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Math Help - Regression Proof Question

  1. #1
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    Regression Proof Question

    Show that Ybar = Yhat bar, that is the average of actual Y values and the average of estimated Y values are the same.

    No clue where to start...any help greatly appreciated!
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  2. #2
    MHF Contributor matheagle's Avatar
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    This is not clearly stated. I assume you're using the simple linear regression model y=\beta_0+\beta_1 x+\epsilon.

    Multiplying by n, what you want to show is that  \sum_{i=1}^ny_i=\sum_{i=1}^n\hat y_i.

    Or, the sum of the residuals,  \sum_{i=1}^ne_i=\sum_{i=1}^ny_i-\hat y_i=0.

    The residuals sum to zero in the model y=\beta_0+\beta_1 x+\epsilon.

    But they don't sum to zero if you drop the \beta_0 term in your model, because the vector of all ones
    is no longer in the column space of the design matrix.

    Here's a simple proof in this case....
    Just substitute \hat\beta_0= \bar y-\hat\beta_1\bar x into the model...

    \hat y_i=\hat\beta_0 +\hat \beta_1 x_i =(\bar y-\hat\beta_1\bar x )+\hat \beta_1 x_i =\bar y +\hat\beta_1\bigl(x_i-\bar x\bigr) .

    Now take the sum of this and note that

    \sum_{i=1}^n\bigl(x_i-\bar x\bigr)=0 leads you to your conclusion.
    Last edited by matheagle; April 28th 2009 at 03:40 PM.
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