |

4. CBD Model with a cohort effect (M6) | ${\kappa}_{t}^{(1,p)}+{\kappa}_{t}^{(2,p)}(x-\overline{x})+{\gamma}_{t-x}^{(p)}$ |

5. CBD Model with quadratic and cohort effects (M7) | ${\kappa}_{t}^{(1,p)}+{\kappa}_{t}^{(2,p)}(x-\overline{x})+{\kappa}_{t}^{(3,p)}({(x-\overline{x})}^{2}-{\widehat{\sigma}}_{x}^{2})+{\gamma}_{t-x}^{(p)}$ |

6. CBD Model with an age-dependent cohort effect (M8) | ${\kappa}_{t}^{(1,p)}+{\kappa}_{t}^{(2,p)}(x-\overline{x})+{\gamma}_{t-x}^{(p)}({x}_{c}^{(p)}-x)$ |

7. Plat model (PLAT) | ${\beta}_{x}^{(1,p)}+{\kappa}_{t}^{(1,p)}+{\kappa}_{t}^{(2,p)}(x-\overline{x})+{\gamma}_{t-x}^{(p)}$ |

8. Common Factor Model (CF) | ${\beta}_{x}^{(1,p)}+{\beta}_{x}^{(2)}{\kappa}_{t}$ |

9. Augmented Common Factor Model (ACF) | ${\beta}_{x}^{(1,p)}+{\beta}_{x}^{(2)}{\kappa}_{t}+{\beta}_{x}^{(2,p)}{\kappa}_{t}^{(2,p)}$ |

**Table 2.**Interval forecast accuracy by period. Proportion of cases in which the observed life expectancy falls in the forecasting interval. Dark shades of green and red signify higher interval forecast accuracy, whereas lighter shades denote lower interval forecast accuracy. Rolling test period case.

Training Period | Test Period | LC | CBD | M6 | M7 | M8 | PLAT | RH | CF | ACF | MA E.W | MA PR.W | MA S.M | MA TR |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1950–1979 | 1980–1994 | 83% | 84% | 95% | 84% | 77% | 95% | 71% | 82% | 91% | 100% | 98% | 100% | 96% |

1951–1980 | 1981–1995 | 82% | 89% | 95% | 83% | 81% | 91% | 71% | 81% | 86% | 99% | 98% | 99% | 96% |

1952–1981 | 1982–1996 | 83% | 87% | 90% | 79% | 81% | 88% | 72% | 81% | 86% | 99% | 98% | 99% | 97% |

1953–1982 | 1983–1997 | 81% | 84% | 91% | 83% | 79% | 86% | 81% | 84% | 84% | 99% | 98% | 98% | 96% |

1954–1983 | 1984–1998 | 75% | 81% | 87% | 75% | 76% | 82% | 77% | 77% | 79% | 98% | 97% | 97% | 95% |

1955–1984 | 1985–1999 | 86% | 87% | 92% | 83% | 84% | 92% | 87% | 88% | 86% | 99% | 98% | 99% | 97% |

1956–1985 | 1986–2000 | 80% | 82% | 83% | 77% | 80% | 78% | 86% | 75% | 76% | 98% | 98% | 98% | 95% |

1957–1986 | 1987–2001 | 81% | 82% | 85% | 80% | 80% | 84% | 88% | 77% | 76% | 98% | 97% | 98% | 96% |

1958–1987 | 1988–2002 | 82% | 83% | 87% | 77% | 83% | 82% | 92% | 82% | 80% | 98% | 97% | 98% | 95% |

1959–1988 | 1989–2003 | 79% | 80% | 76% | 74% | 81% | 74% | 90% | 75% | 75% | 96% | 95% | 96% | 92% |

1960–1989 | 1990–2004 | 78% | 84% | 85% | 76% | 80% | 80% | 95% | 77% | 74% | 97% | 97% | 97% | 91% |

1961–1990 | 1991–2005 | 86% | 88% | 79% | 75% | 89% | 82% | 89% | 81% | 81% | 97% | 97% | 97% | 96% |

