The mortality of the Italian population: Smoothing techniques on the Lee–Carter model

Abstract

Several approaches have been developed for forecasting mortality using the stochastic model. In particular, the Lee-Carter model has become widely used and there have been various extensions and modifications proposed to attain a broader interpretation and to capture the main features of the dynamics of the mortality intensity. Hyndman-Ullah show a particular version of the Lee-Carter methodology, the so-called Functional Demographic Model, which is one of the most accurate approaches as regards some mortality data, particularly for longer forecast horizons where the benefit of a damped trend forecast is greater. The paper objective is properly to single out the most suitable model between the basic Lee-Carter and the Functional Demographic Model to the Italian mortality data. A comparative assessment is made and the empirical results are presented using a range of graphical analyses. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Statistics, 2011, Vol. 5, No. 2A, 705-724. This reprint differs from the original in pagination and typographic detail. 1 2 V. D'AMATO, G. PISCOPO AND M. RUSSOLILLOits expected values. These changes clearly affect pricing and reserve allocation for life annuities and represent one of the major threats to a social security system that has been planned on the basis of a more modest life expectancy. The risk is of using mortality tables that do not take these trends into account, thus underestimating the survival probability and determining inappropriate premiums. To face this risk, it is necessary to build projected tables including this trend. Thus, reasonable mortality forecasting techniques have to be used to consistently predict the trends [Brouhns, Denuit and Vermunt (2002)]. In that respect, over the years a number of approaches have been proposed for forecasting mortality using the stochastic model, however, the Lee-Carter model [Lee and Carter (1992)] unquestionably represents a milestone in the literature. This methodology has become widely used and there have been various extensions and modifications proposed to attain a broader interpretation and to capture the main features of the dynamics of the mortality intensity [e.g., Booth, Maindonald and Smith (2002); Haberman and Renshaw (2003, 2008); Hyndman and Ullah (2007); Renshaw and Haberman (2003a, 2003b)].The main statistical tool of LC is least-squares estimation via singular value decomposition of the matrix of the log age-specific observed death rates. In fact, the mortality data (death counts and exposures-to-risk) have to fill a rectangular matrix. Henceforth, we will denote with m x,t the observed death rates at age x during calendar year t, obtained by the ratio between the number of deaths, D x,t , recorded at age x during year t, from an exposure-to-risk E x,t , that is, the number of person years from which D x,t occurred. As regards the Italian population data set on the basis of the death rates, classified by gender and individual ...

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