# When does Least Square Regression (LSQ) line equal to Least Absolute Deviation (LAD) line?

I have the following question at hand.

Suppose $(x_1,y_1),(x_2,y_2),\cdots,(x_{10},y_{10})$ represents a set of bi-variate observations on $(X,Y)$ such that $x_2=x_3=\cdots =x_{10}\ne x_1.$ Under what conditions will the Least Square Regression line of $Y$ on $X$ be identical to the Least Absolute Deviation line?

I know that say we want to find $\hat{\alpha}$ and $\hat\beta$ such that $Y=\hat\alpha+\hat\beta X$; the LSQ method will give $$\hat\beta={\sum\limits_{i=1}^{10} (x_i-\bar x)y_i\over \sum\limits_{i=1}^{10}(x_i-\bar x)x_i}$$ and hence $\hat\alpha$. Can someone help me proceed?

• Intuitively, there's a trivial case and a non trivial case where pairs of points stabilize a line with equal errors. Aug 5, 2016 at 18:15

1. Make up or generate some data consistent with the conditions in the question. Try $$x_1=0, y_1=0$$ and $$x_2..x_{10}=1$$ (choosing some values for $$y_i$$, $$i=2,...,10$$). Where do the lines pass relative to the first point?

2. Now start as above but try placing $$y_i$$, $$i=2,...,10$$ at say 1,2,3,4,5,6,7,8,9 respectively. Where do the lines go? 1. Now place $$y_i$$, $$i=2,...,10$$ at say 1,2,3,4,5,6,7,8,99 respectively. Where do the lines go? What is special/interesting about the fitted values for the two lines at $$x=1$$?

(If it's not clear try some other values for $$y_{10}$$.) 1. Can you prove this is the case more generally?

This ultimately brings us to a slightly simpler question, which relates to when means and medians are equal in the univariate case. (There's a simple, obvious condition that's sufficient, but not necessary.)

There are a number of posts on site that discuss the other case. There are some interesting examples here