Leah Guichard

Ph, Anesthésiste-Algologue Valeria Martinez

Douleur

Surtitre prise en charge Prévenir la chronicisation de la douleur postopératoire Auteurs

Le dépistage des patients vulnérables à la douleur dès la consultation d'anesthésie doit être une priorité pour prévenir le fait que celle-ci ne devienne une douleur chronique postopératoire. Les évolutions chirurgicales et pharmacologiques, le développement de l'analgésie locorégionale et le dépistage précoce des douleurs neuropathiques devraient permettre une prise en charge adaptée afin d'éviter la chronicisation. © 2018 Mots clés -analgésie locorégionale ; dépistage ; douleur chronique ; prévention La douleur chronique postchirurgicale (DCPC) est définie comme une douleur apparaissant après la chirurgie, persistant plus de 2 mois, et sans lien avec une complication postchirurgicale ni avec une douleur préopératoire. C'est une pathologie iatrogénique, fréquente et invalidante [1,2]. En France, les douleurs chroniques postopératoires (DCPO) constitueraient environ 20 % des motifs de consultation des centres d'évaluation et de traitement de la douleur. Chaque année, des milliers de patients opérés présentent une DCPC et mettent en échec un acte chirurgical considéré comme réussi. Plusieurs facteurs de risques de la DCPO sont aujourd'hui bien connus. Certains moyens de prévention sont recommandés et peuvent être mis en place tout au long du parcours du patient en périopératoire. T1 Vulnérabilité préopératoire à la douleur Il est reconnu aujourd'hui que certains patients sont plus vulnérables que d'autres à développer une DCPC. La recherche de ces facteurs est essentielle afin d'adapter et cibler davantage la prise en charge.

T2 Présence d'une douleur préopératoire et prise d'opioïdes

La présence d'une douleur préopératoire, au niveau du site chirurgical, est souvent associée à la survenue d'une DCPC. Cela a été montré dans plusieurs modèles chirurgicaux. Les caractéristiques de la douleur préopératoire telles que la durée et les caractéristiques neuropathiques semblent également avoir une implication dans la survenue de la DCPC. L'existence d'une douleur chronique dans une région, différente du site opératoire, a aussi été identifiée comme un facteur de risque. La prise de morphiniques en préopératoire, reflet de la présence de douleur, semble également être un vecteur de risque de DCPC dans plusieurs modèles chirurgicaux. Ces deux facteurs sont responsables d'une sensibilisation du système nerveux central.

T2 Vulnérabilité psychologique

Une revue méthodique et critique de la littérature a montré que la dépression, la vulnérabilité psychologique et le stress étaient les facteurs psychologiques les plus à risque d'induire une DCPC [START_REF] Hinrichs-Rocker | Psychosocial predictors and correlates for chronic post-surgical pain (CPSP) -a systematic review[END_REF]. Or, ces facteurs ne sont pas les seuls mis en cause. D'autres études, principalement prospectives, ont exploré l'influence du catastrophisme [START_REF] Sullivan | Psychological determinants of problematic outcomes following Total Knee Arthroplasty[END_REF] et de l'état d'anxiété préopératoire [START_REF] Theunissen | Preoperative Anxiety and Catastrophizing: A Systematic Review and Meta-analysis of the Association With Chronic Postsurgical Pain[END_REF].

T2 Prédisposition génétique

Notre capital génétique influence la modulation de la douleur et l'évolution vers une douleur chronique après un traumatisme chirurgical. Des facteurs génétiques liés aux polymorphismes des canaux sodiques, calciques ou potassiques prédisposant à la douleur neuropathique émergent des études expérimentales. Certains de ces polymorphismes se confirment aussi chez l'homme : le polymorphisme des canaux calciques CACNG2 prédispose notamment aux douleurs neuropathiques après chirurgie du sein, et celui des canaux potassiques KCNS1 à la douleur neuropathique après une lésion nerveuse traumatique [START_REF] Costigan | Multiple chronic pain states are associated with a common amino acid-changing allele in KCNS1[END_REF].

T1 Lésion nerveuse peropératoire L'existence d'une lésion nerveuse paraît être un facteur important dans le développement de la DCPC [START_REF] Kehlet | Persistent postsurgical pain: risk factors and prevention[END_REF]. La part de la douleur neuropathique dans ce cadre est importante et varie de 6 à 69 % selon la chirurgie [START_REF] Haroutiunian | The neuropathic component in persistent postsurgical pain: a systematic literature review[END_REF]. Une lésion nerveuse est souvent au premier plan dans les modèles chirurgicaux les plus à risque d'induire une DCPC. On peut citer la lésion du nerf intercostal pour la thoracotomie, les lésions du nerf intercosto-brachial pour la chirurgie du sein avec curage ganglionnaire, etc. Cependant, de nombreux travaux, qui ont mesuré la fonction nerveuse par examen sensoriel quantifié ou électrophysiologie, ont montré que la lésion nerveuse n'était pas toujours suivie de douleur. Les données de la littérature permettent d'estimer le rapport entre douleur neuropathique et lésion nerveuse à un tiers. Les traumatismes nerveux au cours de la chirurgie peuvent être de différents ordres : section, étirement, écrasement. Pour certains, la section franche du nerf conduirait à moins de douleur dans la chirurgie de la hernie inguinale, mais ceci n'est pas confirmé par tous. La DCPC se rencontre dans tous les types de chirurgie, mais celles qui entraînent le plus de douleurs chroniques sont la thoracotomie avec une prévalence de 14 à 83 %, la chirurgie mammaire avec 24 à 84 %, la sternotomie 17-30 %, et enfin, le prélèvement de crête avec 12-35 % [START_REF] Kehlet | Persistent postsurgical pain: risk factors and prevention[END_REF][START_REF] Haroutiunian | The neuropathic component in persistent postsurgical pain: a systematic literature review[END_REF][START_REF] Duale | Neuropathic aspects of persistent postsurgical pain: a French multicenter survey with a 6-month prospective follow-up[END_REF] (figure 1).

T1 Caractéristiques prédictives de la douleur postopératoire

Plusieurs études ont mis en évidence que certaines caractéristiques de la douleur postopératoire sont prédictives de chronicisation. Trois émergent de la littérature : une intensité élevée, une résolution lente, des caractéristiques neuropathiques précoces. Une relation entre l'intensité de la douleur aiguë et la présence d'une DCPC a été rapportée dans plusieurs chirurgies : sein, thorax, amputation, cure de hernie inguinale, arthroplastie de la hanche et du genou. Ces chirurgies sont plus ou moins douloureuses et impliquent un traumatisme tissulaire et nerveux très variable. Le temps passé avec une douleur sévère dans les premières 24 heures postopératoires s'est avéré un facteur prédictif de chronicisation. La présence de douleurs neuropathiques aiguës durant la période postopératoire immédiate est significativement associée au développement de douleurs neuropathiques chroniques survenues trois mois après une chirurgie thoracique majeure et un prélèvement de crête [START_REF] Martinez | Risk factors predictive of chronic postsurgical neuropathic pain: the value of the iliac crest bone harvest model[END_REF][START_REF] Searle | Can chronic neuropathic pain following thoracic surgery be predicted during the postoperative period?[END_REF]. Ces études ont également montré que les patients présentant des scores élevés aux outils de dépistage de la douleur neuropathique (questionnaire DN4 ou LAANS) avaient une plus forte probabilité de développer une douleur neuropathique chronique importante. Cependant, il apparaît que la valeur prédictive la plus élevée se situe à un mois, lors de la première consultation postopératoire avec le chirurgien [START_REF] Martinez | Risk factors predictive of chronic postsurgical neuropathic pain: the value of the iliac crest bone harvest model[END_REF].

T1 Prévention primaire de la DCPO

Il est important de connaître les facteurs de risque de la DCPC afin de les rechercher dès la consultation d'anesthésie. Cela permettra de proposer une stratégie chirurgicale, anesthésique et un suivi postopératoire adaptés.

T2 Consultation d'anesthésie

Lors de la consultation préopératoire, l'intensité et la durée de l'évolution de la douleur au niveau du site concerné ou à distance sont manifestement les signes d'une plus grande vulnérabilité à la douleur. Les antécédents de douleur neuropathique ou de syndrome douloureux régional complexe (SDRC) post-traumatiques doivent être renseignés. La vulnérabilité psychologique est un élément à prendre également en considération. Une évaluation préopératoire de l'anxiété et du besoin d'information (score Apais) avec une prise en charge adaptée permet de diminuer le risque de douleur sévère postopératoire et par conséquent de DCPC [START_REF] Theunissen | Preoperative Anxiety and Catastrophizing: A Systematic Review and Meta-analysis of the Association With Chronic Postsurgical Pain[END_REF][START_REF] Aubrun | Réactualisation de la recommandation sur la douleur postopératoire[END_REF]. Au sein de notre équipe, nous avons mis en place un tampon "vulnérabilité douleur" sur le dossier d'anesthésie, lors de la consultation, permettant de cibler la prise en charge de ces patients et de les suivre de façon plus spécifique en postopératoire.

T2 Épargne nerveuse et expérience du chirurgien

Le chirurgien joue un rôle important dans la prévention de la DCPC. Le choix de la technique, de la voie d'abord ainsi que sa dextérité auront des répercussions directes sur la limitation des lésions nerveuses et des attritions tissulaires.

