Consider the standard linear model y = X + and suppose that bi is the least squares estimate obtained by omitting the ith observation yi , that is, bi = argmin X j6=i (yj (X)j ) 2 . Let \hat{y}i,i = (Xbi)i be the ith fitted value for a model fitted without the ith observation. Define b to be the least squares estimate based on the response data y^T = (y1, . . . , yi1, ybi,i , yi+1, . . . , yn). Prove that b = bi , that is, the linear model obtained from fitting all responses except the ith is the same as the one obtained from fitting the data y.

I have already found b_* to be (X^TX)^{-1}X^Ty_*, I was wondering what to do next