# Optimal Neighborhood Selection for AR-ARCH Random Fields with Application to Mortality

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Random Fields Memory Models for Mortality Improvement Rates

#### 2.1. Classic Mortality Models

#### 2.2. Through a Random Field Framework

- (i)
- Mortality Surface

- (ii)
- Random Field Memory Models

- A-1
- $\parallel F({x}_{0},\theta ,\xi ){\parallel}_{p}<\infty $ for some ${x}_{0}\in {\mathbb{R}}^{V}$,
- A-2
- $\parallel F({x}^{\prime},\theta ,\xi )-F(x,\theta ,\xi ){\parallel}_{p}<{\sum}_{v\in V}{\alpha}_{v}\parallel {x}_{v}^{\prime}-{x}_{v}\parallel $ for all $x={\left({x}_{v}\right)}_{v\in V},{x}^{\prime}={\left({x}_{v}^{\prime}\right)}_{v\in V}\in {\mathbb{R}}^{V}$, where the coefficients ${\alpha}_{v}$ are such that ${\sum}_{v\in V}{\alpha}_{v}<1$.

#### 2.3. AR-ARCH-Type Random Fields

**A-1**and

**A-2**needed to ensure the existence and uniqueness of a stationary solution. Indeed, note that the function F is given by

## 3. Estimation, Asymptotic Inference, and Model Selection

#### 3.1. Quasi-Maximum Likelihood Estimator (QMLE)

**Hypothesis 1**

**(H1)**.

**Hypothesis 2**

**(H2)**.

**Hypothesis 3**

**(H3)**.

**Hypothesis 4**

**(H4)**.

**Theorem**

**1**.

- (i)
- Consistency. If the assumptions H1, H2, and H3 hold, then the QMLE estimator ${\widehat{\theta}}_{T}$ in Equation (11) is consistent in the sense that for each ${\left({\beta}_{v}\right)}_{v\in {V}_{1}},{\left({\beta}_{v}^{\prime}\right)}_{v\in {V}_{1}},{\left({\alpha}_{v}\right)}_{v\in {V}_{2}}$, and ${\left({\alpha}_{v}^{\prime}\right)}_{v\in {V}_{2}}$, we have$${\widehat{\theta}}_{T}\stackrel{\mathbb{P}}{\u27f6}{\theta}^{0}.$$
- (ii)
- Asymptotic normality. Under assumptions H1, H2, H3, and H4,

#### 3.2. Optimal Model Selection

- (i)
- Likelihood ratio test statistic (LRTS)

- (ii)
- Penalized QMLE

- I-1
- ${\mathcal{V}}_{1}$, ${\mathcal{V}}_{2}$ (maximal possible neighborhoods) are a finite set and ${V}_{1}^{0}\subset {\mathcal{V}}_{1}$, ${V}_{2}^{0}\subset {\mathcal{V}}_{2}$.
- I-2
- ${a}_{T}(.)$ is increasing, ${a}_{T}\left({k}_{1}\right)-{a}_{T}\left({k}_{2}\right)$ goes to infinity as T tends to go to infinity as soon as ${k}_{1}>{k}_{2}$, and ${a}_{T}\left(k\right)/T$ tends to 0 as T tends to go to infinity for any $k\in \mathbb{N}$.

**Theorem**

**3.**

**Proof.**

- (iii)
- Bayesian Information Criterion

- (iv)
- Search Algorithm

`R`, the non-linear optimization routines in most packages will be sufficient to find the values of the parameters $\theta $ that maximize the quasi-likelihood. This can be, for instance, implemented based on some derivative-free algorithms, such as the Nelder–Mead, using heuristics to search for the maximizers under the constraint of stationarity (8) or (9). However, these methods have excessive run-times and the characterization of the optimal model can take some days or even months. Therefore, in order to enhance this step, we can supply the optimization algorithm with the closed form formulas of the gradient and Hessian of the likelihood function g; see Appendix A. In our case, the execution time for implementing the neighborhood selection procedure on the maximal neighborhoods in Figure 2 using

`R`was expected to take too much time. Therefore, we have solved this by considering parallel calculations and by making use of external compiled languages to speed up the selection routine.

