# A Parametric Factor Model of the Term Structure of Mortality

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Lee–Carter Model

## 3. Stylized Facts of the Mortality Curve

**The age dimension:**To illustrate the age dimension properties, which a good mortality model should be able to capture, we show the log mortality on 10 year intervals from 1950 to 2010 for men and women in Figure 1a–d for the US and France. The mortality curve shows a similar shape over the ages, but the level of mortality tends to decline over time; the shape is very similar across both genders and countries. The infant mortality is seen to decline rapidly during early childhood. In the late teens, the mortality rate experiences a rapid increase often termed the ‘accident hump’, which appears either as a distinct hump or as a flattening out of the death rates; see Heligman and Pollard (1980). After the accident hump, the mortality rates are gradually increasing with age (log-linearly). Thus, for a model to produce realistic results, these three facts should hold for each year. The three properties could also be interpreted in terms of biological reasonableness as described by Cairns et al. (2006), which rules out patterns that are biologically unreasonable such as a decreasing mortality curve for the older as well as the crossing over of age-specific mortality rates.

**The time dimension:**When investigating the time dynamics in the development of the log age-specific death rates, the review paper by Wong-Fupuy and Haberman (2004, p. 56) notes that “There is a broad consensus across the resulting projections: (1) an approximately log-linear relationship between mortality rates and time, (2) decreasing improvements according to age". The first point helps to explain the success of the LC model where the common time-varying factor is found to evolve almost linearly in most applications—see, e.g., Lee and Miller (2001) and Callot et al. (2016). The log-linear development of death rates over time is illustrated in Figure 2a–d. The second observation of decreasing improvements in mortality with respect to age can be described by the so called compensation effect of mortality—see, e.g., Gavrilov and Gavrilova (1979, 1991).3 In Figure 2a–d, this effect is seen by a slope of the log mortality-time plot that decreases with age.

## 4. The Parametric Factor Model for the Term Structure of Mortality

## 5. Estimation Procedure for the Parametric Factor Model

#### 5.1. The Two-Step Estimation Procedure

#### 5.2. One-Step Estimation

## 6. Empirical Analysis

#### 6.1. Estimates Using the Two-Step Procedure

#### 6.2. Cointegrating Analysis of the Factors

#### 6.3. Estimates Using the One-Step Procedure

#### 6.4. Model Fit

## 7. Forecast Evaluation

**France, men.**For French men, the MCS using a 1-year forecast horizon includes the RWD and FDA specifications. However, when expanding the forecast horizon, the MCS now includes three variants of the PFM and, in fact, for a twenty-year forecast horizon, the MCS excludes the RWD and FDA specifications. It is interesting to observe that, in this forecast competition, the LC model is never included in the MCS. The same applies for the PFM model specification where the factors are modelled as a VAR(1) in levels. This is not surprising because all the factors were found to have unit roots.

**USA, men.**The pattern observed for French men generally applies for US men as well. However, for the 20-year horizon, only two of the PFM models are included in the MCS.

**France, women.**For French women and a forecasting horizon of one year, the results are rather similar to those of French men and in particular the RWD specification and the FDA model are the ones included in the MCS. For a 10-year horizon, the MCS also includes a single PFM specification and, for a 20-year horizon, only the PFM with a VAR(1) in levels is not included in the MCS.

**USA, women.**For US women, the RWD model is always in the MCS. For a 10-year horizon, the results are similar to French women and, for a 20-year horizon, the MCS is slightly smaller than for French women and includes in particular the two PFM specifications the FDA and the RWD specifications.

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | Following Nielsen and Nielsen (2014), the choice of restrictions is of no importance for the resulting forecasts. Other normalizations could be considered; however, this gives an intuitive interpretation of ${\alpha}_{x}$. |

2 | We restrict the data to 1950 and onwards as this removes outliers, and we avoid structural changes in the exposure; see Lee and Miller (2001) and Booth et al. (2002). To avoid uncertainty about the death rates, due to a few observations, we further restrict the ages to cover the ages 0 to 95 as is standard in the mortality forecasting literature. |

3 | This is also called the Strehler-Mildvan correlation due to Strehler and Mildvan (1960). |

4 | This is similar to Lee and Carter (1992) who assumed a homoskedastic error term for the LC model. The i.i.d. homoscedasticity assumption is necessary for the analysis of the present paper, but the assumption may be critical in certain cases—see, e.g., Doz et al. (2011). |

5 | This corresponds to the partial correlation squared between the fitted and observed values. |

6 | The MCS approach is implemented via the Ox-package Mulcom 3.0 by Hansen and Lunde (2014) in Oxmetrics 7, see Doornik (2013). |

