Semi-parametric extensions of the Cairns–Blake–Dowd model: A one-dimensional kernel smoothing approach

Over the last few decades, there has been an enormous growth in mortality modeling as the field of mortality risk and longevity risk has attracted great attention from academic, government and private sectors. In this paper, we propose a time-varying coefficient (TVC) mortality model aiming to combine the good characteristics of existing models with efficient model calibration methods. Nonparametric kernel smoothing techniques have been applied in the literature of mortality modeling and based on the findings from Li et al.’s (2015) study, such techniques can significantly improve the forecasting performance of mortality models. In this study we take the same path and adopt a kernel smoothing approach along the time dimension. Since we follow the model structure of the Cairns–Blake–Dowd (CBD) model, the TVC model we propose can be seen as a semi-parametric extension of the CBD model and it gives specific model design according to different countries’ mortality experience. Our empirical study presented here includes Great Britain, the United States, and Australia amongst other developed countries. Fitting and forecasting results from the empirical study have shown superior performances of the model over a selection of well-known mortality models in the current literature.     

Incorporating crossed classification credibility into the Lee–Carter model for multi-population mortality data

Recent developments in actuarial literature have shown that credibility theory can serve as an effective tool in mortality modelling, leading to accurate forecasts when applied to single or multi-population datasets. This paper presents a crossed classification credibility formulation of the Lee–Carter method particularly designed for multi-population mortality modelling. Differently from the standard Lee–Carter methodology, where the time index is assumed to follow an appropriate time series process, herein, future mortality dynamics are estimated under a crossed classification credibility framework, which models the interactions between various risk factors (e.g. genders, countries). The forecasting performances between the proposed model, the original Lee–Carter model and two multi-population Lee–Carter extensions are compared for both genders of multiple countries. Numerical results indicate that the proposed model produces more accurate forecasts than the Lee–Carter type models, as evaluated by the mean absolute percentage forecast error measure. Applications with life insurance and annuity products are also provided and a stochastic version of the proposed model is presented.     

How can a cause-of-death reduction be compensated for by the population heterogeneity? A dynamic approach

In the context of widening socioeconomic inequalities in mortality, it has become crucially important to understand the impact of population heterogeneity and its evolution on mortality. In particular, recent developments in multi-population mortality have raised a number of questions, among which is the issue of evaluating cause-of-death reduction targets set by national and international institutions in the presence of heterogeneity. The aim of this paper is to show how the population dynamics framework contributes to addressing these issues, relying on English population data and cause-specific number of deaths by socioeconomic circumstances, over the period 1981–2015.The analysis of the data first highlights the complexity of recent demographic developments, characterized by significant compositional changes, with considerable variations according to the age class or cohort, along with a widening of socioeconomic inequalities. We then introduce a dynamic framework for studying the impact of composition changes on the mortality of the global population. In particular, we show how a cause-of-death reduction could be compensated for in the presence of heterogeneity, which could lead to misinterpretations when assessing public policies impacts and/or for the forecasting of future trends.     

A semiparametric panel approach to mortality modeling

During the past twenty years, there has been a rapid growth in life expectancy and an increased attention on funding for old age. Attempts to forecast improving life expectancy have been boosted by the development of stochastic mortality modeling, for example the Cairns–Blake–Dowd (CBD) 2006 model. The most common optimization method for these models is maximum likelihood estimation (MLE) which relies on the assumption that the number of deaths follows a Poisson distribution. However, several recent studies have found that the true underlying distribution of death data is overdispersed in nature (see Cairns et al. 2009 and Dowd et al. 2010). Semiparametric models have been applied to many areas in economics but there are very few applications of such models in mortality modeling. In this paper we propose a local linear panel fitting methodology to the CBD model which would free the Poisson assumption on number of deaths. The parameters in the CBD model will be considered as smooth functions of time instead of being treated as a bivariate random walk with drift process in the current literature. Using the mortality data of several developed countries, we find that the proposed estimation methods provide comparable fitting results with the MLE method but without the need of additional assumptions on number of deaths. Further, the 5-year-ahead forecasting results show that our method significantly improves the accuracy of the forecast.     