1962–1991 | 1992–2006 | 85% | 89% | 81% | 75% | 85% | 82% | 87% | 79% | 78% | 96% | 97% | 97% | 96% |

1963–1992 | 1993–2007 | 86% | 85% | 79% | 75% | 83% | 81% | 84% | 76% | 82% | 98% | 97% | 97% | 90% |

1964–1993 | 1994–2008 | 76% | 79% | 61% | 59% | 83% | 63% | 86% | 66% | 73% | 91% | 92% | 92% | 86% |

1965–1994 | 1995–2009 | 84% | 86% | 70% | 70% | 82% | 75% | 87% | 71% | 81% | 99% | 99% | 99% | 95% |

1966–1995 | 1996–2010 | 82% | 85% | 60% | 60% | 84% | 65% | 90% | 70% | 76% | 97% | 97% | 97% | 87% |

1967–1996 | 1997–2011 | 81% | 82% | 61% | 60% | 81% | 66% | 90% | 67% | 76% | 97% | 98% | 98% | 96% |

1968–1997 | 1998–2012 | 85% | 81% | 60% | 62% | 79% | 72% | 89% | 70% | 79% | 98% | 98% | 98% | 93% |

1969–1998 | 1999–2013 | 82% | 79% | 53% | 59% | 74% | 66% | 88% | 68% | 76% | 96% | 95% | 96% | 89% |

1970–1999 | 2000–2014 | 74% | 75% | 33% | 41% | 73% | 54% | 86% | 61% | 73% | 94% | 95% | 95% | 88% |

1971–2000 | 2001–2015 | 88% | 83% | 45% | 59% | 77% | 68% | 72% | 65% | 80% | 96% | 94% | 96% | 91% |

1972–2001 | 2002–2016 | 90% | 86% | 49% | 70% | 79% | 68% | 62% | 64% | 86% | 98% | 98% | 98% | 94% |

1973–2002 | 2003–2017 | 87% | 82% | 35% | 63% | 74% | 67% | 70% | 62% | 85% | 95% | 94% | 95% | 78% |

1974–2003 | 2004–2018 | 75% | 82% | 25% | 52% | 68% | 56% | 61% | 59% | 79% | 92% | 90% | 92% | 82% |

1975–2004 | 2005–2019 | 90% | 77% | 51% | 83% | 77% | 83% | 41% | 68% | 87% | 99% | 99% | 98% | 95% |

Average | 82% | 83% | 70% | 71% | 80% | 76% | 81% | 73% | 80% | 97% | 97% | 97% | 93% |

**Table 3.**Interval forecast accuracy by country. Proportion of cases in which the observed life expectancy falls in the forecasting interval. Dark shades of green and red signify higher interval forecast accuracy, whereas lighter shades denote lower interval forecast accuracy. Rolling test period case.

LC | CBD | M6 | M7 | M8 | PLAT | RH | CF | ACF | MA E.W | MA PR.W | MA S.M | MA TR | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AUSTRALIA | 90% | 94% | 82% | 95% | 96% | 84% | 91% | 92% | 91% | 99% | 99% | 99% | 97% |

CANADA | 61% | 62% | 49% | 78% | 59% | 54% | 58% | 62% | 52% | 99% | 96% | 99% | 78% |

ENGLAND AND WALES | 74% | 75% | 78% | 69% | 71% | 70% | 89% | 63% | 75% | 99% | 99% | 99% | 99% |

FRANCE | 97% | 96% | 34% | 63% | 89% | 80% | 88% | 87% | 97% | 100% | 99% | 100% | 97% |

ITALY | 84% | 88% | 70% | 59% | 88% | 76% | 83% | 88% | 85% | 97% | 97% | 98% | 95% |

JAPAN | 99% | 87% | 87% | 62% | 75% | 96% | 80% | 50% | 99% | 100% | 100% | 100% | 96% |

NETHERLANDS | 56% | 61% | 77% | 71% | 57% | 76% | 80% | 67% | 58% | 86% | 87% | 87% | 86% |

SPAIN | 98% | 98% | 85% | 70% | 95% | 94% | 95% | 88% | 99% | 100% | 100% | 100% | 95% |

SWEDEN | 73% | 80% | 62% | 42% | 78% | 65% | 71% | 69% | 66% | 91% | 88% | 90% | 84% |

USA | 91% | 92% | 71% | 96% | 89% | 67% | 71% | 67% | 80% | 100% | 100% | 100% | 98% |

Average | 82% | 83% | 70% | 71% | 80% | 76% | 81% | 73% | 80% | 97% | 97% | 97% | 93% |

