The Partial Regression Coefficients

where the expression for the RSS is obtained by simple algebraic manipulations of (7.4.1).

208 PART ONE: SINGLE-EQUATION REGRESSION MODELS

The most straightforward procedure to obtain the estimators that will minimize (7.4.2) is to differentiate it with respect to the unknowns, set the resulting expressions to zero, and solve them simultaneously. As shown in Appendix 7A, Section 7A.1, this procedure gives the following normal equations [cf. Eqs. (3.1.4) and (3.1.5)]:

E YiX2i = $1 E X2i + 02 E X2i + 03 E X2iX3i (7.4.4) E ^X3i = 01 E X3i + 02 E X2iX3i + 03 E X2i (7.4.5)

which is the OLS estimator of the population intercept p1.

Following the convention of letting the lowercase letters denote deviations from sample mean values, one can derive the following formulas from the normal equations (7.4.3) to (7.4.5):

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