Hi Purely indeterministic (PI) is defined in the CH12 summary as meaning that knowledge of earlier values of X is progressively less useful at predicting the value of X(subscript N) as N tends to infinity. I don't currently see how this fits with the white noise definition and the given fact that white noise is purely indeterministic. Isn't it the case that for white noise knowledge of earlier values of X is equally "useful" (ie not useful at all), regardless of how large N becomes? If that's right, should the PI definition read "progressively less useful, or no more useful". Or something like that. Far more likely is that I've missed something - please help!
Purely indeterministic basically means that Cov(Xt, Xt+k) gets smaller as the lag k gets bigger. In a trivial case, let's consider an MA(1): Xt=et+et-1. Cov(Xt,Xt)=2(sigma squared) Cov(Xt, Xt-1)=sigma squared Cov(Xt, Xt-2)=0 (and in fact the covariance will always be 0 for lags greater than 1) So you can see, as the lag inreased, the covariance decreased. In the most extreme trivial case, let's consider Xt=et. (So we're only worrying about one term of white noise, as you do in your question.) Cov(Xt,Xt)=sigma squared Cov(Xt, Xt-1)=0 So again, the covariance has decreased as the lag increased from 0 to 1.
The question does not ask you to find that. Indeed it would not be possible without a plot of he PACF.
