Age-coherent extensions of the Lee–Carter model

Accurate forecasts of mortality rates are essential to various types of demographic research such as population projection, and the pricing of insurance products such as pensions and annuities. Recent studies have considered a spatial temporal autoregressive (STAR) model for the mortality surface, where mortality rates for each age depend (temporally) on their historical values as well as (spatiality) on those of neighboring age cohorts. This model has sound statistical properties including cointegrated dependent variables and the existence of closed‐form solutions. Despite its improved forecasting performance over the famous Lee–Carter (LC) model, the constraint that only the effects of the same and neighboring cohorts are significant can be too restrictive. In this study, we adopt a data‐driven adaptive weighted structure and propose the adaptive STAR (ASTAR) model. Retaining all the desirable features of the STAR, our model uniformly outperforms the LC and STAR counterparts in terms of forecasting accuracy, when mortality data for ages 0–100 from the UK, France, Italy, Spain, and Japan over the period 1950–2016 are considered. Two sensitivity tests and additional simulation results also lead to robust conclusions. The proposed ASTAR model could therefore be a widely useful tool for modeling and forecasting mortality rates in other contexts, and may be extended to multipopulation modeling.

Multivariate Long Memory Cohort Mortality Models

Motivated by a recent discovery that the two-step inference for the Lee–Carter mortality model may be inconsistent when the mortality index does not follow from a nearly integrated AR(1) process, we propose a test for a unit root in a Lee–Carter model with an AR(p) process for the mortality index. Although testing for a unit root has been studied extensively in econometrics, the method and asymptotic results developed in this paper are unconventional. Unlike a blind application of existing R packages for implementing the two-step inference procedure in Lee and Carter (1992) to the U.S. mortality rate data, the proposed test rejects the null hypothesis that the mortality index follows from a unit root AR(1) process, which calls for serious attention on using the future mortality projections based on the Lee–Carter model in policy making, pricing annuities and hedging longevity risk. A simulation study is conducted to examine the finite sample behavior of the proposed test too.

Modeling and forecasting human mortality are significant research topics in several disciplines because mortality rates are fundamental in planning and policy decisions. Among various techniques, Lee Carter (LC) model is one of the most popular stochastic method in human mortality modeling. The original LC model was fuzzified to eliminate the assumptions related with homoscedasticity. The existing fuzzy model makes use of ordinary least squares (OLS) technique, which prevents the model to capture the existing fluctuations in data. In this study, a revised version of fuzzy LC model utilizing singular value decomposition (SVD) technique is proposed to overcome this issue. After modeling the mortality rates, their future values are forecasted by a modified first order fuzzy time series technique. For illustration purposes, proposed method is applied to mortality data of Finland. Numerical outputs show that proposed method is statistically better in modeling mortality compared to the existing fuzzy method. In addition, the modified fuzzy time series technique generates better forecasts than the original version.

Forecasting mortality has been a vital issue in demography and actuarial science.It also has profound implications for pension plan and long-term economic forecasts of the nation.In the present paper we examine various forecasting methods for mortality in the framework of cointegrated time series analysis.The Lee-Carter (LC) method has been regarded as the benchmark for forecasting mortality.However, its forecasting accuracy has been known to be particularly poor for short-term forecasts, while it is well for long-term forecasts.Recently, a new methods called the multivariate time series variance component (MTV) method has been proposed which explicitly satisfies cointegration restrictions of the series.It overcomes weak points of the LC method.In the present paper we propose two new methods.The first one is the modified MTV (mMTV) method which modifies the MTV method in order to get more accurate forecast of the trend component of the method.The second is the all-component Lee-Carter (LCA) method which generalizes the Lee-Carter method, by using all principal components, in order to improve short-term forecasts of the LC method.However, it may be noted that the LCA method does not satisfy cointegration restrictions.We analytically compare forecasting accuracy of the proposed methods with the Lee-Carter method and the MTV method in the framework of cointegrated time series.We further compare them in a Monte Carlo experiment and in an empirical application of forecasting mortality for Japanese male.It is shown that the mMTV method is generally the most accurate in the Monte Carlo experiment and in Japanese data.The MTV method works almost as well.However, since the drift estimator is inefficient, it is slightly less accurate than the mMTV method in some occasions.The forecast accuracy of the LCA method is reasonably high and can be equivalent to the mMTV method in some occasions, but is generally inferior to the MTV method and the mMTV method.As expected, the LC method is the worst among methods examined in the present study.The mMTV method is recommended for practical use.

Mortality forecasting plays a central role in both demographic and actuarial studies. Classical approaches, such as the Lee-Carter (LC) model, typically rely on mortality rates as the primary measure. In recent years, compositional data analysis (CoDA), which respects summability and non-negativity constraints, has gained increasing attention for mortality forecasting. While the centred log-ratio (CLR) transformation is commonly used to map compositional data to real space, the α-transformation, a generalisation of log-ratio transformations, offers greater flexibility and adaptability. This study introduces the α-transformation as an alternative to the CLR transformation within a CoDA framework under a discrete-age setting for all-cause mortality forecasting, which has not been previously investigated. To enable a fair comparison of transformation choices, zero values in the data are imputed, although the α-transformation can inherently accommodate them. Using age-specific life table death counts for males and females in 31 selected European countries/regions from 1983 to 2018, the proposed method demonstrates comparable performance to the CLR transformation in most cases, with improved forecast accuracy in some instances. These findings highlight the potential of the α-transformation to enhance mortality forecasting within the CoDA framework.

