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Math Help - var/covar of Linear Lease Square estimates

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    var/covar of Linear Lease Square estimates

    Goal: Find, Var(\hat B_0) and Covar(\hat B_1, \hat B_0)

    (Note: y_i = B_0 + x_i B_1 +e_i, with e_i ~ N(0,\sigma ^2) So, everything is fixed, but e_i and y_i. Now I found unbiased estimates \hat B_0 and \hat B_1. Also, I found the variance of \hat B_1.)

    \hat B_0 = \bar y - {\bar x} \hat B_1 and,

    \hat B_1 = \frac{\sum x_i y_i -{\bar y} \sum x_i}{\sum x_i ^2 - {\bar x} \sum x_i}

    Work:

    E[\hat B_0 ^2] = E[(\bar y - {\bar x}\hat B_1)^2] = E[\bar y ^2] - 2{\bar x}E[{\bar y}\hat B_1] + \bar x E[\hat B_1^2]

    Each part is easy enough, I just don't know how to start E[{\bar y}\hat B_1]....?

    Lastly, how to approach E[\hat B_0\hat B_1]....?
    Last edited by Anonymous1; April 29th 2010 at 04:40 PM.
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