**Table 4.**Interval forecast accuracy by period. Proportion of cases in which the observed Gini index falls in the forecasting interval. Dark shades of green and red signify higher interval forecast accuracy, whereas lighter shades denote lower interval forecast accuracy. Rolling test period case.

Training Period | Test Period | LC | CBD | M6 | M7 | M8 | PLAT | RH | CF | ACF | MA E.W | MA PR.W | MA S.M | MA TR |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1950–1979 | 1980–1994 | 92% | 81% | 97% | 96% | 74% | 95% | 79% | 90% | 91% | 100% | 100% | 100% | 98% |

1951–1980 | 1981–1995 | 89% | 88% | 97% | 96% | 74% | 95% | 79% | 86% | 86% | 99% | 99% | 99% | 99% |

1952–1981 | 1982–1996 | 90% | 82% | 92% | 93% | 73% | 94% | 79% | 85% | 86% | 99% | 99% | 99% | 97% |

1953–1982 | 1983–1997 | 89% | 77% | 97% | 98% | 71% | 93% | 86% | 88% | 87% | 99% | 99% | 100% | 98% |

1954–1983 | 1984–1998 | 81% | 76% | 87% | 86% | 68% | 96% | 83% | 80% | 76% | 98% | 97% | 97% | 95% |

1955–1984 | 1985–1999 | 92% | 83% | 94% | 96% | 73% | 100% | 89% | 90% | 89% | 100% | 100% | 100% | 99% |

1956–1985 | 1986–2000 | 85% | 78% | 85% | 95% | 72% | 97% | 87% | 78% | 78% | 98% | 97% | 98% | 93% |

1957–1986 | 1987–2001 | 86% | 79% | 85% | 95% | 69% | 98% | 90% | 81% | 78% | 99% | 99% | 99% | 97% |

1958–1987 | 1988–2002 | 86% | 79% | 91% | 95% | 66% | 98% | 92% | 86% | 81% | 99% | 98% | 98% | 96% |

1959–1988 | 1989–2003 | 84% | 79% | 82% | 91% | 72% | 95% | 90% | 77% | 74% | 96% | 96% | 96% | 94% |

1960–1989 | 1990–2004 | 84% | 75% | 87% | 93% | 61% | 97% | 94% | 79% | 76% | 99% | 99% | 99% | 94% |

1961–1990 | 1991–2005 | 90% | 84% | 85% | 93% | 68% | 95% | 88% | 75% | 84% | 98% | 98% | 98% | 96% |

1962–1991 | 1992–2006 | 87% | 83% | 88% | 93% | 63% | 93% | 84% | 75% | 81% | 98% | 98% | 98% | 95% |

1963–1992 | 1993–2007 | 84% | 81% | 85% | 90% | 58% | 89% | 82% | 74% | 76% | 95% | 94% | 95% | 91% |

1964–1993 | 1994–2008 | 78% | 82% | 69% | 81% | 63% | 91% | 87% | 68% | 67% | 95% | 95% | 96% | 90% |

1965–1994 | 1995–2009 | 90% | 83% | 79% | 96% | 62% | 93% | 83% | 74% | 78% | 99% | 99% | 99% | 96% |

1966–1995 | 1996–2010 | 82% | 83% | 69% | 87% | 64% | 93% | 89% | 75% | 68% | 100% | 99% | 100% | 94% |

1967–1996 | 1997–2011 | 85% | 81% | 71% | 86% | 63% | 92% | 88% | 71% | 69% | 99% | 98% | 99% | 95% |

1968–1997 | 1998–2012 | 86% | 81% | 70% | 86% | 59% | 90% | 84% | 74% | 77% | 99% | 98% | 99% | 96% |

