Understanding the Lee-Carter Mortality Forecasting Method

We demonstrate here several previously unrecognized or insufficiently appreciated properties of the Lee-Carter mortality forecasting approach, a method used widely in both the academic literature and practical applications. We show that this model is a special case of a considerably simpler, and less often biased, random walk with drift model, and prove that the age profile forecast from both approaches will always become less smooth and unrealistic after a point (when forecasting forward or backwards in time) and will eventually deviate from any given baseline. We use these and other properties we demonstrate to suggest when the model would be most applicable in practice. The method proposed in Lee and Carter (1992) has become the “leading statistical model of mortality [forecasting] in the demographic literature” (Deaton and Paxson, 2004). It was used as a benchmark for recent Census Bureau population forecasts (Hollmann, Mulder and Kallan, 2000), and two U.S. Social Security Technical Advisory Panels recommended its use, or the use of a method consistent with it (Lee and Miller, 2001). In the last decade, scholars have “rallied” (White, 2002) to this and closely related approaches, and policy analysts forecasting all-cause and cause-specific mortality in countries around the world have followed suit (Booth, Maindonald and Smith, 2002; Deaton and Paxson, 2004; Haberland and Bergmann, 1995; Lee, Carter and Tuljapurkar, 1995; Lee and Rofman, 1994; Lee and Skinner, 1999; Miller, 2001; NIPSSR, 2002; Perls et al., 2002; Preston, 1993; Tuljapurkar and Boe, 1998; Tuljapurkar, Li and Boe, 2000; Wilmoth, 1996, 1998a,b). Lee and Carter developed their approach specifically for U.S. mortality data, 1933-1987. However, the method is now being applied to all-cause and cause-specific mortality data from many countries and time periods, all well beyond the application for which it was designed. It thus appears to be a good time to reassess the approach, as the issues these new applications pose could not have been foreseen by the original authors. See Lee (2000a) for an earlier effort along these lines. In this paper, we demonstrate several previously unrecognized or insufficiently appreciated properties of the Lee-Carter model, and use these properties to suggest where and when the model would be most applicable. Section 1 describes the forecasting method in some detail. Since the method can be seen as a special case of a principal components method (Bozik and Bell, 1987; Bell and Monsell, 1991) with a single component , Section 2 summarizes a diverse array of 240 mortality data sets (24 causes of death from 10 countries each) to give a sense of where the method has a chance of working well. Then, in Section 3 we show that the Lee-Carter model is equivalent to a special type of multivariate random walk with drift (RWD) model, in which the covariance matrix depends on the drift vector. The implication of this special structure is that while the RWD estimator is unbiased for data generated by the Lee-Carter model and other types of data, the Lee-Carter model is biased for data generated by the general RWD model with arbitrary covariance matrix. The Lee-Carter estimator is more efficient when data are known to be drawn from the Lee-Carter model. These observations suggest that, since the RWD does not make any assumption about the structure of the covariance matrix, while the Lee-Carter approach does, the Lee-Carter estimator will be preferable to the RWD only when we have high confidence in its underlying assumptions. The similarity of the two models means that the much