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Math Help - Time Series Covariance Problem

  1. #1
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    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!
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  2. #2
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    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))
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