Fixed Point Theorems by D. R. Smart

By D. R. Smart

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8) by xtÀk and taking expectations yields k ˆ 0kÀ1 Y for k b 1 whilst for k ˆ 0 and k ˆ 1 we obtain, respectively 0 ˆ 01 ˆ ' 2 À …0 À †' 2 and 1 À 00 ˆ À' 2 Eliminating ' 2 from these two equations allows the ACF of the ARMA(1,1) process to be given by &1 ˆ …1 À 0†…0 À † 1 ‡ 2 À 20 and &k ˆ 0&kÀ1 Y for k b 1 The ACF of an ARMA(1,1) process is therefore similar to that of an AR(1) process, in that the autocorrelations decay exponentially at a rate 0. Unlike the AR(1), however, this decay starts from &1 rather than from &0 ˆ 1.

Using this idea of being `close to', Poskitt and Tremayne (1987) introduce the concept of a model portfolio. Models are compared to the selected …p1 Y q1 † process by way of the statistic, using AIC for illustration ! 1 N ˆ exp À TfAIC…p1 Y q1 † À AIC…pY q†g 2 Although N has no physical meaning, its value may be used to `grade the decisiveness of the evidence' against a particular model. Poskitt and Univariate linear stochastic models: basic concepts 37 p Tremayne (1987) suggest that a value of N less than 10 may be thought of as being a close competitor to …p1 Y q1 †, with the set of closely competing models being taken as the model p portfolio.

19) is as 0…B†w~ t ˆ …B†at where w~ t ˆ wt À "w . 12 plots generated data for Á2 xt ˆ 2 ‡ at , where again at $ NID…0Y 9† and x0 ˆ x1 ˆ 10. The inclusion of the deterministic quadratic trend has a dramatic effect on the evolution of the series, with the non-stationary `noise' being completely swamped after a few periods. 19) therefore allows both stochastic and deterministic trends to be modelled. 12 `Second difference with drift' model 0 Tˆ 0, the model may be interpreted as representing a deterministic trend (a polynomial in time of order d) buried in non-stationary noise, which will typically be autocorrelated.

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