#
Parsimonious Predictive Mortality Modeling by Regularization and Cross-Validation with and without Covid-Type Effect^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- Standard mortality models have a large number of parameters (easily between 100 and 200 parameters) in comparison with the small number of observations (often around 2000). Regularization techniques allow us to obtain a parsimonious model with very few parameters.
- Model choice is often performed based on the Bayes information criterion (BIC), which measures the ability of the model to fit the historical data (in-sample criterion). Our method, based on cross-validation, will select the model which has the best performance in terms of predictions (out-of-sample criterion).

## 2. Notation and Data

- The mortality rate ${q}_{x,t}$ which represents the probability that an individual aged exactly x at time t dies in the following year, i.e., between t and $t+1$, is given by$${q}_{x,t}=1-{e}^{-{\mu}_{x,t}}=1-{e}^{-{m}_{x,t}}.$$
- If we treat the number of deaths ${d}_{x,t}$ as a random variable ${D}_{x,t}$ and the central exposure ${e}_{x,t}$ as fixed, then ${D}_{x,t}$ follows a Poisson distribution$${D}_{x,t}\sim \mathrm{Poisson}\left({e}_{x,t}{\mu}_{x,t}\right),$$

## 3. Standard Approach to Mortality Modeling

#### 3.1. The General Structure of Mortality Models

- The static age function term ${\alpha}_{x}$ which captures the average shape of the mortality curve.
- N period-age terms ${\beta}_{x}^{\left(i\right)}{\kappa}_{t}^{\left(i\right)}$ ($i=1,\cdots ,N$) which describe the mortality trends.
- The term ${\gamma}_{t-x}$ represents the lifelong cohort effect of certain generations.

#### 3.2. Estimation and Standard Model Choice Criterion

#### 3.3. Forecasting

## 4. Regularized Mortality Modeling

#### 4.1. Polynomial Representation of Mortality Models

#### 4.2. Estimation by Regularization and Cross-Validation

#### 4.3. Forecasting

#### 4.4. Effect of COVID-19

- During the year of the pandemic, $t={t}_{p}$, we assume that there is a death increase of $\beta \left(x\right)$:$${d}_{x,t}^{\mathrm{new}}=(1+\beta \left(x\right)){d}_{x,t}.$$
- The next K years of the pandemic, $t={t}_{p}+1,\cdots ,{t}_{p}+K$, deaths and exposures are reduced to compensate the deaths of COVID-19:$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {d}_{x,t}^{\mathrm{new}}& ={d}_{x,t}-\frac{{d}_{x-m,{t}_{p}}}{K},\hfill \\ \hfill {e}_{x,t}^{\mathrm{new}}& ={e}_{x,t}-{d}_{x-m,{t}_{p}}\frac{K-m+1}{K},\hfill \end{array}\end{array}$$

