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210 PART ONE: SINGLE-EQUATION REGRESSION MODELS

Note the similarity between this estimator of a2 and its two-variable counterpart [a2 = (J] u2)/(n — 2)]. The degrees of freedom are now (n — 3) because in estimating J2 u we must first estimate fi1, fi2, and fi3, which consume 3 df. (The argument is quite general. Thus, in the four-variable case the df will be n — 4.)

The estimator a2 can be computed from (7.4.18) once the residuals are available, but it can also be obtained more readily by using the following relation (for proof, see Appendix 7A, Section 7A.3):

which is the three-variable counterpart of the relation given in (3.3.6).

Properties of OLS Estimators

The properties of OLS estimators of the multiple regression model parallel those of the two-variable model. Specifically:

1. The three-variable regression line (surface) passes through the means Y, X2, and X3, which is evident from (7.4.3) [cf. Eq. (3.1.7) of the two-variable model]. This property holds generally. Thus in the k-variable linear regression model [a regressand and (k — 1) regressors]

Yi = fi1 + fi2 X2i + fo X3i + ••• + fikXki + u (7.4.20)

we have

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