TEG1 Deux études confirment cet aspect, la première concerne la chirurgie thoracique qui montre que l'incidence de la DCPC peut varier du simple au double en fonction du chirurgien prenant en charge le patient [START_REF] Maguire | A study exploring the role of intercostal nerve damage in chronic pain after thoracic surgery[END_REF]. La seconde compare l'incidence de la DCPC postmastectomie de deux types de structures de santé qui se distinguent par le nombre d'interventions réalisées annuellement. Il apparaît clairement que l'incidence de la DCPC et du membre fantôme est moins élevée dans les structures les plus expérimentées [START_REF] Tasmuth | Chronic post-treatment symptoms in patients with breast cancer operated in different surgical units[END_REF]. Les recommandations récentes sur les pratiques chirurgicales visant à réduire l'incidence de la DCPC, notamment après une cure de hernie inguinale, préconisent de visualiser les nerfs et leurs trajets sur le site opératoire et de ne pas les endommager quand cela est possible [START_REF] Alfieri | International guidelines for prevention and management of post-operative chronic pain following inguinal hernia surgery[END_REF]. TEG1 Les chirurgies les plus à risque d'induire de la DCPC ont cependant fait l'objet d'innovations techniques notables au cours de ces dix dernières années. Ainsi, l'exérèse du ganglion sentinelle limite le curage ganglionnaire dans la chirurgie du sein, le développement de la thoracoscopie épargne le nerf intercostal dans la chirurgie pulmonaire. La laparoscopie pour la chirurgie de la hernie inguinale, la fixation de plaques par colle biologique en cas d'abord laparoscopique, la neurectomie en cas d'abord direct, la non fermeture du péritoine pour la césarienne, la chirurgie mini-invasive en orthopédie constituent autant de facteurs qui tendent à réduire l'incidence de la DCPC.

T2 Prévention pharmacologique

Plusieurs études pharmacologiques ont été menées pour évaluer les effets préventifs des médicaments donnés en périopératoire sur la douleur chronique postopératoire. T3 Kétamine À ce jour, seule la kétamine a montré une réduction modeste mais significative de l'incidence de la douleur chronique. L'utilisation à faible dose de kétamine durant l'intervention chirurgicale est recommandée par la Société française d'anesthésie et de réanimation (Sfar) [START_REF] Aubrun | Réactualisation de la recommandation sur la douleur postopératoire[END_REF]. Les dernières recommandations formalisées d'expert (RFE) préconisent en première intention son administration en bolus intraveineux à la dose de 0,5 mg.kg-1 ± administration continue à la dose de 0,125 à 0,25 mg/kg/h lors des chirurgies à risque de douleur aiguë intense ou pourvoyeuse de DCPC, ainsi que chez les patients vulnérables à la douleur. Cette administration doit être réalisée au bloc opératoire sous anesthésie générale afin de limiter les effets psychodysleptiques pour les patients [START_REF] Aubrun | Réactualisation de la recommandation sur la douleur postopératoire[END_REF]. L'utilisation de la kétamine permet une réduction de l'intensité douloureuse et des besoins en opioïdes en postopératoire immédiat, ainsi que le risque de nausées-vomissements.

T3 Lidocaïne

Il est probablement recommandé d'administrer de la lidocaı ̈ne en intraveineux et en continu à la dose de 1 à 2 mg/kg en bolus intraveineux, suivi de 1 à 2 mg/kg/h, chez les patients adultes opeŕeś d'une chirurgie majeure (abdomino-pelvienne, rachidienne) et ne bénéficiant pas d'une analgésie peŕinerveuse ou péridurale concomitante dans le but de diminuer la douleur postopératoire et d'améliorer la réhabilitation [START_REF] Aubrun | Réactualisation de la recommandation sur la douleur postopératoire[END_REF]. À ce jour, nous disposons seulement de 4 études incluant un total de 357 patients évaluant les effets de l'administration périopératoire de la lidocaïne sur la douleur chronique. Les résultats restent contradictoires et le nombre total des patients inclus est inférieur à la taille d'échantillon optimal pour conclure.

T3 Gabapentinoïdes

La gabapentine (Neurontin®) et la prégabaline (Lyrica®) sont des inhibiteurs des canaux calcium voltage dépendant. Leur utilisation n'a pas montré d'effet préventif sur la DCPC. Cette classe thérapeutique utilisée en périopératoire serait à l'origine d'une sédation fréquente et importante contre-indiquant son utilisation en chirurgie courte et/ou ambulatoire. Une récente méta-analyse incluant 15 études (1 874 patients) a montré que la prégabaline donnée en périopératoire, en dose unique ou de façon prolongée, ne prévenait pas la DCPC [START_REF] Martinez | Perioperative pregabalin administration does not prevent chronic postoperative pain: systematic review with a meta-analysis of randomized trials[END_REF].

T3 Protoxyde d'azote

Le protoxyde d'azote est un antagoniste non compétitif des récepteurs NMDA (N-méthyl-Daspartate). Une seule étude clinique a montré une réduction importante de la DCPC après une chirurgie non cardiaque lors d'une administration de protoxyde d'azote durant la chirurgie [START_REF] Chan | Chronic postsurgical pain after nitrous oxide anesthesia[END_REF]. L'effet préventif observé dans cette étude était important avec une réduction de plus de 50 % de l'incidence de la DCPC [START_REF] Chan | Chronic postsurgical pain after nitrous oxide anesthesia[END_REF].

T3 Analgésie locale et locorégionale

Le rôle de l'anesthésie locale et locorégionale dans la prévention de la chronicisation des douleurs est encore sujet à controverse. Les études dans ce domaine restent peu nombreuses et ont des effectifs faibles. Cependant, plusieurs arguments physiologiques et une méta-analyse récente ont montré que certains blocs auraient une efficacité préventive dans le développement de douleurs chroniques dans les deux chirurgies engendrant le plus de douleur neuropathique. Le bloc paravertébral préviendrait la DCPC dans la chirurgie mammaire tandis que la péridurale thoracique préviendrait la douleur chronique post-thoracotomie [START_REF] Andreae | Local anaesthetics and regional anaesthesia for preventing chronic pain after surgery[END_REF]. L'infiltration locale continue pour le prélèvement de greffe iliaque, ainsi que le bloc paravertébral pour la chirurgie mammaire, sont recommandés par la Sfar pour leur innocuité, même si le nombre limité de patients inclus dans ces différentes études ne permet pas de conclure avec certitude à l'efficacité de ces techniques.

T1 Prévention secondaire

La prévention secondaire consiste essentiellement dans un dépistage précoce postopératoire des patients les plus susceptibles de développer une DCPC. Les éléments qui doivent nous alerter sont en premier lieu une trajectoire douloureuse élevée en postopératoire [START_REF] Althaus | Distinguishing between pain intensity and pain resolution: Using acute post-surgical pain trajectories to predict chronic post-surgical pain[END_REF], une durée prolongée de douleur intense [START_REF] Fletcher | Chronic postsurgical pain in Europe: An observational study[END_REF] et des caractéristiques de douleur neuropathique [START_REF] Beloeil | Early postoperative neuropathic pain assessed by the DN4 score predicts an increased risk of persistent postsurgical neuropathic pain[END_REF]. TEG1 Une relation entre l'intensité de la douleur aiguë et la présence d'une DCPC a été rapportée dans plusieurs chirurgies. La douleur aiguë sévère après arthroplastie de la hanche multiplie par 10 le risque de développer une DCPC [START_REF] Puolakka | Persistent pain following knee arthroplasty[END_REF]. L'hyperalgésie secondaire a pu être mesurée en postopératoire chez les patients en calculant la surface d'hypersensibilité autour de la cicatrice. Cette hyperalgésie péricicatricielle a été utilisée comme témoin de la sensibilisation centrale, et ses variations comme preuves de l'effet anti-hyperalgique ou hyperalgésique de nombreuses molécules et techniques d'analgésies périopératoires. Sa surface a été corrélée à l'incidence de la DCPC dans plusieurs études [START_REF] Eisenach | Preventing chronic pain after surgery: who, how, and when?[END_REF]. Il est probable qu'une prise en charge précoce diminuerait la sensibilisation centrale et les conséquences néfastes à long terme. TEG1 De plus, il est important d'organiser un dépistage très précoce de la douleur neuropathique avant la sortie des patients, notamment en utilisant le score DN4 (annexe A) qui est la convergence d'éléments de l'interrogatoire et de l'examen clinique. Ce score est considéré comme positif s'il est ≥ 4/10 avec une sensibilité de 82,9 % et une spécificité de 89,9 %. Les données actuelles montrent que les douleurs neuropathiques postopératoires sont souvent méconnues. Leur diagnostic et leur prise en charge sont trop tardifs. Aucun examen complémentaire à l'outil DN4 n'est alors nécessaire pour reconnaître cette douleur et débuter un traitement analgésique adapté. Notre responsabilité est de mettre en place une organisation permettant d'identifier les patients avec des parcours douloureux postopératoires anormaux. Ces patients devraient pouvoir bénéficier d'un suivi ciblé.

T1 Conclusion

Il est important que toutes les personnes intervenant auprès du patient en périopératoire soient sensibilisées à la DCPO. La place de l'anesthésiste est majeure tout au long du parcours du patient. Elle commence dès la consultation d'anesthésie par la recherche des facteurs de risque de chronicisation de la douleur, et par la prise en soins de l'anxiété périopératoire. Elle se poursuit au bloc par la prise en charge ciblée (kétamine, anesthésie locorégionale, sensibilisation du chirurgien à l'épargne nerveuse…), puis par la détection d'une douleur intense postopératoire, d'autant plus si celle-ci a des composantes neuropathiques. La dernière étape est alors de prendre en charge cette douleur de façon adaptée et précoce.

Déclaration de liens d'intérêts

Les auteurs déclarent ne pas avoir de liens d'intérêts.


