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C.8 TESTING LINEAR RESTRICTIONS: GENERAL F TESTING USING MATRIX NOTATION

In Section 8.7 we introduced the general F test to test the validity of linear restrictions imposed on one or more parameters of the k-variable linear regression model. The appropriate test was given in (8.7.9) [or its equivalent (8.7.10)]. The matrix counterpart of (8.7.9) can be easily derived. Let

ûR = the residual vector from the restricted least-squares regression û UR = the residual vector from the unrestricted least-squares regression

Then

ûRûR = J2 ûR = RSS from the restricted regression ûURû UR = Y* ûUR = RSS from the unrestricted regression m = number of linear restrictions k = number of parameters (including the intercept) in the unrestricted regression n = number of observations

The matrix counterpart of (8.7.9) is then

which follows the F distribution with (m, n — k) df. As usual, if the computed F value from (C.8.1) exceeds the critical F value, we can reject the restricted regression; otherwise, we do not reject it.

C.9 PREDICTION USING MULTIPLE REGRESSION: MATRIX FORMULATION

In Section 8.9 we discussed, using scalar notation, how the estimated multiple regression can be used for predicting (1) the mean and (2) individual values of Y, given the values of the X regressors. In this section we show how to express these predictions in matrix form. We also present the formulas to estimate the variances and standard errors of the predicted values; in Chapter 8 we noted that these formulas are better handled in matrix

APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL 941

notation, for the scalar or algebraic expressions of these formulas become rather unwieldy.

Mean Prediction

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