7 | For the case with only two models the forecast performance could be tested via the Diebold and Mariano (1995) test, which only allows for pairwise comparisons, whereas the MCS procedure allows for joint multiple model evaluation. |

8 | Note that we here use the period life expectancy (within year t), whereas the formula in Brouhns et al. (2002) computes the cohort life expectancy. |

9 | These specifications have often been used in studies applying graduation laws of mortality—see Booth and Tickle (2008); McNown and Rogers (1989, 1992). |

10 | In preliminary experiments, we also found this specification to give a better forecast performance compared with using other ARIMA models. |

11 | The factors are estimated using weighted principal components in the R package Demography—see Hyndman and Ullah (2007) and Hyndman et al. (2014) for further details. All other models are estimated using own codes and the packages ‘tsDyn’, ‘VARS’, and ‘Forecast’ in R (R Core Team 2015) by Pfaff (2008); Stigler (2010) and Hyndman (2015). |

**Figure 1.**The log age-specific death rates for the years $1950,1960,1970,1980,1990,2000,$ and 2010 for men and women in France and the USA.

**Figure 2.**The log age-specific death rates for a range of ages for French and American men and women from 1950–2014.

**Figure 3.**p-Values from Johansen’s Trace test are shown for all pairwise combinations of the (log) age-specific death rates for French and US men and women over the period 1950–2014. The test is performed with a restricted time trend in the cointegrating relation and 1 lag in the VAR specification. The p-Values are obtained via the gamma approximation following Doornik (1998, 1999) and shown for significance levels between 0 and 0.10.

**Figure 4.**Plot of the estimated loading functions for the years 1950–2014 for men and women in France and USA. The loading functions correspond to the level, infant, accident hump, and adult age groups, respectively. The loadings are estimated following the two-step procedure described in Section 5.

**Figure 5.**The factors ${\kappa}_{i,t},\phantom{\rule{0.277778em}{0ex}}i=0,1,2,3$ estimated by the two-step procedure for France using data from 1950–2014. The plots are showing the level factor, infant factor, accident hump factor, and adult factor for both genders, respectively.

**Figure 6.**The factors ${\kappa}_{i,t},\phantom{\rule{0.277778em}{0ex}}i=0,1,2,3$ estimated by the two-step procedure for USA using data from 1950–2014. The plots are showing the level factor, infant factor, accident hump factor, and adult factor for both genders, respectively.

**Figure 7.**The factors ${\kappa}_{i,t},\phantom{\rule{0.277778em}{0ex}}i=0,1,2,3$ estimated by the one-step procedure for USA from 1950–2014 and assuming a VAR(1) model for the first difference of the factors, for both genders.

**Figure 8.**The factors ${\kappa}_{i,t},\phantom{\rule{0.277778em}{0ex}}i=0,1,2,3$ are estimated by the one-step procedure for France using data from 1950–2014 and assuming a VECM specification for the factors, for both genders.

**Figure 9.**The mean of the data and the mean of the parametric factor model estimated using the two-step procedure for both men and women, for France and USA. The estimation period is 1950–2014.

**Figure 10.**The pseudo ${R}^{2}$ (within) for the PFM and the LC model for all ages estimated using the two-step procedure. The ${R}^{2}$ is shown for both men and women in France and the USA, respectively.

**Figure 11.**The partial ${R}^{2}$ for the infant, level, accident hump, and adult factor estimated using the two-step procedure for the years 1950–2014. The relative improvements from each of the factors are shown in excess of a $65\%$ threshold. This is shown for both genders for France and the US, respectively.

**Table 1.**Estimated loading function parameters and standard errors from the first step in the two-step procedure for French and US men and women. The standard errors are calculated using the inverse Fisher information criterion.

Men | ||||||
---|---|---|---|---|---|---|

${\mathit{\lambda}}_{\mathbf{1}}$ | ${\mathit{\lambda}}_{\mathbf{2}}$ | ${\mathit{\lambda}}_{\mathbf{3}}$ | k | ${\mathit{\sigma}}^{\mathbf{2}}$ | ||

