Suppose we have a sample (x i , y i ), i = 1, . . . , n. By completing the square show that Given any 2 points (x i , y i ) and (x j , y j ), the slope of the line joining the first point to the...

Suppose we have a sample (x_i, y_i), i = 1, . . . , n. By completing the square show that

Given any 2 points (x_i, y_i) and (x_j, y_j), the slope of the line joining the first point to the second point is:

(To allow for data sets with repeated values of the x_i, define bi_j
= 0 if xi = x_j.) Show that the ols estimate

  is a weighted combination of the

where the weights
 This weighting scheme gives larger weight to pairs of points that are widely separated,
   is large, and less weight to pairs that are close together.