Quentin Guibert
email: q.guibert@hotmail.fr.

Pierrick Piette
email: pierrick.piette@gmail.com.

Olivier Lopez
email: olivier.lopez0@upmc.fr.

Forecasting mortality rate improvements with a high-dimensional VAR

Keywords: JEL Classication: C18, C32, C52, C53 Mortality forecasting, High-dimensional time series, Vector Autoregression, Elastic-Net

diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Identifying patterns in the mortality dynamics of a population is a hard task due to the complex underlying phenomena that impact the death rates. This problem is of crucial interest for government policies, pension funds and insurance companies. A wide range of models has been developed since the introduction of the famous model proposed by [START_REF] Lee | Modeling and Forecasting U. S. Mortality[END_REF]. Most of these approaches rely on time-series modeling with past data and forecast the main factors inuencing the force of mortality, see among other [START_REF] Booth | Applying Lee-Carter under conditions of variable mortality decline[END_REF], [START_REF] Brouhns | A Poisson log-bilinear regression approach to the construction of projected lifetables[END_REF], [START_REF] Cairns | A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration[END_REF], [START_REF] Cairns | A quantitative comparison of stochastic mortality models using data from England and Wales and the United States[END_REF], [START_REF] Renshaw | On simulation-based approaches to risk measurement in mortality with specic reference to Poisson LeeCarter modelling[END_REF], [START_REF] Plat | On stochastic mortality modeling[END_REF], and [START_REF] Hunt | A General Procedure for Constructing Mortality Models[END_REF]. Some reviews are available in the literature, see e.g. [START_REF] Booth | Mortality Modelling and Forecasting: a Review of Methods[END_REF], [START_REF] Cairns | A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration[END_REF], and [START_REF] Barrieu | Understanding, modelling and managing longevity risk: key issues and main challenges[END_REF]. The purpose of the present paper is to provide a new exible modeling for the evolution of the mortality. Our high-dimensional vector autoregressive (VAR) approach, combined with an elastic-net penalty estimation method, aims to capture complex demographic eects without imposing a too restricting shape for the dynamics.

In recent years, some advances have been made to improve the forecast of mortality rates compared to the traditional factor-based models inspired by [START_REF] Lee | Modeling and Forecasting U. S. Mortality[END_REF]. This innovation has been provoked in particular by practitioners need for managing longevity risk and responding to the Solvency II requirements in insurance. Indeed, traditional models, even when a cohort eect is considered, have a reasonable t, but poorer forecasts, indicating that these models may overt the data. In such a context, one of the major concerns is to avoid the divergence of mortality rates between adjacent ages and dierent countries. Such inconsistency in the forecasting is pointed out for example by [START_REF] Börger | Modeling the Mortality Trend under Modern Solvency Regimes[END_REF], who explain that these low performances are due to the fact that traditional models mainly focus on the central trajectory projection. Several directions have been explored to overcome this issue. [START_REF] Li | Extending the Lee-Carter method to model the rotation of age patterns of mortality-decline for long-term projection[END_REF] develop an approach letting the age coecients rotate over time, based on an expert judgment. [START_REF] Hunt | Robustness and convergence in the LeeCarter model with cohort eects[END_REF] add an additional constraint on the cohort eect extensions of the [START_REF] Renshaw | On simulation-based approaches to risk measurement in mortality with specic reference to Poisson LeeCarter modelling[END_REF] model to overcome the convergence and robustness issues induced by the two-stage tting algorithm for parameters. Regarding mortality trends of multiple populations, a relatively wide literature is organized around the idea of a biological convergence at a long horizon, see [START_REF] Dowd | A Gravity Model of Mortality Rates for Two Related Populations[END_REF], [START_REF] Jarner | Modelling Adult Mortality in Small Populations: The Saint Model[END_REF], [START_REF] Li | Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method[END_REF], [START_REF] Enchev | Multi-population mortality models: tting, forecasting and comparisons[END_REF], and Cairns et al. (2016a) among others. These approaches estimate the mortality model by bringing together the data of several countries. Recently, [START_REF] Bohk-Ewald | Probabilistic mortality forecasting with varying age-specic survival improvements[END_REF] propose to approach the turning points of the mortality problem by combining trends of several countries.

Based on the observation that mortality rates are, in fact, noisy data, other alternative methods have emerged. If no exceptional event occurs, one can assume that the mortality surface is rather smooth over the age and time dimensions. Thus, functional data analysis and nonparametric smoothing techniques have been applied to mortality modeling, leading to a particular family of mortality models (see e.g. [START_REF] Currie | Smoothing and forecasting mortality rates[END_REF][START_REF] Hyndman | Robust forecasting of mortality and fertility rates: A functional data approach[END_REF][START_REF] Li | Two-Dimensional Kernel Smoothing of Mortality Surface: An Evaluation of Cohort Strength[END_REF]Dokumentov et al., 2018). These models are known to have good tting and forecasting performances, however they mostly consider future values as missing ones, making the stochastic generation of multiple prospective mortality scenarios non intuitive.

Other approaches focusing on the age-period dependency have recently been proposed with the constraint of being more data-driven. [START_REF] Christiansen | Dierences in European Mortality Rates: A Geometric Approach on the AgePeriod Plane[END_REF] use spatial statistics to forecast ageperiod mortality rate improvements using a kriging method. Their approach is parsimonious and provides good performances for short-term projection. However, it seems that their longterm results are more questionable. [START_REF] Doukhan | A class of random eld memory models for mortality forecasting[END_REF] also focus on the surface of mortality improvements and model it parsimoniously with an AR-ARCH specication for a random eld memory model. A valuable feature of their approach is that both dependencies between cohorts and the conditional heteroscedasticity of mortality are taken into account. Although they have good forecasting results, it is dicult to justify the size of the neighborhoods used to specify the memory process. [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF] choose a VAR process to consider the spatial dependence of mortality rates between neighboring ages adapted to short-term and long-term perspectives. These authors account for sparsity and stationarity in their VAR model by constraining the shape of the Granger causality matrix [START_REF] Granger | Investigating Causal Relations by Econometric Models and Cross-spectral Methods[END_REF] as a lower triangular. Their model is also able to consider multiple populations.

In this paper, we propose an alternative approach that newly forecasts the age-period dependency using a large VAR specication on the log-mortality improvements. Although a VAR model is suitable for mortality time-series and is able to capture both long-term relationships and short-term shocks (see e.g. [START_REF] Salhi | Basis risk modelling: a cointegration-based approach[END_REF], it is dicult to estimate accurately such models using an ordinary least square (OLS) technique, as these series are highly correlated and histories of data are relatively short. To avoid overparameterization, existing forecasting approaches impose an a priori spatio-temporal dependency structure between mortality rates or mortality improvements, which implies that only some selected series can interact. In contrast, our main contribution is the introduction of an estimation framework allowing for a large and exible VAR structure without excluding potentially relevant relationships. A great feature of such a VAR specication is that all classical mortality models could naturally be included in our specication, especially the so-called cohort and period eects as noted by [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF].

Following recent developments in economics and nance [START_REF] Fan | Sparse High Dimensional Models in Economics[END_REF][START_REF] Furman | VAR Estimation with the Adaptive Elastic Net[END_REF], we develop a penalized VAR method based on the elastic-net [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF], which allows to take into account the sparsity correctly. Indeed, such a VAR model has a sparse structure in high dimension, which requires an accurate estimation method for shrinking zero coecients in the Granger causality matrix. Compared to a classical maximum likelihood estimation approach, the key idea behind the elastic-net is to incorporate a penalty, which constrains the parameters. This penalty is a combination of an L 2 term (as in a ridge regression) to avoid ill-conditioning matrices, and an L 1 term (as in a LASSO regression) to produce a sparse model. By sparse model, we mean that our data-driven automatic selection produces a model with a relatively small number of nonzero parameters. As noted by [START_REF] Furman | VAR Estimation with the Adaptive Elastic Net[END_REF], this is an attractive alternative to Bayesian VAR procedures usually considered in an econometrical framework and developed for example by [START_REF] Hahn | A Bayesian Multi-Population Mortality Projection Model[END_REF] for multiple populations modeling. Indeed, such approaches require to introduce relevant priors and do not address the sparsity's issue. The residuals are modeled as a Gaussian vector where the variance-covariance matrix is described using a parametric form for parsimony purposes.

Similarly to [START_REF] Doukhan | A class of random eld memory models for mortality forecasting[END_REF], but contrary to [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF], our approach models the log of mortality improvements rather than the log of mortality rates. Several empirical elements have been advanced in the recent literature showing the interest of mortality improvements. [START_REF] Haberman | Parametric mortality improvement rate modelling and projecting[END_REF] show that a dual approach based on improvement rates can be followed for usual mortality models. They generally obtain quite comparable (but often better) forecasting results with this alternative route for the Lee-Carter model and its variants. As also noted by [START_REF] Bohk-Ewald | Probabilistic mortality forecasting with varying age-specic survival improvements[END_REF], mortality improvements seem to be easier to analyze, which facilitates the identication of divergences in mortality. As our approach is highly exible, we expect that it can better capture complex patterns of mortality improvements. Another argument is that mortality improvements are generally stationary (see e.g. [START_REF] Chai | A double-exponential GARCH model for stochastic mortality[END_REF], which is required for projection as our approach, contrary to [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF], does not impose constraints for guarantying stationarity.

