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Figure 10-5
A sequence generated
by the recursions
xn 4xn−1(1 − xn−1).
10.3 Nonlinear Models 345
frequently observed in practical time series and are seen also in the sample paths of
nonlinear (and infinite-variance) models. They are rarely seen, however, in the sample
paths of Gaussian linear processes. Other characteristics suggesting deviation from
a Gaussian linear model are discussed by Tong (1990).
Many observed time series, particularly financial time series, exhibit periods
during which they are “less predictable” (or “more volatile”), depending on the past
history of the series. This dependence of the predictability (i.e., the size of the prediction
mean squared error) on the past of the series cannot be modeled with a linear
time series, since for a linear process the minimum h-step mean squared error is
independent of the past history. Linear models thus fail to take account of the possibility
that certain past histories may permit more accurate forecasting than others,
and cannot identify the circumstances under which more accurate forecasts can be
expected. Nonlinear models, on the other hand, do allow for this. The ARCH and
GARCH models considered below are in fact constructed around the dependence of
the conditional variance of the process on its past history.
10.3.2 Chaotic Deterministic Sequences
To distinguish between linear and nonlinear processes, we need to be able to decide in
particular when a white noise sequence is also iid. Sequences generated by nonlinear
deterministic difference equations can exhibit sample correlation functions that are
very close to those of samples from a white noise sequence. However, the deterministic
nature of the recursions implies the strongest possible dependence between successive
observations. For example, the celebrated logistic equation (see May, 1976, and Tong,
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