Time Series Covariance Problem

Hello,

I am having problems with a covariance question in my time series class.

Suppose you have a time series t_0, t_1 ... t_n and have an AR Model such that

r_t = phi_0 + phi_1*r_(t-1) + phi_2*r_(t-2) + a_t

where a_t is a white noise process with mean zero and sd = sigma^2

I am asked to find the variance and I have most of it except for the Cov(r_(t-1) , r_(t-2))

Furthermore what I am confused about is how does one go from

Cov(r_(t-1) , r_(t-2)) = E[ (r(t-1) - mu) (r(t-2) - mu) ] to the following:

phi_1*E[ (r(t-2) - mu) (r_(t-2) - mu) ] = phi_2*E[ (r_(t-3) - mu) (r_(t-2) - mu) ]

I know there is an intermediate step I do not understand here. Would someone please explain it.

Thank you very much for answering my question!

Re: Time Series Covariance Problem

You can use the autocovariance function for an AR(2)

γ(k) = φ1*γ(k-1) + φ2*γ(k-2)

where γ(k) = Cov(r(t), r(t+k))