We compare our high-dimensional VAR model to ve dierent benchmark mortality forecasting models: the usual Lee-Carter model [START_REF] Lee | Modeling and Forecasting U. S. Mortality[END_REF] and the M7 model developed by [START_REF] Cairns | A quantitative comparison of stochastic mortality models using data from England and Wales and the United States[END_REF] which are standard factor-based models; a reference model in smoothing methodologies developped by [START_REF] Hyndman | Robust forecasting of mortality and fertility rates: A functional data approach[END_REF] and the more recent smoothing RESPECT model introduced by Dokumentov et al. (2018); and nally the STAR model, based like ours on a highdimensional VAR method, developped by [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF]. Using the root mean squared error measure, we show that our approach leads to general better tting (in-sample) and forecasting (out-of-sample) of the mortality rate time series from the three countries we have focused our analysis on. Moreover, our data-driven model implies more stable errors over dierent countries while the benchmark models tend to have more variable predictive power depending on the considered population.

The remainder of this paper is organized as follows. In Section 2 we describe the VAR model we retained. The high-dimensional estimation of this model is then developed in Section 3. We present the data we used, dierent results that we obtained and a comparison to other standard mortality models in Section 4. Finally, Section 5 proposes an extension of the VAR model to multi-population modeling, and Section 6 considers some ways of improvement and concludes.

2

A Vector Autoregression approach for mortality rate improvements In this section, we introduce an econometric model to describe the mortality improvement dynamics jointly. The mortality models we introduce in the literature in Section 1 are initially based on an analysis of the main factors explaining a common trend of mortality rates. For instance, many models have been developed in the past for capturing the cohort eect, observed in the residuals for improvement rates plots [START_REF] Willets | The cohort eect: insights and explanations[END_REF]. Conversely, our approach only imposes an autoregressive structure, which encountered these classical models1 , as shown for example by [START_REF] Salhi | Basis risk modelling: a cointegration-based approach[END_REF] or [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF]. In particular, the latter authors explain in details how the cohort and period eects can directly be captured in the VAR(1) representation, without dening specic factors explicitly.

Throughout this paper, we focus on the time series y i,t ln pm i,t q, where m i,t is the crude annual death rate at age i and at date t. These rates can be easily computed thanks to annual risk exposures and count of deaths for a country of interest. Those series are usually not stationary, as a trend can be observed in mortality rates and life expectancy. Since we want to apply our vector autoregressive model on stationary series, we compute the rst dierence of the log-mortality rate ∆y i,t y i,t ¡ y i,t¡1 or, in other words, the log-mortality improvement rates. By working on these quantities, we remove a linear trend in the y i,t series.

With this notation, we specify the mortality rate improvement process by a stationary vector autoregressive model of temporal lag p or a VAR ppq. For a minimum age i min and a maximum age i max , we dene the d-dimensional vector of log-mortality rate improvements, with d i max ¡i min 1, as ∆Y t p∆y i min ,t , ∆y i min 1,t , . . . , ∆y imax,t q t . Next, we assume the following dependence structure dynamic,

∆Y t C p ķ1 A k ∆Y t¡k E t , (2.1)
where, for k 1, . . . , p, A k are d ¢ d-autoregressive matrices, C is a d-dimensional vector of constants (an intercept), and E t is a d-dimensional Gaussian white noise with mean 0 and Σ the related covariance matrix. We denote by i,t its marginals. The matrices A k , k t1, . . . , pu, capture the relationship between current mortality improvements and the k th lag of ∆Y t . In other words, this corresponds to the Granger causality between dierent cohorts for the mortality improvement rates. As a result, for a VAR(1) model, the coecients related to the rst subdiagonal of the Granger causality matrix capture a cohort eect for individuals born in the same year. As also noted by [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF], the terms of the diagonal can be interpreted as a period eect, since they measure an eect for a xed age between periods.

The VAR ppq model allows taking into account a more complex dependence structure than the usual mortality factor models. First, our model enables a larger exibility in the long-term spatio-temporal dependence structure through the autoregressive matrices than the standard factor models. For a given square pi, tq in the Lexis diagram, we let the possibility for the improvement mortality rates ∆y i,t to be dependent of all the ages among the d-dimensional space of ages, and through all the p temporal lags. In particular, we notice that this domain includes a cohort eect for these improvement rates. For each factor ∆y i¡1,t¡1 , . . . , ∆y i¡p,t¡p , this eect is indeed captured by the loading coecients positioned on the k th -subdiagonal of the matrix A k for each k t1, . . . , pu.

Hence, the VAR ppq structure permits for the shocks to propagate through dierent periods. Compared to the model proposed by [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF], the lag order p can take a value greater than 1, allowing to capture a more complex dependence structure.

Furthermore, it has the ability to enlighten some eects that are not captured in the standard mortality literature, e.g. between neighboring cohorts, as we do not impose any constraint on the matrices A k , k t1, . . . , pu. Compared to most of factor models, the second improvement of our model on the dependence exibility is that it captures the long-term co-movement by the autoregressive matrices and the short-term dependence through the covariance matrix at the same time.

Nevertheless, the major issue of our VAR ppq model is that it is a natural high-dimensional problem. The number of parameters for the Granger causality matrices is pd 2 , without considering the covariance matrix and the constant vector. In mortality modeling studies, it is common to focus on the age range from 0 to 100, that is to say d 101, while the historical data for estimation rarely exceed 70 years. Given this, a VAR p3q implies 30, 704 parameters estimated on only 7, 070 observations, which makes the ordinary least-squares estimation not feasible. To avoid over-tting, additional constraints have to be added. In this direction, [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF] impose that some parameters have to be nil for guarantying that the model is sparse and stationary. Conversely, we choose a less arbitrary high-dimensional selection variables technique, developed in the next section, to ensure sparsity.

Similarly, the covariance matrix estimation is also a high-dimensional problem with d pd 1q 2 parameters. In order to estimate prediction intervals, we consider an additional specication for the residuals. Although some high-dimensional techniques do exist for covariance estimation (see e.g. [START_REF] Schäfer | A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics[END_REF][START_REF] Opgen-Rhein | From correlation to causation networks: a simple approximate learning algorithm and its application to high-dimensional plant gene expression data[END_REF][START_REF] Bickel | Covariance regularization by thresholding[END_REF][START_REF] Bien | Sparse estimation of a covariance matrix[END_REF], we rather choose a simple parametric form presented in the following part to reduce the number of parameters.

3

High-dimensional estimation of the VAR model

As highlighted in the previous section, the VAR ppq model estimation is a high-dimensional problem, especially with mortality data. The estimation can be decomposed into two parts: rst, we estimate the pd 2 -dimensional autoregressive matrices, then the d 2 -dimensional covariance matrix. The dimension reduction in the autoregressive matrices is treated through an elastic penalization in Section 3.1. We tackle the problem of the covariance through the choice of a parametric form in Section 3.2.

Elastic-net

We now described the extension of the elastic-net regularization and variable selection method, proposed by [START_REF] Zou | Regularization and variable selection via the elastic net[END_REF], for the high-dimensional estimation of our autoregressive matrices. This technique can be seen as the combination of the LASSO L 1 -penalty, introduced by [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF], and the ridge L 2 -penalty developed by [START_REF] Hoerl | Ridge regression: Biased estimation for nonorthogonal problems[END_REF]. Elastic-net has similar properties of variable selection as the LASSO. Moreover, it provides a grouping eect: highly correlated variables tend to be selected or dropped together. LASSO and elastic-net have already been extended to VAR model [START_REF] Gefang | Bayesian doubly adaptive elastic-net Lasso for VAR shrinkage[END_REF][START_REF] Basu | Regularized estimation in sparse high-dimensional time series models[END_REF], mostly with an economic application (see e.g. [START_REF] Song | Large vector auto regressions[END_REF][START_REF] Furman | VAR Estimation with the Adaptive Elastic Net[END_REF].

Therefore, we estimate the VAR ppq model presented in Equation (2.1) with T observations of the process ∆Y t for t t min , . . . , t max by minimizing the criterion

L pC, A 1 , . . . , A p q 1 T ¡ p tmax ţt min p }∆Y t ¡ C ¡ p ķ1 A k ∆Y t¡k } 2 2 ¡ αλ p ķ1 }A k } 1 ¡ p1 ¡ αq λ 2 p ķ1 }A k } 2 2 , (3.1)
where we dene for a d-dimensional

vector b pb i q 1¤i¤d }b} 2 2 d i1 |b i | 2 ,
and for a d ¢ d-dimensional matrix B pb i,j q 1¤i¤d,1¤j¤d

}B} 1 d i1 d j1 |b i,j | and }B} 2 2 d i1 d j1 |b i,j | 2 .
The parameter α r0, 1s is a hyper-parameter which determines the mix between ridge and LASSO penalties. We use a 10-folds cross-validation method to choose the penalty coecient λ. It determines the strength of the penalties, for example in the LASSO case, the higher λ gets, the fewer number of variables are selected. The algorithm we used is described in [START_REF] Friedman | Regularization Paths for Generalized Linear Models via Coordinate Descent[END_REF]. In theory, the LASSO L 1 -penalty part forces most of the coecients to 0. Nevertheless, in a more practical approach, the algorithm employed does not lead to exact zeroes. Thus, the R-package glmnet [START_REF] Friedman | Regularization Paths for Generalized Linear Models via Coordinate Descent[END_REF] applies a threshold on the coecients. Furthermore, following Chatterjee and Lahiri ( 2011), the sparsevar R-package [START_REF] Vazzoler | sparsevar: A Package for Sparse VAR/VECM Estimation[END_REF] enables to apply a more tailor-made threshold for time-series estimation which equals to 1 c pd ln T , that we retain for our model.

