This appendix presents the classical linear regression model involving k variables (Y and X2, X3, ... , Xk) in matrix algebra notation. Conceptually, the k-variable model is a logical extension of the two- and three-variable models considered thus far in this text. Therefore, this appendix presents very few new concepts save for the matrix notation.1
A great advantage of matrix algebra over scalar algebra (elementary algebra dealing with scalars or real numbers) is that it provides a compact method of handling regression models involving any number of variables; once the k-variable model is formulated and solved in matrix notation, the solution applies to one, two, three, or any number of variables.
C.1 THE ^-VARIABLE LINEAR REGRESSION MODEL
If we generalize the two- and three-variable linear regression models, the k-variable population regression model (PRF) involving the dependent variable Y and k — 1 explanatory variables X2, X3, . . . , Xk may be written as
PRF: Y = Pi + 02X2i + faX3i + ••• + faXki + Ui i = 1, 2, 3,..., n
where 01 = the intercept, 02 to 0k = partial slope coefficients, u = stochastic disturbance term, and i = ith observation, n being the size of the population. The PRF (C.1.1) is to be interpreted in the usual manner: It gives the mean or expected value of Y conditional upon the fixed (in repeated sampling) values of X2, X3,..., Xk, that is, E(Y | X2i, X3i,..., Xki).
1Readers not familiar with matrix algebra should review App. B before proceeding any further. Appendix B provides the essentials of matrix algebra needed to follow this appendix.
APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL 927
Equation (C.1.1) is a shorthand expression for the following set of n simultaneous equations:
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