Parameter estimation and forecasts for an integrated Lee-Carter model

Abstract

The model to represent mortality that Lee and Carter proposed is still widely used. They proposed a method to estimate the parameters in the model with mortality data using the singular value decomposition. In addition, in the forecasting of future mortality, the time-dependent parameter is linearly estimated using ARIMA (0,1,0). This method treats the parameters as nonstochastic for part of the model construction using observed data, while it is stochastic for the forecasting of future mortality. This results in inconsistencies throughout the whole model. Girosi and King interpreted the parameters in the Lee-Carter model as random variables and provided an integrated expression for the two parts, which derived a single stochastic model; however, the covariance matrix in the single stochastic model is not the one by an ordinal ARIMA(0,1,0) and an estimation method was not clearly discussed, and estimating the parameters in their model appears to be difficult owing to the complicated covariance matrix. Therefore, this study proposes a new integrated model of the Lee-Carter model based on Girosi and King’s interpretation. Our model is defined as a stochastic model that does not conflict with the concepts of Lee and Carter’s existing model with the covariance matrix deduced by Girosi and King. Furthermore, we provide an estimation method for the parameters in our model by applying the idea of conditional distributions used in classical AR and MA models.