The hyper-parameter α is determined through a grid search. For every value α h of a predened grid tα 1 , . . . , α H u, we estimate the parameters of the VAR model 3 Ĉ, Â1 , . . . , Âp , λA α h , as explained just before, and deduce the residuals Êα h ,t for t tt min p, . . . , t max u.

In the applications, we estimate the tuning parameters by minimizing the prediction error, that we obtain by computing the root-mean-square error

RMSE pα h q g f f e 1 d pT ¡ pq tmax ţt min p } Êα h ,t } 2 2 .
(3.2)

In our application, we considerate the grid t0.5, 0.6, 0.7, 0.8, 0.9, 1u in order to impose a larger weight to the LASSO penalty for sparsity purposes.

The choice of the lag order p for our VAR elastic-net (VAR-ENET) model diers signicantly from the usual lag order selection in the standard VAR models. The parameter p does not fully determine the number of parameters, since the LASSO penalty force the less signicant coecients to zero. By increasing the lag order, some non-null coecients can be forced to zero in favor of other coecients in autoregressive matrices of higher lag order. Moreover, if there is no signicant coecient above a certain lag order, all autoregressive matrice above the limit order are largely forced to zero. Thus, we chose a relatively large p to capture eventual high order lag eects, without being worried of over-tting.

3.2

Variance-covariance estimation

The autoregressive matrices are not the only high-dimensional problem of the VAR ppq model, the variance-covariance matrix estimation has dpd 1q 2 parameters. This number can quickly get higher than the number of observations while dealing with mortality modeling, and then can cause overtting, as noted e.g. by [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF].

To overcome this issue, we propose an approach to estimate the covariance matrix with a parametric covariance function, in a manner similar to [START_REF] Spodarev | Extrapolation of stationary random elds[END_REF]. Firstly, for each age i, we estimate the standard empirical variance σ2 i of the residual, and for each couple of ages pi, jq ti min , . . . , i max u 2 , we estimate the empirical correlation ri,j σi,j σi σj ,

where σi,j is the empirical covariance. Then, guided by the form of the empirical correlation matrices and by the approach of [START_REF] Christiansen | Dierences in European Mortality Rates: A Geometric Approach on the AgePeriod Plane[END_REF], we use a parametric form close to the stable family of covariance functions r i,j βe ¡pa i a j q¢|i¡j| ¢ 1 ti$ju 1 tiju ,

with β ¥ 0 and a i ¥ 0 for each age i. We t the model based on the empirical correlation as the OLS solution. Thus, after determining β and pâ i min , . . . , âimax q, we compute our parametric correlation ri,j given by Equation (3.3). Finally, for each couple of ages pi, jq we estimate the covariance by σi,j ri,j ¢ σi σj .

Empirical analysis

In this section, we apply our high-dimensional VAR-ENET model to real data and show its strengths in estimating and forecasting populations. Dierent populations are considered and we analyze both our in-sample and out-sample results compared to those obtained with retained benchmark models.

In the following, the computations are carried out with the R software (R Core Team, 2019). Our scripts are available upon request.

4.1

Data

The datasets that we analyze comes from the Human Mortality Database (2019). We choose to illustrate our approach with historical mortality data from the England and Wales (UK), the United States (US) and France (FR), as these populations have been largely studied, but have specic features. At rst, the overall population is considered, and then both males and females are segregated. We select the age-period observation t45, . . . , 99u¢t1950, . . . , 2016u which was available for these 3 countries when the data was extracted. We begin our analysis by a visual inspection of our data. Figure 1 describes the shape of the period log-mortality improvements for populations on a Lexis diagram where the trajectory of one cohort follows a 45 degree line. Dierent cohort eects can be observed for these countries with pink (resp. green) shades for positive (resp. negative) improvements, indicating a lower (resp. higher) survival. On the center, UK improvement rates do exhibit some signicant diagonal patterns corresponding to the so-called cohort eects. A diagonal stands out, more precisely for individuals aged 45 in 1965. Several vertical patterns corresponding to period eects are also observed, especially for the older ages. Diagonal and vertical structures associated with cohort and period eects also stand out in the American data, even if the patterns are less marked than in the English data. Contrary to English and American mortality, the French mortality data doesn't clearly display any diagonal structures, but only period eects. We note that the cohort eects (also observable on residual plots), which used to appear in the French data, were strongly reduced with the correction developed by [START_REF] Boumezoued | Improving HMD mortality estimates with HFD fertility data[END_REF], thanks to fertility rates. Female and male data are displayed in Appendix 1.

We have chosen to apply the VAR-ENET to the rst dierence of log-mortality rates series because they are known in the literature as stationary time series. In order to verify this point, we perform a Phillips-Perron test [START_REF] Perron | Trends and random walks in macroeconomic time series: Further evidence from a new approach[END_REF] and an augmented Dickey-Fuller test [START_REF] Said | Testing for unit roots in autoregressive-moving average models of unknown order[END_REF] on every age mortality series for each of the nine populations of interest. All of these time series satisfy the Phillips-Perron test at a condence level of 1%, and 93% of them are considered as stationary by the augmented Dickey-Fuller test at a level of 5%. These results strengthen our choice to focus on the rst dierence of time-series.

Benchmark models

In this section, we present the benchmark mortality models that we compare to the VAR-ENET. First, we retain two models from the standard factor-based family: the usual Lee-Carter (LC) model [START_REF] Lee | Modeling and Forecasting U. S. Mortality[END_REF], estimated with the approach of [START_REF] Brouhns | A Poisson log-bilinear regression approach to the construction of projected lifetables[END_REF] and given by

y i,t α i β i κ t , (4.1)
with the hyper-parameters α i and β i , and the mortality trend κ t , the M7 model developed by [START_REF] Cairns | A quantitative comparison of stochastic mortality models using data from England and Wales and the United States[END_REF], which considers a quadratic and a cohort eect, i.e.

y i,t κ p1q t pi ¡ īq κ p2q t κ p3q t ¡ pi ¡ īq 2 ¡ σ2 i © γ t¡i , (4.2)
where κ pjq t , j 1, 2, 3, are period eects, γ t¡i is a cohort eect, ī is the average age in the data, and σ2

i is the average value of pi ¡ īq 2 .

These last models are estimated using the R-package StMoMo [START_REF] Villegas | StMoMo: An R Package for Stochastic Mortality Modelling[END_REF] following their usual two-stage tting procedure: rst, we estimate the factor coecients of each model, and then we forecast it using univariate ARIMA processes, automatically selected by the R-package using an AIC criterion. To be comparable with our one-stage tting approach, the results for these models are those obtained after tting the time-series parameters.

We also consider two recent models based on smoothing methodologies : the classical model proposed by [START_REF] Hyndman | Robust forecasting of mortality and fertility rates: A functional data approach[END_REF] (HU) which estimates a nonparametric smoothing function f t piq for every period t that smooths mortality rates over the age dimension, and is then decomposed

f t piq µpiq K ķ1 β t,k φ k piq, (4.3)
where µpiq is a measure of location of f t piq, pφ k piqq i1,...,K is a set of orthonormal basis functions of dimension K ¥ 1. This model is applied thanks to the R-package demography [START_REF] Hyndman | demography: Forecasting Mortality, Fertility, Migration and Population Data[END_REF] based on weighted penalized regression splines for smoothing.

The recent RESPECT model, developed by Dokumentov et al. (2018) and implemented in the R-package smoothAPC (Dokumentov and Hyndman, 2018), which uses L 1 -regularized bivariate smoothing over the age and period dimensions, and further allows identication of period and cohort eects on the smoothing residuals.

Finally, we retain a model closer to the methodology of the VAR-ENET, which is based on a spatial-temporal autoregressive framework: the STAR method, introduced by [START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF]. It models the dynamic of the log mortality rates through a large rst-order VAR, of which autoregressive matrix's parameters are forced to a sparse estimation by the following constraints: y i min ,t 1 y i min ,t m i min , (4.4) y i min 1,t 1 p1 ¡ α i min 1 qy i min 1,t α i min 1 y i min ,t m i min 1 , (4.5) and

y i 1,t 1 p1 ¡ α i 1 ¡ β i 1 qy i 1,t α i 1 y i,t β i 1 y i¡1,t m i 1 , (4.6)
for i ti min 2, . . . , i max u, m i 1 a parameter, and α i 1 and β i 1 two positive parameters that are smaller than 1. The model is estimated using the benchmark ordinary least square proposed by the authors.

4.3

In-sample analysis

In this section, we present the results of our empirical estimations with the VAR-ENET model for each population. We especially focus on the study of the estimated Granger causality matrices that describe the long-term underlying mortality dynamic of the model. The goodness of t is analyzed by comparing the in-sample results with the benchmark models presented in Section 4.2.

Parameters estimation

Let us present our estimated results on the period 1950 ¡ 2016. The parameters are estimated as described in Section 3. For the lag order p, we choose the value 7, which represents between 10% and 15% of the observation, depending if we analyze the in-sample or, as in the latter sections, outsample. Table 1 reports the list of the estimated hyper-parameters for each population of interest.

We note that, for some populations, we retain the value 1 for α, i.e. we estimate the model with the LASSO constraint only. The rst Granger causality matrix A 1 for each population is displayed in Figure 2. These estimated matrices are sparse, i.e. most of the coecients are estimated to 0 while minimizing the criterion given in Equation (3.1). We identify two main structures by observing the non-zero coecients. We interpret these patterns in terms of demographic eects, basing our explanations on the underlying mortality dynamic of the model induced by the matrices and described in Equation (2.1). First, an expected cohort eect, induced by individuals belonging to the same generation, is highlighted by allocated coecients on the k th subdiagonal for A k for k t1, . . . , pu. Indeed, those coecients in the VAR model describe the Granger causality of ∆y i¡k,t¡k on ∆y i,t . This eect appears positively mainly for the younger ages of our dierent samples. It is more diuse between the ages of 65 and 85 years old. In Figure 2, the dierence between countries appears clearly and the so-called cohort eect is relatively strong for the US population, compared to the FR and the UK. The cohort eect is also clearly visible for k t2, . . . , 7u (not shown here).

