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Suppose I have the linear regression model $ \hat{y_i} = a + b x_i $ for $a,b$ obtained via OLS. How does one regress $y$ back on the residuals $\hat{e}_i = y_i - \hat{y}_i$? If we write $ \hat{\hat{y_i}} = c + d \hat{e}_i$ and attempt to use the regression coefficient formulas, I'm unsure how to write $c$ and $d$ in terms of $a,b$.

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    I'm still having some trouble. I understand the orthogonality property, but the equations I am finding doesn't seem correct.2012-12-28

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