- (v)
- Spatial Autocorrelation Function

## 4. Numerical and Empirical Analyses

#### 4.1. Simulation Study

#### 4.2. Real-World Data Application

- (i)
- Model Selection

- (ii)
- Diagnostic Checks

- (iii)
- Predictive Performance

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Appendix The Gradient and the Hessian of the Quasi-Likelihood Function $\mathit{g}$

## Appendix B. Additional Figures

#### Appendix B.1. Diagnostic Checks of Models’ Residuals

**Figure A1.**Autocorrelation: Inspection of spatial autocorrelation of the residuals ${\widehat{\xi}}_{s}$ using the definition of the spatial autocorrelation function given in Equation (16).

**Figure A2.**QQ-plots: Visual inspection of the normality of the residuals. Drawing of the QQ-plots is based on 100 Monte-Carlo simulations and comparisons between the obtained empirical quantiles of ${\xi}_{s}$ with the theoretical quantiles of a centered Gaussian i.i.d. random field.

**Figure A3.**Histograms of residuals: Comparison of the historical residuals’ densities (histograms) and a standard Gaussian density (lines).

#### Appendix B.2. Out-of-Sample Analysis and Predictive Performance

**Figure A4.**Life expectancy: Forecast life expectancy over the period of 2000–2030 at ages $65,75$, and 85 for the France, England and Wales, and US populations produced by both the selected AR-ARCH (black) model and Lee–Carter (red) model. The 95% prediction intervals are also included. The crude mortality over the observed period is also presented (black circles).

**Figure A5.**Mortality rates: Forecast mortality rates over the period of 2000–2030 at ages $65,75$, and 85 for the France, England and Wales, and US populations produced by both the selected AR-ARCH (black) model and Lee–Carter (red) model. The 95% prediction intervals are also included. The crude mortality over the observed period is also presented (black circles).