1969–1998 | 1999–2013 | 84% | 78% | 64% | 81% | 59% | 91% | 86% | 69% | 72% | 97% | 97% | 97% | 95% |

1970–1999 | 2000–2014 | 74% | 81% | 42% | 67% | 63% | 88% | 87% | 59% | 58% | 97% | 96% | 97% | 90% |

1971–2000 | 2001–2015 | 86% | 82% | 58% | 87% | 66% | 86% | 72% | 66% | 73% | 97% | 96% | 97% | 91% |

1972–2001 | 2002–2016 | 89% | 80% | 60% | 91% | 67% | 85% | 58% | 67% | 83% | 100% | 100% | 100% | 91% |

1973–2002 | 2003–2017 | 82% | 79% | 49% | 85% | 68% | 89% | 70% | 60% | 76% | 95% | 95% | 95% | 85% |

1974–2003 | 2004–2018 | 73% | 86% | 38% | 69% | 63% | 86% | 64% | 55% | 67% | 95% | 95% | 95% | 83% |

1975–2004 | 2005–2019 | 89% | 72% | 61% | 95% | 64% | 72% | 40% | 73% | 84% | 100% | 100% | 100% | 93% |

Average | 85% | 81% | 76% | 89% | 66% | 92% | 81% | 75% | 77% | 98% | 98% | 98% | 94% |

**Table 5.**Interval forecast accuracy by country. Proportion of cases in which the observed Gini index falls in the forecasting interval. Dark shades of green and red signify higher interval forecast accuracy, whereas lighter shades denote lower interval forecast accuracy. Rolling test period case.

LC | CBD | M6 | M7 | M8 | PLAT | RH | CF | ACF | MA E.W | MA PR.W | MA S.M | MA TR | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AUSTRALIA | 93% | 97% | 83% | 99% | 95% | 89% | 90% | 90% | 85% | 100% | 100% | 100% | 97% |

CANADA | 69% | 51% | 61% | 78% | 38% | 80% | 55% | 67% | 57% | 100% | 98% | 100% | 82% |

ENGLAND AND WALES | 83% | 75% | 78% | 84% | 84% | 94% | 88% | 64% | 67% | 100% | 100% | 100% | 96% |

FRANCE | 94% | 89% | 42% | 92% | 68% | 97% | 86% | 86% | 92% | 100% | 100% | 100% | 97% |

ITALY | 88% | 95% | 84% | 91% | 78% | 95% | 82% | 87% | 79% | 99% | 99% | 99% | 98% |

JAPAN | 93% | 83% | 89% | 96% | 56% | 98% | 84% | 46% | 88% | 100% | 100% | 100% | 100% |

NETHERLANDS | 59% | 56% | 80% | 83% | 47% | 91% | 82% | 75% | 66% | 91% | 90% | 91% | 85% |

SPAIN | 98% | 98% | 88% | 92% | 71% | 97% | 95% | 91% | 97% | 100% | 100% | 100% | 100% |

SWEDEN | 81% | 82% | 78% | 79% | 59% | 89% | 80% | 75% | 66% | 93% | 92% | 93% | 89% |

USA | 93% | 82% | 79% | 98% | 67% | 89% | 70% | 72% | 78% | 99% | 99% | 99% | 95% |

Average | 85% | 81% | 76% | 89% | 66% | 92% | 81% | 75% | 77% | 98% | 98% | 98% | 94% |

**Table 6.**Interval forecast accuracy by period. Proportion of cases in which the observed life expectancy falls in the forecasting interval. Dark shades of green and red signify higher interval forecast accuracy, whereas lighter shades denote lower interval forecast accuracy. Fixed test period case.

