# Fixed Point Theorems by D. R. Smart

By D. R. Smart

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Example text

8) by xtÀk and taking expectations yields k  0kÀ1 Y for k b 1 whilst for k  0 and k  1 we obtain, respectively 0  01  ' 2 À 0 À ' 2 and 1 À 00  À' 2 Eliminating ' 2 from these two equations allows the ACF of the ARMA(1,1) process to be given by &1  1 À 00 À  1  2 À 20 and &k  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  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  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  Bat where w~ t  wt À "w . 12 plots generated data for Á2 xt  2  at , where again at \$ NID0Y 9 and x0  x1  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 T 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.