By Howell Tong (auth.)
In the final years or so, i used to be so much lucky in being given possibilities of lecturing on a brand new technique to a number of audiences in Britain, China, Finland, France and Spain. regardless of my virtually Confucian perspective of who prefer speaking (i.e. a brief list) to writing (i.e. an enduring record), the nice and cozy encouragement of associates has resulted in the consequent notes. i'm additionally basically too aware of the infancy of the technique brought in those notes. besides the fact that, it really is my honest desire that publicity to a much wider viewers will speed up its adulthood. Readers are assumed to be accustomed to the elemental conception of time sequence research. The booklet through Professor M.B. Priestley (1981) can be utilized as a common reference. bankruptcy One is addressed to the overall query: "why will we want non-linear time sequence models?" After describing a few major merits of linear types, it singles out numerous significant boundaries of linearity. after all, the choice displays my own view at the topic, that is merely at its very starting, even supposing there does appear to be a common contract within the literature that point irr'eversibility and restrict cycles are one of the so much obvious.
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Additional resources for Threshold Models in Non-linear Time Series Analysis
Example text
Brillinger and Rosenblatt. 210). A stationary time series {X t } is time-reversible if for every positive integer n. • t n £ l. • X_t ) have the same joint 1 2 n 1 2 n distributions. A stationary time series which is not time-reversible is said to be time-;'rreversible. 12) Yt : g(X t ). for each t. where 9 is a one-to-one function. Now. it is obvious that {Y t } is stationary and time-reversible if and only it {Xt } is stationary and time-reversible. Thus. it would be inappropriate to fit ARMA models to time-irreversible data even after applying a one-to-one transformation g.
In the former case. limitations (2). (3). (4). to some extent. by a judicious choice of the distribution of the £t' As a typical illustration. 1) 32 (-1:3. /3). After some where Et has a uniform distribution on manipulation E(XtIXt_l) is shown to be non-linear as illustrated in Figs. 8. E ext! )JJ Fig. /3); (a::. 1). ------Pk--t=t:;---;-I"77--~ " Fi g. 8 Regression function of lag(-l) of model Xt = £t - aEt_l' Et uniformly distributed on (-/3. 13); (a < 1). Shepp. Slepian and Wyner (1980) have given a detailed study of the reg~ession functions.
5 + + + + + + + Fig. 5 Sample regression function of Xt on Xt _l for data in Fig. 4. 79. Sample estimate of YOPj2/Var{E(XtIXt_l)} is roughly 1. 5 FIG. 6 Sample regression function of Xt on Xt - 3 for data in Fig. 4. 13. 04. 10}. That is. We call ~in [YO Pj2/var{E{XtIXt_j)}] the second order index of J£Z linearity. When var{E{XtIXt_j)} to this is defined to be 1. = 0 for some j. the ratio of Yo Pj2 Let this index be denoted by~. It involves only the bivariate distribution. and 0 sl sLIt may be roughly interpreted as a practical measure of how close the probabilistic structure of {X t } is to normality (see Tong.