#### 4.5. Model Uncertainty and Prediction Intervals

- In order to extract a period index, we assume that mortality rates improve at a general improvement rate ${\lambda}_{t}$ per year:$${\mu}_{x,t+1}={\lambda}_{t}\phantom{\rule{4pt}{0ex}}{\mu}_{x,t},\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{any}\phantom{\rule{4.pt}{0ex}}x={x}_{1},\cdots ,{x}_{k}.$$
- We estimate past improvement rates with the simple estimator:$${\widehat{\lambda}}_{t}=\frac{{\sum}_{x}{\widehat{\mu}}_{x,t+1}}{{\sum}_{x}{\widehat{\mu}}_{x,t}}$$
- We fit a random walk with drift (RWD) to past improvement rates: ${\widehat{\lambda}}_{t}$ for $t={t}_{1},\cdots ,{t}_{{n}_{1}}$.
- After the RWD fit, we can then obtain prediction intervals for future death rates:$$I{C}_{1-\alpha}\left({\mu}_{x,t+1}\right)=\left(\right)open="["\; close="]">{\lambda}_{t,\alpha /2}\phantom{\rule{4pt}{0ex}}{\widehat{\mu}}_{x,t};{\lambda}_{t,1-\alpha /2}\phantom{\rule{4pt}{0ex}}{\widehat{\mu}}_{x,t}$$
- Prediction intervals centered around the extrapolated central death rates can then be obtained with a shift of the mean:$$I{C}_{1-\alpha}^{c}\left({\mu}_{x,t+1}\right)=\left(\right)open="["\; close="]">{\lambda}_{t,\alpha /2}{\widehat{\mu}}_{x,t}-{\overline{\lambda}}_{t}\phantom{\rule{4pt}{0ex}}{\widehat{\mu}}_{x,t}+{\widehat{\mu}}_{x,t+1};{\lambda}_{t,1-\alpha /2}{\widehat{\mu}}_{x,t}-{\overline{\lambda}}_{t}\phantom{\rule{4pt}{0ex}}{\widehat{\mu}}_{x,t}+{\widehat{\mu}}_{x,t+1}$$

## 5. Numerical Analysis and Comparison of Models

#### 5.1. Smoothing and Forecasting Central Death Rates for France

**glmnet**(Friedman et al. (2010)).

#### 5.2. Comparison with Standard Age-Period-Cohort Mortality Models

**StMoMo**(Millossovich et al. (2018)) in the log-Poisson setting with appropriate identifiability constraints.

- The RH model is very sensitive to the fitted period. The model can go from the worst to the best model (cf. USA) and from the best to the worst model (cf. Belgium). Cairns et al. (2011) already pointed out that lack of robustness in this model as parameter estimates jump to a qualitatively different solution when we use more data.
- The regularized model and the Lee-Carter model appear to be reasonably robust relative to changes in the period of data used either in terms of MAE values or ranking positions. In particular, models without cohort effects tend to be more robust towards the sample data used as they do not require the projections of cohort terms.

#### 5.3. Comparison with the P-Spline Model

**MortalitySmooth**(Camarda et al. (2012)). Hereafter, we highlight similarities and differences with our approach. Moreover, 10-year forecasts of both methods are also compared for different countries and extrapolation issues in the P-spline approach are also discussed. In particular, we found that the P-spline approach is sensitive to small perturbations and provides unrealistic forecasts for the USA and in the context of COVID-19.

- The basis: In the P-spline model, the basis consists of cubic B-splines, which are bell-shaped curves composed of smoothly joined polynomial pieces of degree 3. By construction, B-splines are defined in a local manner, i.e. each B-spline is non-zero only in a certain neighborhood of the mortality surface. On the other hand, our regularized mortality model uses a global two-dimensional polynomial basis, see (6). More flexibility can be included either by increasing the number of knots in the B-spline approach or by increasing the degree order in the global polynomial basis.
- The penalty: To smooth the mortality surface and avoid overfitting, a penalty term on the regression coefficients is introduced. In the P-spline model, the following penalty matrix is defined:$$\mathit{P}=\mathit{P}({\lambda}_{a},{\lambda}_{y})={\lambda}_{a}{\mathit{I}}_{{c}_{y}}\otimes {\mathit{D}}_{a}^{\u22ba}{\mathit{D}}_{a}+{\lambda}_{y}{\mathit{I}}_{{c}_{a}}\otimes {\mathit{D}}_{y}^{\u22ba}{\mathit{D}}_{y},$$On the other hand, in our regularized model, the penalty term is given by the elastic-net regularization:$$\mathit{P}=\mathit{P}(\lambda ,\alpha )=\lambda \left(\right)open="("\; close=")">\left(\right)open="("\; close=")">1-\alpha ,$$

