12. In each case, indicate whether the statement regarding the relationship Y xb + is...

12. In each case, indicate whether the statement regarding the relationship Y ¼ xb + « is true or false, and justify your answer.

(a) Let the random (n 1) vector Y represent a random sample from some composite experiment, where E(«) ¼ 0, and E(««0

) ¼ s2 I. Suppose the x-matrix has full column rank, but that the first and second columns of x are nearly linearly dependent and, as a result, the determinant of x0 x is near zero, equaling

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. In this case, although b^ ¼ x0 ð Þ x 1 x0 Y is still an unbiased estimator, it is no longer BLUE (i.e., it loses its “best” property of having the smallest covariance matrix in the linear unbiased class of estimators).

(b) The ei 0

s are homoskedastic and jointly independent with E(ei) ¼ d 6¼ 0 8i. Also, xb ¼ b1i + b2Z where i is an (n 1) column vector of 1’s, and Z is a (n 1)

column vector of explanatory variable values.

Then if b^ is the least-squares estimator of

b, ^

b2 is the BLUE of b2.

(c) The disturbance terms are related as et ¼ ret1 + Vt, where the Vt 0

s are iid with E(Vt) ¼ 0 and var(Vt) ¼ s2 8t, and |r| < 1. The least squares estimator is both BLUE and consistent.