Question

show that regression lines intersect at x bar and y bar

Answer 1

Answer:

If we have that y is a dependant variable on x, then the least squares regression line is given as

y−y¯=Cov[x,y]Var[x](x−x¯)

We can derive this formula by considering the optimization problem of minimizing the square of the residuals; more formally if we have a set of points (x1,y1),(x2,y2),…,(xn,yn) then the least squares regression line minimizes the function

D(a,b)=∑i=1nϵ2i=∑i=1n(yi−[a+bxi])

so by solving for ∂D/∂a=0 and ∂D/∂b=0 simulataneously we get the above form.

Given the above form of the least squares regression, it should then become apparent why if we then take a least squares regression where x is dependant on y why the two lines intersect at (x¯,y¯) - if it isn't immediate obvious, then recall the definition of Cov[x,y] and wlog. consider the case where x¯=y¯=0.