<Dl> <Dd> β ^ = (X T X) − 1 X T y . (\ displaystyle (\ hat (\ beta)) = (X ^ (T) X) ^ (- 1) X ^ (T) y .) </Dd> </Dl> <Dd> β ^ = (X T X) − 1 X T y . (\ displaystyle (\ hat (\ beta)) = (X ^ (T) X) ^ (- 1) X ^ (T) y .) </Dd> <P> The residual vector e ^ (\ displaystyle (\ hat (e))) is y − X β ^ = y − X (X T X) − 1 X T y (\ displaystyle y-X (\ hat (\ beta)) = y-X (X ^ (T) X) ^ (- 1) X ^ (T) y), so the residual sum of squares e ^ T e ^ (\ displaystyle (\ hat (e)) ^ (T) (\ hat (e))) is, after simplification, </P> <Dl> <Dd> R S S = y T y − y T X (X T X) − 1 X T y . (\ displaystyle RSS = y ^ (T) y-y ^ (T) X (X ^ (T) X) ^ (- 1) X ^ (T) y .) </Dd> </Dl>

Error sum of squares (ess) is computed as