## References

- Barrieu, P.; Bensusan, H.; El Karoui, N.; Hillairet, C.; Loisel, S.; Ravanelli, C.; Salhi, Y. Understanding, modelling and managing longevity risk: Key issues and main challenges. Scand. Actuar. J.
**2012**, 2012, 203–231. [Google Scholar] [CrossRef] [Green Version] - Lee, R.D.; Carter, L.R. Modeling and forecasting US mortality. J. Am. Stat. Assoc.
**1992**, 87, 659–671. [Google Scholar] - Hunt, A.; Villegas, A.M. Robustness and convergence in the Lee–Carter model with cohort effects. Insur. Math. Econ.
**2015**, 64, 186–202. [Google Scholar] [CrossRef] - Doukhan, P.; Pommeret, D.; Rynkiewicz, J.; Salhi, Y. A Class of Random Field Memory Models for Mortality Forecasting. Insur. Math. Econ.
**2017**, 77, 97–110. [Google Scholar] [CrossRef] [Green Version] - Doukhan, P.; Truquet, L. A fixed point approach to model random fields. ALEA: Lat. Am. J. Probab. Math. Stat.
**2007**, 3, 111–132. [Google Scholar] - Cairns, A.J.G.; Blake, D.; Dowd, K. A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. J. Risk Insur.
**2006**, 73, 687–718. [Google Scholar] [CrossRef] - Renshaw, A.E.; Haberman, S. A cohort-based extension to the Lee–Carter model for mortality reduction factors. Insur. Math. Econ.
**2006**, 38, 556–570. [Google Scholar] [CrossRef] - Cairns, A.J.G.; Blake, D.; Dowd, K.; Coughlan, G.D.; Epstein, D.; Ong, A.; Balevich, I. A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. N. Am. Actuar. J.
**2009**, 13, 1–35. [Google Scholar] [CrossRef] - Dowd, K.; Cairns, A.J.G.; Blake, D.; Coughlan, G.D.; Epstein, D.; Khalaf-Allah, M. Evaluating the goodness of fit of stochastic mortality models. Insur. Math. Econ.
**2010**, 47, 255–265. [Google Scholar] [CrossRef] - Mavros, G.; Cairns, A.J.G.; Kleinow, T.; Streftaris, G. Stochastic Mortality Modelling: Key Drivers and Dependent Residuals. N. Am. Actuar. J.
**2016**, 21, 343–368. [Google Scholar] [CrossRef] - Giacometti, R.; Bertocchi, M.; Rachev, S.T.; Fabozzi, F.J. A comparison of the Lee–Carter model and AR–ARCH model for forecasting mortality rates. Insur. Math. Econ.
**2012**, 50, 85–93. [Google Scholar] [CrossRef] - Chai, C.M.H.; Siu, T.K.; Zhou, X. A double-exponential GARCH model for stochastic mortality. Eur. Actuar. J.
**2013**, 3, 385–406. [Google Scholar] [CrossRef] - Lee, R.; Miller, T. Evaluating the performance of the Lee-Carter method for forecasting mortality. Demography
**2001**, 38, 537–549. [Google Scholar] [CrossRef] - Gao, Q.; Hu, C. Dynamic mortality factor model with conditional heteroskedasticity. Insur. Math. Econ.
**2009**, 45, 410–423. [Google Scholar] [CrossRef] - Brouhns, N.; Denuit, M.; Vermunt, J.K. A Poisson log-bilinear regression approach to the construction of projected lifetables. Insur. Math. Econ.
**2002**, 31, 373–393. [Google Scholar] [CrossRef] [Green Version] - Li, J.; Pitt, D.; Li, H. Dispersion modelling of mortality for both sexes with Tweedie distributions. Scand. Actuar. J.
**2021**, 1–19. [Google Scholar] [CrossRef] - Willets, R.C. The cohort effect: Insights and explanations. Br. Actuar. J.
**2004**, 10, 833–877. [Google Scholar] [CrossRef] - Hitaj, A.; Mercuri, L.; Rroji, E. Lévy CARMA models for shocks in mortality. Decis. Econ. Financ.
**2019**, 42, 205–227. [Google Scholar] [CrossRef] - Loisel, S.; Serant, D. In the Core of Longevity Risk: Hidden Dependence in Stochastic Mortality Models and Cut-Offs in Prices of Longevity Swaps. 2007. Available online: https://hal.archives-ouvertes.fr/hal-00201393/ (accessed on 18 December 2016).
- Jevtić, P.; Luciano, E.; Vigna, E. Mortality surface by means of continuous time cohort models. Insur. Math. Econ.
**2013**, 53, 122–133. [Google Scholar] [CrossRef] - Lazar, D.; Denuit, M.M. A multivariate time series approach to projected life tables. Appl. Stoch. Model. Bus. Ind.
**2009**, 25, 806–823. [Google Scholar] [CrossRef] - Li, H.; Lu, Y. Coherent Forecasting of Mortality Rates: A Sparse AutoRegression Approach. ASTIN Bull.
**2017**, 47, 1–24. [Google Scholar] [CrossRef] - Gaille, S.; Sherris, M. Modelling mortality with common stochastic long-run trends. Geneva Pap. Risk Insur. Issues Pract.
**2011**, 36, 595–621. [Google Scholar] [CrossRef] [Green Version] - Salhi, Y.; Loisel, S. Basis risk modelling: A co-integration based approach. Statistics
**2017**, 51, 205–221. [Google Scholar] [CrossRef] [Green Version] - Millossovich, P.; Biffis, E. A bidimensional approach to mortality risk. Decis. Econ. Financ.
**2006**, 29, 71–94. [Google Scholar] - Olivieri, A.; Pitacco, E. Life tables in actuarial models: From the deterministic setting to a Bayesian approach. Adv. Stat. Anal.
**2012**, 2012, 127–153. [Google Scholar] [CrossRef] - Loubaton, P. Champs Stationnaires au Sens Large sur Z
^{2}: Propriétés Structurelles et Modèles Paramétriques; Ecole Nationale Supérieure des Télécommunications: Paris, France, 1989. [Google Scholar] - Guyon, X. Random Fields on a Network: Modeling, Statistics, and Applications; Springer Science & Business Media: Berlin/Heidelberg, Germany, 1995. [Google Scholar]
- Engle, R.F. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econom. J. Econom. Soc.
**1982**, 50, 987–1007. [Google Scholar] [CrossRef] - Chen, H.; MacMinn, R.; Sun, T. Multi-population mortality models: A factor copula approach. Insur. Math. Econ.
**2015**, 63, 135–146. [Google Scholar] [CrossRef] - Van der Vaart, A. Asymptotic Statistics; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Vuong, Q.H. Likelihood ratio Tests for model selection and non-nested hypotheses. Econometrica
**1989**, 57, 307–333. [Google Scholar] [CrossRef] [Green Version] - Razali, N.M.; Wah, Y.B. Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests. J. Stat. Model. Anal.
**2011**, 2, 21–33. [Google Scholar] - Cairns, A.; Blake, D.; Dowd, K.; Kessler, A. Phantoms never die: Living with unreliable population data. J. R. Stat. Soc. Ser. A
**2016**, 179, 975–1005. [Google Scholar] [CrossRef] [Green Version] - Lee, R. The Lee-Carter method for forecasting mortality, with various extensions and applications. N. Am. Actuar. J.
**2000**, 4, 80–91. [Google Scholar] [CrossRef] - Liu, Y.; Li, J.H. The Locally Linear Cairns–Blake–Dowd Model: A Note On Delta–Nuga Hedging of Longevity Risk. ASTIN Bull.
**2017**, 47, 79–151. [Google Scholar] [CrossRef] - Gneiting, T.; Raftery, A.E. Strictly proper scoring rules, prediction, and estimation. J. Am. Stat. Assoc.
**2007**, 102, 359–378. [Google Scholar] [CrossRef] - Shang, H.L. Point and interval forecasts of age-specific life expectancies: A model averaging approach. Demogr. Res.
**2012**, 27, 593–644. [Google Scholar] [CrossRef] [Green Version] - Doukhan, P. Mathematics and Applications; Stochastic Models for Time Series; Springer: Berlin/Heidelberg, Germany, 2018; Volume 80. [Google Scholar]
- Ferland, R.; Latour, A.; Oraichi, D. Integer-valued GARCH processes. J. Time Ser. Anal.
**2006**, 27, 923–942. [Google Scholar] [CrossRef]