A person's life expectancy is one of the statistical measures used to determine how long they will live. However, in Sri Lanka, there is a lack of long-term, data-driven mortality forecasting, which limits the ability to make informed decisions in public health, insurance, and policy planning. This study addresses this gap by applying the Lee-Carter (LC) mortality estimation method to improve life expectancy analysis and mortality forecasting. In general, a life table that details the effects of mortality is used to calculate life expectancy. As a result, mortality predictions and life table analysis are important in the insurance, actuarial science, government, and pension planning sectors. The Registrar General's Office and the Department of Census and Statistics in Sri Lanka provided the mid-year population data and deaths as historical data for this study from 1992 to 2014. The Lee-Carter (LC) mortality estimation method is used to forecast mortality using actual death rates and mid-year populations from 1992 to 2014. ax (the average age-specific log mortality rate over time), bx (the sensitivity of mortality at each age to changes in the mortality index), and kt (the overall mortality index over time) parameters are estimated using the Singular Value Decomposition (SVD) method in the Lee-Carter methodology between 1992 and 2014. kt is forecasted for the years 2015–2035 using the traditional time series approach, and no specific time series model such as ARIMA or Exponential Smoothing was used. In 2035, the probability of dying as a 55-year-old woman and man are 0.023 and 0.051, respectively, according to the research. The following is a statement of the study's results: South Asia's average life expectancy is 69.41 years, while Sri Lanka's is 74.5 years. A country's future can be improved by thoroughly investigating demographics and life expectancy, as these factors directly affect financial stability.

BackgroundPrecise predictions of incidence and mortality rates due to breast cancer (BC) are required for planning of public health programs as well as for clinical services. A number of approaches has been established for prediction of mortality using stochastic models. The performance of these models intensely depends on different patterns shown by mortality data in different countries.MethodsThe BC mortality data is retrieved from the Global burden of disease (GBD) study 2017 database. This study include BC mortality rates from 1990 to 2017, with ages 20 to 80+ years old women, for different Asian countries. Our study extend the current literature on Asian BC mortality data, on both the number of considered stochastic mortality models and their rigorous evaluation using multivariate Diebold-Marino test and by range of graphical analysis for multiple countries.ResultsStudy findings reveal that stochastic smoothed mortality models based on functional data analysis generally outperform on quadratic structure of BC mortality rates than the other lee-carter models, both in term of goodness of fit and on forecast accuracy. Besides, smoothed lee carter (SLC) model outperform the functional demographic model (FDM) in case of symmetric structure of BC mortality rates, and provides almost comparable results to FDM in within and outside data forecast accuracy for heterogeneous set of BC mortality rates.ConclusionConsidering the SLC model in comparison to the other can be obliging to forecast BC mortality and life expectancy at birth, since it provides even better results in some cases. In the current situation, we can assume that there is no single model, which can truly outperform all the others on every population. Therefore, we also suggest generating BC mortality forecasts using multiple models rather than relying upon any single model.

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

Existing literature argues that the mortality rate of a specific age is affected not only by its own lags but by the lags of neighbouring ages, known as cohort effects. Although these effects are assumed constant in most studies, they can be dynamic over a long timespan. Consequently, popular mortality models with time-invariant age-dependent coefficients, including the Lee-Carter (LC) and vector autoregression (VAR) models, are incapable of modelling these dynamic cohort effects. To capture such dynamic patterns, we propose a time-varying coefficient spatial-temporal autoregressive (TVSTAR) model that allows for flexible time-dependent parameters. The proposed TVSTAR model is compatible with multi-population modelling and enjoys sound statistical properties. Using empirical results of mortality data from the United Kingdom (UK) and France over the period 1950–2016, we show that the TVSTAR model consistently outperforms the LC (Li-Lee, or LL) and the original STAR model under the single-population (multi-population) modelling framework. Finally, our empirical results suggest that cohort effects strengthen over time for very old ages in both the UK and France. Using simulation evidence, we argue that this observed upward trend can be caused by the overall advancement in the mortality evolution of the same cohort.

