Gauss–Markov Model

So far we have approached the linear model mostly as a method of mathematical approximation. In this chapter, we pose the Gauss-Markov model, which embodies the most common assumptions for the statistical approach to the linear model, leading to the Gauss-Markov Theorem. The Gauss-Markov model takes the form

y = Xb+ e (4.1) where y is the (N × 1) vector of observed responses and X is the (N × p) known design matrix. As before, the coefficient vector b is unknown and to be determined or estimated. The main features of the Gauss-Markov model are the assumptions on the error e:

E(e) = 0 and Cov (e) = σ 2IN . (4.2) The notation for expectation and covariances above can be rewritten component by component:

(E(e))i = i th component of E(e) = E(ei ) (Cov (e))i j = i, j th element of covariance matrix = Cov (ei , e j )

so that the Gauss-Markov assumptions can be rewritten as

E(ei ) = 0, i = 1, . . . , N

Cov(ei , e j ) = {

σ 2 for i = j 0 for i = j

,

that is, the errors in the model have a zero mean, constant variance, and are uncorrelated. An alternative view of the Gauss-Markov model does not employ the error vector e:

E(y) = Xb,Cov(y) = σ 2IN . The assumptions in the Gauss-Markovmodel are easily acceptable for many practical problems, and deviations from these assumptions will be considered in more detail later.