Although in principle U'U can be computed from the estimated residuals, in practice it can be obtained directly as follows. Recalling that ^u^ (= RSS) = TSS — ESS, in the two-variable case we may write e u = E y2—$2 e x2 (3.3.6)
and in the three-variable case e u = e y2 — $2 E yxi — $3 E y^ (7.4.19)
By extending this principle, it can be seen that for the k-variable model
ESS: P2J2yiX2i + ■■■ + P^yiXki = (3'X'y — nY2 (C.3.17)
936 APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL
where the term nY2 is known as the correction for mean.6 Therefore, u u = y'y - p'X'y (C.3.18)
Once u'u is obtained, a2 can be easily computed from (C.3.14), which, in turn, will enable us to estimate the variance-covariance matrix (C.3.13). For our illustrative example,
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