By Fitzmaurice G.M., Laird N.M.

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7), attains the minimum for any solution of X, Xb = X'y. 13) and note that Xb = = X(X'X)-X'y+X(1 - (X'X)-X'X)w X(X' X)- X'y (which is independent of w). 81. 11) > (y - Xb)'(y - Xb) = 8(b) = y'y - 2y'Xb+ b'X'Xb = y'y - b'X'Xb = y'y - fj'fj. 3 Geometric Properties of OL8 For the T x K-matrix X, we define the column space n(x) = {(I: (I = X(3, (3 E n K }, which is a subspace of nT. If we choose the norm IIxli = (x'x)1/2 for x E nT, then the principle of least squares is the same as that of minimizing II y - (I II for (I E n(X).

3 Mean Dispersion Error The quadratic risk is closely related to the matrix-valued criterion of the mean dispersion error (MDE) of an estimator. The MDE is defined as the matrix M({J, {3) = E({J - {3)({J - {3)'. 42) We will denote the covariance matrix of an estimator (J by V({J): V({J) = E({J - E({J))(/3 - E(/3))'. If E(/3) = {3, then /3 will be called unbiased (for {3). If E(/3) called biased. The difference between E({J) and {3 is Bias({J, {3) = E({J) - {3. If {J is unbiased, then obviously Bias({J, {3) = 0 .

XKi)' is a K-vector 3. 2): Yt = x~{3 + et , t = 1, ... 4) where {3' = ({31, ... 5) where e' = (ell ... , eT). We consider the problems of estimation and testing of hypotheses on {3 under some assumptions. 6) t=l for a suitably chosen function M, some examples of which are M(x) = Ixl and x 2 • More generally, one could minimize a global function of e such as maxt Iet lover t. First we consider the case M (x) = x 2 , which leads to the least-squares theory, and later, introduce other functions that may be more appropriate in some situations.