Imparting structural instability to mortality forecasts: Testing for sensitive dependence on initial conditions with innovations

This article explores a nontraditional approach to examining the problem of forecast uncertainty in extrapolative demographic models. It builds on prior research on stochastic time series forecast models, but diverges to examine their deterministic counterparts. The focus here is an examination of the structural integrity of the Lee‐Carter (1992) method applied to mortality forecasts. I investigate the nonlinear dynamics of the Lee‐Carter method, particularly its sensitive dependence of the forecasts on the initial conditions of the model. I examine the Lee‐Carter nonlinear demographic model, mx,t — exp (ax+ bxkt + ex,t), which is decomposed using SVD to derive a single time‐varying linear index of mortality, kt. From a 90 year time series of kt, forty nine 40 year realizations are sampled. These realizations are modeled and estimated using Box‐Jenkins techniques. The estimated parameters of these realizations and the first case of each of the samples are the initial conditions for the iterations of nonlinearized transformation of k, to exp (kt). The terminal year for each of the 49 iterated series is 2065. The deterministic nonlinear dynamics of this system of 49 iterated series is investigated by testing its Lyapunov exponents as a nonparametric diagnostic of a one dimensional dynamical system. The exponents are all negative, indicating that chaos is not prevalent in this system. The nonexistence of chaos suggests stability in the model and reaffirms the predictability of this one dimensional map. Augmenting the iterations of the initial conditions with additive stochastic innovations, {et, t ≥ 1}, creates a stochastic dynamical system of the form, k _t = k _t,−1 — c + ϕ flu +et. Here, et is treated as a surrogate for some unanticipated time series event (e.g. an epidemic) that impacts the deterministic map. Gaussian white noise innovations do not move the iterations far from equilibrium and only for short time intervals. So, stepping the mean of the innovations by .01 produces stable Lyapunov exponents until the mean equals .35 where some of the exponents are positive. At this point, deterministic chaos is evident, implying instability in the forecasts. The substantive implications of this instability are discussed.

Notes

This article is a revised version of papers presented at the 1997 Annual Meeting of the International Symposium on Forecasting and the 1998 Annual Meeting of the Population Association of America. I thank the editor for his very thoughtful comments.