Baseline models of spatial population dynamics

To understand human population dynamics fully, before considering complex human agency it may be useful to construct baseline models to see where such agency may and may not be necessary. In fact, the dynamics of human populations may be amenable to mathematical modeling with relatively parsimonious mechanisms. We review some of the more prominent of such models, namely, the spatial Galton-Watson (GW) model, modifications of the GW model that add migration and immigration, and the Bolker-Pacala model, in which mortality (or birth rate) is affected by competition. We show that change in the distribution of population density over the last century for 12 American rural states may be captured by the simplest of the models, the spatial GW model.     

Coherent mortality forecasting with generalized linear models: A modified time-transformation approach

In this paper, we propose an alternative approach for forecasting mortality for multiple populations jointly. Our contribution is developed upon the generalized linear models introduced by Renshaw et al., (1996) and Sithole et al., (2000), in which mortality forecasts are generated within the model structure, without the need of additional stochastic processes. To ensure that the resulting forecasts are coherent, a modified time-transformation is developed to stipulate the expected mortality differential between two populations to remain constant when the long-run equilibrium is attained. The model is then further extended to incorporate a structural change, an important property that is observed in the historical mortality data of many national populations. The proposed modeling methods are illustrated with data from two different pairs of populations: (1) Swedish and Danish males; (2) English and Welsh males and U.K. male insured lives.     

Multivariate dynamic modeling for Bayesian forecasting of business revenue

Forecasting enterprise-wide revenue is critical to many companies and presents several challenges and opportunities for significant business impact. This case study is based on model developments to address these challenges for forecasting in a large-scale retail company. Focused on multivariate revenue forecasting across collections of supermarkets and product categories, hierarchical dynamic models are natural: these are able to couple revenue streams in an integrated forecasting model, while allowing conditional decoupling to enable relevant and sensitive analysis together with scalable computation. Structured models exploit multi-scale modeling to cascade information on price and promotion activities as predictors relevant across categories and groups of stores. With a context-relevant focus on forecasting revenue 12 weeks ahead, the study highlights product categories that benefit from multi-scale information, defines insights into when, how, and why multivariate models improve forecast accuracy, and shows how cross-category dependencies can relate to promotion decisions in one category impacting others. Bayesian modeling developments underlying the case study are accessible in custom code for interested readers.     

区域经济发展智能预测方法

分析了影响区域经济发展的各种因素,指出由于这些因素相互制约、相互影响,使得传统的经济预测方法越来越难以胜任区域经济发展预测的需要.论述了核方法在处理非线性、不确定性和不精确性数据上存在的优势,建立了基于核方法三种经济预测模型,并将这三种预测模型与其它两种预测方法一起,对区域经济的发展进行组合预测.最后,采用数据融合的方法将各个体模型的预测结果进行集成,作为最终的输出.实际的结果表明,基于核方法的组合预测技术能取得较为理想的预测效果.     

Recursive estimation: A unified approach to the identification estimation,and forecasting of hydrological systems

Mathematical models of hydrological and water-resource systems have been formulated in many different ways and with various levels of complexity. There are advantages to be gained, therefore, by trying to unify some of the more common models within a statistical framework which will allow for more objective methods of model calibration. In this paper, we consider the general class of linear, dynamic models, as applied to the characterisation of flow and dispersion behavior in rivers, and show how these can be unified within the context of recursive time-series analysis and estimation. This allows not only for more objective, data-based approaches to stochastic model structure identification, but also for improved statistical estimation and the development of both constant parameter and self-adaptive, Kalman-filter-based forecasting procedures. The unified approach presented in the paper is being applied successfully in other environmental areas, such as soil science, climatic data analysis, meterological forecasting, and plant physiology.     

Explaining young mortality

Stochastic modeling of mortality rates focuses on fitting linear models to logarithmically adjusted mortality data from the middle or late ages. Whilst this modeling enables insurers to project mortality rates and hence price mortality products it does not provide good fit for younger aged mortality. Mortality rates below the early 20’s are important to model as they give an insight into estimates of the cohort effect for more recent years of birth. It is also important given the cumulative nature of life expectancy to be able to forecast mortality improvements at all ages. When we attempt to fit existing models to a wider age range, 5-89, rather than 20-89 or 50-89, their weaknesses are revealed as the results are not satisfactory. The linear innovations in existing models are not flexible enough to capture the non-linear profile of mortality rates that we see at the lower ages. In this paper, we modify an existing 4 factor model of mortality to enable better fitting to a wider age range, and using data from seven developed countries our empirical results show that the proposed model has a better fit to the actual data, is robust, and has good forecasting ability.     