Fr | Estimate | 0.553 | 11.981 | 1.093 | 20.308 | 0.020 |

Std. Err | 0.013 | 0.024 | 0.004 | 0.002 | 0.018 | |

US | Estimate | 0.624 | 10.813 | 1.103 | 20.016 | 0.017 |

Std. Err | 0.013 | 0.020 | 0.003 | 0.002 | 0.018 | |

Women | ||||||

Fr | Estimate | 0.649 | 18.092 | 1.453 | 19.492 | 0.023 |

Std. Err | 0.013 | 0.060 | 0.003 | 0.005 | 0.018 | |

US | Estimate | 0.607 | 19.029 | 1.295 | 18.675 | 0.013 |

Std. Err | 0.010 | 0.042 | 0.003 | 0.004 | 0.018 |

USA | ||||
---|---|---|---|---|

Men | Women | |||

Rank | Trace-Test | p-Value | Trace-Test | p-Value |

0 | 52.240 | [0.323] | 47.939 | [0.512] |

1 | 27.748 | [0.641] | 30.689 | [0.468] |

2 | 13.926 | [0.668] | 15.243 | [0.562] |

3 | 1.1522 | [0.992] | 3.0677 | [0.858] |

France | ||||

0 | 106.790 | [0.000] ** | 108.730 | [0.000] ** |

1 | 56.044 | [0.001] ** | 47.599 | [0.014] * |

2 | 28.089 | [0.024] * | 24.893 | [0.064] |

3 | 3.642 | [[0.788] | 9.2236 | [0.171] |

USA | France | |||
---|---|---|---|---|

Men and Women | Men and Women | |||

Rank | Trace-Test | p-Value | Trace-Test | p-Value |

0 | 286.700 | [0.000] ** | 287.730 | [0.000] ** |

1 | 194.530 | [0.000] ** | 215.400 | [0.000] ** |

2 | 139.460 | [0.001] ** | 158.870 | [0.000] ** |

3 | 89.751 | [0.041] * | 116.050 | [0.000] ** |

4 | 52.267 | [0.321] | 76.354 | [0.002] ** |

5 | 34.795 | [0.257] | 47.410 | [0.015] * |

6 | 19.806 | [0.240] | 24.016 | [0.082] |

7 | 7.897 | [0.268] | 8.524 | [0.218] |

Men | Women | |||
---|---|---|---|---|

USA and France | USA and France | |||

Rank | Trace-Test | p-Value | Trace-Test | p-Value |

0 | 225.130 | [0.000] ** | 221.980 | [0.000] ** |

1 | 158.190 | [0.016] * | 152.790 | [0.036] * |

2 | 111.620 | [0.113] | 107.810 | [0.178] |

3 | 75.429 | [0.311] | 70.517 | [0.490] |

4 | 47.924 | [0.513] | 44.325 | [0.679] |

5 | 27.275 | [0.668] | 27.285 | [0.667] |

6 | 11.364 | [0.850] | 13.676 | [0.688] |

7 | 3.867 | [0.759] | 3.686 | [0.783] |

**Table 5.**Estimated loading function parameters and standard errors from the one-step procedure for French and US men and women. For US, the VAR(1) model in first difference is assumed for the transition dynamics and, for France, a VECM with two cointegrating relations is assumed. The standard errors are calculated using the inverse Fisher information criterion.

Men | ||||||
---|---|---|---|---|---|---|

${\mathit{\lambda}}_{\mathbf{1}}$ | ${\mathit{\lambda}}_{\mathbf{2}}$ | ${\mathit{\lambda}}_{\mathbf{3}}$ | k | ${\mathit{\sigma}}^{\mathbf{2}}$ | ||

Fr | Estimate | 0.556 | 12.151 | 1.094 | 20.324 | 0.021 |

Std. Err | 0.013 | 0.020 | 0.004 | 0.002 | 0.010 | |

US | Estimate | 0.624 | 10.809 | 1.103 | 20.014 | 0.018 |

Std. Err | 0.013 | 0.020 | 0.003 | 0.002 | 0.019 | |

Women | ||||||

Fr | Estimate | 0.648 | 18.711 | 1.453 | 19.450 | 0.024 |

Std. Err | 0.013 | 0.048 | 0.003 | 0.004 | 0.018 | |

US | Estimate | 0.608 | 18.930 | 1.295 | 18.703 | 0.013 |

Std. Err | 0.010 | 0.043 | 0.003 | 0.004 | 0.005 |

**Table 6.**Forecasting life expectancy 1, 10, and 20 years ahead with mean-squared error criterion for US men and women evaluated using the Model Confidence Set. The mean squared error along with p-values for the estimated model confidence set for life expectancy. The models included in the set of best models are marked in boldface. The first five rows show the results for the parametric factor model assuming different specifications for the factor dynamics, whereas the last three rows show results for the benchmark models. The VAR1 in levels and ARIMA specifications are used for comparison.