Second, negative period eects are observed on the main diagonals in just about any population, especially between the ages of 85 and 95. Females in France are more impacted by this eect also for younger ages, whereas almost no cohort eect appears for this group. An opposite situation emerges for the US, where the period eect remains limited to the very older ages.

Third, we notice some age-specic eects corresponding to vertical structure of non-zero co-ecients. This third type of patterns reveal non-trivial interactions between dierent cohorts which are non-necessary within a close neighboring. More concretely, a vertical pattern on the i th column of A 1 reveals a persistent eect from the term ∆y i,t¡1 on t∆y j 1 ,t , . . . , ∆y j l ,t u, with tj 1 , . . . j l u ti min , . . . , i max u. It means that the mortality improvement of a single specic age i seems to impact the mortality improvement on a group of ages tj 1 , . . . j l u one year after. Similar structures can be observed on the matrices A k for k t2, . . . , 7u (not shown here). This latter eect, underlined by our data-driven approach, has not been well documented in the literature yet to our knowledge. These patterns are quite dicult to interpret and to explain within the demographic framework with the available datasets. Disaggregated data would be very useful to explore these eects further. Indeed, these patterns could result from biological, environmental or societal causes, unless it is due to some anomalies in the HMD [START_REF] Cairns | Phantoms never die: living with unreliable population data[END_REF][START_REF] Boumezoued | Improving HMD mortality estimates with HFD fertility data[END_REF]. At this point, we are unable to conclude on the very causes of such age-eects.

On the contrary, the observed cohort and period eects have already been well studied in the literature. However, our model highlights this result in a more data-driven way. Indeed, the existing models either detect these eects in the residuals (e.g. [START_REF] Lee | Modeling and Forecasting U. S. Mortality[END_REF]Dokumentov et al., 2018), or force the estimation of specic parameters (e.g. [START_REF] Cairns | A quantitative comparison of stochastic mortality models using data from England and Wales and the United States[END_REF][START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF]. In our case, we notice these eects by analyzing the parameters estimated without imposing any specic constraints. To exhibit how our data-driven approach can adapt to these eects, we estimate two VAR-ENETp1q models on the French male population over the period 1950-2012, but with the data downloaded from the HMD at two dierent dates: 2 nd October 2017 and 28 th January 2019. Indeed, between these two dates, the HMD data had been updated, following the work of [START_REF] Cairns | Phantoms never die: living with unreliable population data[END_REF] and Boumezoued (2016) using fertility rates. With this correction, the residual plots display a cohort eect which is substantially lessened. The two Granger causality matrices estimated are displayed in Figure 3. In the old version, we clearly remark a subdiagonal in the estimated parameters. In the new estimation, where many false cohort eects had been removed, the pattern on the rst subdiagonal is virtually nil, whereas the negative period eect on the main diagonal is only slightly reduced. Figure 3: The Granger causality matrix A 1 for the French males on the age range 45-99 estimated with a VAR-ENETp1q over the period 1950-2012, on the HMD data downloaded on the 2 nd October 2017 (Before correction) and the 28 th January 2019 (After correction).

In-sample model comparison

We now study how our approach ts well and captures better the mortality pattern on these subsets compared to the benchmark models we retained in Section 4.2. The RMSE value is also computed for each age and each year, and the results are displayed respectively on Figures 4 and5 for all of the overall populations. First, we note that, on top of having one the lowest RMSE with the smoothing models, the VAR-ENET leads to a more stable error over the age. This is clearly noticeable on the higher ages, especially for the French and English data. For example we observe that the M7's tting error drastically increases for ages above 95. More generally, on these two populations, all the benchmark model tend to have an increasing agemarginal RMSE starting from 90 years old, while our model's tting error stays relatively stable. On the American data, the results are more nuanced: whereas we still notice an increase of RMSE values at higher ages for the stochastic benchmark models (LC, M7 and STAR), the smoothing ones and the VAR-ENET lead to stable errors. The RMSE patterns across periods are more erratic. While the errors of smoothing models (HU and RESPECT) are quite stable over the periods, many peaks are observed for all the stochastic models. More particularly, we note that these peaks tend to occur at the same period for all the concerned models, especially on the French and American data. We remark that, among the stochastic models, the VAR-ENET is the one producing the peaks of the lowest amplitude, and is the closest of the smoothing models in terms of goodness-of-t. Finally, we notice that, on the American data, the errors of the VAR-ENET and the STAR model are highly correlated, mainly starting from 1985. This shows the methodological closeness of these two methods. The results for females and males are given in Appendix 2 with similar ndings. Some of the period peaks may be explained by specic events that have an unexpected impact on the mortality rates, such as an inuenza epidemic or a heat wave [START_REF] Huynen | The impact of heat waves and cold spells on mortality rates in the Dutch population[END_REF]. Indeed, this type of exogenous stresses is dicult to predict with only mortality rate series, which explains why the peaks are observable for the three models. For example, we try to explain the relatively high RMSE in France in 2004. In 2003 a heat wave led to one of the hottest summer ever recorded in France and, as a direct consequence, to higher mortality rates during this year, especially for the elderly. Then, the mortality was much lower in 2004 due to the so-called harvesting eect [START_REF] Toulemon | The mortality impact of the August 2003 heat wave in France: Investigating the `harvesting' eect and other long-term consequences[END_REF][START_REF] Izraelewicz | L'eet moisson -l'impact des catastrophes vie sur la mortalité à long terme -Exemple de la canicule de l'été 2003[END_REF]. On the contrary, in the calculation of the RMSE, the mortality rates of the year 2004 are forecasted from the observation of the 2003 rates in accordance with the temporal dynamics we have imposed; in this way, the 2004 mortality was expected to be relatively high. This must explain why we observe a RMSE peak in 2004 for the French population.

4.4

Out-of-sample performance

For risk management in insurance or more generally for demographers or public policy purposes, mortality rates require being predicted based on the past information. A quite usual test for accuracy is to analyze how the model is able to reproduce the mortality rates correctly. Note that this objective is more demanding than measuring the prediction power on the residual life expectancy. A reasonable model should be able to predict a kind of convergence for mortality at a similar level.

We focus on the prediction power of the VAR-ENET model compared to the benchmark models through an analysis of the out-sample forecasting performance on the same age-period space. To this end, we rst estimate each model based on the observations from 1950 to 2000, then we forecast the mortality rates for the period 2001-2016. Note of course that the mortality rates can be easily calculated using the VAR model, based on the predicted improvement rates and the initial values known for the last year of the training sample. We choose a similar measure to the one taken for the in-sample analysis, the root mean squared forecast error (RMSFE) that we dene for a projection horizon h as

RMSFE g f f e 1 dh imax ii min t 0 h ţt 0 1 py i,t ¡ ŷi,t q 2 , (4.7)
where t 0 is the year 2000 and h equals to 16 years in our study.

We compare the predictive power of the dierent models on the period 2001 ¡ 2016 for the 3 populations (overall, female, male) of the 3 countries of interest. The results are displayed in Table 4 and 5 (we also display the results for dierent estimation years but with the same forecasting period in Appendix 3). We note in Table 5 that the average RMSFE is smaller with the VAR-ENET than with the benchmark models, indicating that the former one has higher predictive power in general. However, it is locally outperformed by other models for some populations. Thus, we note for example that the RESPECT model slightly outperforms the VAR-ENET on the French data, and the STAR's RMSFE value on the US male population (5%) is signicantly lower than the one obtained with the VAR-ENET (11.6%) and the other models.

Furthermore, we observe that, in the VAR-ENET applications, all the forecasting errors are of the same order of magnitude, no matter the selected population. This point is highlighted by the standard deviation of RMSFE values over the 9 populations displayed in Table 5 for each model. Although the RESPECT and STAR models locally outperform the VAR-ENET, they are more sensible to population considered, leading to a signicantly higher standard deviation (respectively 9.4% and 11.3% against only 1.4% for our model). The M7, and to a lesser extent the LC and HU models, also tend to have more variable forecasting errors, depending on the considered population. These results highlight the stability of the prediction error of the VAR-ENET over dierent populations compared to the other benchmark models due to the more data-driven approach of the rst one, allowing it to better capture the features of each populations' mortality dynamic. By analyzing the results displayed in Appendix 3, we also notice a better stability of the VAR-ENET over dierent estimation periods. for the HU, the LC, the M7, the RESPECT, the STAR and the VAR-ENET models estimated on the period 1950 ¡ 2000. We compare this indicator for males, females and the overall populations for FR, UK and US.

We plot the RMSFE in Figure 6 for the overall population of our three countries of interest. The results for female and male populations are postponed in Appendix 3. First, we note that the forecasting errors from most models tend to converge with the projection horizon, suggesting that obtaining a signicant enhancement of the forecasting accuracy on the long term seems very challenging. However, we observe that for specic population, some models fail to capture the mortality We now focus on the forecasting error by age groups. We choose to separate the age dimension into 5 classes and compute the RMSFE at a projection horizon of 10 years. By doing so, we compare the predictive power over the dierent ages. Indeed, depending on the purpose of the mortality forecasting application, one could be more interested in producing accurate predictions for some specic ages. We show the results of the three models in Figure 7.