#### 5.3.1. The Case of the Usa

**MortalitySmooth**chose $({\lambda}_{a},{\lambda}_{y})=(0.01,0.003)$ for a $BIC=5270$. As a consequence, the regression coefficients at the age of 50 are not smoothed and present a lot of variability as shown on the right panel of Figure 5. The erratic behaviour of the coefficients at age 50 is then spread to adjacent coefficients and to the entire extrapolated mortality surface. One way to circumvent this issue is to increase the smoothing parameter on the year dimension ${\lambda}_{y}$, which will slightly decrease the goodness-of-fit but improve the extrapolation and the corresponding forecasts. The left panel of Figure 5 depicts the forecast of death rates in the USA with ${\lambda}_{y}=20$ instead of ${\lambda}_{y}=0.003$. For ${\lambda}_{y}=20$, $BIC$ slightly increases to 5335 but offers much more realistic forecasts.

#### 5.3.2. The Case of COVID-19

#### 5.4. Model Uncertainty and Prediction Intervals

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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1. | This type of analysis was for instance performed in Section 6.1.1. in Cairns et al. (2006) or Section 3.3 in Haberman and Renshaw (2011). |

2. | Moreover, we consider the orthogonal polynomial basis through the polym R function to ensure numerical stability. |

3. | The estimated improvement rate on crude death rates is very noisy. The use of smoothed death rates allows one to have a more regular pattern for the improvement rate. |

4. | Except Hong Kong, as data from Hong Kong are only available from 1986 onwards. |

**Figure 1.**Surface of the logarithm of the raw death rates for age from 50 to 90 and year from 1970 to 2017 for French males.

**Figure 3.**Logarithm of crude death rates from year 1970 to 2017 and logarithm of extrapolated death rates from 2018 to 2050.

**Figure 4.**Logarithm of the fitted death rates and forecast death rates for USA by ages (50–90) over years (1986–2017). On the

**left**: forecast by the P-spline model. On the

**right**: forecast by our regularized elastic-net model.

**Figure 5.**On the

**left**: forecast by the P-spline model with ${\lambda}_{y}=20$. On the

**right**: regression coefficients in the P-spline model at age 50 with ${\lambda}_{y}=0.003$ and ${\lambda}_{y}=20$ and at age 90.

**Figure 6.**Logarithm of the fitted death rates and forecast death rates for France by ages (50–90) over years (1986–2030), after a pandemic shock in 2013 compensated by two years with lower mortality. On the

**left**: forecast by the P-spline model. On the

**right**: forecast by our regularized elastic-net model.

**Figure 7.**Logarithm of the fitted death rates and forecast death rates for France by ages (50–90) over years (1986–2030) after a pandemic shock in 2013, followed by a continuous mortality increase. On the

**left**: forecast by the P-spline model. On the

**right**: forecast by our regularized elastic-net model.

**Figure 8.**Fan charts for death rates ${\mu}_{x,t}$ at ages $x=65,x=75,x=85$ from the Lee–Carter model on the left and from the regularized elastic net model on the right to the French population. The dots show historical death rates from 1970–2007 and shades in the fan represent prediction intervals at the 50%, 80% and 95% level.

Mortality Model | Dynamics |
---|---|

Lee–Carter (LC) | $log{\mu}_{x,t}={\alpha}_{x}+{\beta}_{x}{\kappa}_{t}^{\left(1\right)}$ |

Cairns–Blake–Dowd (CBD) | $log{\mu}_{x,t}={\kappa}_{t}^{\left(1\right)}+(x-\overline{x}){\kappa}_{t}^{\left(2\right)}$ |

Age-Period-Cohort (APC) | $log{\mu}_{x,t}={\alpha}_{x}+{\kappa}_{t}^{\left(1\right)}+{\gamma}_{t-x}$ |

Renshaw–Haberman (RH) | $log{\mu}_{x,t}={\alpha}_{x}+{\beta}_{x}{\kappa}_{t}^{\left(1\right)}+{\gamma}_{t-x}$ |

