Struggling w/ Econometric probs: We have the following AR(2) model : Yt = B0 + B1Yt-1+B2Yt-2 + Ɛt Ɛt ~ iid (0,σ2)

[Alex_Of_Darkness]

Hi everyone,

I have some difficulties with two problems. It's really important that I understand them for my exam later this month so I would be more than grateful if someone can help me resolve them.

Problem #1 :
X_t is a second-order time series with auto-covariance γ_j and j ≥ 0. Is Y_t = X_t + X₀ stationary at the second-order?
We also know that X₀ represents the value of X_t when t=0. So X₀ is a random variable.

Problem #2 :
We have the following AR(2) model :
Y_t = B₀ + B₁Y_t-1+B₂Y_t-2 + Ɛ_t
Ɛ_t ~ iid (0,σ²)

a) Show that Ɛ_t is independent of Y_t-1 and Y_t-2. What is the consequence for the OLS estimation of the model?
b) Calculate the expected value of Y_t.
c) While using the following notation :
γ_j = cov(Y_t,Y_t-j) for j = 0,1,2... Show that :
γ₀= B₁γ₁+B₂γ₂+σ²,
γ₁= B₁γ₀+B₂γ₁,
γ₀= B₁γ₁+B₂γ₀.
d) Give the variance of Y_t in term of B₁, B₂ and σ².

What I know for Problem #1 :
I know that Y_t stationary at the second-order if its expected value = u does not depend on t and that cov(Y_t,Y_t-j)=γ₀ does not depend on t neither. I'm just not sure how can I demonstrate what is asked in the question.

What I know for Problem #2 :
a) I really don't know what to use to test the independence...
b) c) d) I assume I'll need the result in a)? I'm just completely lost with problem #2 to be completely honest with all of you.

Thanks a lot for taking time to help me if you can. You would save my life.

 

[Alex_Of_Darkness]

No one can help me?