A. What is the variance of Y_t as a function of â and t?

B. What is ñ(y_t,y_t-k) as a function of â, k, and t?

See equations 5.1.4 and 5.1.5 on page 89.

{Note: This homework assignment has been replaced because of an unclear equation in the textbook; see the new homework assignment. If you have submitted this assignment already, you will be given credit.}

I am following your method and have Var(Y_t) = sigma^2( 1 + b^2 + b^4 + b^6 + ... + b^2(t-1) + b^2t x 0) = sigma^2( 1 + b^2 + b^4 + b^6 + ... + b^2(t-1)) = sigma^2( (1/(1-b^2)) - (b^2t + b^2(t+1) + b^2(t+2)+...). This second part of the equation is what we have to subtract out since it is not an infinite series of 1 + b^2 + b^4 + ...

Next I get: = sigma^2( (1/(1-b^2)) - b^2t(1 + b^2 + b^4 + b^6 + ...)) = sigma^2( (1/(1-b^2)) - b^2t(1/(1-b^2))). There is a common denominator of 1- b^2, and in the numerator it's 1 - b^2t. Therefore, we get sigma^2((1-b^2t)/(1-b^2)).

I am so close, and if I multiply by (-1)/(-1) on both sides of the equation, am I good? So many people were saying they had a negative sign problem, but this doesn't cause a problem, does it?

Var(Y_t) * (-1/-1) = sigma^2((1-b^2t)/(1-b^2) * (-1/-1)
Var(Y_t) ) = sigma^2((b^2t-1)/(b^2-1)

Someone please confirm, I have been working on this for way too long.