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which are precisely the normal equations of the least-squares theory, as can be seen from Appendix 7A, Section 7A.1. Therefore, the ML estimators, the j's, are the same as the OLS estimators, the j's, given previously. But as noted in Chapter 4, Appendix 4A, this equality is not accidental.

Substituting the ML (= OLS) estimators into the (K + 1)st equation just given, we obtain, after simplification, the ML estimator of a2 as a2 = 1 Y(Vi — ji - 02X2i-----faXki)2

As noted in the text, this estimator differs from the OLS estimator a2 = J]«2/(n — k). And since the latter is an unbiased estimator of a2, this conclusion implies that the ML estimator a2 is a biased estimator. But, as can be readily verified, asymptotically, a2 is unbiased too.

CHAPTER SEVEN: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF ESTIMATION 247

7A.5 SAS OUTPUT OF THE COBB-DOUGLAS PRODUCTION FUNCTION (7.9.4) DEP VARIABLE: Y1

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