**Figure 1.**Causal random field: A bi-dimensional representation of the random field ${X}_{s}={\mathrm{IR}}_{s}-\overline{\mathrm{IR}}$ with $s=(a,t)$. The grayed area represents the causal neighbor V needed to characterize the evolution of ${X}_{s}$.

**Figure 2.**Maximal neighborhoods: Initial (maximal) candidate neighborhoods ${\mathcal{V}}_{1}\equiv {\mathcal{V}}_{2}$ given as a collection of points $(i,j)\in \left\{\right(1,0);(1,1);(0,1);(1,2);(2,1);(2,2);(0,2);(2,0\left)\right\}$.

**Figure 3.**Autocorrelation: Autocorrelation of the random field and its squared values given in absolute value following the definition in Equation (16).

**Figure 4.**Model selection: Estimation of the parameters for the optimal model, maximizing the penalized QMLE. The grayed area corresponds to parameters that were not selected by the procedure and are excluded from the neighborhoods.

**Figure 5.**Standardized residuals: Heatmaps for the standardized residuals produced by the Lee–Carter (

**left**panel) and the selected AR-ARCH model (

**right**panel) for the French, England and Wales, and United States male populations.

**Figure 6.**Mortality rates: Forecast mortality rates at age $65,75$, and 85 for France, England and Wales, and US populations produced by both the selected AR-ARCH (black) model and Lee–Carter (red) model. The 95% prediction intervals are also included.