Training Period | Test Period | LC | CBD | M6 | M7 | M8 | PLAT | RH | CF | ACF | MA E.W | MA PR.W | MA S.M | MA TR |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1966–2004 | 2005–2019 | 91% | 84% | 48% | 62% | 83% | 78% | 62% | 69% | 84% | 98% | 99% | 97% | 93% |

1967–2004 | 2005–2019 | 91% | 84% | 49% | 63% | 82% | 77% | 59% | 70% | 85% | 97% | 99% | 98% | 93% |

1968–2004 | 2005–2019 | 91% | 82% | 52% | 67% | 80% | 79% | 55% | 71% | 88% | 96% | 97% | 97% | 92% |

1969–2004 | 2005–2019 | 90% | 81% | 52% | 72% | 79% | 80% | 54% | 71% | 88% | 97% | 99% | 99% | 93% |

1970–2004 | 2005–2019 | 91% | 81% | 50% | 70% | 79% | 79% | 50% | 70% | 88% | 98% | 99% | 98% | 94% |

1971–2004 | 2005–2019 | 91% | 80% | 49% | 75% | 79% | 79% | 47% | 69% | 88% | 99% | 99% | 99% | 94% |

1972–2004 | 2005–2019 | 91% | 80% | 47% | 75% | 78% | 78% | 45% | 66% | 87% | 99% | 99% | 99% | 93% |

1973–2004 | 2005–2019 | 91% | 78% | 50% | 78% | 77% | 81% | 43% | 68% | 88% | 98% | 99% | 98% | 89% |

1974–2004 | 2005–2019 | 91% | 79% | 51% | 81% | 78% | 82% | 44% | 68% | 88% | 99% | 98% | 98% | 90% |

1975–2004 | 2005–2019 | 90% | 77% | 50% | 83% | 78% | 84% | 40% | 69% | 87% | 97% | 99% | 99% | 94% |

1976–2004 | 2005–2019 | 90% | 78% | 52% | 86% | 78% | 85% | 39% | 70% | 88% | 99% | 98% | 98% | 95% |

1977–2004 | 2005–2019 | 91% | 79% | 49% | 85% | 77% | 84% | 40% | 72% | 89% | 96% | 98% | 97% | 94% |

1978–2004 | 2005–2019 | 90% | 79% | 51% | 87% | 78% | 85% | 37% | 75% | 90% | 97% | 98% | 97% | 96% |

1979–2004 | 2005–2019 | 90% | 81% | 49% | 89% | 78% | 84% | 37% | 76% | 90% | 97% | 98% | 98% | 94% |

1980–2004 | 2005–2019 | 89% | 79% | 52% | 89% | 77% | 85% | 35% | 73% | 88% | 98% | 98% | 97% | 99% |

1981–2004 | 2005–2019 | 89% | 79% | 53% | 91% | 77% | 85% | 35% | 74% | 88% | 98% | 98% | 98% | 100% |

1982–2004 | 2005–2019 | 89% | 80% | 56% | 90% | 78% | 84% | 36% | 75% | 88% | 98% | 98% | 98% | 98% |

1983–2004 | 2005–2019 | 86% | 80% | 57% | 90% | 77% | 82% | 33% | 76% | 86% | 99% | 99% | 99% | 95% |

1984–2004 | 2005–2019 | 86% | 82% | 57% | 93% | 78% | 86% | 34% | 76% | 87% | 99% | 99% | 99% | 97% |

1985–2004 | 2005–2019 | 82% | 78% | 62% | 93% | 80% | 85% | 34% | 75% | 85% | 99% | 99% | 99% | 94% |

Average | 90% | 80% | 52% | 81% | 79% | 82% | 43% | 72% | 87% | 98% | 98% | 98% | 94% |

**Table 7.**Interval forecast accuracy by country. Proportion of cases in which the observed life expectancy falls in the forecasting interval. Dark shades of green and red signify higher interval forecast accuracy, whereas lighter shades denote lower interval forecast accuracy. Fixed test period case.

