Q&A bank 4.16???

[12345]

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!

 

[John Lee]

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.