Market risk forecasting for high dimensional portfolios via factor copulas with GAS dynamics

In this paper we propose forecasting market risk measures, such as Value at Risk (VaR) and Expected Shortfall (ES), for large dimensional portfolios via copula modeling. For that we compare several high dimensional copula models, from naive ones to complex factor copulas, which are able to simultaneously tackle the curse of dimensionality and introduce a high level of complexity into the model. We explore both static and dynamic copula fitting. In the dynamic case we allow different levels of flexibility for the dependence parameters which are driven by a GAS (Generalized Autoregressive Scores) model, in the spirit of Oh and Patton (2015). Our empirical results, for assets negotiated at Brazilian BOVESPA stock market from January, 2008 to December, 2014, suggest that, compared to the other copula models, the GAS dynamic factor copula approach has a superior performance in terms of AIC (Akaike Information Criterion) and a non-inferior performance with respect to VaR and ES forecasting.     

Triple seasonal methods for short-term electricity demand forecasting

Online short-term load forecasting is needed for the real-time scheduling of electricity generation. Univariate methods have been developed that model the intraweek and intraday seasonal cycles in intraday load data. Three such methods, shown to be competitive in recent empirical studies, are double seasonal ARMA, an adaptation of Holt–Winters exponential smoothing for double seasonality, and another, recently proposed, exponential smoothing method. In multiple years of load data, in addition to intraday and intraweek cycles, an intrayear seasonal cycle is also apparent. We extend the three double seasonal methods in order to accommodate the intrayear seasonal cycle. Using six years of British and French data, we show that for prediction up to a day-ahead the triple seasonal methods outperform the double seasonal methods, and also a univariate neural network approach. Further improvement in accuracy is produced by using a combination of the forecasts from two of the triple seasonal methods.     

Modeling multi-country mortality dependence and its application in pricing survivor index swaps—A dynamic copula approach

This paper introduces mortality dependence in multi-country mortality modeling using a dynamic copula approach. Specifically, we use time-varying copula models to capture the mortality dependence structure across countries, examining both symmetric and asymmetric dependence structures. In addition, to capture the phenomenon of a heavy tail for the multi-country mortality index, we consider not only the setting of Gaussian innovations but also non-Gaussian innovations under the Lee–Carter framework model. As tests of the goodness of fit of different dynamic copula models, the pattern of mortality dependence, and the distribution of the innovations, we used empirical mortality data from Finland, France, the Netherlands, and Sweden. To understand the effect of mortality dependence on longevity derivatives, we also built a valuation framework for pricing a survivor index swap, then investigated the fair swap rates of a survivor swap numerically. We demonstrate that failing to consider the dynamic copula mortality model and non-Gaussian innovations would lead to serious underestimations of the swap rates and loss reserves.     

模糊时间数列的分析与预测:以台湾地区加权股价指数为例

随着我国经济快速成长，衍生性金融商品的投资分析，已成为国内财务数学研究热门课题。以股票市场而言，人们总希望比别人早一步掌握行情的脉动，以获取最高的报酬率，然而，影响股市加权股价指数波动的因素众多，要如何进行趋势分析与预测，是很多学者相当感兴趣与研究的主题。本文考虑以模糊统计方法，作模糊时间数列的趋势分析与预测。其望应用模糊统计分析方法比传统的时间数列分析方法能得到更合理的解释，且预测结果可以提供决策者更多的信息，做出正确的决策。最后以台湾地区加权股票指数为例，做一实证上的详细探讨。     

Fuzzy adaptive network in presidential elections

Several scientific forecasting models for presidential elections have been suggested. However, most of these models are based on traditional statistics approaches. Since the system is linguistic, vague, and dynamic in nature, the traditional rigorous mathematical approaches are inappropriate for the modeling of this kind of humanistic system. This paper presents a combined neural fuzzy approach, namely a fuzzy adaptive network, to model and forecast the problem of a presidential election. The fuzzy adaptive network, which is ideally suited for the modeling of vaguely defined humanistic systems, combines the advantages of the representation ability of fuzzy sets and the learning ability of a neural network. To illustrate the approach, experiments were carried out by first formulating the problem, then training the network, and, finally, predicting the election results based on the trained network. The experimental results show that a fuzzy adaptive network is an ideal approach for the modeling and forecasting of national presidential elections.     

