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X2n X3n

Yi Y2 Y3

Yn y

5These equations can be remembered easily. Start with the equation Y = f>i + P2 X2i + P3X3i + ■ ■ ■ + PkXu. Summing this equation over the n values gives the first equation in (C.3.8); multiplying it by X2 on both sides and summing over n gives the second equation; multiplying it by X3 on both sides and summing over n gives the third equation; and so on. In passing, note that the first equation in (C.3.8) gives at once f>i = Y- p2X2-----pkXk [cf. (7.4.6)].

APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL 933

Note these features of the (X X) matrix: (1) It gives the raw sums of squares and cross products of the X variables, one of which is the intercept term taking the value of 1 for each observation. The elements on the main diagonal give the raw sums of squares, and those off the main diagonal give the raw sums of cross products (by raw we mean in original units of measurement). (2) It is symmetrical since the cross product between X2i and X3i is the same as that between X3i and X2i. (3) It is of order (k x k), that is, k rows and k columns.

In (C.3.10) the known quantities are (X X) and (Xy) (the cross product between the X variables and y) and the unknown is p. Now using matrix algebra, if the inverse of (X X) exists, say, (X X)-1, then premultiplying both sides of (C.3.10) by this inverse, we obtain

(X'X)-1(X'X)p = (X'X)-1X'y But since (X'X)-1(X'X) = I, an identity matrix of order k x k, we get

I p = (X'X)-1X'y or p = (X'X)-1 X' y k x 1 k x k (k x n)(n x 1)

Equation (C.3.11) is a fundamental result of the OLS theory in matrix notation. It shows how the p vector can be estimated from the given data. Although (C.3.11) was obtained from (C.3.9), it can be obtained directly from (C.3.7) by differentiating uu with respect to p. The proof is given in Appendix CA, Section CA.2.

An Illustration

As an illustration of the matrix methods developed so far, let us rework the consumption-income example of Chapter 3, whose data are reproduced in (C.1.6). For the two-variable case we have

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