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Suppose we are given a set of observations on income $X_{i}$ and consumption $Y_{i}$ and we plot all the $(X_{i},Y_{i})$ on a graph. We want to draw a sample regression function as close as possible to the existing yet unknown population regression function. Why is it considered wrong to take $\sum \hat u_{i}$ as small as possible to draw our regression function?

My study book says that this method assigns "equal weights" to the $\hat u_{i}$ which is not accurate. However, I also agree with the least order square method.

Anyone have a specific example to show me that the normal summation method leads to bad estimation?

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  • $\begingroup$ You might also want to entertain the notion of Least Absolute Deviation regression. Others have thought similar thoughts- though it turns out to be difficult to implement. $\endgroup$ – RegressForward Sep 13 '16 at 20:09
  • $\begingroup$ You certainly mean $\sum \lvert \hat u_{i} \rvert$, I assume. Then that's not considered wrong---it just turned out $\sum\hat u_{i}^2 $ has some particularly nice properties. $\endgroup$ – snoram Nov 12 '16 at 16:41
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Thought about a very simple example: Suppose we have two distinct points $(X_{1},Y_{1})$ and $(X_{2},Y_{2})$. The best regression line for these two points would be the line that passes through both points. However, any line passing through $(\frac{X_{1}+X_{2}}{2},\frac{Y_{1}+Y_{2}}{2})$ would let $\hat u_{1}+\hat u_{2}=0$, just like the perfect regression line.

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