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Math Help - Alternate least squares estimate

  1. #1
    Junior Member
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    Sep 2010
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    63

    Alternate least squares estimate

    I am trying to show that:

    \frac{\sum_i x_i y_i - n \bar{x} \bar{y}}{\sum_i x_{i}^2 -n   \bar{x}^2}=\frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}{\sum_i (x_i -   \bar{x}^2)}

    This is how far I got:


    \frac{\sum_i x_i y_i - n \bar{x} \bar{y}}{\sum_i x_{i}^2 -n \bar{x}^2}

    = \frac{ \sum_i x_i y_i - n \sum_i \frac{x_i}{n} \sum_i   \frac{y_i}{n} }{\sum_i x_{i}^2 -n \sum_i \frac{x_i}{n} \sum_i   \frac{x_i}{n}}

     = \frac{ \sum_i x_i y_i - \frac{1}{n} \sum x_i \sum y_i }{\sum_i  s_{i}^2 - \frac{1}{n} \left( \sum_i x_i \right)^2} ...

    Am I doing this right? Not sure what to do next.
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  2. #2
    Junior Member
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    Sep 2010
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    = \frac{\sum_i x_i y_i - \sum_i x_i \bar{y}}{\sum_i x_{i}^2 - \sum_i x_i \bar{x}}

    = \frac{\sum_i x_i (y_i - \bar{y})}{\sum_i x_i (x_i - \bar{x})} ...

    stuck again...
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  3. #3
    Junior Member
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    Sep 2010
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    Okay i did it, nevermind. More of an algebra problem I guess. Oops
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