E4Ex3i Ex2ix3i2

which give the OLS estimators of the population partial regression coefficients p2 and p3, respectively.

In passing, note the following: (1) Equations (7.4.7) and (7.4.8) are symmetrical in nature because one can be obtained from the other by interchanging the roles of X2 and X3; (2) the denominators of these two equations are identical; and (3) the three-variable case is a natural extension of the two-variable case.

Variances and Standard Errors of OLS Estimators

Having obtained the OLS estimators of the partial regression coefficients, we can derive the variances and standard errors of these estimators in the manner indicated in Appendix 3A.3. As in the two-variable case, we need the standard errors for two main purposes: to establish confidence intervals and

6This estimator is equal to that of (7.3.5), as shown in App. 7A, Sec. 7A.2.

CHAPTER SEVEN: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF ESTIMATION 209

to test statistical hypotheses. The relevant formulas are as follows:7

1 XjE 4 + X2E 4 - 2 XX2 XX 3 E X2iX3i n E xi- E x| - ( E xiiXsO2

or, equivalently,

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