Forecasting mortality rates with a coherent ensemble averaging approach

Abstract

Modeling and forecasting of mortality rates are closely related to a wide range of actuarial practices, such as the designing of pension schemes. To improve the forecasting accuracy, age coherence is incorporated in many recent mortality models, which suggests that the long-term forecasts will not diverge infinitely among age groups. Despite their usefulness, misspecification is likely to occur for individual mortality models when applied in empirical studies. The reliableness and accuracy of forecast rates are therefore negatively affected. In this study, an ensemble averaging or model averaging (MA) approach is proposed, which adopts age-specific weights and asymptotically achieves age coherence in mortality forecasting. The ensemble space contains both newly developed age-coherent and classic age-incoherent models to achieve the diversity. To realize the asymptotic age coherence, consider parameter errors, and avoid overfitting, the proposed method minimizes the variance of out-of-sample forecasting errors, with a uniquely designed coherent penalty and smoothness penalty. Our empirical data set include ten European countries with mortality rates of 0–100 age groups and spanning 1950–2016. The outstanding performance of MA is presented using the empirical sample for mortality forecasting. This finding robustly holds in a range of sensitivity analyses. A case study based on the Italian population is finally conducted to demonstrate the improved forecasting efficiency of MA and the validity of the proposed estimation of weights, as well as its usefulness in actuarial applications such as the annuity pricing.

Should Selection of the Optimum Stochastic Mortality Model Be Based on the Original or the Logarithmic Scale of the Mortality Rate?

“…Alternative stochastic mortality models can be used as reference in the construction of the optimal mortality model. Here, most extensions of the LC model define the reference mortality model assuming a Gaussian error structure of log mortality rates (Chang and Shi 2022;Gao and Shi 2021;Li and Lu 2017;Li and Shi 2021;SriDaran et al 2022) or a Poisson distribution of deaths (Barigou et al 2021;Chen and Millossovich 2018;Enchev et al 2017;Hunt and Blake 2014;Li 2013;Li et al 2016Pitt et al 2018;Wong et al 2020;Yang et al 2016). A less common option for the reference mortality model is to assume a binomial distribution of annual death probabilities (Atance et al 2020) or gamma distribution for mortality rates (Huang et al 2022).…”

“…Selection criteria frequently rely on measures based on squared errors (Chang and Shi 2022;Enchev et al 2017;Gao and Shi 2021;Li and Lu 2017;Li and Shi 2021), absolute errors (Li et al 2016, maximum likelihood (Pitt et al 2018;Yang et al 2016) or a combination of these measures (Atance et al 2020;Chen and Millossovich 2018;Huang et al 2022;Li 2013;Wong et al 2020). Additionally, even the same selection criteria measures are often defined based on either mortality rate predictions (estimates) (Atance et al 2020;Chen and Millossovich 2018) or log mortality rate predictions (estimates) (Chang and Shi 2022;Enchev et al 2017;Gao and Shi 2021;Li and Lu 2017;Li and Shi 2021;Li and Lee 2005;Wong et al 2020). Elsewhere, others have used a combination of measures based on mortality rates expressed on both original and log scales (Li 2013;Li et al 2016).…”

Forecasting Mortality Rates with a Two-Step LASSO Based Vector Autoregressive Model

“…More specifically, age coherence is defined as the condition that forecast mortality rates will not diverge between any two ages in the long run, and it is a desirable feature from the perspective of biological reasonableness (Li and Lu 2017). To tackle these issues, vector autoregressive (VAR) models have been proposed, with a major advantage of having more flexible temporal modeling of mortality rates (see, for example, Chang and Shi 2021, 2022a, 2022bGuibert et al 2019;Yang and Wang 2013, among others. ) In this study, we take a step further and propose a two-step LASSO based VAR (2-LVAR) model, and this new approach represents a significant contribution to the expanding family of VAR-type mortality models.…”