Mortality rates are often disaggregated by different attributes, such as sex, state, education, religion, or ethnicity. Forecasting mortality rates at the national and sub-national levels plays an important role in making social policies associated with the national and sub-national levels. However, base forecasts at the sub-national levels may not add up to the forecasts at the national level. To address this issue, we consider the problem of reconciling mortality rate forecasts from the viewpoint of grouped time-series forecasting methods (Hyndman et al. in, Comput Stat Data Anal 55(9):2579–2589, 2011). A bottom-up method and an optimal combination method are applied to produce point forecasts of infant mortality rates that are aggregated appropriately across the different levels of a hierarchy. We extend these two methods by considering the reconciliation of interval forecasts through a bootstrap procedure. Using the regional infant mortality rates in Australia, we investigate the one-step-ahead to 20-step-ahead point and interval forecast accuracies among the independent and these two grouped time-series forecasting methods. The proposed methods are shown to be useful for reconciling point and interval forecasts of demographic rates at the national and sub-national levels, and would be beneficial for government policy decisions regarding the allocations of current and future resources at both the national and sub-national levels.

Mortality forecasting has always been a target of study by academics and practitioners. Since the introduction and rising significance of securitization of risk in mortality and longevity, more in-depth studies regarding mortality have been carried out to enable the fair pricing of such derivatives. In this article, a comparative analysis is performed on the mortality forecasting accuracy of four mortality models. The methodology employs the Age-Period-Cohort model, the Cairns-Blake-Dowd model, the classical Lee-Carter model and the Kou-Modified Lee-Carter model. The Kou-Modified Lee-Carter model combines the classical Lee-Carter with the Double Exponential Jump Diffusion model. This paper is the first study to employ the Kou model to forecast French mortality data. The dataset comprises death data of French males from age 0 to age 90, available for the years 1900–2015. The paper differentiates between two periods: the 1900–1960 period where extreme mortality events occurred for French males and the 1961–2015 period where no significant jump is observed. The Kou-modified Lee-Carter model turns out to give the best mortality forecasts based on the RMSE, MAE, MPE and MAPE metrics for the period 1900–1960 during which the two World Wars occurred. This confirms that the consideration of jumps and leptokurtic features conveys important information for mortality forecasting.

It is well known that accurate forecasts of mortality rates are essential to various demographic research topics, such as population projections and the pricing of insurance products such as pensions and annuities. In this study, we argue that including the lagged rates of neighbouring ages cannot further improve mortality forecasting after allowing for autocorrelations. This is because the sample cross-correlation function cannot exhibit meaningful and statistically significant correlations. In other words, rates of neighbouring ages are usually not leading indicators in mortality forecasting. Therefore, multivariate stochastic mortality models like the classic Lee–Carter may not necessarily lead to more accurate forecasts, compared with sophisticated univariate models. Using Australian mortality data, simulation and empirical studies employing the Lee–Carter, Functional Data, Vector Autoregression, Autoregression-Autoregressive Conditional Heteroskedasticity and exponential smoothing (ETS) state space models are performed. Results suggest that ETS models consistently outperform the others in terms of forecasting accuracy. This conclusion holds for both female and male mortality data with different empirical features across various forecasting error measurements. Hence, ETS can be a widely useful tool to model and forecast mortality rates in actuarial practice.

The Lee–Carter model could be considered as one of the most important mortality prediction models among stochastic models in the field of mortality. With the recent developments of machine learning and deep learning, many studies have applied deep learning approaches to time series mortality rate predictions, but most of them only focus on a comparison between the Long Short-Term Memory and the traditional models. In this study, three different recurrent neural networks, Long Short-Term Memory, Bidirectional Long Short-Term Memory, and Gated Recurrent Unit, are proposed for the task of mortality rate prediction. Different from the standard country level mortality rate comparison, this study compares the three deep learning models and the classic Lee–Carter model on nine divisions’ yearly mortality data by gender from 1966 to 2015 in the United States. With the out-of-sample testing, we found that the Gated Recurrent Unit model showed better average MAE and RMSE values than the Lee–Carter model on 72.2% (13/18) and 67.7% (12/18) of the database, respectively, while the same measure for the Long Short-Term Memory model and Bidirectional Long Short-Term Memory model are 50%/38.9% (MAE/RMSE) and 61.1%/61.1% (MAE/RMSE), respectively. If we consider forecasting accuracy, computing expense, and interpretability, the Lee–Carter model with ARIMA exhibits the best overall performance, but the recurrent neural networks could also be good candidates for mortality forecasting for divisions in the United States.

This study investigated the performance of the Lee-Carter (LC) method and it variants in modeling and forecasting Malaysia mortality. These include the original LC, the Lee-Miller (LM) variant and the Booth-Maindonald-Smith (BMS) variant. These methods were evaluated using Malaysia’s mortality data which was measured based on age specific death rates (ASDR) for 1971 to 2009 for overall population while those for 1980–2009 were used in separate models for male and female population. The performance of the variants has been examined in term of the goodness of fit of the models and forecasting accuracy. Comparison was made based on several criteria namely, mean square error (MSE), root mean square error (RMSE), mean absolute deviation (MAD) and mean absolute percentage error (MAPE). The results indicate that BMS method was outperformed in in-sample fitting for overall population and when the models were fitted separately for male and female population. However, in the case of out-sample forecast accuracy, BMS method only best when the data were fitted to overall population. When the data were fitted separately for male and female, LCnone performed better for male population and LM method is good for female population.