LC | CBD | M6 | M7 | M8 | PLAT | RH | CF | ACF | MA E.W | MA PR.W | MA S.M | MA TR | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Australia | 96% | 100% | 71% | 100% | 100% | 98% | 37% | 95% | 99% | 100% | 100% | 100% | 97% |

Canada | 85% | 87% | 28% | 94% | 84% | 52% | 30% | 32% | 65% | 99% | 98% | 99% | 84% |

England and Wales | 99% | 100% | 66% | 100% | 98% | 79% | 32% | 64% | 96% | 100% | 100% | 100% | 100% |

France | 90% | 54% | 18% | 89% | 64% | 97% | 33% | 98% | 92% | 100% | 100% | 100% | 99% |

Italy | 91% | 73% | 87% | 96% | 69% | 93% | 24% | 97% | 95% | 100% | 100% | 100% | 83% |

Japan | 99% | 48% | 58% | 73% | 50% | 100% | 53% | 60% | 100% | 100% | 100% | 100% | 100% |

Netherlands | 55% | 62% | 47% | 45% | 58% | 50% | 80% | 48% | 51% | 80% | 87% | 82% | 97% |

Spain | 100% | 90% | 39% | 65% | 77% | 93% | 99% | 84% | 100% | 100% | 100% | 100% | 92% |

Sweden | 100% | 100% | 52% | 81% | 100% | 89% | 21% | 93% | 99% | 100% | 100% | 100% | 100% |

USA | 82% | 88% | 48% | 66% | 87% | 71% | 21% | 46% | 79% | 100% | 100% | 100% | 92% |

Average | 90% | 80% | 52% | 81% | 79% | 82% | 43% | 72% | 87% | 98% | 98% | 98% | 94% |

**Table 8.**Interval forecast accuracy by period. Proportion of cases in which the observed Gini index falls in the forecasting interval. Dark shades of green and red signify higher interval forecast accuracy, whereas lighter shades denote lower interval forecast accuracy. Fixed test period case.

Training Period | Test Period | LC | CBD | M6 | M7 | M8 | PLAT | RH | CF | ACF | MA E.W | MA PR.W | MA S.M | MA TR |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1966–2004 | 2005–2019 | 91% | 80% | 60% | 88% | 64% | 85% | 62% | 73% | 87% | 99% | 100% | 99% | 93% |

1967–2004 | 2005–2019 | 91% | 78% | 62% | 91% | 67% | 85% | 61% | 73% | 87% | 99% | 100% | 99% | 93% |

1968–2004 | 2005–2019 | 91% | 75% | 63% | 91% | 59% | 80% | 56% | 73% | 87% | 99% | 100% | 99% | 94% |

1969–2004 | 2005–2019 | 91% | 74% | 62% | 92% | 62% | 80% | 56% | 73% | 87% | 99% | 100% | 99% | 91% |

1970–2004 | 2005–2019 | 91% | 76% | 61% | 92% | 62% | 81% | 51% | 72% | 87% | 99% | 99% | 99% | 91% |

1971–2004 | 2005–2019 | 90% | 74% | 60% | 93% | 62% | 82% | 50% | 73% | 87% | 99% | 100% | 99% | 92% |

1972–2004 | 2005–2019 | 91% | 76% | 57% | 93% | 60% | 77% | 47% | 72% | 88% | 100% | 100% | 100% | 93% |

1973–2004 | 2005–2019 | 90% | 71% | 61% | 93% | 61% | 75% | 43% | 72% | 85% | 99% | 99% | 99% | 94% |

1974–2004 | 2005–2019 | 90% | 72% | 60% | 94% | 60% | 73% | 44% | 73% | 85% | 100% | 100% | 100% | 96% |

1975–2004 | 2005–2019 | 89% | 73% | 60% | 95% | 66% | 73% | 41% | 73% | 84% | 100% | 100% | 100% | 93% |

1976–2004 | 2005–2019 | 89% | 75% | 60% | 94% | 62% | 68% | 40% | 74% | 84% | 100% | 100% | 100% | 94% |

1977–2004 | 2005–2019 | 88% | 73% | 59% | 92% | 64% | 72% | 38% | 73% | 83% | 99% | 99% | 99% | 91% |

1978–2004 | 2005–2019 | 88% | 72% | 61% | 91% | 67% | 70% | 36% | 75% | 82% | 99% | 99% | 99% | 93% |

1979–2004 | 2005–2019 | 88% | 75% | 56% | 91% | 70% | 73% | 37% | 74% | 83% | 99% | 99% | 99% | 92% |

