follows the t distribution with n — 2 df. Substitution of (1) and (2) into (3) gives Eq. (5.3.2).
5A.3 DERIVATION OF EQUATION (5.9.1)
Equation (1) shows that Z1 ~ N(0, 1). Therefore, by Theorem 5.3, the preceding quantity
follows the x2 distribution with 1 df. As noted in Section 5A.1,
also follows the x2 distribution with n — 2 df. Moreover, as noted in Section 4.3, Z2 is distributed independently of Z1. Then from Theorem 5.6, it follows that
Z2/1 =(ft — 02)2(E x2) Z2/(n — 2) E u2/(n — 2)
follows the F distribution with 1 and n — 2 df, respectively. Under the null hypothesis H0:02 = 0, the preceding F ratio reduces to Eq. (5.9.1).
5.A.4 DERIVATIONS OF EQUATIONS (5.10.2) AND (5.10.6) Variance of Mean Prediction
Given Xi = X0, the true mean prediction E(Yq | X0) is given by
We estimate (1) from
Taking the expectation of (2), given X0, we get
6For proof, see J. Johnston, Econometric Methods, McGraw-Hill, 3d ed., New York, 1984, pp. 181-182. (Knowledge of matrix algebra is required to follow the proof.)
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Econometrics, Fourth Regression Models Regression: Interval Companies, 2004 Edition Estimation and Hypothesis
CHAPTER FIVE: TWO VARIABLE REGRESSION: INTERVAL ESTIMATION AND HYPOTHESIS TESTING 163
because 1 and 2 are unbiased estimators. Therefore,
That is, Y0 is an unbiased predictor of E(Y0 | X0).
Now using the property that var (a + b) = var (a) + var(b) + 2 cov(a, b), we obtain
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