Me again, having a little trouble with the covariance between random variables and white noise, e.g. in this question COV(Xs,Zs). The solution sets this equal to COV(aXs-1 + Zs +BZs-2,Zs) but initially I set this to COV(Xs,Xs - aXs-1 - BZs-2) and couldn't get the same answer. Am I doing something fundamentally incorrect here or could this work? I would say this equals y0 - ay1 - COV(Xs,BZs-2)? As an aside, in general what is the covariance of random variables and white noise such as COV(Xs,BZs-2) here? Is it dependent on the actual equation of the time series? a= alpha, B = beta and y= gamma in the above Apologies for the ramble, hopefully someone can clear this up!
The reason we replace the X is because it's easier to find covariances of X's with Z's and Z's with Z's than it is to find X's with X's (as they'll always give a gamma term). So whilst your method should give an answer you'd have to solve the autocovariance first to get it - but since solving the autocovariance requires this term - you'll be in trouble! So don't! X's with future Z's are zero (as future completley random variation won't affect today's result). X's with current or past Z's will need to be calculated - as the current or past Z's will affect X's in the past which will have a knock on effect. In summary when doing an ARMA(p, q) You'll need to first calculate cov(Xt, Zt) cov(Xt, Zt-1) ... cov(Xt, Zt-q) (as the formula has up to Zt-q terms in it) Then calcualte the autocovariance gamma0 = cov (Xt, Xt) gamma1 = cov (Xt, Xt-1) gamma2 = cov (Xt, Xt-2) etc by replacing the Xt term only with the defining equation (and then using your exciting Xt/Zt combinations where appropriate) Hope this helps.
