What happens if we try to minimize the sum of residuals rather than square or absolute value? Maybe a duplicate-- but I am not asking about least squares vs minimizing absolute deviations. Just simply, what if I chose to minimize the sum of residuals:
Min $\sum_i^n (Y_i-\beta_0 - \beta_1X_i)$ ?
If I try to minimize this, the second order conditions are all zero. Is there a solution to this? Would the strategy be to pick $\beta$'s so that they make the sum of residuals negative infinity?

$\sum_i^n (Y_i-\beta_0 - \beta_1X_i) = 0$ to equivalent a regression line where $\bar Y=\beta_0 + \beta_1 \bar X$ i.e. any (non-vertical) line which passes through the mean point.
One such line is the ordinary least squares result.  But all the others do so too.
Another is $\beta_1=0, \beta_0=\bar Y$, i.e. estimating a constant value for $\hat Y_i$ that ignores the $X_i$, but you can usually get a better fit than that.
You could aim to minimise the sum of the absolute deviations subject to passing through the mean point if you wished, but it would be analytically more complicated.
