To facilitate the subsequent analysis, we shall write (1.1.2) in matrix notation as
у = ХД + u,
where у = (Уі, У2. • • • >УтУ, и = . . . ,ит) and Х =
(х,, х2,. . . , хт)’. In other words, X is the TX К matrix, the /th row of which is x’t. The elements of the matrix X are described as
*11 *12 • • • *1JC
*21 *22 • • • *2AT
_*Г1 *77 • • • *7X.
If we want to focus on the columns of X, we can write X = [x(1), X(2),…. Х(*)], where each x(j) is a Г-vector. If there is no danger of confusing Х(0 with x„ we can drop the parentheses and write simply x(. In matrix notation the assumptions on X and u can be stated as follows: X’X is nonsingular, which is equivalent to stating rank (X) = К if T Ш K Eu = 0; and Euu’ = a2T, where Ir is the T X Tidentity matrix. (Whenever the size of an identity matrix can be inferred from the context, we write it simply as I.)
In the remainder of this chapter we shall no longer use the partition P’ — (A>>$); instead, the elements of P will be written as /? = (/?i, p2,. . . , fiK) Similarly, we shall not necessarily assume that x(1) is the vector of ones, although in practice this is usually the case. Most of our results will be obtained simply on the assumption that X is a matrix of constants, without specifying specific values.