I don't agree with the answer to A posted here. It looks like you did 9/(1-.2^2) which would be the answer if it weren't an alternating series (i.e if it was 9*(1+.2^2+.2^4+...). However, this series is 9*(1+.2^2-.2^4+.2^6+...) I think, which shouldn't converge to the same value, right? I'm not sure if there is an exact formula for solving the alternating series, but I summed the first several terms and rounded and I get a different answer.

Edit: Never mind, I get it. Forgetting how to get a variance!

No. That is not relevant to what I asked.

LIAPP,

If you work out what minnie53053 posted just above, it works out as she has written:

rho2=COV(Yt,Yt-2)/(VAR(Yt)*VAR(Yt-2))^0.5=-0.345/9.375=-0.0368.

I worked out the covariance to be phi^2*Var(e)*(-1+phi^2+phi^4+phi^6.....) and then solved.  Does this help?

Question on part C...

Wouldn't the conventional equation for correlation not work in this scenario since we cannot assume stationarity? Stationarity, by definition, depends only on lag and not absolute time, but we can see that because the terms do not follow the same pattern for every lag of 2, then the correlation assumption doesn't hold.

Given the explanation for Psi on page 55, the coefficient on the first expression isn't non-existent, but is simply Phi^0. This would lead to a coefficient pattern of +, +, -, +, -, +, -, ... , wherein the correlation between e(t) and e(t-2) would be negative, but all other intervals would be positive.

I'm not sure whether I'm missing something, or if the equation is given later in the book, or if the answer is undefined. Any help is appreciated.

yeah, I get -0.0368 too.

COV(Yt,Yt-2)=-0.345.

rho2=COV(Yt,Yt-2)/(VAR(Yt)*VAR(Yt-2))^0.5=-0.345/9.375=-0.0368

MY BAD!  Yep, you're right.  I see sigma^2 and get all excited.  sigma^2 <> var(Y_t)

   -    Dave

@dom2114: Thank you. I had figured that out myself, but appreciate you mentioning it. I was making a different mistake and started to doubt that part that I had done right.

@djfobster: I got the same answer for C.

@DMW: It's not -0.345/9. gamma0 does NOT equal 9. sigma-squared = 9.

I agree with djfobster except the -.345/9 does not equal what he or she says it equals.  But I could be losing it.

   -    Dave

For anyone that actually looks @ these forums for the fall course:

I was having the same problem as the quoted text...my issue was that I was not summing the geometric progression correctly...This is the case for both parts B & C.

The trick is that the first term cannot be part of the geometric sum as the sign is different. E.g. in Part B you get to:

phi*Var(et-1)) - phi^3*Var(et-2) - phi^5*Var(et-3)...

The proper sum is:

Var(et)*[phi - (phi^3 / (1-phi^2) ) ]

i.e. the first time is not part of the geometric sum.

Hope this helps someone.

PMActuary, the derivation on p.56 is for the particular case when all summands are positive. In our HW we have alternating signs, so we have to do a different calculation.

Basically, due to alternating signs, the terms in the covariance sum will enter one with +, the other with -, like this:
for the term -phi^2 x e(t-2) the other one will be +phi x e(t-2), and so on.
(However, the term e(t-1) enters both times with +).