Adding an observation, a data set consists of n observations on Xn and yn. The least squares estimator based on these n observations is bn = (X'n Xn)??1 X??n yn. Another observation, xs and ys, becomes available. Prove that the least squares estimator computed using this additional observation is Note that the last term is es, the residual from the prediction of ys using the coefficients based on Xn and bn. 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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bns = bn + 1 1+ X(XXn)-¹Xs (XXn)-¹xs (ys-xb).