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Math Help - Time series question

  1. #1
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    Time series question

    Suppose I have two series:

    x_t=0.3x_{t-1}+w_t

    y_t=x_t+0.5w_t

    where w_t\sim WN(0,1).

    I try to find the autocovariance function of y_t as followed:

    \gamma(h)=Cov(y_{t-h},y_t)=Cov(x_{t-h}+0.5w_{t-h},x_t+0.5w_t)

    =Cov(x_{t-h},x_t)+0.5Cov(w_{t-h},x_t)+0.5Cov(w_t,x_{t-h})+0.5^2Cov(w_{t-h},w_t)

    Then \gamma(0)=1/(1-0.3^2)+0.5^2

    \gamma(1)=0.3/(1-0.3^2)

    \gamma(2)=0.3^2/(1-0.3^2) and so on..

    My question is: am I right to assume both 0.5Cov(w_{t-h},x_t) and 0.5Cov(w_t,x_{t-h}) are 0 at any lag h?

    By the way, is the series y_t weakly stationary such that \gamma(h) is independent of t?
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  2. #2
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    Re: Time series question

    My question is: am I right to assume both and are 0 at any lag h?
    try evaluating both of the covariances at lag 1 and you will almost certainly find that one of them is 0 and one of them isn't. Try to understand why.

    Spoiler:

    : x_{t-h} is not correlated to future values of the white noise, because by definition future white noise is independent of the current x value.

    : : x is correlated to past values of white noise, because it is correlated to its own history. **Unless there is a tremendous fluke of algebra, i haven't actually worked it out for your particular series.


    Looking at that series i expect it is stationary but i dont know how to prove it without doing the whole question for you. Have a go if you like and post if you get stuck.
    Last edited by SpringFan25; November 19th 2011 at 03:11 PM.
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  3. #3
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    Re: Time series question

    Ok it seems that

    \gamma(0)=1/(1-0.3^2)+0.5^2+1

    \gamma(1)=0.3/(1-0.3^2)+0.5(0.3)

    \gamma(2)=0.3^2/(1-0.3^2)+0.5(0.3)(0.3)

    ...

    It seems to me that E[y_t] and \gamma(h) are not governed by t..
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