Inference and Forecasting for Chaotic Nonlinear Time Series

John Geweke
Duke University

Aug 31 1988

Inference and forecasting for the "tent map" process
xt = a (1 - 2|xt-1 - .5|); .5 are considered for the case in which x1 is observed subject to an additive IIDN(O, σ2) measurement error. This simple model provides a paradigm for more complex and realistic chaotic nonlinear models. Several novel findings are reported.

  1. The likelihood function is characterized by local maxima whose separations are on the order of (2a)-N. Yet virtually all of the mass is concentrated in the neighborhoods of a few of these points whose areas are on the order of (2a)-2N. Graphical presentations indicate the influence of sample size, the parameter a, and the variance of measurement error on the likelihood function.
  2. Steepest ascent and conventional grid methods cannot locate mass points of the likelihood function. New adaptive grid methods developed specifically for these models succeed in doing so.
  3. The adaptive grid methods permit the construction of exact posterior densities for the unobserved signals {xt, t=1, ... , N} and predictive densities for {xt, t = N+1, ... }. The interquartile ranges are proportional to σ2(2a)t-N, which has important implications for sampling and forecasting.


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