ABSTRACT
Toward the end of the seventies of the last century there was an increasing demand to model more complex time series features than those given by a linear autoregressive moving average (ARMA) structure. One drawback of the stationary ARMA model with Gaussian noise at is that it is unable to capture time irreversibility. Time irreversibility is one of the major features exhibited by a nonlinear or non-Gaussian time series model. A stationary time series Xt is time reversible if for any integer n > 0, and any t1, t2, . . . , tn that are integers, the vectors (Xt1 , Xt2 , . . . , Xtn) and (X−t1 , X−t2 , . . . , X−tn) have the same multivariate distribution. A stationary time series that is not time reversible is said to be time irreversible. The result of Weiss (1975) showed that stationary ARMA processes with a nontrivial AR component are time reversible if and only if they are Gaussian. The technical report by Tong and Zhang (2003) gave more results on the conditions of time reversibility. Figure 5.1 shows the time series plot of the Canadian Lynx data 1821-1934 as listed by Elton and Nicholson (1942). It can be seen that the time series take more time to reach the peaks than to come down from the peaks to the troughs. This suggested that the above definition of reversibility would not hold for the Lynx data. Another way of seeing this is to place a mirror on the y-axis and for the mirror image it will take less time to climb up to the peaks than to come down from the peaks. Naturally, new nonlinear models are required to capture these kinds of features. There are, of course, other features arising from nonlinearity that cannot be mimicked by the linear Gaussian ARMA models. One of these is the limit cycles exhibited by a nonlinear difference equation. A limit cycle is a set of points {x1, . . . , xT } with a mapping f(x) such that f(xi) = xi+1, i = 1, . . . , T − 1, and xT+i = xi, i = 1, 2, . . .. Suppose a time series is defined by Xt = g(Xt−1, at), where at is a zero mean white noise process independent of Xt−1. Then we say that Xt admits a limit cycle, when at is set to its mean zero, the mapping Xt = g(Xt−1, 0) induces a recursion Xt = f(Xt−1) that has a limit cycle as t→∞ (Chan
and Tong, 1990). A stationary ARMA model can only have a limit cycle in the trivial case T = 1.