France | Men | Women | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 Year | 10 Year | 20 Year | 1 Year | 10 Year | 20 Year | ||||||||

MSE | Pval | MSE | Pval | MSE | Pval | MSE | Pval | MSE | Pval | MSE | Pval | ||

PFM | VAR1 | 0.098 | 0.000 | 2.493 | 0.022 | 13.480 | 0.000 | 0.164 | 0.000 | 1.719 | 0.000 | 7.529 | 0.000 |

Arima | 0.103 | 0.001 | 0.967 | 1.000 | 4.147 | 1.000 | 0.071 | 0.002 | 0.144 | 0.233 | 0.370 | 0.920 | |

$\Delta $VAR1 | 0.095 | 0.006 | 0.985 | 0.931 | 4.611 | 0.146 | 0.086 | 0.001 | 0.157 | 0.013 | 0.444 | 0.415 | |

VECM2 | 0.095 | 0.002 | 0.986 | 0.931 | 4.511 | 0.087 | 0.082 | 0.000 | 0.138 | 0.233 | 0.439 | 0.484 | |

VECM2SS | 0.234 | 0.000 | 1.331 | 0.284 | 4.393 | 0.712 | 0.359 | 0.000 | 0.741 | 0.005 | 1.041 | 0.036 | |

RWD | 0.032 | 0.611 | 1.135 | 0.284 | 5.439 | 0.000 | 0.032 | 1.000 | 0.103 | 1.000 | 0.367 | 1.000 | |

LC | 0.099 | 0.000 | 1.479 | 0.001 | 6.367 | 0.000 | 0.119 | 0.001 | 0.229 | 0.050 | 0.460 | 0.329 | |

FDA | 0.030 | 1.000 | 1.085 | 0.875 | 5.436 | 0.002 | 0.037 | 0.301 | 0.410 | 0.134 | 1.581 | 0.329 |

**Table 7.**Forecasting life expectancy 1, 10, and 20 years ahead with mean-squared error criterion for US men and women evaluated using the Model Confidence Set. Mean squared error along with p-values for the estimated model confidence set for life expectancy. The models included in the set of best models are marked in boldface.The first five rows show the results for the parametric factor model assuming different specifications for the factor dynamics, whereas the last three rows show results for the benchmark models. The VAR1 in levels and ARIMA specifications are used for comparison.

USA | Men | Women | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 Year | 10 Year | 20 Year | 1 Year | 10 Year | 20 Year | ||||||||

MSE | Pval | MSE | Pval | MSE | Pval | MSE | Pval | MSE | Pval | MSE | Pval | ||

PFM | VAR1 | 0.113 | 0.011 | 1.607 | 0.018 | 5.915 | 0.004 | 0.069 | 0.004 | 1.514 | 0.157 | 11.630 | 0.024 |

Arima | 0.112 | 0.013 | 0.787 | 1.000 | 2.493 | 1.000 | 0.054 | 0.011 | 0.519 | 0.225 | 1.378 | 0.017 | |

$\Delta $VAR1 | 0.110 | 0.010 | 0.897 | 0.140 | 3.094 | 0.038 | 0.057 | 0.011 | 0.562 | 0.005 | 1.369 | 0.002 | |

VECM2 | 0.127 | 0.013 | 1.081 | 0.104 | 2.765 | 0.455 | 0.052 | 0.011 | 0.329 | 1.000 | 0.593 | 1.000 | |

$\Delta $VAR1SS | 0.117 | 0.013 | 1.244 | 0.024 | 3.662 | 0.006 | 0.084 | 0.001 | 0.575 | 0.124 | 1.316 | 0.024 | |

RWD | 0.035 | 1.000 | 1.240 | 0.104 | 4.016 | 0.000 | 0.023 | 1.000 | 0.435 | 0.225 | 1.243 | 0.011 | |

LC | 0.138 | 0.013 | 1.771 | 0.018 | 5.375 | 0.001 | 0.080 | 0.006 | 0.682 | 0.069 | 1.790 | 0.003 | |

FDA | 0.044 | 0.154 | 1.577 | 0.104 | 4.824 | 0.006 | 0.026 | 0.032 | 0.495 | 0.225 | 1.441 | 0.024 |

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Haldrup, N.; Rosenskjold, C.P.T.
A Parametric Factor Model of the Term Structure of Mortality. *Econometrics* **2019**, *7*, 9.
https://doi.org/10.3390/econometrics7010009

**AMA Style**

Haldrup N, Rosenskjold CPT.
A Parametric Factor Model of the Term Structure of Mortality. *Econometrics*. 2019; 7(1):9.
https://doi.org/10.3390/econometrics7010009

**Chicago/Turabian Style**

Haldrup, Niels, and Carsten P. T. Rosenskjold.
2019. "A Parametric Factor Model of the Term Structure of Mortality" *Econometrics* 7, no. 1: 9.
https://doi.org/10.3390/econometrics7010009