**Figure 7.**Life expectancy: Forecast life expectancy at age $65,75$, and 85 for the France, England and Wales, and US populations produced by both the selected AR-ARCH (black) model and Lee–Carter (red) model. The 95% prediction intervals are also included.

**Figure 8.**Confidence interval forecast score: The mean interval score across ages for different forecasting years for the LC, AR-ARCH, and CBD two-factor models.

**Table 1.**Model selection: Monte-Carlo assessment of relative performance of the selection method across data sizes based on 1000 replications.

${\mathit{V}}_{2}$ | ${\mathit{V}}_{1}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

(1,1) | (2,2) | (0,1) | (1,0) | ${\alpha}_{\mathbf{0}}$ | (1,1) | (2,2) | (0,1) | (1,0) | ||

$\mathit{I}=\mathbf{30},\phantom{\rule{0.166667em}{0ex}}\mathit{J}=\mathbf{100}$ | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 64.8% |

1 | 1 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 11.1% | |

1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 7.2% | |

1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 6.1% | |

$\mathit{I}=\mathbf{30},\phantom{\rule{0.166667em}{0ex}}\mathit{J}=\mathbf{40}$ | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 42.30% |

1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 12.00% | |

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 8.1% | |

1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 8.1% | |

1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 6.4% | |

1 | 1 | 0 | 0 | 1 | 1 | 0 | 1 | 0 | 5.4% |

**Table 2.**Normality tests: We perform normality tests of the residuals ${\widehat{\xi}}_{s}$ using the Shapiro–Wilk (SW) test, Anderson–Darling statistics (AD), the Cramér-von Mises test (CVM), Pearson-${\chi}^{2}$ (PC) test, and the Shapiro–Francia (CF) test. The table shows the p values of the tests.

SW | AD | CVM | PC | SF | |
---|---|---|---|---|---|

USA | 0.3428 | 0.7048 | 0.7991 | 0.8523 | 0.3947 |

FRA | 0.6171 | 0.5212 | 0.5438 | 0.6279 | 0.4322 |

GBP | 0.6114 | 0.7626 | 0.8548 | 0.9309 | 0.4827 |

**Table 3.**Point forecast accuracy: Forecast average errors between the LC, AR-ARCH, and CBD (see [6]) two-factor model both for mortality rates and life expectancies over the period of 2000–2016.

AR-ARCH | LC | CBD | |||
---|---|---|---|---|---|

USA | Mortality rate | RMSFE | 1.51 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 5.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.68 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

MAFE | 2.47 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 3.99 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 4.14 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | ||

Life expectancy | RMSFE | 1.13 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.69 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.72 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | |

MAFE | 2.75 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 3.42 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 4.34 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | ||

FRA | Mortality rate | RMSFE | 1.07 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.30 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 5.04 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

MAFE | 2.19 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 2.84 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 4.49 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | ||

Life expectancy | RMSFE | 1.66 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 2.11 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 4.01 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | |

MAFE | 3.28 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 3.79 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 5.24 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | ||

GBR | Mortality rate | RMSFE | 4.11 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 8.34 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 8.01 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

MAFE | 4.36 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 5.83 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 5.72 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | ||

Life expectancy | RMSFE | 7.29 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 8.64 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 8.70 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | |

MAFE | 7.03 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 7.87 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 7.89 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Doukhan, P.; Rynkiewicz, J.; Salhi, Y.
Optimal Neighborhood Selection for AR-ARCH Random Fields with Application to Mortality. *Stats* **2022**, *5*, 26-51.
https://doi.org/10.3390/stats5010003

**AMA Style**

Doukhan P, Rynkiewicz J, Salhi Y.
Optimal Neighborhood Selection for AR-ARCH Random Fields with Application to Mortality. *Stats*. 2022; 5(1):26-51.
https://doi.org/10.3390/stats5010003

**Chicago/Turabian Style**

Doukhan, Paul, Joseph Rynkiewicz, and Yahia Salhi.
2022. "Optimal Neighborhood Selection for AR-ARCH Random Fields with Application to Mortality" *Stats* 5, no. 1: 26-51.
https://doi.org/10.3390/stats5010003