Adata set consists of n observations on X_nand y_n. The least squares estimator based on these n observations is b_n= (X'_nX_n)^??1X'_ny_n. Another observation, x_sand y_s, becomes available. Prove that the least squares estimator computed using this additional observation is

The last term is e_s, the residual from the prediction of y_s using the coefficients based on X_n and b_n. Conclude that the new data change the results of least squares only if the new observation on y cannot be perfectly predicted using the information already in hand.

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1 b,s = b, + -(X,X„)-'x, (; – x,b,). %3D 1+ x (X,X„)-!xs