Info

k x 1

where y = n x 1 column vector of observations on the dependent variable Y X = n x k matrix giving n observations on k — 1 variables X2 to Xk, the first column of 1's representing the intercept term (this matrix is also known as the data matrix) P = k x 1 column vector of the unknown parameters fa, fa,..., fa u = n x 1 column vector of n disturbances ui

Using the rules of matrix multiplication and addition, the reader should verify that systems (C.1.2) and (C.1.3) are equivalent.

System (C.1.3) is known as the matrix representation of the general (k-variable) linear regression model. It can be written more compactly as

y

=X

P + u

n x 1

n x k

Where there is no confusion about the dimensions or orders of the matrix X and the vectors y, P, and u, Eq. (C.1.4) may be written simply as y = X p + u (C.1.5)

As an illustration of the matrix representation, consider the two-variable consumption-income model considered in Chapter 3, namely, Y = fa + faXi + Ui, where Y is consumption expenditure and X is income. Using the

2Fo]lowing the notation introduced in App. B, we shall represent vectors by lowercase boldfaced letters and matrices by uppercase boldfaced letters.

928 APPENDIX C: THE MATRIX APPROACH TO LINEAR REGRESSION MODEL

data given in Table 3.2, we may write the matrix formulation as

70" 65 90 95 110 115 120 140 155 _ 150 _

=

"1 80" 1 100 1 120 1 140 1 160 1 180 1 200 1 220 1 240 1 260

r 01" Lß2.

+

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