TITLE: Why work with squares of error in regression analysis?
QUESTION [8 upvotes]: In regression analysis one finds a line that fits best by minimizing the sum of squared errors.
But why squared errors? Why not use the absolute value of the error? 
It seems to me that with squared errors the outlyers gain more weight. Why is that justified? And if it is justified to give the outlyers more weight, then why give them exactly this weight? Why not for example take the least sum of exponetial errors?

Edit: I am not so much interested in the fact that it might be easier to calculate. Rather the question is: does squaring the errors result in a better fitting line compared to using the absolute value of the error?
Furthermore I am looking for an answer in layman's terms that can enhance my intuitive understanding.

REPLY [3 votes]: Basically, you can ask the same question in the much simpler setting of finding the "best" average of values $x_1,\ldots,x_n$, where I here refer to average in the general sense of finding a single value to represent them such as the (arithmetic) mean, geometric mean, median, or $l_p$-mean (not sure if that's the right name).
For data that actually come from a normal distribution, the mean will be the most powerful estimator of the true mean. However, if the distribution is long-tailed (or has extreme values) the median will be more robust.
You can also use the $l_p$ norm and find the $l_p$-mean, $u$, that minimises $\sum_i |x_i-u|^p$ for any $p\ge1$. (For $p<1$ this need no longer be unique.) For $p=2 $ we have the traditional square distance, while for $p=1$ we get the median (almost). I once found $p=1.5$ to behave well in terms of both power and robustness.
So, switching from least square regression ($l_2$-norm) to using absolute distance ($l_1$-norm) corresponds to switching from mean to median. Which is better depends on the data, and also on the context of the analysis: what you are actually looking for.
The mean does have the advantage that it is an unbiased estimator of the true mean no matter what the underlying distribution is, but usually accuracy is more important than unbiasedness.