MULTIPLE REGRESSION MODEL

We shall rewrite (12.1.1) in vector and matrix notation in two steps. Define the ^-dimensional row vector x( = (хл, xt2, . . . , xtK) and the X-dimensional column vector p = (P], p2,. . . , Px)’- Then (12.1.1) can be written as

(12.1.2) yt = x(‘p + ut, t = 1, 2, . . . , T.

Although we have simplified the notation by going from (12.1.1) to

(12.1.2) , the real advantage of matrix notation is that we can write the T equations in (12.1.2) as a single vector equation.

Define the column vectors у = (yi, Уч, ■ ■ ■, Jr)’ and u = (iq, щ, . . . , uT)’ and define the T X К matrix X whose fth row is equal to x( so that X’ = (x1; x2, . . . , X7-). Then we can rewrite (12.1.2) as

#21 X22 X2 к

(12.1.3) у = XP + u, where X =

Xt Xt2 ’ * ‘ %TK

We assume rank(X) = K. Note that in the bivariate regression model this assumption is equivalent to assuming that xt is not constant for all t.

We denote the columns ofX by х(1), x(2), . . . , x(K). Thus, X = [x(1), x(2), . . . , x(X)]. The assumption rank(X) = К is equivalent to assuming that X(!), X(2), . . . , X(jf) are linearly independent. Another way to express this assumption is to state thatX’X is nonsingular. (See Theorem 11.4.8.)

The assumption that X is full rank is not restrictive, because of the following observation. Suppose rank(X) = K < K. Then, by Definition 11.4.4, we can find a subset of Kx columns of X which are linearly inde­pendent. Without loss of generality assume that the subset consists of the first Ki columns of X and partition X = (X1; X2), where Xj is T X Kl and X2, T X K2. Then we can write X2 = X]A for some K} X K2 matrix A, and hence X = Xj (I, A). Therefore we can rewrite the regression equation

(12.1.3) as

у = XjPi + u,

where P] = (I, A) P and Xj is full rank.

In practice, X(i) is usually taken to be the vector consisting of T ones. But we shall not make this assumption specifically as part of the linear regression model, for most of our results do not require it.

Our assumptions on {ut) imply in terms of the vector u

(12.1.4) £u = 0 and

(12.1.5) £uu’ = a2I.

In (12.1.4), 0 denotes a column vector consisting of T zeroes. We shall denote a vector consisting of only zeroes and a matrix consisting of only zeroes by the same symbol 0. The reader must infer what 0 represents from the context. To understand (12.1.5) fully, the reader should write out the elements of the TXT matrix uu’. The identity matrix on the right-hand side of (12.1.5) is of the size T, which the reader should also infer from the context. Note that u’u, a row vector times a column vector, is a scalar and can be written in the summation notation as Taking

the trace of both sides of (12.1.5) and using Theorem 11.5.8 yields £u’u = a2T.