<p>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.</p>

CC BY-SA 3.0
2016-09-13T15:21:20.187Z
2016-09-13T15:42:21.023Z
2016-09-13T15:42:21.023Z
Omrane
<p>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?</p>

<p>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.</p>

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

CC BY-SA 3.0
2016-09-13T20:09:19.983Z
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.
RegressForward
CC BY-SA 3.0
2016-11-12T16:41:26.677Z
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.
snoram
<econometrics>
Regression Line Method
Omrane
# Regression Line Method

(Asked by: username_0 on 2016-09-13)

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?

**Tags:** econometrics
> **Comments:**
> **username_1**: 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.  
> **username_2**: 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.  

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## 1 answers

### Answer by username_0 on 2016-09-13. Score: 1

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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