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Math Help - Simple linear regression model

  1. #1
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    Simple linear regression model

    The OLS estimator for the simple linear regression model can be written as
    <br />
b_2 = \beta_2 + \Sigma w_te_t where w_t = \frac{(x_t - \overline{x})}{\Sigma(x_t - \overline{x})^2}<br />

    In the case for a sample size of three observations (T=3), this estimator can be written as:
    <br />
b_2 = \beta_2 + w_1e_1 + w_2e_2 + w_3e_3 where w_t = \frac{(x_t - \overline{x})}{\Sigma(x_t - \overline{x})^2}, t = 1,2,3<br />

    Assume that the full set of assumption of the SLRM hold.
    Without using summation notation, use the fact that E[b_2]=\beta_2 to find V[b_2].

    Explain how the assumptions of the model are used to arrive at your answer.

    i started by equating b_2 = E[ (b_2 - E(b_2) )^2] (definition of variance),
    then using known results to

    = E(b_2) - [E(b_2)]^2

    would greatly appreciate if anyone could help me derive V(b_2) without using any summation notation.

    i suppose the end resut of V(b_2) [using a proof that uses summation notation) would be

    \frac{(\sigma^2)}{\Sigma(x_t - \overline{x})^2}

    and,

    E[(e_t)^2] = \sigma^2

    and w_t = \frac{(x_t - \overline{x})}{\Sigma(x_t - \overline{x})^2}
    Last edited by cheeaun; April 6th 2009 at 07:37 AM.
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