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Math Help - AR(1): Variance of Sample Mean

  1. #1
    lpd
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    AR(1): Variance of Sample Mean

    Hi.

    I have an auto-regressive process of order 1, AR(1). Assuming sample size n.

    i) I would like to find, Var(\bar{X}) of AR(1). Would someone kindly show a source for this result? The correlation is denoted as corr(X_i , X_j) = \rho^{|i-j|}

    ii) If the sample auto-correlation (lag 1) was about 0.25, how much an impact does that have on the standard error of \bar{X}, relative to the indpendence case?

    Thank-you!
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  2. #2
    MHF Contributor
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    Re: AR(1): Variance of Sample Mean

    Hey lpd.

    The wiki article:

    Correlation and dependence - Wikipedia, the free encyclopedia

    gives a definition in terms of covariance and individual random variable variance.

    Now if you have these terms you need to include the covariance terms when you are finding the variances of correlated random variables. There is an inductive definition of this, but basically it involves calculation covariance terms for all the appropriate pairs of covariance terms.

    Since you are able to get a definition for the covariance of any two random variables for a given rho (that greek p), if you know the standard deviations of each random variable you can get the covariance term from these two and then calculate the variance of the sample mean of all observations taking into account all covariance terms to get the variance of the sum of all the random variables, and then divide by n^2 to get the variance of the sample mean.
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  3. #3
    lpd
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    Re: AR(1): Variance of Sample Mean

    Is it possible to show how its done, I am still a bit confused...
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  4. #4
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    Re: AR(1): Variance of Sample Mean

    Well as an example if you have X_bar = [X1 + X2]/n then the Var(X_bar) = (1/2^2)*[Var(X1) + Var(X2) + 2*Cov(X1,X2)]. Using the definition of rho, you have rho*sigma_x*sigma_y = COV(X,Y) so you plug that in to get the variance of the sample mean.

    You just extend this idea for N random variables as opposed to two.
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  5. #5
    lpd
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    Re: AR(1): Variance of Sample Mean

    Quote Originally Posted by chiro View Post
    Well as an example if you have X_bar = [X1 + X2]/n then the Var(X_bar) = (1/2^2)*[Var(X1) + Var(X2) + 2*Cov(X1,X2)]. Using the definition of rho, you have rho*sigma_x*sigma_y = COV(X,Y) so you plug that in to get the variance of the sample mean.

    You just extend this idea for N random variables as opposed to two.
    Okay, I went and tried and got,

    Var(\bar{X})= \frac{cov(X_i,X_j)}{n}

    I'm not quite sure how to do the next one... if the sample auto-correlation (lag 1) was about 0.25, how much an impact does that have on the standard error of \bar{X}, relative to the indpendence case?
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  6. #6
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    Re: AR(1): Variance of Sample Mean

    That doesn't look right: you should have variance and covariance terms where the covariance terms have the standard deviations and the correlation coeffecients.

    Can you show us how you derived that formula for Var(X_bar)?
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