1980–2004 | 2005–2019 | 88% | 72% | 60% | 89% | 67% | 69% | 35% | 74% | 83% | 99% | 99% | 99% | 94% |

1981–2004 | 2005–2019 | 88% | 74% | 61% | 89% | 68% | 71% | 35% | 74% | 82% | 99% | 99% | 99% | 93% |

1982–2004 | 2005–2019 | 88% | 75% | 60% | 90% | 71% | 71% | 36% | 74% | 83% | 99% | 100% | 99% | 95% |

1983–2004 | 2005–2019 | 84% | 71% | 62% | 85% | 71% | 66% | 35% | 71% | 78% | 98% | 97% | 98% | 88% |

1984–2004 | 2005–2019 | 85% | 73% | 62% | 90% | 73% | 68% | 38% | 72% | 80% | 99% | 99% | 99% | 93% |

1985–2004 | 2005–2019 | 81% | 71% | 66% | 88% | 68% | 62% | 37% | 70% | 77% | 98% | 98% | 98% | 86% |

Average | 89% | 74% | 61% | 91% | 65% | 74% | 44% | 73% | 84% | 99% | 99% | 99% | 93% |

**Table 9.**Interval forecast accuracy by country. Proportion of cases in which the observed Gini index falls in the forecasting interval. Dark shades of green and red signify higher interval forecast accuracy, whereas lighter shades denote lower interval forecast accuracy. Fixed test period case.

LC | CBD | M6 | M7 | M8 | PLAT | RH | CF | ACF | MA E.W | MA PR.W | MA S.M | MA TR | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

AUSTRALIA | 99% | 100% | 77% | 100% | 84% | 32% | 37% | 96% | 97% | 100% | 100% | 100% | 75% |

CANADA | 64% | 95% | 48% | 82% | 91% | 69% | 29% | 46% | 42% | 100% | 100% | 100% | 81% |

ENGLAND AND WALES | 99% | 100% | 58% | 100% | 97% | 79% | 33% | 70% | 96% | 100% | 100% | 100% | 100% |

FRANCE | 93% | 23% | 27% | 97% | 49% | 87% | 31% | 97% | 91% | 100% | 100% | 100% | 96% |

ITALY | 84% | 54% | 96% | 93% | 50% | 43% | 22% | 85% | 81% | 97% | 98% | 98% | 89% |

JAPAN | 99% | 37% | 84% | 84% | 27% | 96% | 67% | 46% | 98% | 99% | 100% | 99% | 99% |

NETHERLANDS | 52% | 72% | 53% | 64% | 74% | 100% | 81% | 50% | 45% | 96% | 97% | 96% | 86% |

SPAIN | 98% | 87% | 50% | 96% | 50% | 92% | 95% | 89% | 96% | 99% | 100% | 100% | 99% |

SWEDEN | 100% | 94% | 62% | 98% | 58% | 74% | 21% | 93% | 99% | 100% | 100% | 100% | 100% |

USA | 100% | 79% | 53% | 96% | 74% | 69% | 25% | 57% | 96% | 100% | 100% | 100% | 100% |

Average | 89% | 74% | 61% | 91% | 65% | 74% | 44% | 73% | 84% | 99% | 99% | 99% | 93% |

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## Share and Cite

**MDPI and ACS Style**

De Mori, L.; Millossovich, P.; Zhu, R.; Haberman, S.
Two-Population Mortality Forecasting: An Approach Based on Model Averaging. *Risks* **2024**, *12*, 60.
https://doi.org/10.3390/risks12040060

**AMA Style**

De Mori L, Millossovich P, Zhu R, Haberman S.
Two-Population Mortality Forecasting: An Approach Based on Model Averaging. *Risks*. 2024; 12(4):60.
https://doi.org/10.3390/risks12040060

**Chicago/Turabian Style**

De Mori, Luca, Pietro Millossovich, Rui Zhu, and Steven Haberman.
2024. "Two-Population Mortality Forecasting: An Approach Based on Model Averaging" *Risks* 12, no. 4: 60.
https://doi.org/10.3390/risks12040060