M7 | $log{\mu}_{x,t}={\kappa}_{t}^{\left(1\right)}+(x-\overline{x}){\kappa}_{t}^{\left(2\right)}+({(x-\overline{x})}^{2}-{\overline{\sigma}}_{x}^{2}){\kappa}_{t}^{\left(3\right)}+{\gamma}_{t-x}$ |

**Table 2.**Mean Absolute Error for USA, UK, France, Belgium (BEL), Japan (JAP) and Hong Kong (HKG) for the different models of Table 1 and our regularized model (Reg). Fit based on historical data from 1986 to 2007. In bolds: Lowest Mean Absolute Error per Country and in italics: lowest BIC model (excluding Reg).

Mean Absolute Error (10-Year Forecast) | ||||||
---|---|---|---|---|---|---|

USA | UK | France | BEL | JAP | HKG | |

LC | 0.00369 | 0.00225 | 0.00120 | 0.00246 | 0.00213 | 0.00265 |

CBD | 0.00279 | 0.00205 | 0.00505 | 0.00388 | 0.00313 | 0.00261 |

APC | 0.00161 | 0.00251 | 0.00206 | 0.00338 | 0.00250 | 0.00168 |

RH | 0.00368 | 0.00463 | 0.00233 | 0.00308 | 0.00194 | 0.00275 |

M7 | 0.00222 | 0.00152 | 0.00259 | 0.00200 | 0.00097 | 0.00143 |

Reg | 0.00194 | 0.00212 | 0.00274 | 0.00244 | 0.00267 | 0.00214 |

**Table 3.**Similar table as Table 2 but based on historical data from 1970 to 2007.

Mean Absolute Error (10-Year Forecast) | |||||
---|---|---|---|---|---|

USA | UK | France | BEL | JAP | |

LC | 0.00432 | 0.00318 | 0.00147 | 0.00345 | 0.00213 |

CBD | 0.00262 | 0.00193 | 0.00490 | 0.00335 | 0.00355 |

APC | 0.00201 | 0.00265 | 0.00261 | 0.00370 | 0.00328 |

RH | 0.00197 | 0.00313 | 0.00293 | 0.00622 | 0.00167 |

M7 | 0.00407 | 0.00195 | 0.00227 | 0.00142 | 0.00298 |

Reg | 0.00288 | 0.00307 | 0.00262 | 0.00287 | 0.00310 |

**Table 4.**Mean Absolute Error for USA, UK, France, Belgium (BEL), Japan (JAP) and Hong Kong (HKG) for the P-spline model in Currie et al. (2004) and our regularized model, based on historical data from 1986 to 2007. In bold: Lowest MAE per Country.

Mean Absolute Error (10-Year Forecast) | ||||||
---|---|---|---|---|---|---|

USA | UK | France | BEL | JAP | HKG | |

P-splines | 0.01405 | 0.00486 | 0.00232 | 0.00695 | 0.00384 | 0.00271 |

Regularized | 0.00194 | 0.00212 | 0.00274 | 0.00244 | 0.00267 | 0.00214 |

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**MDPI and ACS Style**

Barigou, K.; Loisel, S.; Salhi, Y.
Parsimonious Predictive Mortality Modeling by Regularization and Cross-Validation with and without Covid-Type Effect. *Risks* **2021**, *9*, 5.
https://doi.org/10.3390/risks9010005

**AMA Style**

Barigou K, Loisel S, Salhi Y.
Parsimonious Predictive Mortality Modeling by Regularization and Cross-Validation with and without Covid-Type Effect. *Risks*. 2021; 9(1):5.
https://doi.org/10.3390/risks9010005

**Chicago/Turabian Style**

Barigou, Karim, Stéphane Loisel, and Yahia Salhi.
2021. "Parsimonious Predictive Mortality Modeling by Regularization and Cross-Validation with and without Covid-Type Effect" *Risks* 9, no. 1: 5.
https://doi.org/10.3390/risks9010005