Yet again, we observe that the VAR-ENET is the most stable model over the age classes, when analyzing the forecasting errors at a 15 years projection horizon. While the M7 consistently fails to capture the mortality dynamic at higher ages, the other models have more local issues. For example, we note poorer predictions for the STAR on the French age class 85-94. On the English mortality, the two models LC and HU, and the RESPECT methodology, have respectively higher RMSFE on the age classes 65-74 and 85-94. This point highlights the capacity of our model to uniformly forecast the mortality rates over the age dimension for any of the considered populations. The results presented in this section suggest that our model slightly outperforms the benchmark methodologies in average for the considered data, even though it doesn't lead to the best accuracy for every population. In all the tested situation, it is at least a credible competitor compared to the best model. More importantly, the VAR-ENET seems to be more stable according to the selected population. This last point is the most noticeable dierence between our in and out-ofsample results. Whereas, in our in-sample study, the models' t seems to be relatively equally stable in respect of the considered population, the out-of-sample analysis emphasizes a noticeable heterogeneity in the outcomes depending on the selected dateset. In that regard, the VAR-ENET tends to provide signicantly more consistent forecasts.

4.5

Forecasting application

Figure 8 displays the median forecasts of the log of death rates for ages in t45, 65, 85, 95u using the VAR-ENET model from 2017 to 2066. We note that the trends seem rather realistic. We remark that the male and female mortality rates tend to converge rapidly for the UK population. This forecasting result has already been observed in the literature with other models with a similar estimation period (see e.g. Bohk-Ewald and Rau ( 2017)). We note on the forecasted series that there are some limited shocks during the rst projection years, followed by a linear trend. This eect is characteristic of the VAR model and shows how it can propagate innovation shocks among a cohort for example.

Figure 9 compares the median forecasts of the log of death rates of two popular models LC, the HU model with the VAR-ENET model from 2013 to 2062. First, we note that for many of the forecasted series, the three models produce very similar projections, especially on the female populations at higher ages. On the contrary, for the British male mortality dynamics, the forecasts are noticeably dierent. While the LC and HU models predict a stabilization of the mortality rates, and even a slight increase at age 45 for the smoothing methodology, our VAR-ENET forecasts a decrease consistent with the average longevity improvement over the last decades. The two benchmark models seems to be more impacted by the slowdown of this enhancement observed during the very recent years. To a lesser extent, we also notice a comparable results on the French male population. 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 1960 1970 1980 1990 2000 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 1960 1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 1960 1970 1980 1990 2000 The observed and the projected log of death rates for British (UK), American (US) and French (FR) females and males. This gure compares trends obtained with the HU, the LC and the VAR-ENET models.

A multi-population extension

Some of the standard mortality forecasting models can be extended to multi-population. However, many of these extensions suer from limits. One of the recurrent limits in multi-population mortality modeling is the restriction of the extension to only 2 populations. For example, we can note the GRAVITY model of [START_REF] Dowd | A Gravity Model of Mortality Rates for Two Related Populations[END_REF] or the Bayesian model of [START_REF] Cairns | Bayesian Stochastic Mortality Modelling for Two Populations[END_REF]. Another restriction imposed by some existing multi-population models is the necessity to determine a dominant population and sub-populations, see e.g. the SAINT model of [START_REF] Jarner | Modelling Adult Mortality in Small Populations: The Saint Model[END_REF], or a common trend for the dierent populations like in [START_REF] Li | Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method[END_REF]. In this section, we explore the possibility of extending our model for multi-population mortality forecasting and we give the needed details.

We denote M the number of selected populations and y m,i,t the log of mortality rates for the m th population. We suppose that the pair pi min , i max q is the same for all the populations to avoid exaggerated notations, although we could have chosen M dierent pairs of age limits. Thus, we dene M dierent d-dimensional vectors ∆Y m,t that we concatenate into a single M d-dimensional vector ∆Y t p∆y 1,i min ,t , . . . , ∆y 1,imax,t , ∆y 2,i min ,t , . . . , ∆y M,imax,t q t . We then apply the same model as in Equation (2.1) except that the dimension equals now to M d.

The pMdq ¢ pMdq-dimensional autoregressive matrices and the M d-dimensional vector of con-stants are estimated through the same elastic-net methodology as in the single population problem. However, the covariance matrix estimation needs to be extended since its structure may change signicantly compared to the single population case.

In the multi-population context, the covariance matrix Σ is pMdq ¢ pMdq-dimensional. Firstly, we propose to consider this matrix as a block matrix broken into M 2 dierent d ¢ d-dimensional submatrices noted Σ m,n for each population couple pm, nq t1, . . . , M u 2 . Then, for each population m, we estimate the diagonal submatrix Σ m,m through the same methodology as in the single population, obtaining in this way Σm,m . Since Σm,m is a positive-denite matrix, we dene its Cholesky decomposition Σm,m Rt m Rm , where Rm is an upper triangular matrix. Finally, for a couple of populations pm, nq we estimate the covariance submatrix as

Σm,n ρ m,n ¢ Rt m Rn ,
where ρ m,n r¡1, 1s is the empirical Pearson's correlation coecient between the observations of residuals p m,i,t q pi,tq and p n,i,t q pi,tq for pi, tq ti min , . . . , i max u ¢ tt min p, . . . , t max u.

Thus, our extended covariance model has only M p2d 1q M pM¡1q 2 parameters. The number of ages, d, being generally much larger than the number of populations in mortality modeling, we note that M p2d 1q 4 M pM¡1q 2 , meaning that we do not add many parameters for covariance estimation while modeling M populations together compared to tting M single models.

Conclusions

In this paper, we have proposed a vector-autoregression elastic-net (VAR-ENET) model on the dierentiated log-mortality, leading to three key results. First, this new high-dimensional time series analysis outperforms in tting the mortality rate series of each of the nine populations we considered, compared with the three stochastic benchmark models (LC, M7 and STAR). Moreover, in average, it leads to in-sample errors of same order as the two smoothing benchmark models (HU and RESPECT). Even though our model doesn't produce the most accurate forecasts on each population, it leads to relatively close results compared to the best model each time. In addition, the average RMSFE over the 9 population is lower than the one obtained with any other benchmark models. Furthermore, thanks to its data-driven approach, the VAR-ENET leads to more stable errors than the benchmark models over populations, showing its power of adaptability to the specic mortality dynamics of dierent populations. Compared to the usual strategy which requires to compare a variety of possible models and then select the best for a particular age-period population, our approach gives directly and with little eort a serious candidate for a consistent modeling of the mortality, regardless of the population features. The second key result is that, although we let a large freedom in the spatio-temporal dependence structure without imposing a priori constraints, the VAR-ENET model enlightens three main eects: the so-called cohort and period eects and a specic age eect. While the rst two models have already been well studied in many papers on mortality modeling, we develop in this paper a new ways for detect the eects for any population. The last eect is less known or possibly even unknown in the literature. Future researches are needed, probably on a nest dataset to understands such a phenomena. Finally, the proposed extension of the VAR-ENET to multi-population mortality modeling seems a priori straightforward, without raising unavoidable issues on the number of populations or on the hierarchy between them, considering the estimation process.

Some points should however be improved and need further researches. The rst one concerns the interpretation of the results given by our VAR-ENET model. Although it seems to have a better forecasting and adaptability power than the standard factor-based models, the last ones do benet from a greater interpretability. Indeed, even if most of the coecients in the autoregressive matrices are estimated to zero in the VAR-ENET and that the non-null coecients seems to form specic patterns, the comprehension of the underlying dynamics remains complex. On the contrary, it is much easier to understand the mortality dynamics in terms of period, age and cohort eects, which are directly visible through the use of the classical factor-based models.

Second, we are also aware that some of the hyper-parameter selection techniques we applied can be improved. Firstly, we imposed the lag order p equals to 7 for all the population. In a sensibility analysis, we have noted that according to the considered population, the highest predictive power of the VAR-ENET(p) model is not reached at the same lag order p. These results suggest that an optimization on the hyper-parameter p could be developed. The second hyper-parameter to be improved is the mixing weight parameter α between the LASSO and ridge penalties. In the general context of elastic-net regression, it is usually selected with a grid search. However, [START_REF] Friedman | Regularization Paths for Generalized Linear Models via Coordinate Descent[END_REF] propose to optimize it through a cross-validation by following the same methodology as the λ selection.

Third, the log-mortality rates series y i,t are known to not be stationary, but also to be cointegrated (see e.g. [START_REF] Chen | Multi-population mortality models: A factor copula approach[END_REF][START_REF] Salhi | Basis risk modelling: a cointegration-based approach[END_REF][START_REF] Li | Coherent Forecasting of Mortality Rates: A Sparse Vector-Autoregression Approach[END_REF]. In our paper, we choose to study the rst dierence in the log mortality-rates and, by doing so, we lose some information about the long-term co-movement. Another way that we can deal with the non-stationarity and cointegration is to rather select the Vector Error Correcting Model (VECM). Nevertheless, although high-dimensional VAR model has been relatively well studied and recently documented especially in nancial econometrics, VECM sparse estimation with elastic-net or other techniques seems to be a new eld (see e.g. [START_REF] Wilms | Forecasting using sparse cointegration[END_REF], and could be developed further for the mortality projection. A major improvement of our model would be to implement the elastic-net procedure to VECM estimation and apply it to the log-mortality series.