AN ENCOMPASSING APPROACH TO MODELING FISHERY DYNAMICS: MODELING DYNAMIC NONLINEAR SYSTEMS

Uncertainty about the nature and significance of nonlinearities and the manner in which dynamics affect future realizations makes model specification the most difficult aspect of modeling dynamic systems. By interpreting several popular fishery models as subcases of a nesting dynamic Taylor series approximation, we isolate the specification differences between these models in a way that accounts for commonalities. On the argument that the differences due to alternative nonlinear forms are likely to be small compared to more mundane considerations such as delay difference and general dynamic lag specification, we propose an alternative model that uses the terms from the first order approximation common to all models combined with a data-based determination of the appropriate lags using the methods of state space time series analysis. Finally, the success of the alternative models is judged in an application to Pacific halibut data.     

A combined analytical singularity subtraction method and compact implicit differencing scheme for aerodynamic mixing

In spite of the rapid advances in both scalar and parallel computational tools, the large number and breadth of variables involved in both design and inverse problems make the use of sophisticated fluid flow models impractical. With this restriction, it may be concluded that an important family of methods for mathematical/computational development are reduced or approximate models. In this study, a combined perturbation/numerical modeling methodology is developed. The numerical portion of the model uses a compact finite difference scheme, while analytical solutions are used to resolve singular behavior that is inherent to this flow. Solutions are presented to illustrate the efficiency of this methodology.     

Dynamic mortality factor model with conditional heteroskedasticity

In most methods for modeling mortality rates, the idiosyncratic shocks are assumed to be homoskedastic. This study investigates the conditional heteroskedasticity of mortality in terms of statistical time series. We start from testing the conditional heteroskedasticity of the period effect in the naïve Lee-Carter model for some mortality data. Then we introduce the Generalized Dynamic Factor method and the multivariate BEKK GARCH model to describe mortality dynamics and the conditional heteroskedasticity of mortality. After specifying the number of static factors and dynamic factors by several variants of information criterion, we compare our model with other two models, namely, the Lee-Carter model and the state space model. Based on several error-based measures of performance, our results indicate that if the number of static factors and dynamic factors is properly determined, the method proposed dominates other methods. Finally, we use our method combined with Kalman filter to forecast the mortality rates of Iceland and period life expectancies of Denmark, Finland, Italy and Netherlands.     

Time series forecasting with neural network ensembles: an application for exchange rate prediction

This paper investigates the use of neural network combining methods to improve time series forecasting performance of the traditional single keep-the-best (KTB) model. The ensemble methods are applied to the difficult problem of exchange rate forecasting. Two general approaches to combining neural networks are proposed and examined in predicting the exchange rate between the British pound and US dollar. Specifically, we propose to use systematic and serial partitioning methods to build neural network ensembles for time series forecasting. It is found that the basic ensemble approach created with non-varying network architectures trained using different initial random weights is not effective in improving the accuracy of prediction while ensemble models consisting of different neural network structures can consistently outperform predictions of the single ‘best’ network. Results also show that neural ensembles based on different partitions of the data are more effective than those developed with the full training data in out-of-sample forecasting. Moreover, reducing correlation among forecasts made by the ensemble members by utilizing data partitioning techniques is the key to success for the neural ensemble models. Although our ensemble methods show considerable advantages over the traditional KTB approach, they do not have significant improvement compared to the widely used random walk model in exchange rate forecasting.     

A comparison of the Lee-Carter model and AR-ARCH model for forecasting mortality rates

With the decline in the mortality level of populations, national social security systems and insurance companies of most developed countries are reconsidering their mortality tables taking into account the longevity risk. The Lee and Carter model is the first discrete-time stochastic model to consider the increased life expectancy trends in mortality rates and is still broadly used today. In this paper, we propose an alternative to the Lee-Carter model: an AR(1)-ARCH(1) model. More specifically, we compare the performance of these two models with respect to forecasting age-specific mortality in Italy. We fit the two models, with Gaussian and t-student innovations, for the matrix of Italian death rates from 1960 to 2003. We compare the forecast ability of the two approaches in out-of-sample analysis for the period 2004-2006 and find that the AR(1)-ARCH(1) model with t-student innovations provides the best fit among the models studied in this paper.