Finally, even though we introduce an extension to multi-population mortality forecasting of our model, we don't show any empirical studies on that subject in this paper, which rather focuses on the single population case. Many points need to be analyzed in greater detail to correctly assess the behavior of the VAR-ENET model applied to multi-population. It notably includes the examination of a broader list of countries, the specic case of sub-regional populations and the comparison of forecasts to recent multi-population models. Further more specic studies should be conducted to examine the multi-population model. Note: This table reports the out-of-sample performance via the RMSFE values for the HU, the LC, the M7, the RESPECT, the STAR and the VAR-ENET p4q models estimated on the period 1970¡2000. We compare this indicator for males, females and the overall populations for FR, UK and US.

Figure 1 :

 1 Figure 1: The period log-mortality improvements for England and Wales (UK), the United States (US) and France (FR) on the age-period observation t45, . . . , 99u ¢ t1950, . . . , 2016u for overall populations.

Figure 2 :

 2 Figure 2: The Granger causality matrix A 1 for England and Wales (UK), the United States (US)and France (FR) on the age range 45-99 for females, males and the overall population.

Figure 5 :

 5 Figure 5: The RMSE for England and Wales (UK), the United States (US) and France (FR)grouped by period for the overall populations.

Figure 6 :

 6 Figure 6: The RMSFE for England and Wales (UK), the United States (US) and France (FR)grouped by period for the overall populations.

Figure 8 :

 8 Figure8: The observed and the projected log of death rates for British (UK), American (US) and French(FR) females and males with the 97.5% prediction intervals, obtained from the VAR-ENET model.

  Figure9: The observed and the projected log of death rates for British (UK), American (US) and French (FR) females and males. This gure compares trends obtained with the HU, the LC and the VAR-ENET models.

Table 1 :

 1 The estimated VAR-ENET hyper-parameters.

	Country	Population	α	λ
	FR	Female	0.8	0.0012
	FR	Male	0.6	0.0003
	FR	Total	0.8	0.0006
	US	Female	0.6	0.0010
	US	Male	1.0	0.0005
	US	Total	1.0	0.0005
	UK	Female	0.6	0.0004
	UK	Male	1.0	0.0003
	UK	Total	0.8	0.0005
	Note: This table displays the estimated
	hyper-parameters α and λ in Equa-tion (3.1) for the VAR p7q models. We
	consider males, females and the overall
	populations for FR, UK and US.

Table 2 :

 2 Table 2 contains the values of the RMSE, as dened in Equation (3.2) for each model and each population. Table 3 displays summary statistics of RMSE values over all the populations considered for each model. Benchmark models and the VAR-ENET have quite comparable results. Although the VAR-ENET has not the lowest value for each population, it globally leads to one of the best in-sample results with the HU and RESPECT model (around 2% in average). More particularly, the RESPECT model outperforms the VAR-ENET on the US populations. The RMSE of the VAR-ENET and benchmark models.

	Country	Model	RMSE Female	RMSE Male	RMSE Overall
	FR	VAR	0.032	0.013	0.021
	FR	HU	0.022	0.024	0.017
	FR	LC	0.054	0.050	0.041
	FR	M7	0.076	0.071	0.065
	FR	RESPECT	0.021	0.025	0.022
	FR	STAR	0.042	0.045	0.038
	UK	VAR	0.014	0.015	0.018
	UK	HU	0.024	0.029	0.021
	UK	LC	0.052	0.059	0.050
	UK	M7	0.058	0.049	0.044
	UK	RESPECT	0.022	0.027	0.016
	UK	STAR	0.040	0.044	0.035
	US	VAR	0.020	0.017	0.017
	US	HU	0.017	0.016	0.014
	US	LC	0.045	0.049	0.042
	US	M7	0.048	0.050	0.046
	US	RESPECT	0.010	0.009	0.006
	US	STAR	0.025	0.024	0.023

Note: This table reports the RMSE values obtained after tting the VAR-ENET and the considered benchmark models. We compare this indicator for males, females and the overall populations for FR, UK and US.

Table 3 :

 3 Summary statistics for the RMSE of the VAR-ENET and the benchmark models.Note: This table reports statistics of the RMSE values obtained after tting the VAR-ENET and the considered benchmark models over the males, females and the overall populations for FR, UK and US.

	Model	Mean	Standard Deviation	Minimum	Maximum
	VAR	0.019	0.006	0.013	0.032
	HU	0.020	0.005	0.014	0.029
	LC	0.049	0.006	0.041	0.059
	M7	0.056	0.012	0.044	0.076
	RESPECT	0.018	0.008	0.006	0.027
	STAR	0.035	0.009	0.023	0.045

Table 4 :

 4 The RMSFE of the VAR and the benchmark models estimated on the period 1950 ¡ 2000.Note: This table reports the out-of-sample performance via the RMSFE values

	Country	Model	RMSFE Female	RMSFE Male	RMSFE Overall
	FR	VAR	0.088	0.110	0.078
	FR	HU	0.082	0.112	0.067
	FR	LC	0.111	0.113	0.067
	FR	M7	0.676	0.193	0.257
	FR	RESPECT	0.083	0.091	0.071
	FR	STAR	0.098	0.127	0.417
	UK	VAR	0.095	0.087	0.080
	UK	HU	0.109	0.138	0.122
	UK	LC	0.142	0.141	0.138
	UK	M7	0.228	0.099	0.145
	UK	RESPECT	0.281	0.230	0.296
	UK	STAR	0.083	0.115	0.156
	US	VAR	0.078	0.116	0.078
	US	HU	0.061	0.141	0.085
	US	LC	0.085	0.122	0.087
	US	M7	0.237	0.144	0.135
	US	RESPECT	0.110	0.081	0.075
	US	STAR	0.071	0.050	0.049

Table 5 :

 5 Summary statistics for the RMSFE of the VAR-ENET and the benchmark models.Note: This table reports statistics of the out-of-sample performance via the RMSFE values for the VAR-ENET and the considered benchmark models models estimated on the period 1950 ¡ 2000 over the males, females and the overall populations for FR, UK and US. dynamics, therefore the RMSFE strongly diverges, see e.g. M7 and STAR on the French data or HU on the English data. Nevertheless, our model doesn't suer from this drawback. Furthermore, it always belongs among the best models for any population and any period considered in this study.

										Model					Mean		Standard Deviation		Minimum		Maximum
										VAR					0.090						0.014						0.078			0.116
										HU						0.102						0.030						0.061			0.141
										LC						0.112						0.027						0.067			0.142
										M7						0.235						0.174						0.099			0.676
										RESPECT			0.146						0.094						0.071			0.296
										STAR					0.129						0.113						0.049			0.417
									FR																UK																US
		0.40
		0.38
		0.36		VAR
		0.34		HU LC
		0.32		M7
		0.30		RESPECT
	rmsfe (log mortality improvements)	0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28		STAR
		0.12
		0.10
		0.08
		0.06
		0.04
		0.02
		0.00
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
																									years

Table 6 :

 6 The RMSFE of the VAR and the benchmark models estimated on the period 1970 ¡ 2000.

	Country	Model	RMSFE Female	RMSFE Male	RMSFE Overall
	FR	VAR	0.088	0.092	0.068
	FR	HU	0.081	0.085	0.058
	FR	LC	0.085	0.086	0.055
	FR	M7	0.266	0.137	0.139
	FR	RESPECT	0.077	0.085	0.074
	FR	STAR	0.101	0.110	0.172
	UK	VAR	0.074	0.091	0.067
	UK	HU	0.116	0.134	0.126
	UK	LC	0.133	0.146	0.129
	UK	M7	0.148	0.111	0.097
	UK	RESPECT	0.321	0.232	0.260
	UK	STAR	0.073	0.083	0.069
	US	VAR	0.105	0.116	0.092
	US	HU	0.105	0.118	0.087
	US	LC	0.133	0.123	0.090
	US	M7	0.192	0.133	0.133
	US	RESPECT	0.148	0.081	0.077
	US	STAR	0.120	0.070	0.079

Table 7 :

 7 Summary statistics for the RMSFE of the VAR-ENET and the benchmark models estimated on the period 1970 ¡ 2000. Note: This table reports statistics of the out-of-sample performance via the RMSFE values for the VAR-ENET and the considered benchmark models models estimated on the period 1970 ¡ 2000 over the males, females and the overall populations for FR, UK and US.Figure 15: The RMSE for England and Wales (UK), the United States (US) and France (FR) grouped by period for the male populations.Figure 16: The RMSFE for England and Wales (UK), the United States (US) and France (FR) grouped by period for the female populations.Figure 17: The RMSFE for England and Wales (UK), the United States (US) and France (FR) grouped by age for the female populations.Figure 18: The RMSFE for England and Wales (UK), the United States (US) and France (FR) grouped by period for the male populations.Figure 19: The RMSFE for England and Wales (UK), the United States (US) and France (FR) grouped by age for the male populations.

	Model	Mean	Standard Deviation	Minimum	Maximum
	VAR	0.088	0.017	0.067	0.116
	HU	0.101	0.025	0.058	0.134
	LC	0.109	0.031	0.055	0.146
	M7	0.151	0.051	0.097	0.266
	RESPECT	0.151	0.096	0.074	0.321
	STAR	0.097	0.033	0.069	0.172

More precisely, it is their alternatives using mortality improvements, as documented by[START_REF] Haberman | Parametric mortality improvement rate modelling and projecting[END_REF].

Acknowledgments

This work beneted from the nancial support of the ANR project "LoLitA" (ANR-13-BS01-0011). The authors are thankful to the anonymous referee for valuable advice and suggestions provided in relation to the rst draft of this paper.

Note: This table reports the out-of-sample performance via the RMSFE values for the HU, the LC, the M7, the RESPECT, the STAR and the VAR-ENET p4q models estimated on the period 1980¡2000. We compare this indicator for males, females and the overall populations for FR, UK and US.